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- Dombrowski, S.C., McGill, R.J., Watkins, M.W., Canivez, G.L., Pritchard, A.E. and Jacobson, L.A. (2024). Conjectures and Refutations in Structural Ability Research: Insights from Bayesian Structural Equation Modeling.
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Abstract
The use of Bayesian structural equation modeling (BSEM) provided additional insight into the WISC–V theoretical structure beyond that offered by traditional factor analytic approaches (e.g., exploratory factor analysis and maximum likelihood confirmatory factor analysis) through the specification of all cross loadings and correlated residual terms. The results indicated that a five-factor higher-order model with a correlated residual between the Visual-Spatial and Fluid Reasoning group factors provided a superior fit to the four bifactor model that has been preferred in prior research. There were no other statistically significant correlated residual terms or cross loadings in the measurement model. The results further suggest that the WISC–V ten subtest primary battery readily attains simple structure and can be interpreted as indicated in the WISC–V’s scoring and interpretive manual. Additionally, BSEM may help attenuate the replication crises in school psychology within the area of cognitive assessment structural validity research given that it is a potentially versatile and powerful method that permits systematic evaluation of more complex structural relationships obviating the need for CFA based post hoc specification searches which can be prone to confirmation bias and capitalization on chance.
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- Asparouhov, T., Muthén, B. & Morin, A. J. S. (2015). Bayesian structural equation modeling with cross-loadings and residual covariances: Comments on Stromeyer et al. Journal of Management, 41, 1561-1577. DOI: 10.1177/0149206315591075
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Abstract
A recent article in the Journal of Management gives a critique of a Bayesian approach to factor analysis proposed in Psychological Methods. This commentary responds to the authors’ critique by clarifying key issues, especially the use of priors for residual covariances. A discussion is also presented of cross-loadings and model selection tools. Simulated data are used to illustrate the ideas. A re-analysis of the example used by the authors reveals a superior model overlooked by the authors.
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- De Bondt, N, & Van Petegem, P. (2015). Psychometric evaluation of the overexcitability questionnaire-two applying Bayesian structural equation modeling (BSEM) and multiple-Group BSEM-based alignment with approximate measurement invariance. Frontiers in Psychology 6:1963. DOI: 10.3389/fpsyg.2015.01963
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- Muthén, B. & Asparouhov, T. (2013). BSEM measurement invariance analysis. Mplus Web Notes: No. 17
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This paper concerns measurement invariance analysis for situations with many groups or time points. A BSEM (Bayesian Structural Equation Modeling) approach is proposed for detecting non-invariance that is similar to modification indices with maximum-likelihood estimation, but unlike maximum-likelihood is applicable also for high-dimensional latent variable models for categorical variables. Under certain forms of non-invariance, BSEM gives proper comparisons of factor means and variances using only approximate measurement invariance and without relaxing the invariance specifications or deleting non-invariant items. To ensure correct estimation, a two-step Bayesian analysis procedure is proposed, where step 1 uses BSEM to identify non-invariant parameters and step 2 frees those parameters.
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- van de Schoot, R., Tummers, L., Lugtig, P., Kluytmans, A., Hox, J. & Muthén, B. (2013). Choosing between Scylla and Charybdis? A comparison of
scalar, partial and the novel possibility of approximate measurement invariance. Frontiers in Psychology, 4, 1-15. DOI: 10.3389/fpsyg.2013.00770.
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Abstract
Measurement invariance (MI) is a pre-requisite for comparing latent variable scores across groups. The current paper introduces the concept of approximate MI building on the work of Muthén and Asparouhov and their application of Bayesian Structural Equation Modeling (BSEM) in the software Mplus. They showed that with BSEM exact zeros constraints can be replaced with approximate zeros to allow for minimal steps away from strict MI, still yielding a well-fitting model. This new opportunity enables researchers to make explicit trade-offs between the degree of MI on the one hand, and the degree of model fit on the other. Throughout the paper we discuss the topic of approximate MI, followed by an empirical illustration where the test for MI fails, but where allowing for approximate MI results in a well-fitting model. Using simulated data, we investigate in which situations approximate MI can be applied and when it leads to unbiased results. Both our empirical illustration and the simulation study show approximate MI outperforms full or partial MI In detecting/recovering the true latent mean difference when there are (many) small differences in the intercepts and factor loadings across groups. In the discussion we provide a step-by-step guide in which situation what type of MI is preferred. Our paper provides a first step in the new research area of (partial) approximate MI and shows that it can be a good alternative when strict MI leads to a badly fitting model and when partial MI cannot be applied.
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- Muthén, B. & Asparouhov, T. (2012).
Bayesian SEM: A more flexible representation of substantive theory.
Psychological Methods, 17(3), 313-335. DOI: 10.1037/a0026802
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Download the 1st version dated September 29, 2010 containing a MIMIC section, more tables, and the corresponding Mplus inputs, data, and outputs.
The seven web tables correspond to tables 8, 10, 17, 18, 19, 20, and 21 of the first version.
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Abstract
"This paper proposes a new approach to factor analysis and structural equation modeling using Bayesian analysis. The new approach replaces parameter specifications of exact zeros with approximate zeros based on informative, small-variance priors. It is argued that this produces an analysis that better rejects substantive theories. The proposed Bayesian approach is particularly beneficial in applications where parameters are added to a conventional model such that a non-identified model is obtained if maximum-likelihood estimation is applied. This approach is useful for measurement aspects of latent variable modeling such as with CFA and the measurement part of SEM. Two application areas are studied, cross-loadings and residual correlations in CFA. The approach encompasses three elements: Model testing, model estimation, and model modi cation. Monte Carlo simulations and real data are analyzed using Mplus."
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- Muthén, B. & Asparouhov, T. (2012). Rejoinder to MacCallum, Edwards, and Cai (2012) and Rindskopf (2012): Mastering a new method. Psychological Methods, Vol 17(3), 346-353. DOI: 10.1037/a0029214
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"This rejoinder discusses the general comments on how to use Bayesian structural equation modeling (BSEM) wisely and how to get more people better trained in using Bayesian methods. Responses to specific comments cover how to handle sign switching, nonconvergence and nonidentification, and prior choices in latent variable models. Two new applications are included. The first one revisits the Kaplan (2009) science model by considering priors on primary parameters. The second one applies BSEM to the bifactor model that was hypothesized in the original Holzinger and Swineford (1939) study."
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- Asparouhov, T. & Muthén, B. (2024). Continuous Time Dynamic Structural Equation Models. Version 5. October 18, 2024.
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- Muthén, B., Asparouhov, T. & Shiffman, S. (2024). Dynamic Structural Equation Modeling with Floor Effects. Version 2. Forthcoming in Psychological Methods.
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Abstract
Intensive longitudinal data analysis, commonly used in psychological studies, often
concerns outcomes that have strong floor effects, that is, a large percentage at its lowest
value. Ignoring a strong floor effect, using regular analysis with modeling assumptions
suitable for a continuous-normal outcome, is likely to give misleading results. This
paper suggests that two-part modeling may provide a solution. It can avoid potential
biasing effects due to ignoring the floor effect. It can also provide a more detailed
description of the relationships between the outcome and covariates allowing different
covariate effects for being at the floor or not and the value above the floor. A smoking
cessation example is analyzed to demonstrate available analysis techniques.
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- Muthén, B., Asparouhov, T. & Keijsers, L. (2024). Dynamic Structural Equation Modeling with Cycles. Structural Equation Modeling: A Multidisciplinary Journal. DOI: 10.1080/10705511.2024.2406510
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Cyclical phenomena are commonly observed in many areas of repeated measurements, especially with intensive longitudinal data. A typical example is circadian
(24-hour) rhythm of physical measures such as blood pressure, heart rate, glucose
level, and alertness. This paper focuses on positive affect which is a common measure
in psychological studies and for which circadian rhythm has been observed but not
analyzed by modern statistical methods. The paper demonstrates that a large new
analysis arsenal is available for analysis of cyclical features in intensive longitudinal
data. This can help researchers extract more information from their data to get more
valid estimates of coupled processes and to get new theoretical insights into circadian
rhythms of mood. To assist in this effort, the analyses are based on general models
with a rich set of features while still being accessible without an unduly steep learning
curve. Scripts for the Mplus software are available for all the analyses presented.
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- Asparouhov, T. & Muthén, B. (2024). Practical Aspects of Dynamic Structural Equation Models. Technical Report. Version 4. December 2, 2024.
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- Asparouhov, T. & Muthén, B. (2022). Residual Structural Equation Models. Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2022.2074422
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- Hamaker, E.L., Asparouhov, T, & Muthén, B. (2021). Dynamic structural equation modeling as a combination of time series modeling, multilevel modeling, and structural equation modeling. To be published as Chapter 31 in: The Handbook of Structural Equation Modeling (2nd edition); Rick H. Hoyle (Ed.); Publisher: Guilford Press.
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- McNeish, D. & Hamaker, E.L. (2020). A primer on two-level dynamic structural equation models for intensive longitudinal data in Mplus. Psychological Methods, 25(5), 610–635. https://doi.org/10.1037/met0000250
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- McNeish, D. (2019). Two-Level dynamic structural equation models with small samples. Structural Equation Modeling: A Multidisciplinary Journal, 26(6), 948-966. DOI: 10.1080/10705511.2019.1578657
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- Mun, C.J., Suk, H.W., Davis, M.C., Karoly, P., Finan, P., Tennen, H., & Jensen, M.P. (2019). Investigating intraindividual pain variability: Methods, applications, issues, and directions. Pain. DOI: 10.1097/j.pain.0000000000001626
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- Lundgren, B. & Schultzberg, M. (2019). Application of the economic theory of self-control to model energyconservation behavioral change in households. Energy. DOI: 10.1016/j.energy.2019.05.217
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Abstract
Smart meters and in-house displays hold a promise of energy conservation for those who invest in such technology. Research has shown that households only have a limited interest in such technology and information is thus often neglected, with rather limited energy savings. Surprisingly few empirical investigations have a theoretical foundation that may explain what is going on from a behavioral perspective. In this study the economic theory of self-control is used to model energy-efficient behavior in middle-income households in Sweden. Our results show that different levels of energy-efficient behavior do not really have any impact on the actual consumption levels of electricity. Instead, different beliefs exist of being energy-efficient, but the households do not act accordingly. We recommend to policy makers that the payment time period should be changed to pre-paid electricity to stimulate the monitoring of bills and to introduce a gaming strategy to change incentives for energy conservation.
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- Öhrlund, I., Schultzberg, M. & Bartusch, C. (2019). Identifying and estimating the effects of a mandatory billing demand charge. Applied Energy, 237, 885-895. DOI: 10.1016/j.apenergy.2019.01.028
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- Schultzberg, M. (2019). Using high frequency pre-treatment outcomes to identify causal effects in non-experimental data. Doctoral Dissertation, Paper 1, Department of Statistics, Uppsala University.
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- Armstrong, B., Covington, L.B., Unick, G.J., & Black, M.M. (2018). Bidirectional effects of sleep and sedentary behavior qmong toddlers: A dynamic multilevel modeling approach. Journal of Pediatric Psychology, 44(3), 275-285. DOI: 10.1093/jpepsy/jsy089
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- Joly-Burra, E., Van der Linden, M. & Ghisletta, P. (2018). Intraindividual variability in inhibition and prospective memory in healthy older adults: Insights from response regularity and rapidity. Journal of Intelligence, 6(1), 13. DOI: 10.3390/jintelligence6010013
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- Hamaker, E.L., Asparouhov, T., Brose, A., Schmiedek, F. & Muthen, B. (2018). At the frontiers of modeling intensive longitudinal data: Dynamic structural equation models for the affective measurements from the COGITO study. Multivariate Behavioral Research, DOI: 10.1080/00273171.2018.1446819
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Abstract
Dynamic structural equation modeling (DSEM) is a newly emerging class of techniques by which we can model the dynamic patterns in intensive longitudinal data. When using DSEM to analyse the time series of multiple individuals, we
specify a time series model at the within-person level and allow for individual di?erences at the between-person level in the parameters that describe the dynamics. We use DSEM to analyze a?ective data from the COGITO study, which consists of two samples
of over one hundred individuals each who were measured for one hundred days each. We use composite scores of positive and negative a?ect and apply a multilevel vector autoregressive model to investigate individual di?erences
in means, autoregressions and cross-lagged e?ects. Then we extend the model with random residual variances, and finally we investigate whether the random e?ects mediate the e?ect of prior depression on later depression scores. We point out some
additional options, and discuss several unresolved issues.
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- Schultzberg, M. & Muthén, B. (2018). Number of subjects and time points needed for multilevel time series analysis: A simulation study of dynamic structural equation modeling. Structural Equation Modeling: A Multidisciplinary Journal, 25:4, 495-515, DOI:10.1080/10705511.2017.1392862.
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Dynamic Structural Equation modeling (DSEM) is a novel intensive longitudinal data (ILD) analysis framework. DSEM uses two-level modeling with time on level 1 and individuals on level 2. It models intra-individual changes over time and allows the parameters of these processes to vary across
individuals using random effects. DSEM merges time series, structural equation, multilevel, and time-varying effects models. Despite the well-known properties of these analysis areas by themselves, it is unclear how their sample size requirements and recommendations transfer to the DSEM
framework. ILD are sampled in two dimensions, across subjects and across repeated measures within subjects. This paper presents the results of a simulation study that examines the estimation quality of univariate two-level autoregressive models of order one, AR(1), using Bayesian analysis
in Mplus Version 8. Three features are varied in the simulations: complexity of the model, number of subjects, and number of time points per subject. The models cover empty random mean-only models and models using a random AR(1) mean, a random autoregressive coefficient, and a random residual
variance as mediators on level 2. The number of subjects and number of time points per subject are varied between 10 and 200, in various combinations. Special attention is given to the power and accuracy of the level 2 regression slopes. The results are summarized with sample size guidelines
for each model. Samples with many subjects and few time points are showed to perform substantially better than samples with few subjects and many time points.
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- Asparouhov, T. & Muthén, B. (2021). Expanding the Bayesian structural equation, multilevel and mixture models to logit, negative-binomial and nominal variables. Structural Equation Modeling: A Multidisciplinary Journal, 28:4, 622-637, DOI: 10.1080/10705511.2021.1878896
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Recent work on the Polya-Gamma distribution provides a break-through for the Bayesian modeling of logit, count and nominal variables. We describe how the methodology is incorporated in the Mplus modeling framework and illustrate it with several examples: logistic latent growth models, multilevel IRT, multilevel time-series models for count data, multilevel nominal regression, and nominal factor analysis.
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- Asparouhov, T. & Muthén, B. (2021). Bayesian estimation of single and multilevel models with latent variable interactions. Structural Equation Modeling: A Multidisciplinary Journal, 28:2, 314-328, DOI: 10.1080/10705511.2020.1761808
(*NOTE: Scripts refer to section numbers from the Mplus Web Note 23 version of this paper that are not present in the current version.)
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In this article we discuss single and multilevel SEM models with latent variable interactions. We describe the Bayesian estimation for these models and show through simulation studies that the Bayesian
method outperforms other methods such as the maximum-likelihood method. We show that multilevel moderation models can easily be estimated with the Bayesian method.
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- Asparouhov, T. & Muthén, B. (2021). Advances in Bayesian model fit evaluation for structural equation models, Structural Equation Modeling: A Multidisciplinary Journal, 28:1, 1-14, DOI: 10.1080/10705511.2020.1764360
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In this article, we discuss the Posterior Predictive P-value (PPP) method in the presence of missing data,
the Bayesian adaptation of the approximate fit indices RMSEA, CFI and TLI, as well as the Bayesian
adaptation of the Wald test for nested models. Simulation studies are presented. We also illustrate how
these new methods can be used to build BSEM models.
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- Zitzmann, S. & Hecht, M. (2019). Going beyond convergence in Bayesian estimation: Why precision matters too and how to assess it. Structural Equation Modeling: A Multidisciplinary Journal, 26:4, 646-661, DOI: 10.1080/10705511.2018.1545232
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- Lüdtke, O., Robitzsch, A., & Wagner, J. (2018). More stable estimation of the STARTS model: A Bayesian approach using Markov chain Monte Carlo techniques. Psychological Methods, 23(3), 570-593. DOI: 10.1037/met0000155.
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- Shi, D., Song, H., DiStefano, C., Maydeu-Olivares, A., McDaniel H.L. & Jiang, Z. (2018). Evaluating factorial invariance: An interval estimation approach using Bayesian structural equation modeling. Multivariate Behavioral Research, 54(2):224-245. DOI: 10.1080/00273171.2018.1514484
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- Enders, Craig K.& Mansolf, M. (2018). Assessing the fit of structural equation models with multiply imputed data. Psychological Methods 23(1), 76–93. DOI: 10.1037/met0000102
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- van de Schoot, R., Sijbrandij, M., Depaoli, S., Winter, S.D., Olff M. & van Loey, N.E. (2018). Bayesian PTSD-trajectory analysis with informed priors based on a systematic literature search and expert elicitation. Multivariate Behavioral Research. 53(2), 267-291, DOI: 10.1080/00273171.2017.1412293
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- Dombrowski, S.C., Golay, P., McGill, R.J., & Canivez, G.L. (2018). Investigating the theoretical structure of the DAS-II core battery at school age using Bayesian structural equation modeling. Psychology in the Schools. 55(2), 190-207, DOI: 10.1002/pits.22096
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- Nestler, S. & Back, M.D. (2017). Using cross-classified structural equation models to examine the accuracy of personality judgements. Psychometrika, 82(2), 475-497. DOI: 10.1007/s1136-015-9485-6
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- Helm, J.L., Castro-Schilo, L. & Oravecz, Z. (2016). Bayesian versus maximum likelihood estimation of multitrait–multimethod confirmatory factor models.
Structural Equation Modeling: A Multidisciplinary Journal, 24(1), 17-30, DOI: 10.1080/10705511.2016.1236261
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- Zitzmann, S., Lüdtke, O., Robitzsch A. & Marsh, H.W. (2016). A Bayesian approach for estimating multilevel latent contextual models,
Structural Equation Modeling: A Multidisciplinary Journal, 23:(5), 661-679, DOI: 10.1080/10705511.2016.1207179
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- Zercher F., Schmidt P., Cieciuch J. & Davidov E. (2015). The comparability of the universalism value over time and across countries in the European Social Survey: exact vs. approximate measurement invariance. Frontiers in Psychology. 6:733. DOI: 10.3389/fpsyg.2015.00733
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- Zyphur, M. J., Zammuto, R. F., & Zhang, A. (2016). Multilevel latent polynomial regression for modeling (in)congruence across organizational groups: The case of organizational culture research. Organizational Research Methods, 19(1), 53-79. DOI: 10.1177/1094428115588570
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This paper addresses (in)congruence across different kinds of organizational respondents or ‘organizational groups’ — such as managers versus non-managers or women versus men — and the effects of congruence on organizational outcomes. We introduce a novel multilevel latent polynomial regression model (MLPM) that treats standings of organizational groups as latent ‘random intercepts’ at the organization level, while subjecting these to latent interactions that
enable response surface modeling to test congruence hypotheses. We focus on the case of organizational culture research, which usually samples managers and excludes non-managers. Re-analyzing data from 67 hospitals with 6,731 managers and non-managers, we find that non-managers perceive their organizations’ cultures as less humanistic and innovative and more controlling than managers, and less congruence between managers and non-managers in these perceptions is associated with lower levels of quality improvement in organizations. Our results call into question the validity of findings from organizational culture and other research that tends to sample one organizational group to the exclusion of others. We discuss our findings and the MLPM, which can be extended to estimate latent interactions for tests of multilevel moderation/interactions.
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- Asparouhov, T. & Muthén, B. (2016). General random effect latent variable modeling: Random subjects, items, contexts, and parameters. In Harring, J. R., & Stapleton, L. M., & Beretvas, S. N. (Eds.), Advances in multilevel modeling for educational research: Addressing practical issues found in real-world applications (pp. 163-192). Charlotte, NC: Information Age Publishing, Inc.
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- Bruyneel, L., Baoyue, L., Squires, A., Spotbeen, S., Meuleman, B., Lesaffre, E., & Sermeus, W. (2014).
Bayesian multilevel MIMIC modeling for studying measurement invariance in cross-group comparisons.
Medical Care, 55(4):e25-e35. DOI: 10.1097/MLR.0000000000000164
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- Kim, S., Suh, Y., Kim, J., Albanese, M., & Langer, M. (2013). Single and multiple ability estimation in the SEM framework: A non-informative Bayesian estimation approach.
Multivariate Behavioral Research, 48(4), 563–591. DOI:10.1080/00273171.2013.802647.
- Fong, T. C. T., & Ho, R. T. H. (2013). Factor analysis of the hospital depression and anxiety scale: A Bayesian structural equation modeling approach.
Quality of Life Research, 22(10):2857-63. DOI: 10.1007/s11136-013-0429-2.
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Purpose: The latent structure of the Hospital Anxiety and Depression Scale (HADS) has caused inconsistent results in the literature. The HADS is frequently analyzed via maxi- mum likelihood confirmatory factor analysis (ML-CFA). However, the overly restrictive assumption of exact zero cross-loadings and residual correlations in ML-CFA can lead to poor model fits and distorted factor structures. This study applied Bayesian structural equation modeling (BSEM) to evaluate the latent structure of the HADS. Methods Three a priori models, the two-factor, three- factor, and bifactor models, were investigated in a Chinese community sample (N = 312) and clinical sample (N = 198) using
ML-CFA and BSEM. BSEM specified approximate zero cross-loadings and residual correlations through the use of zero-mean, small-variance informative priors. The model comparison was based on the Bayesian information criterion (BIC).
Results Using ML-CFA, none of the three models pro- vided an adequate fit for either sample. The BSEM two- factor model with approximate zero cross-loadings and residual correlations fitted both samples well with the lowest BIC of the three models and displayed a simple and parsimonious factor-loading pattern. Conclusions: The study demonstrated that the two-factor structure fitted the HADS well, suggesting its usefulness in assessing the symptoms of anxiety and depression in clinical practice. BSEM is a sophisticated and flexible statistical technique that better reflects substantive theories and locates the source of model misfit. Future use of BSEM is recommended to evaluate the latent structure of other psychological instruments.
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- van de Schoot, R., Kaplan, D., Denissen, J., Asendorpf , J.B., Neyer, F.J. & van Aken, M.A.G. (2013). A gentle introduction to bayesian analysis:
Applications to research in child development. Child Development, 85(3):842-860. DOI: 10.1111/cdev.12169.
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Bayesian statistical methods are becoming ever more popular in applied and fundamental research. In this study a gentle introduction to Bayesian analysis is provided. It is shown under what circumstances it is attractive to use Bayesian estimation, and how to interpret properly the results. First, the ingredients underlying Bayesian methods are introduced using a simplifed example. Thereafter, the advantages and pitfalls of the specification of prior knowledge are discussed. To illustrate Bayesian methods explained in this study, in a second example a series of studies that examine the theoretical framework of dynamic interactionism are considered. In the Discussion the advantages and disadvantages of using Bayesian statistics are reviewed, and guidelines on how to report on Bayesian statistics are provided.
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- Zyphur, M. & Oswald, F. (2015). Bayesian estimation and inference: A user's guide. Journal of Management, 41(2), 390–420
DOI: 10.1177/0149206313501200
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This paper introduces the “Bayesian revolution” that is sweeping across multiple disciplines but has yet to gain a foothold in organizational research. The foundations of Bayesian estimation and inference are first reviewed. Then, two empirical examples are provided to show how Bayesian methods can overcome limitations of frequentist methods: (a) a structural equation model of testosterone’s effect on status in teams, where a Bayesian approach allows directly testing a traditional null hypothesis as a research hypothesis and allows estimating all possible residual covariances in a measurement model, neither of which are possible with frequentist methods; and (b) an ANOVA-style model from a true experiment of ego depletion’s effects on performance, where Bayesian estimation with informative priors allows results from all previous research (via a meta-analysis and other previous studies) to be combined with estimates of study effects in a principled manner, yielding support for hypotheses that is not obtained with frequentist methods. Data are available from the first author, code for the program Mplus is provided, and tables illustrate how to present Bayesian results. In conclusion, the many benefits and few hindrances of Bayesian methods are discussed, where the major hindrance has been an easily solvable lack of familiarity by organizational researchers.
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- Asparouhov, T. & Muthén, B. (2012). Comparison of computational methods for high dimensional item factor analysis.
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In this article we conduct a simulation study to compare several
methods for estimating confirmatory and exploratory item factor analysis using the software
programs Mplus and IRTPRO. When the num- ber of factors is bigger than three or four the standard
numerical integration methodology used for computing the maximum-likelihood estimates is
intractable due to the exponentially large number of in- tegration points needed to compute the
likelihood. Several methods have been developed recently to overcome these computational problems
however they have not been directly compared previously. In this paper we present a
simulation study to compare maximum likelihood estimation based on Montecarlo integration,
maximum likelihood estimation based on Metropolis-Hastings Robbins-Monro algorithm, maximum
likelihood estimation based on two-tier integration,
Bayesian estimation and the weighted least square estimation.
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- Golay, P., Reverte, I., Rossier, J., Favez, N., & Lecerf, T. (2012, November 12). Further
insights on the French WISC–IV factor structure through Bayesian structural equation
modeling. Psychological Assessment, 25(2):496-508. DOI: 10.1037/a0030676
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- Kaplan & Depaoli (2012). Bayesian Structural Equation Modeling, excerpt from Handbook of Structural Equation Modeling, The Guilford Press.
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- Van de Schoot, R., Verhoeven, M. & Hoijtink, H. (2012): Bayesian evaluation of informative hypotheses in
SEM using Mplus: A black bear story, European Journal of Developmental Psychology, 10(1), 81-98. DOI:10.1080/17405629.2012.732719
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- Van de Schoot, R., Hoijtink, H., Hallquist, M. N., & Boelen, P.A. (2012). Bayesian evaluation of inequality-constrained hypotheses in SEM models using Mplus. Structural Equation Modeling, 19(4), 593-609. DOI: 10.1080/10705511.2012.713267
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- Asparouhov, T. & Muthén, B. (2023). Bayesian analysis using Mplus: Technical implementation. Technical Report. Version 4. February 13, 2023.
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- Asparouhov, T. & Muthén, B. (2022). Multiple imputation with Mplus. Technical Report. Version 4, March 8, 2022.
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- Asparouhov, T. & Muthén, B. (2021). Bayesian analysis of latent variable models using Mplus. Technical Report. Version 5. September 18, 2021.
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Mplus inputs, data, and outputs
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- Asparouhov, T. & Muthén, B. (2010). Plausible values for latent variables using Mplus. Technical Report. August 21, 2010.
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- Muthén, B. (2010). Bayesian analysis in Mplus: A brief introduction. Technical Report. Version 3.
DownloadMplus inputs, data, and outputs used in this paper.
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Abstract
"This paper uses a series of examples to give an introduction to how Bayesian analysis is carried out in Mplus. The examples are a mediation model with estimation of an indirect e ect, a structural equation
model, a two-level regression model with estimation of a random intercept variance, a multiple-indicator binary growth model with a large number of latent variables, a two-part growth model, and a mixture model. It is shown
how the use of Mplus graphics provides information on estimates, convergence, and model t. Comparisons are made with frequentist estimation using maximum likelihood and weighted least squares. Data and Mplus scripts are available on the Mplus website."
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- Revuelta, J., Maydeu-Olivares, A. & Ximénez, C. (2019). Factor analysis for nominal (first choice) data. DOI: 10.1080/10705511.2019.1668276
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Abstract
We show how a factor analysis model can be fitted to nominal data using the computer program Mplus. The model is akin to multinomial logistic regression with unobserved predictors (the common factors) and was initially proposed by Bock (1972) in the one-factor case. Recently, extensions to multiple factors and alternative parameterizations to facilitate parameter interpretation have been proposed. We present four examples in which several versions of the model are estimated using Mplus: a) a one-factor model applied to situational items measuring assertiveness, b) an exploratory factor analysis applied to attitudinal data, c) a confirmatory factor analysis applied to educational data with testlets, and d) the newest parameterization of the model applied to an emotional stability scale. All data files and computer codes are provided as supplementary materials.
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- Barendse, M.T., Oort, F.J., & Timmerman, M.E. (2014).
Using exploratory factor analysis to determine the dimensionality of discrete responses.
Structural Equation Modeling: A Multidisciplinary Journal, 22(1), 87-101. DOI: 10.1080/10705511.2014.934850
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- Myers, N.D., Ahn, S. & Ying, J. (2011).
Sample size and power estimates for a confirmatory factor analytic model in exercise and sport: A Monte
Carlo approach.
Research Quarterly for Exercise and Sport (Measurement and Evaluation section), 82(3), 412-423. DOI: 10.1080/02701367.2011.10599773
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Mplus inputs, data, and outputs
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Abstract
"Monte Carlo methods can be used in data analytic situations (e.g., validity studies) to make decisions
about sample size and to
estimate power. The purpose of using Monte Carlo methods in a validity study
is to improve the methodological approach within a
study where the primary focus is on construct
validity issues and not on advancing statistical theory. The purpose of this study is to
demonstrate
how Monte Carlo methods can be used to determine sample size and to estimate power for a confirmatory
factor analytic
model under model-data conditions commonly encountered in exercise and sport. Because
the purpose is pursued by way of demonstration
with the Coaching Efficacy Scale II–High School
Teams, related sample size recommendations are provided: N ? 200 for the
theoretical model; N ? 300
for the population model. Technical terms (e.g., coverage) are defined when necessary."
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- Flora, D.B. & Curran P.J. (2004).
An empirical evaluation of alternative methods of estimation for confirmatory factor analysis with ordinal
data. Psychological Methods, 9(4), 466-491. DOI: 10.1037/1082-989X.9.4.466
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Abstract
"Confirmatory factor analysis (CFA) is widely used for examining hypothesized relations among ordinal
variables (e.g., Likert-type items). A theoretically appropriate method fits the CFA model to polychoric
correlations using either weighted least squares (WLS) or robust WLS. Importantly, this approach
assumes that a continuous, normal latent process determines each observed variable. The extent to
which violations of this assumption undermine CFA estimation is not well-known. In this article, the
authors empirically study this issue using a computer simulation study. The results suggest that estimation
of polychoric correlations is robust to modest violations of underlying normality. Further,
WLS performed adequately only at the largest sample size but led to substantial estimation difficulties
with smaller samples. Finally, robust WLS performed well across all conditions."
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- Muthén, B., Kaplan, (1985). A comparison of some methodologies for the factor analysis of non-normal Likert variables. British Journal of Mathematical and Statistical Psychology, 38, 171-189. 10.1111/j.2044-8317.1985.tb00832.x
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Abstract
"'This paper considers the problem of applying factor analysis to non-normal categorical
variables. A Monte Carlo study is conducted where five prototypical cases of non-normal variables are
generated. Two normal theory estimators, ML and GLS, are compared to :Browne's (1982) ADF
estimator. A categorical variable methodology (CVM) estimator of Muth{m (1984) is also
considered for the most severely skewed case. Results show that ML and GLS chi-square tests are quite robust but
obtain too large values for variables that are severely skewed and kurtotic. ADF, however,
performs well. Parameter estimate bias appears non-existent for all estimators. Results also
show that ML and GLS estimated standard errors are biased downward. For ADF no such standard
error bias was found. The CVM estimator appears to work well when applied to severely skewed
variables that had been dichotomized. ML and GLS results for a kurtosis only case showed no
distortion of chi-square or parameter estimates and only a slight downward bias in estimated
standard errors. The results are compared to those of other related studies."
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- Muthén, B. (1978).
Contributions to factor analysis of dichotomous variables. Psychometrika, 43, 551-560.
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Abstract
"A new method is proposed for the factor analysis of dichotomous variables. Similar to the method of
Christoffersson this uses information from the first and second order proportions to fit a multiple factor model.
Through a transformation into a new set of sample characteristics, the estimation is considerably simplified.
A generalized least-squares estimator is proposed, which asymptotically is as efficient as the corresponding
estimator of Christoffersson, but which demands less computing time."
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- Klopack, E.T.& Wickrama, K.A.S. (2020) Modeling latent change score analysis and extensions in Mplus: A practical guide for
researchers. Structural Equation Modeling: A Multidisciplinary Journal, 27(1), 97-110, DOI: 10.1080/10705511.2018.1562929
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- McArdle, J., Hamagami, F., Chang, J.Y., and Hishinuma, E.S. (2014). Longitudinal dynamic analyses of depression and academic achievement in the Hawaiian high schools health
survey using contemporary latent variable change models. Structural Equation Modeling: A Multidisciplinary Journal, 21(4), 608-629. DOI:10.1080/10705511.2014.919824
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- Grimm, K., Zhang Z., Hamagami, F. & Mazzocco, M. (2013). Modeling nonlinear change via latent change and latent acceleration frameworks: Examining velocity and acceleration of growth trajectories
Multivariate Behavioral Research, DOI: 10.1080/00273171.2012.755111.
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Abstract
We propose the use of the latent change and latent acceleration frameworks
for modeling nonlinear growth in structural equation models. Moving to these frameworks allows
for the direct identification of rates of change and accelera- tion in latent growth
curves—information available indirectly through traditional growth curve models when change
patterns are nonlinear with respect to time. To illustrate this approach, exponential growth
models in the three frameworks are fit to longitudinal response time data from the Math Skills
Development Project (Mazzocco & Meyers, 2002, 2003). We highlight the additional information gained
from fitting growth curves in these frameworks as well as limitations and extensions
of these approaches.
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- Stapleton, L.M. (2008). Variance estimation using replication methods in structural equation modeling with complex sample data. Structural Equation Modeling, 15(2), 183-210. DOI: 10.1080/10705510801922316
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Abstract
"This article discusses replication sampling variance estimation techniques that are
often applied in analyses using data from complex sampling designs: jackknife
repeated replication, balanced repeated replication, and bootstrapping. These techniques
are used with traditional analyses such as regression, but are currently not
used with structural equation modeling (SEM) analyses. This article provides an
extension of these methods to SEM analyses, including a proposed adjustment to
the likelihood ratio test, and presents the results from a simulation study suggesting
replication estimates are robust. Finally, a demonstration of the application of these
methods using data from the Early Childhood Longitudinal Study is included.
Secondary analysts can undertake these more robust methods of sampling variance
estimation if they have access to certain SEM software packages and data
management packages such as SAS, as shown in the article."
hide abstract
- Asparouhov, T. & Muthén, B. (2007).
Testing for informative weights and weights trimming in multivariate modeling with survey data.
Proceedings of the 2007 JSM meeting in Salt Lake City, Utah, Section on Survey Research Methods.
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Abstract
"Analyzing the informativeness of the sampling weights can lead to significant improvement in the precision of
model estimation with survey data. A test for weights ignorability was proposed in Pfeffermann’s (1993). We propose
a modification of this test which improves its performance for small and medium sample size problems. We
also generalize the test to a test of equivalence between two different sets of sampling weights, which can be used
to test the informativeness of individual weight components. We evaluate the performance of these techniques
in simulation studies based on linear regression and multivariate factor analysis models. We also apply the test
of equivalence to the problem of finding the optimal level of weight trimming and illustrate this approach with a
practical example. We describe the implementation of these techniques in the software package Mplus."
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- Asparouhov, T. (2006).
General multi-level modeling with sampling weights.
Communications in Statistics: Theory and Methods, Volume 35(3), 439-460. DOI: 10.1080/03610920500476598
An earlier version of this paper appeared as Mplus Web Notes: No. 8 with the title Weighting for unequal probability of selection in multilevel modeling. Refer to Mplus
Web Notes: No. 8 for more details.
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Abstract
"In this article we study the approximately unbiased multilevel pseudo maximum likelihood (MPML) estimation
method for general multilevel modeling with sampling weights. We conduct a simulation study to
determine the effect various factors have on the estimation method. The factors we included in this
study are scaling method, size of clusters, invariance of selection, informativeness of selection,
intraclass correlation and variability of standardized weights. The scaling method is an indicator of
how the weights are normalized on each level. The invariance of the selection is an indicator of whether
or not the same selection mechanism is applied across clusters. The informativeness of the selection
is an indicator of how biased the selection is. We summarize our findings and recommend a multistage
procedure based on the MPML method that can be used in practical applications."
hide abstract
- Asparouhov, T. & Muthén, B. (2006).
Comparison of estimation methods for complex survey data analysis.
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Abstract
"Recently structural equation modeling software packages have implemented more accurate statistical methodology
for analyzing complex survey data. The computational algorithms however vary across the packages
and produce different results even for simple models. In this note we conduct simulation studies
to compare the performance of the methods implemented in Mplus and LISREL. The Mplus algorithm produced
more accurate results."
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- Asparouhov, T. & Muthén, B. (2006).
Multilevel modeling of complex survey data.
Proceedings of the Joint Statistical Meeting in Seattle, August 2006. ASA section on Survey Research Methods, 2718-2726.
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Abstract
"We describe a multivariate, multilevel, pseudo maximum likelihood estimation method for multistage stratified
cluster
sampling designs, including finite population and unequal probability sampling. Multilevel
models can
be estimated with this method while incorporating the sampling design in the standard
error computation. Design
based adjustment of the likelihood ratio test (LRT) statistic is proposed.
We also discuss multiple group
and subpopulation analysis in this context. Simulation studies
are conducted to evaluate the performance of the
proposed estimator and test statistic. We also compare
the estimators and the LRT adjustments implemented in
Mplus and LISREL in simulation studies."
hide abstract
- Asparouhov, T. (2005).
Sampling weights in latent variable modeling.
Structural Equation Modeling, 12, 411-434. DOI: 10.1207/s15328007sem1203_4
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Abstract
"In this paper we review several basic statistical tools needed for modeling data with sampling weights
that are implemented in Mplus Version 3. We illustrate these tools in simulation studies for several
latent variable models including factor analysis with continuous and categorical indicators, latent
class analysis and growth models. We review the pseudo maximum likelihood (PML) estimation method
and we show how it is used with stratifed cluster sampling. We also show how the weighted least squares
(WLS) method for estimating structural equation models with categorical and continuous outcomes
implemented in Mplus is extended to incorporate sampling weights. The performance of several chi-square
tests under unequal probability sampling is evaluated. Simulation studies compare the methods used
in several statistical packages such as Mplus, HLM, SAS Proc Mixed, MLwiN and the weighted sample
statistics method used in other software packages."
hide abstract
- Asparouhov, T. & Muthén, B. (2005). Multivariate statistical modeling with survey data. Proceedings of the Federal Committee on Statistical Methodology (FCSM) Research Conference.
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Abstract
"We describe an extension of the pseudo maximum likelihood (PML) estimation method developed by Skinner
(1989) to multistage strati¯ed cluster sampling designs, including ¯nite population and unequal probability
sampling. We conduct simulation studies to evaluate the performance of the proposed estimator.
The estimator is also compared to the general estimating equation (GEE) method for linear regression
implemented in SUDAAN. We investigate the distribution of the likelihood ratio test (LRT) statistic
based on the pseudo log-likelihood value and describe an adjustment that gives correct chi-square
distribution. The performance of the adjusted LRT is evaluated with a simulation study based on
the Behrens-Fisher problem in a strati¯ed cluster sampling design."
hide abstract
- Asparouhov, T. (2004).
Stratification in multivariate modeling.
Mplus Web Notes: No. 9.
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Abstract
"In this note we illustrate stratified complex sampling with several simulation studies implemented in
Mplus 3.1 and discuss the effect of stratification on parameter and variance estimation. We compare
the results obtained by Mplus with those obtained by SUDAAN on linear and logistic regression models."
hide abstract
- Muthén, B. & Satorra, A. (1995).
Complex sample data in structural equation modeling.
Sociological Methodology, 25, 267-316.
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Abstract
"Large-scale surveys using complex sample designs are frequently carried out by government agencies. The
statistical analysis technology available for such data is, however, limited in scope. This study
investigates and further develops statistical methods that could be used in software for the analysis
of data collected under complex sample designs. First, it identifies several recent methodological
lines of inquiry which taken together provide a powerful and general statistical basis for a complex
sample, structural equation modeling analysis. Second, it extends some of this research to new situations
of interest. A Monte Carlo study that empirically evaluates these techniques on simulated data
comparable to those in large-scale complex surveys demonstrates that they work well in practice.
Due to the generality of the approaches, the methods cover not only continuous normal variables but
also continuous non-normal variables and dichotomous variables. Two methods designed to take into account
the complex sample structure were investigated in the Monte Carlo study. One method, termed aggregated
analysis, computes the usual parameter estimates but adjusts standard errors and goodness-of-fit
model testing. The other method, termed disaggregated analysis, includes a new set of parameters
reflecting the complex sample structure. Both of the methods worked very well. The conventional
method that ignores complex sampling worked poorly, supporting the need for development of special methods
for complex survey data."
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- Planalp, E.M., Du, H., Braungart-Rieker, J.M., & Wang, W. (2016). Growth curve modeling to studying change: A comparison of approaches using longitudinal dyadic data with distinguishable dyads. Structural Equation Modeling: A
Multidisciplinary Journal, DOI: 10.1080/10705511.2016.1224088
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- Whittaker, T.A., Beretvas, S.N., & Falbo, T. (2014) Dyadic curve-of-factors model: An introduction and illustration of a model for longitudinal nonexchangeable dyadic data, Structural
Equation Modeling: A Multidisciplinary Journal, 21:2, 303-317, DOI: 10.1080/10705511.2014.882695
contact first author
- James L. Peugh, David DiLillo & Jillian Panuzio (2013)
Analyzing mixed-dyadic data using structural equation models.
Structural Equation Modeling. Pages: 314-337
DOI: 10.1080/10705511.2013.769395
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Abstract
"Mixed-dyadic data, collected from distinguishable (nonexchangeable) or indistinguishable (ex-
changeable) dyads, require statistical analysis techniques that model the variation within dyads
and between dyads appropriately. The purpose of this article is to provide a tutorial for perform-
ing structural equation modeling analyses of cross-sectional and longitudinal models for mixed
independent variable dyadic data, and to clarify questions regarding various dyadic data analysis
specifications that have not been addressed elsewhere. Artificially generated data similar to
the Newlywed Project and the Swedish Adoption Twin Study on Aging were used to illustrate analysis
models for distinguishable and indistinguishable dyads, respectively. Due to their widespread use
among applied researchers, the AMOS and Mplus statistical analysis software packages were used to
analyze the dyadic data structural equation models illustrated here. These analysis models are
presented in sufficient detail to allow researchers to perform these analyses using their preferred
statistical analysis software package."
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- Guo, J., Marsh, H. W., Parker, P. D., Dicke, T., Lüdtke, O., & Diallo, T. M. O. (2019). A Systematic Evaluation and Comparison Between Exploratory Structural Equation Modeling and Bayesian Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal. DOI: 10.1080/10705511.2018.1554999
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Abstract
In this study, we contrast two competing approaches, not previously compared, that balance the rigor of
CFA/SEM with the flexibility to fit realistically complex data. Exploratory SEM (ESEM) is claimed to provide an
optimal compromise between EFA and CFA/SEM. Alternatively, a family of three Bayesian SEMs (BSEMs)
replace fixed-zero estimates with informative, small-variance priors for different subsets of parameters: crossloadings
(CL), residual covariances (RC) or CLs and RCs (CLRC). In Study 1, using three simulation studies,
results showed that (1) BSEM-CL performed more closely to ESEM; (2) BSEM-CLRC did not provide more
accurate model estimation compared with BSEM-CL; (3) BSEM-RC provided unstable estimation; (4) different
specifications of targeted values in ESEM and informative priors in BSEM have significant impacts on model
estimation. The real data analysis (Study 2) showed that the differences in estimation between different models
were largely consistent with those in Study1 but somewhat smaller.
hide abstract
- Howard, J., Gagné, M., Morin, A.J.S., Wang, Z.N., & Forest, J. (2016). Using bifactor exploratory structural equation modeling to test for a continuum structure of motivation. Journal of Management, 44(7), 2638-2664. DOI: 0.1177/0149206316645653
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Abstract
"This paper explores the nature of workplace motivation by testing the continuum structure of
motivation proposed by self-determination theory (SDT) through the application of relatively new
and advanced methodological techniques. Specifically we demonstrate the usefulness of the
overarching bifactor exploratory structural equation modeling (B-ESEM) framework in organizational
psychology and discuss implications of such models over more traditional confirmatory factor
analyses. This framework is applied to responses obtained from 1124 Canadian employees who
completed a multidimensional measure of workplace motivation. The results support a continuum of
self-regulation but furthermore indicate the importance of accounting for both quality of
motivation in addition to its global quantity. Indeed, the results showed that specific types of
motivation explained variance in covariates over and above the variance already explained by the
global quantity of self-determination. The current study further demonstrates the limitation of the
commonly used relative autonomy index and offers alternate conceptualizations of human motivation."
hide abstract
- Vazsonyi, A. T., Ksinan, A., Mikuska, J. & Jiskrova, G. (2015). The Big Five and adolescent adjustment: An empirical test across six cultures. Personality and Individual Differences, 83, 234-244. DOI:10.1016/j.paid.2015.03.049
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- Morin, A.J.S., Arens, A.K., & Marsh, H.W. (2015). A Bifactor Exploratory Structural Equation Modeling framework for the identification of distinct sources of construct-relevant psychometric multidimensionality. Structural Equation Modeling, A Multidisciplinary Journal. DOI: 10.1080/10705511.2014.961800
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download associated webnote.
- Aichholzer, J. (2014).
Random intercept EFA of personality scales.
Journal of Research in Personality, 53: 1-4.
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- Herrmann, A., & Pfister, H. (2013).
Simple measures and complex structures: Is it worth employing a more complex model of personality in Big Five inventories?
Journal of Research in Personality. DOI: 10.1016/j.jrp.2013.05.004
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Abstract
"The poor performance of five-factor personality inventories in confirmatory factor analyses (CFAs)
prompted some to question their construct validity. Others doubted the CFA’s suitability and
suggested applying Exploratory Structural Equation Modeling (ESEM). The question arises as to
what impact the application of either method has on the construct validity of personality
inventories. We addressed this question by applying ESEM and CFA to construct better-fitting,
though more complex models based on data from two questionnaires (NEO PI-R and 16PF). Generally,
scores derived from either method did not differ substantially. When applying ESEM, convergent
validity declined but discriminant validity improved. When applying CFA, convergent and
discriminant validity decreased. We conclude that using current personality questionnaires that
utilize a simple structure is appropriate."
hide abstract
- Morin, A.J.S., Marsh, H.W., & Nagengast, B. (2013).
Chapter 10. Exploratory Structural Equation Modeling. In Hancock, G. R., & Mueller, R. O. (Eds.). (2013).
Structural equation modeling: A second course (2nd ed.).
Charlotte, NC: Information Age Publishing, Inc.
Supplementary materials used in this chapter can be found here and here.
contact first author
- Marsh, H. W., Nagengast, B. & Morin, A. J. S. (2012).
Measurement invariance of big-five factors over the life span: ESEM tests of gender, age, plasticity,
maturity, and La
Dolce Vita effects.
Developmental Psychology. DOI: 10.1037/a0026913
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Abstract
"This substantive-methodological synergy applies evolving approaches to factor analysis to substantively
important
developmental issues of how five-factor-approach (FFA) personality measures vary with gender,
age,
and their interaction. Confirmatory factor analyses (CFAs) conducted at the item level often
do not
support a priori FFA structures, due in part to the overly restrictive assumptions of CFA
models. Here we
demonstrate that exploratory structural equation modeling (ESEM), an integration of
CFA and exploratory
factor analysis, overcomes these problems with the 15-item Big Five Inventory
administered as part of the
nationally representative British Household Panel Study (N 14,021; age:
15–99 years,Mage 47.1). ESEM
fitted the data substantially better and resulted in much more differentiated
(less correlated) factors than did
CFA. Methodologically, we extended ESEM (introducing ESEM-within-CFA
models and a hybrid of
multiple groups and multiple indicators multiple causes models),
evaluating full measurement invariance and
latent mean differences over age, gender, and their interaction.
Substantively the results showed that women
had higher latent scores for all Big Five factors
except for Openness and that these gender differences were
consistent over the entire life span. Substantial
nonlinear age effects led to the rejection of the plaster
hypothesis and the maturity principle
but did support a newly proposed la dolce vita effect in old age. In later
years, individuals
become happier (more agreeable and less neurotic), more self-content and self-centered (less
extroverted
and open), more laid back and satisfied with what they have (less conscientious, open, outgoing
and
extroverted), and less preoccupied with productivity."
hide abstract
- Marsh, H. W., Lüdtke, O., Muthén, B., Asparouhov, T., Morin, A. J. S., Trautwein, U. & Nagengast, B. (2010).
A new look at the big-five factor structure through exploratory structural equation modeling.
Psychological Assessment, 22, 471-491.
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Abstract
"NEO instruments are widely used to assess big-five personality factors, but confirmatory factor analyses
(CFA) conducted at the item-level do not support their a priori structure, due in part, to the overly
restrictive CFA assumptions. We demonstrate that exploratory structural equation modeling (ESEM),
an integration of CFA and EFA, overcomes these problems with responses (n= 3390) to the 60-item
NEO-FFI: (a) ESEM fits the data better and results in substantially more differentiated (less correlated)
factors than CFA; (b) tests of gender invariance with the 13-model ESEM taxonomy of full measurement
invariance of factor loadings, factor variance-covariances, item uniquenesses, correlated uniquenesses,
item intercepts, differential item functioning, and latent means show that women score higher
on all NEO big-five factors; (c) longitudinal analyses support measurement invariance over time
and the maturity principle (decreases in Neuroticism, increases in Agreeableness, Openness and Conscientiousness).
Based on ESEM, we addressed substantively important questions with broad applicability
to personality research that could not be appropriately addressed with either traditional EFA or
CFA approaches.
"
hide abstract
- Asparouhov, T. & Muthén, B. (2009).
Exploratory structural equation modeling.
Structural Equation Modeling, 16, 397-438.
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Abstract
"Exploratory factor analysis (EFA) is a frequently used multivariate analysis technique in statistics.
Jennrich
and Sampson (1966) solved a significant EFA factor loading matrix rotation problem
by deriving
the direct Quartimin rotation. Jennrich was also the first to develop standard errors
for rotated
solutions, although these have still not made their way into most statistical software
programs.
This is perhaps because Jennrich’s achievements were partly overshadowed by the
subsequent development
of confirmatory factor analysis (CFA) by Jöreskog (1969). The strict
requirement of zero cross-loadings
in CFA, however, often does not fit the data well and has
led to a tendency to rely on extensive
model modification to find a well-fitting model. In such
cases, searching for a well-fitting measurement
model may be better carried out by EFA (Browne,
2001). Furthermore, misspecification of zero
loadings usually leads to distorted factors with overestimated
factor correlations and subsequent
distorted structural relations. This article describes an
EFA-SEM (ESEM) approach, where in addition
to or instead of a CFA measurement model, an
EFA measurement model with rotations can be used in a
structural equation model. The ESEM
approach has recently been implemented in the Mplus program. ESEM
gives access to all the
usual SEM parameters and the loading rotation gives a transformation of structural
coefficients
as well. Standard errors and overall tests of model fit are obtained. Geomin and
Target rotations
are discussed. Examples of ESEM models include multiple-group EFA with measurement
and
structural invariance testing, test–retest (longitudinal) EFA, EFA with covariates and direct
effects,
and EFA with correlated residuals. Testing strategies with sequences of EFA and CFA models
are
discussed. Simulated and real data are used to illustrate the points."
hide abstract
- Marsh, H.W., Muthén, B., Asparouhov, A., Lüdtke, O., Robitzsch, A., Morin, A.J.S., & Trautwein, U. (2009).
Exploratory structural equation modeling, integrating CFA and EFA: Application to students’ evaluations
of university teaching.
Structural Equation Modeling, 16, 439-476.
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Abstract
"This study is a methodological-substantive synergy, demonstrating the power and flexibility of
exploratory
structural equation modeling (ESEM) methods that integrate confirmatory and exploratory
factor
analyses (CFA and EFA), as applied to substantively important questions based on
multidimentional students’
evaluations of university teaching (SETs). For these data, there is a well
established ESEM
structure but typical CFA models do not fit the data and substantially inflate correlations among the
nine SET factors (median rs D .34 for ESEM, .72 for CFA) in a way that
undermines discriminant validity
and usefulness as diagnostic feedback. A 13-model taxonomy of
ESEM measurement invariance is
proposed, showing complete invariance (factor loadings, factor
correlations, item uniquenesses, item
intercepts, latent means) over multiple groups based on the
SETs collected in the first and second
halves of a 13-year period. Fully latent ESEM growth models
that unconfounded measurement error from
communality showed almost no linear or quadratic
effects over this 13-year period. Latent multiple
indicators multiple causes models showed that
relations with background variables (workload/difficulty,
class size, prior subject interest, expected
grades) were small in size and varied systematically
for different ESEM SET factors, supporting
their discriminant validity and a construct validity interpretation
of the relations. A new approach to
higher order ESEM was demonstrated, but was not fully
appropriate for these data. Based on ESEM
methodology, substantively important questions were addressed
that could not be appropriately
addressed with a traditional CFA approach."
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- Kam, C. C. S. & Cheung, S.F. (2023). A constrained factor mixture model for detecting careless responses that is simple to implement. Organizational Research Methods. DOI: 10.1177/10944281231195298
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- Kam, C. C. S., & Fan, X. (2020). Investigating response heterogeneity in the context of positively and negatively worded items by using factor mixture modeling. Organizational Research Methods, 23(2) 322-341. DOI: 10.1177/1094428118790371
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Abstract
"Factor mixture modeling was used to investigate potential response incongruity between positively and negatively worded items. Survey respondents (N = 591) answered
questions about job satisfaction and dissatisfaction. Results revealed two classes of respondent: a majority class, who generally do not have problems answering positively
and negatively worded items; and a minority class, who have serious trouble with negatively worded items. With the exclusion of the minority class, job satisfaction and
dissatisfaction were found to be essentially unidimensional, rather than bidimensional as previous research had suggested. These results not only challenge previous findings
regarding the bidimensionality of job satisfaction, but also question the widespread research practice of assuming population homogeneity in survey responses. A flow diagram
illustrating the analytic procedure and an Mplus syntax program are provided so that researchers can conduct similar investigations on constructs of interest."
hide abstract
- ten Have, M., Lamers, F., Wardenaar, K., Beekman, A., de Jonge, P., van Dorsselaer, S., Tuithof, M., Kleinjan, M. & de Graaf, R. (2016). The identification of symptom-based subtypes of depression: A nationally representative cohort study. Journal of Affective Disorders, 190, 395-406. DOI 10.1016/j.jad.2015.10.040
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- McLarnon, M. J. W., Carswell, J. J., & Schneider, T. J. (2015). A case of mistaken identity? Latent profiles in vocational interests. Journal of Career Assessment, 23, 166-185. DOI:10.1177/1069072714523251
view abstract
contact first author.
- Morin, A.J.S., & Marsh, H.W. (2014). Disentangling shape from levels effects in person-
centered analyses: An illustration based university teacher multidimensional profiles of
effectiveness. Structural Equation Modeling: A Multidisciplinary Journal, 21: 1–21.
contact first author
- Wall, M. M., Guo, J., & Amemiya, Y. (2012).
Mixture factor analysis for approximating a nonnormally distributed continuous latent factor with continuous
and dichotomous observed variables.
Multivariate Behavioral Research,
47:2, 276-313.
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Abstract
"Mixture factor analysis is examined as a means of flexibly estimating nonnor-
mally distributed continuous
latent factors in the presence of both continuous
and dichotomous observed variables. A simulation
study compares mixture factor
analysis with normal maximum likelihood (ML) latent factor modeling.
Different
results emerge for continuous versus dichotomous outcomes. For dichotomous
outcomes, normal
ML path estimates have bias that worsens as latent factor
skew/kurtosis increases and does not
diminish as sample size increases, whereas
the mixture factor analysis model produces nearly unbiased
estimators as sample
sizes increase (500 and greater) and offers near nominal coverage probability.
For
continuous outcome variables, both methods produce factor loading estimates with
minimal bias
regardless of latent factor skew, but the mixture factor analysis is more
efficient. The method is
demonstrated using data motivated by a study on youth with cystic fibrosis examining predictors of treatment
adherence. In summary,
mixture factor analysis provides improvements over normal ML estimation
in the
presence of skewed/kurtotic latent factors, but due to variability in the estimator
relating
the latent factor to dichotomous outcomes and computational issues, the
improvements were only fully
realized, in this study, at larger sample sizes (500
and greater)."
hide abstract
- Tueller, S.J,, Drotar, S. & Lubke, G.H. (2011). Addressing the problem of switched class labels in latent variable mixture model simulation studies. Structural Equation Modeling, 18, 110-131.
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Abstract
"The discrimination between alternative models and the detection of latent classes in the context of latent variable mixture modeling depends on sample size, class separation, and other aspects that are related to power. Prior to a mixture analysis it is useful to investigate model performance in
a simulation study that reflects the research settings. Multiple data sets are generated under 1 or more models, and alternative models are fitted to the data. The aggregation of results over multiple data
sets is complicated by the fact that mixture models are only identified up to a permutation of the class labels. Estimated class labels are arbitrary, with the effect that the estimated parameters for
Class 1 could be incorrectly labeled as Class 2, Class 3, and so forth, relative to their data generating labels. In a simulation study, the detection of switched labels needs to be automated. Switched
class labels are not necessarily simple to detect. This article describes different possible scenarios of switched class labels, and develops an algorithm implemented in R that (a) detects switched
labels, and (b) provides information that can be used to either correct class labels or to discard a particular data set from a simulation if class labels are ambivalent. The algorithm is useful in
Monte Carlo simulations involving latent variable mixture models."
hide abstract
- Walton, K.E., Ormel, J. & Krueger, R.F. (2011). The dimensional nature of externalizing behaviors in adolescence: Evidence from a direct comparison of categorical, dimensional, and hybrid models. Journal of Abnormal Child Psychology, 39(4): 553-61. DOI: 10.1007/s10802-010-9478-y
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Abstract
"Researchers have recognized the importance of developing an accurate classification system for externalizing
disorders, though much of this work has been framed by a priori preferences for categorical
vs. dimensional constructs. Newer statistical technologies now allow categorical and dimensional models
of psychopathology to be compared empirically. In this study, we directly compared the fit of categorical
and dimensional models of externalizing behaviors in a large and representative community
sample of adolescents at two time points separated by nearly 2.5 years (N = 2027; mean age at Time 1
= 11.09 years; 50.8% female). Delinquent and aggressive behaviors were assessed with child and parent
Child Behavior Checklist reports. Latent trait, latent class, and factor mixture models were fit
to the data, and at both time points, the latent trait model provided the best fit to the data. The
item parameters were inspected and interpreted, and it was determined that the items were differentially
sensitive across all regions of the dimension. We conclude that classification models can be based
on empirical evidence rather than a priori preferences, and while current classification systems
conceptualize externalizing problems in terms of discrete groups, they can be better conceptualized
as dimensions."
hide abstract
- Clark, S.L. (2010).
Mixture modeling with behavioral data.
Doctoral dissertation, University of California, Los Angeles.
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Abstract
"United States schools and students suffer from problems associated with student
behavioral disorders.
There is a need for innovate statistical methods to analyze data to
which will help inform the development
of new strategies to deal with the issues
associated with behavioral problems. The three papers
in this dissertation focus on
explicating certain mixture models which have shown promise in analyzing
behavioral
data. An important interest in mixture modeling is the investigation of what types
of
individuals belong to each latent class by relating classes to auxiliary variables.
The first paper
presents results from real data examples and simulations to show how
various factors, such as sample
size, can impact the estimates and standard errors of
auxiliary variable effects and testing mean
equality across classes. Based on the results of
the examples and simulations, suggestions are made
about how to select auxiliary
variables for a latent class analysis (LCA). The second paper discusses
the factor mixture
model (FMM) which uses a hybrid of both categorical and continuous latent variables.
The
FMM is a good model for the underlying structure of behavioral disorders because
the use
of both categorical and continuous latent variables allows the structure to be
simultaneously categorical
and dimensional. The use of the FMM in behavioral research
is not prevalent because there is
little research about how the FMM should be applied in
practice. This paper explores the FMM by studying
two real data examples: conduct
disorder and attention-deficit hyperactivity disorder. Through
these examples, this paper
aims to explain the different formulations of the FMM, the various steps
in building a
FMM, as well as how to decide between a FMM and alternative models. The third paper
explores
of the use of two mixture model as potential phenotypes in ACE analysis: LCA
and FMM. The use
of these models as phenotypes is demonstrated through an example
concerning conduct disorder in a
sample of Finnish twins. A discussion about extending
the models in this dissertation to be applicable
to longitudinal data or include gene by
environment (GxE) interactions is also presented."
hide abstract
- Leite, W. & Cooper, L. (2010).
Detecting social desirability bias using factor mixture models.
Multivariate Behavioral Research, 45:2, 271-293.
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Abstract
"Based on the conceptualization that social desirable bias (SDB) is a discrete event
resulting from an
interaction between a scale’s items, the testing situation, and the
respondent’s latent trait on a
social desirability factor, we present a method that
makes use of factor mixture models to identify
which examinees are most likely to
provide biased responses, which items elicit the most socially desirable
responses,
and which external variables predict SDB. Problems associated with the common
use
of correlation coefficients based on scales’ total scores to diagnose SDB and
partial correlations
to correct for SDB are discussed. The method is demonstrated
with an analysis of SDB in the Attitude
toward Interprofessional Service-Learning
scale with a sample of students from health-related fields."
hide abstract
- Masyn, K. E. & Henderson, C.E. (2010).
Exploring the latent structures of psychological constructs in social development using the dimensional-categorical
spectrum.
Social Development, 19, 3, 2010.
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Abstract
"This paper provides an introduction to a recently developed conceptual framework—the dimensional–categorical
spectrum—for utilizing
general factor mixture models to explore the latent structures of psychological
constructs. This framework offers advantages over
traditional latent variable models that
usually employ either continuous latent factors or categorical latent class variables to characterize
the
latent structure and require an a priori assumption about the underlying nature of the construct
as either purely dimension or purely
categorical. The modeling process is discussed in detail and
then illustrated with data on the delinquency items of Achenbach's child
behavior checklist from a
sample of children in the National Adolescent and Child Treatment Study."
hide abstract
- Reynolds, M.R., Keith, T.Z. & Beretvas, S.N. (2010).
Use of factor mixture modeling to capture Spearman's law of diminishing returns.
Intelligence 38, 231–241.
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Abstract
"Spearman's law of diminishing returns (SLODR) posits that at higher levels of general cognitive
ability
the general factor (g) performs less well in explaining individual differences in cognitive
test
performance. Research has generally supported SLODR, but previous research has required
the a priori
division of respondents into separate ability or IQ groups. The present study sought
to obviate this
limitation through the use of factor mixture modeling to investigate SLODR in
the Kaufman Assessment
Battery for Children-Second Edition (KABC-II). A second-order
confirmatory factor model was modeled
as a within-class factor structure. The fit and
parameter estimates of several models with varying
number of classes and factorial invariance
restrictions were compared. Given SLODR, a predictable
pattern of findings should emerge
when factor mixture modeling is applied. Our results were consistent
with these SLODR-based
predictions, most notably the g factor variance was less in higher g mean
classes. Use of factor
mixture modeling was found to provide support for SLODR while improving the model
used to
investigate SLODR."
hide abstract
- Clark, S.L., Muthén, B., Kaprio, J., D’Onofrio, B.M., Viken, R., Rose, R.J., Smalley, S. L. (2009). Models and strategies for factor mixture analysis: Two examples concerning the structure underlying psychological disorders.
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show abstract
Abstract
"The factor mixture model (FMM) uses a hybrid of both categorical and continuous latent
variables. The
FMM is a good model for the underlying structure of psychopathology because
the use of both categorical
and continuous latent variables allows the structure to be
simultaneously categorical and dimensional.
While the conceptualization of the FMM has been
explained in the literature, the use of the
FMM is still not prevalent. One reason is that there is
little research about how such models should
be applied in practice and, once a well fitting
model is obtained, how it should be interpreted. In
this paper, the FMM will be explored in more
detail by studying two real data examples: conduct disorder
and attention-deficit hyperactivity
disorder. By exploring these examples, this paper aims to
explain the different formulations of
the factor mixture model, the various steps in building a factor
mixture model, as well as how to
decide between a factor mixture model and alternative models."
hide abstract
- Kim, Y.K. & Muthén, B. (2009).
Two-part factor mixture modeling: Application to an aggressive behavior measurement instrument.
Structural Equation Modeling, 16, 602-624.
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Abstract
"This study introduces a two-part factor mixture model as an alternative analysis approach to modeling
data where strong floor effects exist in the measured items. This method, which builds upon already
established modeling techniques for longitudinal data, provides the possibility of developing new measurement
models for data where a substantial portion of the sample have not yet experienced the behavior.
It does so by identifying latent classes through a three-step modeling approach. The method
is applied to data from a randomized preventive intervention trial in Baltimore public schools administered
by the Johns Hopkins Center for Early Intervention. The proposed model revealed otherwise unobserved
subpopulations among the children in the study in terms of their tendency towards and level
of aggression. Furthermore, the modeling approach was validated through a Monte Carlo simulation."
hide abstract
- Lubke, G. & Muthén, B. (2007).
Performance of factor mixture models as a function of model size, covariate effects, and class-specific
parameters.
Structural Equation Modeling, 14(1), 26–47.
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Abstract
"Factor mixture models are designed for the analysis of continuous multivariate data, assuming that one
or more common factors capture the common content of the observed variables, and that the population,
from which the data are obtained, consists of several distinct latent classes. Factor mixture modeling
involves obtaining estimates of the model parameters, and assigning subjects to their most likely
latent class. In the present simulation study parameter coverage and correct class membership assignment
are quantified for different factor mixture models with increasing covariate effects and increasing
class separation. The investigated models are the latent profile model, 1-, 2-, 3-factor models,
and the linear growth mixture model. Parameter coverage is good throughout, as are convergence
rates. Correct class assignment is unsatisfactory for small class separation without covariates, but
improves dramatically with increasing separation and/or covariate effects. Model performance does
not depend on the type of factor mixture model, or the number of observed variables."
hide abstract
- Lubke, G., Muthén, B., Moilanen, I., McGough, J., Loo, S., Swanson, J., Yang, M., Taanila, A., Hurtig, T., Jarvelin, M. & Smalley, S. (2007).
Subtypes versus severity differences in the Attention-Deficit/Hyperactivity disorder in the northern
Finnish birth cohort.
Journal of the American Academy of Child and Adolescent Psychiatry, 46, 1584-1593.
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Abstract
"Objective: To investigate whether behaviors of inattention, hyperactivity, and impulsivity among adolescents
in Northern
Finland reflect qualitatively distinct subtypes of ADHD, variants along a single
continuum of severity, or of severity
differences within subtypes. Method: Latent class models, exploratory
factor models, and factor mixture models were
applied to questionnaire data of ADHD behaviors
obtained from the Northern Finland Birth Cohort (NFBC). Latent class
models correspond to qualitatively
distinct subtypes, factor analysis corresponds to severity differences, and factor mixture
analysis
allows for both subtypes and severity differences within subtypes. Results: A comparison of the
different models
shows that models that distinguish between a low scoring majority class (unaffecteds)
and a high scoring minority class
(affecteds), and allow for two factors (inattentive, hyperactive-impulsive)
with severity differences provide the best fit.
Conclusions: The analysis provides support
that a high-scoring minority group (8.8% of males and 6.8% of females) likely
reflects an ADHD
group in the Northern Finland Birth Cohort, whereas the majority of the population falls into a low-scoring
group
of unaffecteds. Distinct factors composed of items of inattention and hyperactivity-impulsivity
are evident for both
sexes with considerable variability in severity within each class.
J. Am. Acad. Child Adolesc. Psychiatry,
2007;46(12):1584Y1593. Key Words: latent class analysis, factor
analysis, factor mixture analysis."
hide abstract
- Lubke, G. & Neale, M. (2006).
Distinguishing between latent classes and continuous factors: Resolution by maximum likelihood?
Multivariate Behavioral Research, 41(4), 499–532.
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Abstract
"Latent variable models exist with continuous, categorical, or both types of latent
variables. The role
of latent variables is to account for systematic patterns in the
observed responses. This article
has two goals: (a) to establish whether, based on
observed responses, it can be decided that an underlying
latent variable is continuous
or categorical, and (b) to quantify the effect of sample size and
class proportions
on making this distinction. Latent variable models with categorical, continuous,
or
both types of latent variables are fitted to simulated data generated under
different types of
latent variable models. If an analysis is restricted to fitting continuous
latent variable models assuming
a homogeneous population and data stem
from a heterogeneous population, overextraction of factors
may occur. Similarly, if
an analysis is restricted to fitting latent class models, overextraction
of classes may
occur if covariation between observed variables is due to continuous factors. For
the
data-generating models used in this study, comparing the fit of different exploratory
factor mixture
models usually allows one to distinguish correctly between
categorical and/or continuous latent
variables. Correct model choice depends on
class separation and within-class sample size."
hide abstract
- Muthén, B. (2006).
Should substance use disorders be considered as categorical or dimensional?
Addiction, 101 (Suppl. 1), 6-16.
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Abstract
"This paper discusses the representation of diagnostic criteria using categorical and dimensional statistical
models. Conventional modeling using categorical or continuous latent variables in the form of
latent class analysis and factor (IRT) analysis has limitations for the analysis of diagnostic criteria.
New hybrid models are discussed which provide both categorical and dimensional representations
in the same model using mixture models. Conventional and new models are applied and compared using
recent data for DSM-IV alcohol dependence and abuse criteria from the National Epidemiologic Survey
on Alcohol and Related Conditions. It is found that new hybrid mixture models are more suitable than
latent class and factor (IRT) models. Classification results from hybrid models are compared to the
DSM-IV approach of using the number of diagnostic criteria fulfilled. Implications for DSM-V are discussed
in terms of reporting results using both categories and dimensions."
hide abstract
- Muthén, B. & Asparouhov, T. (2006).
Item response mixture modeling: Application to tobacco dependence criteria.
Addictive Behaviors, 31, 1050-1066.
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Abstract
"This paper illustrates new hybrid latent variable models that are promising for phenotypical analyses.
The hybrid models combine features of dimensional and categorical analyses seen in the conventional
techniques of factor analysis and latent class analysis. The paper focuses on the analysis of categorical
items, which presents especially challenging analyses with hybrid models and has recently
been made practical in the Mplus program. The hybrid models are typically seen to fit data better
than conventional models of factor analysis (IRT) and latent class analysis. An illustration is given
in the form of analysis of tobacco dependence in a general population survey."
hide abstract
- Lubke, G.H. & Muthén, B. (2005).
Investigating population heterogeneity with factor mixture models.
Psychological Methods, 10, 21-39.
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Abstract
"Sources of population heterogeneity may or may not be known. Factor mixture models can be used to explore
unknown population heterogeneity while integrating known sources as covariates. Different ways
to incorporate covariates are discussed in detail. Advantages of factor mixture modeling are described
in comparison to other methods designed for data stemming from heterogenous populations. A step-by-step
analysis of a subset of data from the Longitudinal Survey of American Youth (LSAY) illustrates
how factor mixture models can be applied in an exploratory fashion to data collected at a single time
point."
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collapse topic
- Litson, K., Thornhill, C., Geiser, C., Burns, G.L., & Servera, M. Applying and interpreting mixture distribution latent state-trait models. (2019). Structural Equation Modeling: A Multidisciplinary Journal 26(6), 931-947. DOI: 10.1080/10705511.2019.1575741
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contact first author.
- O’Neill, T. A., McLarnon, M. J. W., Hoffart, G. C., Woodley, H. J., & Allen, N. A. (2018). The structure and function of team conflict state profiles. Journal of Management, 44(2), 811-836. DOI:10.1177/0149206315581662
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contact first author.
- Morin, A.J.S., & Wang, J.C.K. (2016). A gentle introduction to mixture modeling using physical fitness data. In N. Ntoumanis, & N. Myers (Eds.), An Introduction to Intermediate and Advanced Statistical Analyses for Sport and Exercise Scientists (pp. 183-210). London, UK: Wiley
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view supplement
- Morin, A. J. S. (2016). Person-centered research strategies in commitment research. In J.P. Meyer (Ed.), The handbook of employee commitment. Cheltenham, UK: Edward Elgar.
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- Sterba, S. (2013) Understanding linkages among mixture nodels. Multivariate Behavioral Research, 48:775-815. DOI: 10.1080/00273171.2013.827564
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show abstract
Abstract
The methodological literature on mixture modeling has rapidly expanded in the past
15 years, and mixture models are increasingly applied in practice. Nonetheless, this literature
has historically been diffuse, with different notations, motivations, and parameterizations making
mixture models appear disconnected. This pedagogical review facilitates an integrative
understanding of mixture models. First, 5 proto- typic mixture models are presented in a unified
format with incremental complexity while highlighting their mutual reliance on familiar
probability laws, common assumptions, and shared aspects of interpretation. Second, 2 recent
extensions— hybrid mixtures and parallel-process mixtures—are discussed. Both relax a key
assumption of classic mixture models but do so in different ways. Similarities in construction and
interpretation among hybrid mixtures and among parallel-process mixtures are emphasized. Third, the
combination of both extensions is motivated and illustrated by means of an example on
oppositional defiant and depressive symptoms. By clarifying how existing mixture models relate and
can be combined, this article bridges past and current developments and provides a foundation for
understanding new developments.
hide abstract
- Muthén, B. (2008).
Latent variable hybrids: Overview of old and new models.
In Hancock, G. R., & Samuelsen, K. M. (Eds.), Advances in latent variable mixture models, pp. 1-24. Charlotte, NC: Information Age Publishing, Inc.
Click here for information
about the book.
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contact author
- Muthén, B. (2002).
Beyond SEM: General latent variable modeling.
Behaviormetrika, 29, 81-117.
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show abstract
Abstract
"This article gives an overview of statistical analysis with latent variables. Using traditional structural
equation modeling as a starting point, it shows how the idea of latent variables captures a
wide variety of statistical concepts, including random effects, missing data, sources of variation in
hierarchical data, finite mixtures, latent classes, and clusters. These latent variable applications
go beyond the traditional latent variable useage in psychometrics with its focus on measurement
error and hypothectical constructs measured by multiple indicators. The article argues for the value
of integrating statistical and psychometric modeling ideas. Different applications are discussed
in a unifying framework that brings together in one general model such different analysis types as
factor models, growth curve models, multilevel models, latent class models and discrete- time survival
models. Several possible combinations and extensions of these models are made clear due to the unifying
framework."
hide abstract
- Muthén, B. (2001).
Latent variable mixture modeling.
In G. A. Marcoulides & R. E. Schumacker (eds.), New Developments and Techniques in Structural Equation Modeling (pp. 1-33). Lawrence Erlbaum Associates.
Mplus inputs and outputs used in this paper can be viewed and/or downloaded from the Examples
page.
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show abstract
Abstract
"This chapter gives an overview of latent variable modeling with both categorical and continuous latent
variables. Conventional latent class, structural equation, and growth models are extended and integrated
in a general modeling framework."
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collapse topic
- Tsang, S., Duncan, G.E., Dinescu, D, & Turkheimer, E. (2018) Differential models of twin correlations in skew for body-mass index (BMI). PLoS ONE 13(3): e0194968. DOI:10.1371/journal.pone.0194968
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contact first author
- Paquin, S., Lacourse, E., Brendgen, M., Vitaro, F., Dionne, G., Tremblay, R.E., & Boivin, M. (2017). Heterogeneity in the development of proactive and reactive aggression in childhood: Common and specific genetic - environmental factors. PLoS ONE 12(12): e0188730. DOI: 10.1371/journal.pone.0188730
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- Dinescu, D., Horn, E. E., Duncan, G., & Turkheimer, E. (2016). Socioeconomic modifiers of genetic and environmental influences on body mass index in adult twins. Health Psychology. DOI: 10.1037/hea0000255
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- Lacourse, E., Boivin, M., Brendgen, M., Petitclerc, A., Girard, A., Vitaro, F., Paquin, S., Ouellet-Morin, I., Dionne, G., & Tremblay, R.E. (2014). A longitudinal twin study of physical aggression during early childhood: evidence for a developmentally dynamic genome. Psychological Medicine, 44:12, 2617-2627, DOI: 10.1017/S0033291713003218
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- Zyphur, M. J., Zhang, Z., Barsky, A. P., & Li, W. D. (2013). An ACE in the hole: Twin family models for applied behavioral genetics research.. The Leadership Quarterly, 24(4), 572-594. DOI: 10.1016/j.leaqua.2013.04.001
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- Hunter, A.M., Leuchter, A.F., Power, R.A., Muthén, B., McGrath, P.J., Lewis, C.M., Cook, I.A., Garriock, H.A., McGuffin, P., Uher, R., & Hamilton, S.P. (2013). A genome-wide association study of a sustained pattern of antidepressant response. Journal of Psychiatric Research, 47(9), 1157-1165. DOI: 10.1016/j.jpsychires.2013.05.002
view abstract
- Kerner, B. & Muthén, B. (2009).
Growth mixture modelling in families of the Framingham Heart Study.
BMC Proceedings, 3, 1-5.
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Abstract
"Growth mixture modelling is a less explored method in genetic research to address unobserved heterogeneity
in population samples. Here, we applied this technique to longitudinal data of the Framingham
Heart Study. We examined systolic blood pressure measures in 1060 males from 692 families and detected
three subclasses, which varied significantly in their developmental trajectories over time. The first
class consisted of 60 high-risk individuals with elevated blood pressure early in life and a steep
increase over time. The second group of 131 individuals displayed first normal blood pressure, but
showed a significant increase over time and reached high blood pressure values late in their life
time. The largest group of 869 individuals could be considered a normative group with normal blood
pressure on all exams. In order to identify genetic modulators for this phenotype we tested 2340 single
nucleotide polymorphisms on Chromosome 8 for association with the class membership probabilities
of our model. The probability of being in Class 1 was significantly associated with a very rare variant
(rs1445404) present in only four individuals from four different families located in the coding
region of the gene EYA (eyes absent homolog 1 in Drosophila) (p= 1.39 x 10-13). Mutations in EYA are
known to cause Brachio-Oto-Renal syndrome, as well as isolated renal malformations. Renal malformations
could cause high blood pressure early in life. This result awaits replication; however, it suggests
that analyzing genetic data stratified for high-risk subgroups defined by a unique development
over time could be useful for the detection of rare mutations in common multi-factorial diseases."
hide abstract
- Bartels, M., Cacioppo, J.T., Hudziak, J.J., & Boomsma, D.I. (2008).
Genetic and environmental contributions to stability in loneliness throughout childhood.
American Journal of Medical Genetics Part B, 147B, 385-391.
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- D’Onofrio, B.M., Hulle, C.A., Waldman, I.D., Rodgers, J. L. Harden, K.P., Rathouz, P.J. & Lahey, B.B. (2008).
Smoking during pregnancy and offspring externalizing problems: An exploration of genetic and environmental
confounds.
Development and Psychopathology, 20, 139-164.
Mplus scripts can be obtained from the first author.
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Abstract
"Previous studies have documented that smoking during pregnancy (SDP) is associated with offspring externalizing
problems,
even when measured covariates were used to control for possible confounds. However,
the association may be
because of nonmeasured environmental and genetic factors that increase risk
for offspring externalizing problems. The
current project used the National Longitudinal Survey
of Youth and their children, ages 4–10 years, to explore the relations
between SDP and offspring conduct
problems (CPs), oppositional defiant problems (ODPs), and attention-deficit/
hyperactivity problems
(ADHPs) using methodological and statistical controls for confounds. When offspring were
compared
to their own siblings who differed in their exposure to prenatal nicotine, there was no effect of
SDP on
offspring CP and ODP. This suggests that SDP does not have a causal effect on offspring CP and
ODP. There was a
small association between SDP and ADHP, consistent with a causal effect of SDP,
but the magnitude of the association
was greatly reduced by methodological and statistical controls.
Genetically informed analyses suggest that unmeasured
environmental variables influencing both SDP
and offspring externalizing behaviors account for the previously
observed associations. That is, the
current analyses imply that important unidentified environmental factors account
for the association
between SDP and offspring externalizing problems, not teratogenic effects of SDP."
hide abstract
- Rathouz, P.J., Hulle, C.A., Rodgers, J.L., Waldman, I.D., & Lahey, B.B. (2008).
Specification, testing, and interpretation of gene-by-measured-environment interaction models in the
presence of gene-environment correlations.
Behavior Genetics, 38, 301-315.
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show abstract
Abstract
"Abstract Purcell (Twin Res 5:554–571, 2002) proposed
a bivariate biometric model for testing and quantifying
the
interaction between latent genetic influences and measured
environments in the presence
of gene–environment correlation.
Purcell’s model extends the Cholesky model to
include gene–environment
interaction. We examine a
number of closely related alternative models that do not
involve gene–environment
interaction but which may fit
the data as well as Purcell’s model. Because failure to
consider
these alternatives could lead to spurious detection
of gene–environment interaction, we propose
alternative
models for testing gene–environment interaction in the
presence of gene–environment correlation,
including one
based on the correlated factors model. In addition, we note
mathematical errors
in the calculation of effect size via
variance components in Purcell’s model. We propose a
statistical
method for deriving and interpreting variance
decompositions that are true to the fitted model."
hide abstract
- Boomsma, D., Cacioppo, J., Muthén, B., Asparouhov, T. & Clark, S. (2007).
Longitudinal genetic analysis for loneliness in Dutch twins.
Twin Research and Human Genetics, 10, 267-273.
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Abstract
"In previous studies we obtained evidence that variation in loneliness has a genetic component.
Based
on adult twin data, the heritability estimate for loneliness, which was assessed as an
ordinal trait,
was 48%. These analyses were done on loneliness scores averaged over items (“I
feel lonely” and “Nobody
loves me”) and over time points. In this paper we present a
longitudinal analysis of loneliness
data assessed in 5 surveys (1991 through 2002) in Dutch
twins (N=8,389) for the two separate items
of the loneliness scale.
From the longitudinal growth modelling it was found sufficient to have
non-zero variance for
the intercept only, while the other effects (linear, quadratic and cubic slope)
had zero variance.
For the item “I feel lonely” we observed an increasing age trend up to age 30,
followed by a
decline to age 50. Heritability for individual differences in the intercept was estimated
at 77%.
For the item “Nobody loves me” no significant trend over age was seen; the heritability
of the
intercept was estimated at 70%."
hide abstract
- Harden, K.P., Turkheimer, E. & Loehlin, J.C. (2006).
Genotype by environment interaction in adolescents’ cognitive aptitude.
Behavioral Genetics.
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Abstract
"In a replication of Turkheimer, Haley, Waldron, D’Onofrio, Gottesman II (2003, Socioeconomic status modifies
heritability of IQ in young children. Psychological Science, 14:623-628), we investigate genotype–environment
(G · E) interaction in the cognitive aptitude of 839 twin pairs who completed the
National Merit Scholastic Qualifying Test in 1962. Shared environmental influences were stronger for
adolescents from poorer homes, while genetic influences were stronger for adolescents from more affluent
homes. No significant differences were found between parental income and parental education interaction
effects. Results suggest that environmental differences between middle- to upper-class families
influence the expression of genetic potential for intelligence, as has previously been suggested
by Bronfenbrenner and Ceci’s (1994, Nature-nurture reconceptualized in developmental perspective:
a bioecological Model Psychological Review, 101:568-586) bioecological model."
hide abstract
- Muthén, B., Asparouhov, T. & Rebollo, I. (2006).
Advances in behavioral genetics modeling using Mplus: Applications of factor mixture modeling to twin
data.
Twin Research and Human Genetics, 9, 313-324.
Mplus inputs, outputs, and data are not yet available for this article.
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Abstract
"This article discusses new latent variable techniques developed by the authors. As an illustration, a
new factor mixture model is applied to MZ-DZ twin analysis of binary items measuring alcohol use disorder.
In this model, heritability is simultaneously studied with respect to latent class membership
and within-class severity dimensions. Different latent classes of individuals are allowed to have
different heritability for the severity dimensions. The factor mixture approach appears to have great
potential for genetic analyses of heterogeneous populations. Generalizations for longitudinal data
are also outlined."
hide abstract
- Prescott, C.A. (2004).
Using the Mplus computer program to estimate models for continuous and categorical data from twins.
Behavior Genetics, 34, 17-40.
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Abstract
"Historically, the focus of behavior genetic research was to obtain estimates of the sources of familial
resemblance for a single phenotype. Current research strategies have moved beyond heritability estimates
to the search for physiological and behavioral mechanisms by which genetic risk is translated
into individual differences in behavior and disease liability. Such research questions often require
multivariate designs and complex analytic models, including the analysis of continuous and categorical
dependent variables within the same model. Recent advances in computer software for categorical
data analysis have increased the tools available for researchers in behavior genetics. This paper
describes how to use the Mplus software program (Muthén and Muthén, 1998, 2002) for the
analysis of data obtained from twins. Example analyses include two- and five-group twin models for
univariate and bivariate continuous and categorical variables. Data on alcoholism and age at first
drink drawn from the Virginia Adult Twin Study of Psychiatric and Substance Use Disorders are used to
illustrate how Mplus can be used to analyze multiple-category variables, recode and transform variables,
select subgroups for analysis, handle subjects with incomplete data, include constraints to ensure
non-negative loadings, include model covariates, model sex differences, and test alternative hypotheses
about mediation of genetic risk by measured variables."
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- Jo, B, Hastie, T.J., Zetan, L., Youngstrom, E.A., Findling, R.L. & McCue Horowitz, S. (2023). Multivariate Behavioral Research, 58(6), 1057-1071. DOI: 10.1080/00273171.2023.218275
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Abstract
Despite its potentials benefits, using prediction targets generated based on latent variable (LV) modeling is not a common practice in supervised learning, a dominating framework for developing prediction models. In supervised learning, it is typically assumed that the outcome to be
predicted is clear and readily available, and therefore validating outcomes before predicting them is a foreign concept and an unnecessary step. The usual goal of LV modeling is inference, and therefore using it in supervised learning and in the prediction context requires a major conceptual
shift. This study lays out methodological adjustments and conceptual shifts necessary for integrating LV modeling into supervised learning. It is shown that such integration is possible by combining the traditions of LV modeling, psychometrics, and supervised learning. In
this interdisciplinary learning framework, generating practical outcomes using LV modeling and systematically validating them based on clinical validators are the two main strategies. In the example using the data from the Longitudinal Assessment of Manic Symptoms (LAMS)
Study, a large pool of candidate outcomes is generated by flexible LV modeling. It is demonstrated that this exploratory situation can be used as an opportunity to tailor desirable prediction targets taking advantage of contemporary science and clinical insights.
hide abstract
- Herle, M., Micali, N., Abdulkadir, M., Bryant-Waugh, R., Hubel, C., Bulik, C.M. & De Stavola, B.L. (2020). Identifying typical trajectories in longitudinal data: modelling strategies and interpretations. European Journal of Epidemiology. DOI: 10.1007/s10654-020-00615-6
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- Dong, H., Hayashi, K., Singer, J., Milloy, M.J., DeBeck, K., Wood, E., & Kerr, T. (2019). Trajectories of injection drug use among people who use drugs in Vancouver, Canada, 1996-2017: growth mixture modeling using data from prospective cohort studies.
Addiction. DOI: 10.1111/add.14756
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- Jo, B., Findling, R.L., Wang, C., Hastie, T.J., Youngstrom, E.A., Arnold, L.E., Fristad, M.A., & McCue Horowitz, S. (2016). Statistics in Medicine. Targeted use of growth mixture modeling: a learning perspective. DOI: 10.1002/sim.7152
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- Van de Schoot, R., Sijbrandij, M., Winter, S.D., Depaoli, S., & Vermunt, J.K. (2017). The GRoLTS-Checklist: Guidelines for Reporting on Latent Trajectory Studies, Structural Equation Modeling: A Multidisciplinary Journal, 24:3, 451-467, DOI: 10.1080/10705511.2016.1247646
view abstract
- Serang, S., Grimm, K. J., McArdle, J. J. (2016). Estimation of time-unstructured nonlinear mixed-effects mixture models. Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2016.1197777
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- Diallo, T.M.O, Morin, A.J.S. & Lu, H. (2016). Impact of misspecifications of the latent variance-covariance and residual matrices on the class enumeration accuracy of growth mixture models. Structural Equation Modeling: A Multidisciplinary
Journal, 23(4), 507-531, DOI: 10.1080/10705511.2016.1169188
View abstract
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- Diallo, T.M.O, Morin, A.J.S. & Lu, H. (2016). The impact of total and partial inclusion or exclusion of active and inactive time invariant covariates in growth mixture models. Psychological Methods, 22(1), 166–190. DOI: 10.1037/met0000084
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- Leoutsakos, J.-M.S., Forrester, S.N., Corcoran, C.D., Norton, M.C., Rabins, P.V., Steinberg, M.I., Tschanz, J.T. & Lyketsos, C.G. (2015). Latent classes of course in Alzheimer’s disease and predictors: the Cache County Dementia Progression Study. International Journal of Geriatric Psychiatry, 30: 824–832. DOI: 10.1002/gps.4221
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- Muthén, B. & Asparouhov T. (2015). Growth mixture modeling with non-normal distributions. Statistics in Medicine, 34:6, 1041–1058. DOI: 10.1002/sim.6388
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Abstract
A limiting feature of previous work on growth mixture modeling is the assumption of normally
distributed variables within each latent class. With strongly non-normal outcomes this means
that several latent classes are required to capture the observed variable distributions. Being able
to relax the assumption of within-class normality has the advantage that a non-normal observed
distribution does not necessitate using more than one class to fit the distribution. It would be
valuable to add parameters representing the skewness and the thickness of the tails. A new growth
mixture model of this kind is proposed drawing on recent work in a series of papers using the
skew-t distribution; see, for example, Lee and McLachlan (2014). The new method is illustrated
using the longitudinal development of BMI in two data sets. The first data set is from the
National Longitudinal Survey of Youth covering ages 12 to 23. Here, the development is related to
an antecedent measuring socioeconomic background. The second data set is from the Framingham Heart
Study covering ages 25 to 65. Here, the development is related to the concurrent event of
treatment for hypertension using a joint growth mixture-survival model.
hide abstract
- Depaoli, S. (2013). Mixture class recovery in GMM under varying degrees of class separation: Frequentist versus Bayesian estimation. Psychological Methods, 18(2), 186-219. DOI: 10.1037/a0031609
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Abstract
Growth mixture modeling (GMM) represents a technique that is designed to capture change over time
for unobserved subgroups (or latent classes) that exhibit qualitatively different patterns of growth.
The aim of the current article was to explore the impact of latent class separation (i.e., how
similar growth trajectories are across latent classes) on GMM performance. Several estimation
conditions were compared: maximum likelihood via the expectation maximization (EM) algorithm and
the Bayesian framework implementing diffuse priors, “accurate” informative priors, weakly
informative priors, data-driven informative priors, priors reflecting partial-knowledge of
parameters, and “inaccurate” (but informative) priors. The main goal was to provide insight about
the optimal estimation condition under different degrees of latent class separation for GMM.
Results indicated that optimal parameter recovery was obtained though the Bayesian approach using
“accurate” informative priors, and partial-knowledge priors showed promise for the recovery of the
growth trajectory parameters. Maximum likelihood and the remaining Bayesian estimation conditions
yielded poor parameter recovery for the latent class proportions and the growth trajectories.
hide abstract
- Prince, M. and Maisto, S. (2013). The clinical course of alcohol use disorders: Using joinpoint analysis
to aid in interpretation of growth mixture models. Drug and Alcohol Dependence. DOI: 10.1016/j.drugalcdep.
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Abstract
Background: The clinical course of alcohol use disorders (AUD) is marked by great heterogeneity
both within and between individuals. One approach to modeling this heterogeneity is latent growth
mixture modeling (LGMM), which identi?es a number of latent subgroups of drinkers with drinking
trajectories that are similar within a latent subgroup but different between subgroups. LGMM is
data-driven and uses an iterative process of testing a sequential number researcher-selected of
latent subgroups then selecting the best ?tting model. Despite the advantages of LGMM (e.g.,
identifying subgroups among heterogeneous longitudinal data), one limitation is the lack of
precision of LGMM to model abrupt changes in drinking during treatment that are often observed by
clinicians. Joinpoint analysis (JPA) is a data analysis procedure that is used to identify
discrete change points in longitudinal data (e.g., changes from increasing to decreasing or
decreasing to increasing).
Method: This study presents a demonstration of using JPA as a post hoc procedure for LGMM to
improve accuracy in modeling abrupt changes in clinical course of AUD.
Results: Results from this secondary data analysis of 549 AUD participants participating in the
NIAAA sponsored relapse replication and extension project uncovered four latent classes of drinking
trajectories. Discussion: Within these trajectories the addition of JPA improved precision in
modeling the clinical
course of AUDs.
hide abstract
- Leoutsakos, J.S., Muthén, B.O., Breitner, J.C.S. & Lyketsos, C.G. (2012). Effects of NSAID treatments on cognitive decline vary by phase of pre-clinical Alzheimer disease: Findings from the randomized controlled ADAPT trial.
International Journal of Geriatric Psychiatry, 27(4):364-74. DOI: 10.1002/gps.2723
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Abstract
Objective: We examined effects of non-steroidal anti-inflammatory drugs (NSAID) on cognitive decline as a function of phase of pre-clinical Alzheimer’s disease (AD). Methods: Given recent findings that cognitive decline accelerates as clinical diagnosis
is approached, we used rate of decline as a proxy for phase of pre-clinical Alzheimer’s disease. We fit growth mixture models of Modified Mini-Mental State Examination (3MS) trajectories with data from 2,388 participants in the Alzheimer’s Disease
Antiinflammatory Prevention Trial (ADAPT), and included class-specific effects of naproxen and celecoxib. Results: We identified 3 classes: “no-decline”, “slow-decline”, and “fast-decline”, and examined effects of celecoxib and naproxen on linear slope and rate of change by
class. Inclusion of quadratic terms improved fit of the model (-2 log likelihood difference: 369.23; p<0.001), but resulted in reversal of effects over time. Over four years, participants in the slow-decline class on placebo typically lost 6.6 3MS points, while
those on naproxen lost 3.1 points (p-value for difference: 0.19). Participants in the fastdecline class on placebo typically lost 11.2 points, but those on celecoxib first declined and then gained points (p-value for difference from placebo: 0.04), while those on
naproxen showed a typical decline of 24.9 points (p-value for difference from placebo: <0.0001). Conclusions: Our results appeared statistically robust, but provided some unexpected contrasts in effects of different treatments at different times. Naproxen may attenuate
cognitive decline in slow decliners while accelerating decline in fast decliners. Celecoxib appeared to have similar effects at first but then to attenuate change in fast decliners.
hide abstract
- Shiyko, M., Li, Y., & Rindskopf, D. (2012). Poisson growth mixture modeling of intensive longitudinal data: An application to smoking cessation behavior. Structural Equation Modeling: A Multidisciplinary Journal, 19:1, 65-85. DOI: 10.1080/10705511.2012.634722
view abstract
- Gueorguieva, R., Mallinckrodt, C., & Krystal, J. (2011).
Trajectories of depression severity in clinical trials of Duloxetine.
Arch Gen Psychiatry, 68(12): 1227-1237.
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Abstract
"Context : The high percentage of failed clinical trials in
depression may be due to high placebo response
rates and
the failure of standard statistical approaches to capture
heterogeneity in treatment
response.
Objective: To assess whether growth mixture modeling
can provide insights into antidepressant
and placebo
responses in clinical trials of patients with major
depression.
Design: We reanalyzed
clinical trials of duloxetine to
identify distinct trajectories of Hamilton Scale for Depression
(HAM-D)
scores during treatment. We analyzed
the trajectories in the entire sample and then separately
in
all active arms and in all placebo arms. Effects
of duloxetine hydrochloride, selective serotonin
reuptake
inhibitor (SSRI), and covariates on the probability
of following a particular trajectory
were assessed. Outcomes
in different trajectories were compared using
mixed-effects models.
Setting:
Seven randomized double-blind clinical trials
of duloxetine vs placebo and comparator SSRI.
Patients:
A total of 2515 patients with major depression.
Interventions : Duloxetine and comparator
SSRI.
Main Outcome Measure: Total score on the HAM-D.
Results: In the entire sample and
in the antidepressanttreated
subsample, we identified trajectories of responders
(76.3% of the sample)
and nonresponders (23.7% of
the sample). However, placebo-treated patients were characterized
by
a single response trajectory. Duloxetine and
SSRI did not differ in efficacy, and compared with placebo
they
significantly decreased the odds of following
the nonresponder trajectory. Antidepressant
responders
had significantly better HAM-D scores over time than
placebo-treated patients, but antidepressant
nonresponders
had significantly worse HAM-D scores over time
than the placebo-treated patients.
Conclusions:
Most patients treated with serotonergic antidepressants
showed a clinical trajectory
over time that
is superior to that of placebo-treated patients. However,
some patients receiving
these medications did more poorly
than patients receiving placebo. These data highlight the
importance
of ongoing monitoring of medication risks
and benefits during serotonergic antidepressant treatment.
They
should further stimulate the search for biomarkers
or other predictors of responder status
in guiding
antidepressant treatment.
"
hide abstract
- Muthén, B., Brown, C.H., Hunter, A., Cook, I.A. & Leuchter, A.F. (2011).
General approaches to analysis of course: Applying growth mixture modeling to randomized trials of depression
medication.
In P.E. Shrout (ed.), Causality and Psychopathology: Finding the Determinants of Disorders and their Cures (pp. 159-178). New York: Oxford University Press.
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Abstract
This chapter discusses the use of growth mixture modeling to assess treatment effects
in clinical trials.
The motivation is a study of depression medication in a double-blind
placebo-controlled trial.
Studies of this type typically show placebo response and placebo
non response. Growth mixture modeling
(GMM) is well suited for representing such
heterogeneity among subjects in that it can identify di
erent types of trajectory shapes.
GMM can be seen as a combination of conventional mixed effects modeling
and cluster
analysis, also allowing prediction of class membership and estimation of each individuals
most
likely class membership. GMM has particularly strong potential for analyses of
randomized
trials because it responds to the need to investigate for whom a treatment
is eff ective by allowing
for di erent treatment e ects in di erent trajectory classes. In
this trial, a separate analysis
of the placebo group nds evidence of a placebo response
trajectory class with a strong initial
improvement, followed by a later worsening. A
separate analysis of the medication group shows two types
of responder classes, one with
an initial improvement only and one with a sustained improvement.
A joint analysis
of the placebo and medication groups makes it possible to estimate medication e ects
in
the presence of placebo-response effects and shows bene ts of medication. Analysis
strategies
and alternatives for assessing medication e ects are discussed.
Key words: Randomized trials, growth
modeling, causal e ects, latent variables,
trajectory classes, maximum likelihood.
hide abstract
- Reinecke, J. & Seddig, D. (2011).
Growth mixture models in longitudinal research.
AStA Advances in Statistical Analysis, 95, 415-434.
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Abstract
"Latent growth curve models as structural equation models are extensively
discussed in various research
fields (Curran and Muthén in Am. J. Community Psychol.
27:567–595, 1999; Duncan et al. in An introduction
to latent variable growth
curve modeling. Concepts, issues and applications, 2nd edn., Lawrence
Earlbaum,
Mahwah, 2006; Muthén and Muthén in Alcohol. Clin. Exp. Res. 24(6):882–891,
2000a; in J.
Stud. Alcohol. 61:290–300, 2000b). Recent methodological and statistical
extension are focused on
the consideration of unobserved heterogeneity in
empirical data. Muthén extended the classic structural
equation approach by mixture
components, i.e. categorical latent classes (Muthén in Marcouldies,
G.A., Sckumacker,
R.E. (eds.), New developments and techniques in structural equation modeling,
pp.
1–33, Lawrance Erlbaum,Mahwah, 2001a; in Behaviometrika 29(1):81–117,
2002; in Kaplan, D. (ed.), The
SAGE handbook of quantitative methodology for the
social sciences, pp. 345–368, Sage, Thousand Oaks,
2004). The paper discusses applications
of growth mixture models with data on delinquent behavior
of adolescents
from the German panel study Crime in the modern City (CrimoC) (Boers et al. in Eur.
J.
Criminol. 7:499–520, 2010; Reinecke in Delinquenzverläufe im Jugendalter: Empirische
Überprüfung
von Wachstums- und Mischverteilungsmodellen, Institut für
sozialwissenschaftliche Forschung e.V.,Münster,
2006a; inMethodology 2:100–112,
2006b; in van Montfort, K., Oud, J., Satorra, A. (eds.), Longitudinal
models in the
behavioral and related sciences, pp. 239–266, Lawrence Erlbaum, Mahwah, 2007).
Observed
as well as unobserved heterogeneity will be considered with growth mixture
models. Special
attention is given to the distribution of the outcome variables as counts. Poisson and negative binomial
distributions with zero inflation are considered
in the proposed growth mixture models variables.
Different model specifications will
be emphasized with respect to their particular parameterizations."
hide abstract
- Grimm, K.J., Ram, N. & Estabrook, R. (2010).
Nonlinear structured growth mixture models in Mplus and OpenMx.
Multivariate Behavioral Research, 45, 887-909.
The technical appendix for this paper can be viewed here.
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Abstract
"Growth mixture models (GMMs; B. O. Muthén & Muthén, 2000; B. O. Muthén
& Shedden, 1999) are a combination
of latent curve models (LCMs) and finite
mixture models to examine the existence of latent classes
that follow distinct
developmental patterns. GMMs are often fit with linear, latent basis, multiphase,
or
polynomial change models because of their common use, flexibility in modeling
many types of
change patterns, the availability of statistical programs to fit such
models, and the ease of programming.
In this article, we present additional ways of
modeling nonlinear change patterns with GMMs.
Specifically, we show how LCMs
that follow specific nonlinear functions can be extended to examine
the presence
of multiple latent classes using the Mplus and OpenMx computer programs. These
models
are fit to longitudinal reading data from the Early Childhood Longitudinal
Study–Kindergarten Cohort
to illustrate their use.
hide abstract
- Hunter, A. M., Muthén, B.O., Cook, I.A. & Leuchter, A. F. (2010).
Antidepressant response trajectories and quantitative electroencephalography (QEEG) biomarkers in major
depressive disorder.
Journal of Psychiatric Research, 44, 90-98.
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Abstract
Individuals with Major Depressive Disorder (MDD) vary regarding the rate, magnitude and stability of
symptom
changes during antidepressant treatment. Growth mixture modeling (GMM) can be used to
identify
patterns of change in symptom severity over time. Quantitative electroencephalographic (QEEG)
cordance
within the first week of treatment has been associated with endpoint clinical outcomes but has
not
been examined in relation to patterns of symptom change. Ninety-four adults with MDD were randomized
to
eight weeks of double-blinded treatment with fluoxetine 20 mg or venlafaxine 150 mg
(n = 49)
or placebo (n = 45). An exploratory random effect GMM was applied to Hamilton Depression Rating
Scale
(Ham-D17) scores over 11 timepoints. Linear mixed models examined 48-h, and 1-week changes
in QEEG
midline-and-right-frontal (MRF) cordance for subjects in the GMM trajectory classes. Among
medication
subjects an estimated 62% of subjects were classified as responders, 21% as non-responders,
and
17% as symptomatically volatile—i.e., showing a course of alternating improvement and worsening.
MRF
cordance showed a significant class-by-time interaction (F(2,41) = 6.82, p = .003); as hypothesized,
the
responders showed a significantly greater 1-week decrease in cordance as compared to non-responders
(mean
difference = .76, Std. Error = .34, df = 73, p = .03) but not volatile subjects. Subjects
with a volatile
course of symptom change may merit special clinical consideration and, from a research
perspective, may
confound the interpretation of typical binary endpoint outcomes. Statistical
methods such as GMM are
needed to identify clinically relevant symptom response trajectories.
hide abstract
- Morin, A.J., Mainano, C., Nagengast, B., Marsh, H.W., Morizot, J. & Janosc, M. (2011). General growth mixture analysis of adolescents' developmental trajectories of anxiety: The impact of untested invariance assumptions on substantive interpretations. Structural Equation Modeling: A Multidisciplinary Journal, 18:4,
613-648, DOI: 10.1080/10705511.2011.607714
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Abstract
"Substantively, this study investigates potential heterogeneity in the developmental trajectories of
anxiety
in adolescence. Methodologically, this study demonstrates the usefulness of General Growth
Mixture
Analysis (GGMA) in addressing these issues and illustrates the impact of untested invariance
assumptions
on substantive interpretations. This study relied on data from the Montreal Adolescent
Depression
Development Project (MADDP), a four-year follow-up of over 1000 adolescents who
completed the
Beck Anxiety Inventory each year. GGMA models relying on different invariance
assumptions were empirically
compared. Each of these models converged on a five-class solution, but
yielded different substantive
results. The model with class-varying variance-covariance matrices was
retained as providing
a better fit to the data. These results showed that although elevated levels of
anxiety may fluctuate
over time, they clearly do not represent a transient phenomenon. This model
was then validated
in relation to multiple predictors (mostly related to school violence) and outcomes
(GPA, school dropout,
depression, loneliness and drug-related problems)."
hide abstract
- Pickles, P. & Croudace, T. (2010).
Latent mixture models for multivariate and longitudinal outcomes.
Statistical Methods in Medical Research, 19, 271–289.
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Abstract
"Repeated measures and multivariate outcomes are an increasingly common feature of trials. Their joint
analysis
by means of random effects and latent variable models is appealing but patterns of heterogeneity
in
outcome profile may not conform to standard multivariate normal assumptions. In addition,
there is much
interest in both allowing for and identifying sub-groups of patients who vary in treatment
responsiveness.
We review methods based on discrete random effects distributions and mixture models
for application in
this field."
hide abstract
- Qureshi, I. & Fang, Y. (2010).
Socialization in open source software projects: A growth mixture modeling approach.
Organizational Research Methods, 1-31.
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Abstract
"The success of open source software (OSS) projects depends heavily on the voluntary participation
of
a large number of developers. To remain sustainable, it is vital for an OSS project community to
maintain
a critical mass of core developers. Yet, only a small number of participants (identified here
as
‘‘joiners’’) can successfully socialize themselves into the core developer group. Despite the
importance
of joiners’ socialization behavior, quantitative longitudinal research in this area is
lacking.
This exploratory study examines joiners’ temporal socialization trajectories and their
impacts on
joiners’ status progression. Guided by social resource theory and using the growth
mixture modeling
(GMM) approach to study 133 joiners in 40 OSS projects, the authors found
that these joiners differed
in both their initial levels and their growth trajectories of socialization
and identified four
distinct classes of joiner socialization behavior. They also found that these
distinct latent classes
of joiners varied in their status progression within their communities. The
implications for research
and practice are correspondingly discussed."
hide abstract
- Feldman, B.J., Masyn, K.E. & Conger, R.D. (2009).
New approaches to studying problem behaviors: A comparison of methods for modeling longitudinal, categorical
adolescent drinking data.
Developmental Psychology, 45, 3, 652-676.
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Abstract
"Analyzing problem-behavior trajectories can be difficult. The data are generally categorical and often
quite
skewed, violating distributional assumptions of standard normal-theory statistical models. In
this
article, the authors present several currently available modeling options, all of which make
appropriate
distributional assumptions for the observed categorical data. Three are based on the generalized
linear
model: a hierarchical generalized linear model, a growth mixture model, and a latent
class growth
analysis. They also describe a longitudinal latent class analysis, which requires fewer
assumptions than
the first 3. Finally, they illustrate all of the models using actual longitudinal
adolescent alcohol-use data.
They guide the reader through the model-selection process, comparing the
results in terms of convergence
properties, fit and residuals, parsimony, and interpretability. Advances
in computing and statistical
software have made the tools for these types of analyses readily
accessible to most researchers. Using
appropriate models for categorical data will lead to more accurate
and reliable results, and their
application in real data settings could contribute to substantive
advancements in the field of development
and the science of prevention."
hide abstract
- Grimm, K.J. & Ram, N. (2009).
A second-order growth mixture model for developmental research.
Research in Human Development, 6, 121-143.
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Abstract
"Growth mixture modeling, a combination of growth modeling and finite mixture
modeling, is a flexible,
exploratory method for identifying and describing betweenperson
heterogeneity in change. In this article
we introduce a second-order growth
mixture model that combines a longitudinal common factor model,
measurement
invariance constraints, latent growth model, and mixture model. This approach capitalizes
on
the benefits of multivariate measurement and the flexibility of mixtures
for representing
heterogeneity. We describe the model and illustrate its use with
multi-reporter longitudinal data from
the National Institute of Child Health and
Human Development (NICHD) Study of Early Child Care and
Youth Development
tracking the development of children’s externalizing behaviors through elementary
school."
hide abstract
- Muthén, B. & Asparouhov, T. (2009).
Growth mixture modeling: Analysis with non-Gaussian random effects.
In Fitzmaurice, G., Davidian, M., Verbeke, G. & Molenberghs, G. (eds.), Longitudinal Data Analysis, pp. 143-165. Boca Raton: Chapman & Hall/CRC Press.
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Abstract
"Growth mixture analysis represents unobserved heterogeneity among the subjects in their development using
both random effects and finite mixtures. In particular, the mixture components allow different
means of the random effects, although any parameter in the growth model can vary. This chapter gives
an overview of examples motivating modeling with such trajectory classes. A general latent variable
modeling framework is presented together with its maximum-likelihood estimation. Examples from criminology
and education are analyzed. The choice of a normal or a non-parametric distribution for the
random effects is discussed and investigated using a simulation study. Key words: Growth modeling, finite
mixtures, latent variables, trajectory classes, maximum likelihood, non-parametric distribution.
Complete address of first author: Graduate School of Education & Information Studies, Moore Hall,
Box 951521, Los Angeles CA 90095-1521."
hide abstract
- Muthén, B. & Brown, H. (2009).
Estimating drug effects in the presence of placebo response: Causal inference using growth mixture modeling.
Statistics in Medicine, 28, 3363-3385.
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Abstract
"Placebo-controlled randomized trials for antidepressants and other drugs often show a response for a
sizeable
percentage of the subjects in the placebo group. Potential placebo responders can be assumed
to
exist also in the drug treatment group, making it difficult to assess the drug effect. A key drug
research
focus should be to estimate the percentage of individuals among those who responded to the
drug who
would not have responded to the placebo (‘Drug Only Responders’). This paper investigates
a finite mixture
model approach to uncover percentages of up to four potential mixture components:
Never Responders,
Drug Only Responders, Placebo Only Responders, and Always Responders. Two examples
are used to
illustrate the modeling, a 12-week antidepressant trial with a continuous outcome (Hamilton
D score) and
a 7-week schizophrenia trial with a binary outcome (illness level). The approach
is formulated in causal
modeling terms using potential outcomes and principal stratification. Growth
mixture modeling (GMM)
with maximum-likelihood estimation is used to uncover the different mixture
components. The results
point to the limitations of the conventional approach of comparing marginal
response rates for drug and
placebo groups. It is useful to augment such reporting with the GMM-estimated
prevalences for the four
classes of subjects and the Drug Only Responder drug effect estimate.
Copyright q 2009 John Wiley &
Sons, Ltd."
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- Petras, H. & Masyn, K. "General growth mixture analysis with antecedents and consequences of change." Handbook of Quantitative Criminology. Ed. Alex Piquero, Ed. David Weisburd. New York: Springer-Verlag, 2010. 69-100.
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- Uher, R., Muthén, B., Souery, D., Mors, O., Jaracz, J., Placentino, A., Petrovic, A., Zobel, A., Henigsberg, N., Rietschel, M., Aitchison, K., Farmer, A. & McGuffin, P. (2009).
Trajectories of change in depression severity during treatment with antidepressants.
Psychological Medicine, published online October 29, 2009.
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Abstract
"Background: Response and remission defined by cut-off values on the last observed depression
severity
score are commonly used as outcome criteria in clinical trials, but ignore the time-course of
symptomatic
change and may lead to inefficient analyses. We explore alternative categorisation of
outcome
by naturally occurring trajectories of symptom change.
Methods: Growth mixture models (GMM) were applied
to repeated measurements of depression
severity in 807 participants with major depression treated
for 12 weeks with escitalopram or
nortriptyline in the part-randomized Genome-based Therapeutic
Drugs for Depression (GENDEP)
study. Latent trajectory classes were validated as outcomes in drug efficacy
comparison and
pharmacogenetic analyses.
Results: The final two-piece growth mixture model categorised
participants into a majority (75%)
following a gradual improvement trajectory and the remainder
following a trajectory with rapid
initial improvement. The rapid improvement trajectory was overrepresented
among nortriptylinetreated
participants and showed an antidepressant-specific pattern
of pharmacogenetic associations.
In contrast, conventional response and remission favoured escitalopram
and produced chance results
in pharmacogenetic analyses. Controlling for drop-out reduced drug differences
on response and
remission but did not affect latent trajectory results.
Conclusions: Latent
trajectory mixture models capture heterogeneity in the development of clinical
response after the
initiation of antidepressants and provide an outcome that is distinct from
traditional endpoint measures.
It differentiates between antidepressants with different modes of
action and is robust against
bias due to differential discontinuation."
hide abstract
- Uher, R., Muthén, B., Souery, D., Mors, O., Jaracz, J., Placentino, A., Petrovic, A., Zobel, A., Henigsberg, N., Rietschel, M., Aitchison, K., Farmer, A. & McGuffin, P. (2009).
Technical appendix: Methods and results of growth mixture modelling.
Psychological Medicine, published online October 29, 2009.
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Abstract
"Recent advances in statistical modelling based on mixture extension of the latent growth model
make it
possible to categorise subjects based on temporal patterns of change with latent variable
methods
such as growth mixture modelling (GMM) that can provide unbiased estimates of
trajectories of change
in the presence of missing data (Muthén & Asparouhov 2008; Beunckens et
al. 2008). This appendix describes
in details the application of GMM to data from the the Genomebased
Therapeutic Drugs for Depression
(GENDEP) project, a twelve-weeks part-randomized openlabel
study of depression treatment comparing
two active antidepressant drugs."
hide abstract
- Boscardin, C., Muthén, B., Francis, D. & Baker, E. (2008).
Early identification of reading difficulties using heterogeneous developmental trajectories.
Journal of Educational Psychology, 100, 192-208.
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Abstract
"Serious conceptual and procedural problems associated with current diagnostic methods call for alternative
approaches to assessing and diagnosing students with reading problems. This study presents a new
analytic model to improve the classification and prediction of children’s reading development. 411
children in kindergarten through 2nd grade were administered measures of phonological awareness, word
recognition, and rapid naming skills. The application of growth mixture models provides a more dynamic
view of the learning process and the correlates that affect the rate of reading development.
Growth mixture modeling was used to examine the presence of heterogeneous developmental patterns and
served to identify one group of students with distinct developmental patterns who are most at risk
for reading difficulties. The results indicate that precursor reading skills such as phonological awareness
and rapid naming are highly predictive of later reading development and that developmental profiles
formed in kindergarten are directly associated with development in grades 1 and 2. Students
identified as having reading-related difficulties in kindergarten exhibited slower development of reading
skills in subsequent years of the study. Key words: Reading Development, Screening, Reading Skills,
Achievement, Longitudinal Studies, Models"
hide abstract
- Jung, T. & Wickrama, K.A.S. (2008).
An introduction to latent class growth analysis and growth mixture modeling.
Social and Personality Psychology Compass, 2, 302-317.
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Abstract
"In recent years, there has been a growing interest among researchers in the use
of latent class and growth
mixture modeling techniques for applications in the
social and psychological sciences, in part
due to advances in and availability of
computer software designed for this purpose (e.g., Mplus and
SAS Proc Traj).
Latent growth modeling approaches, such as latent class growth analysis (LCGA)
and
growth mixture modeling (GMM), have been increasingly recognized for
their usefulness for identifying
homogeneous subpopulations within the larger
heterogeneous population and for the identification of
meaningful groups or
classes of individuals. The purpose of this paper is to provide an overview of
LCGA
and GMM, compare the different techniques of latent growth modeling, discuss
current debates
and issues, and provide readers with a practical guide for
conducting LCGA and GMM using the Mplus software."
hide abstract
- Kreuter, F. & Muthén, B. (2008).
Analyzing criminal trajectory profiles: Bridging multilevel and group-based approaches using growth mixture modeling. Journal of Quantitative Criminology,
24, 1-31. Click here to download Mplus input and output files associated with this paper.
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Abstract
"Over the last 25 years, a life-course/developmental perspective on criminal behavior has assumed increasing
prominence in the criminological literature. This theoretical development has been accompanied
by changes in the statistical models used to analyze criminological data. There are two main statistical
modeling techniques currently used to model longitudinal data. These are growth curve models
and latent class growth models, also known as grouped-based trajectory models. This paper is a contribution
to the recent debate on the use of these two models. Using the well known Cambridge data on
criminal conviction, this paper compares the two “classical” models – conventional growth curve model
and group-based trajectory models in terms of their performance with these particular data. It also
introduces two additional models that bridge the gap between conventional growth models and group-based
trajectory models. Sensitivity and substantive conclusions are then discussed for the models with
the best performance. The main goals of this paper are to broaden the set of tools available to
criminologists in analyzing data from a life-course perspective, and to provide a concrete step-by-step
illustration of such an analysis."
hide abstract
- Kreuter, F. & Muthén, B. (2008).
Longitudinal modeling of population heterogeneity: Methodological challenges to the analysis of empirically
derived criminal trajectory profiles.
In Hancock, G. R., & Samuelsen, K. M. (Eds.), Advances in latent variable mixture models, pp. 53-75. Charlotte, NC: Information Age Publishing, Inc.
Click here for information
about the book.
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- Tolvanen, A. (2008).
Latent growth mixture modeling: A simulation study.
Doctoral dissertation, Department of Mathematics, University of Jyvaskyla, Finland.
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Abstract
Latent growth curve modeling (LGM) combined with the latent classes (LGMM)
in the SEM context, is the
method under investigation in this study. This dynamic
way of analyzing longitudinal data takes an
increasingly central position in the
social sciences, e.g. in psychology. Despite twenty years development
of the
theory behind the LGM and LGMM, these are novel methods in analyzing data in
practice.
With limited sample size the functionality of the model is unknown.
The aim of this dissertation was
to examine the functionality of the linear LGM
model with four repeated measurements, which is a typical
case in longitudinal
research. LGMM parameters were estimated using maximum likelihood
estimation
with robust standard errors (MLR). The effect of differences between
latent classes in mean values
of latent components with varying sample sizes is
examined in this study. Other affecting factors
examined are reliability of
observed variables, number of repeated measures, model construct and additional
measurement
points. The functionality of LGMM was approached from three
different viewpoints:
1) problems in estimation of model parameters expressed as
number of failed estimations and as the
number of negative variance estimates, 2)
the ability of AIC, BIC and aBIC information criteria and
VLMR, LMR and
BLRT statistical tests to decide the number of latent classes, and 3) good
parameter
estimation, which was evaluated using four different criteria: MSE,
proportion of bias in MSE, bias
of standard error, and 95 % coverage.
The results of Monte Carlo simulations suggest that from information
criteria AIC,
BIC aBIC and VLMR and LMR tests, BIC is most useful with small sample sizes
(
) and aBIC with large sample sizes ( ). The few results suggest that
the BLRT test could be useful
in any situation. More investigation is needed to
further support the functionality of this test. The
study reveals that the estimation
of LGMM fails only in a few cases, and problems in estimation appear
mainly in
the negative variance estimates. The results of the simulations suggest that it is
possible
to identify the true two-latent classes when SMD is at least 2, in which
case reliability of
observed variables should be high and the sample size should be
relatively large. In that case estimation
produce good parameter estimates. When
SMD is 4 or 5, the probability in identifying the right
two-latent-class solution
instead of the wrong one-class solution is greater than .70 with the smallest
sample
size (n=50) using BIC in models with high reliability. To achieve reliable results
in estimation,
the sample size should be greater than 50.
n < 500 n ? 500
Key words: Latent growth mixture
modeling, Monte Carlo simulation
hide abstract
- Muthén, B. (2006).
The potential of growth mixture modeling. Commentary.
Infant and Child Development, 15, 623-625.
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- Schaeffer, C.M., Petras, H., Ialongo, N., Masyn, K.E., Hubbard, S., Poduska, J., & Sheppard, K. (2006).
A comparison of girl's and boy's aggressive-disruptive behavior trajectories across elementary school:
Prediction to young adult antisocial outcomes.
Journal of Consulting and Clinical Psychology, 74, 500-510.
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Abstract
Multiple group analysis and general growth mixture modeling was used to determine whether
aggressive–
disruptive behavior trajectories during elementary school, and their association with young
adulthood
antisocial outcomes, vary by gender. Participants were assessed longitudinally beginning at
age 6
as part of an evaluation of 2 school-based preventive programs. Two analogous trajectories were
found
for girls and boys: chronic high aggression– disruption (CHAD) and stable low aggression–
disruption
(LAD). A 3rd class of low moderate aggression– disruption (LMAD) for girls and increasing
aggression–
disruption (IAD) for boys also was found. Girls and boys in analogous CHAD classes did not
differ
in trajectory level and course, but girls in the CHAD and LAD classes had lower rates of antisocial
outcomes
than boys. Girls with the LMAD trajectory differed from boys with the IAD trajectory.
hide abstract
- Greenbaum, P.E., Del Boca, F.K., Darkes, J., Wang, C. & Goldman, M.S. (2005).
Variation in the drinking trajectories of freshman college students.
Journal of Consulting and Clinical Psychology, 73, 229-238
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Abstract
Recently, Del Boca, Darkes, Greenbaum, and Goldman (2004) examined temporal variations in drinking during
the freshmen college year and the relationship of several risk factors to these variations. Here,
using the same data, we investigate whether a single growth curve adequately characterizes the variability
in individual drinking trajectories. Latent growth mixture modeling identified five drinking
trajectory classes: Light-Stable, Light-Stable + High Holiday, Medium- Increasing, High-Decreasing,
and Heavy-Stable. In multivariate predictor analyses, gender (i.e., more females) and lower alcohol
expectancies distinguished the Light-Stable from other trajectories; only expectancies differentiated
the High-Decreasing from the Heavy-Stable and Medium-Increasing classes. These findings move us
closer to identification of individuals at risk for developing problematic trajectories and to development
of interventions tailored to specific drinker classes.
hide abstract
- Muthén, B. (2004).
Latent variable analysis: Growth mixture modeling and related techniques for longitudinal data.
In D. Kaplan (ed.), Handbook of quantitative methodology for the social sciences (pp. 345-368). Newbury Park, CA: Sage Publications.
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Abstract
This chapter gives an overview of recent advances in latent variable analysis. Emphasis is placed on
the strength of modeling obtained by using a flexible combination of continuous and categorical latent
variables. To focus the discussion, analysis of longitudinal data using growth models will be considered.
Continuous latent variables are common in growth modeling in the form of random effects
that capture individual variation in development over time. The use of categorical latent variables
in growth modeling is in contrast perhaps less familiar. The aim of this chapter is to show the usefulness
of growth model extensions using categorical latent variables. Examples are drawn from research
on achievement development and high school dropout and from research on delinquency development.
hide abstract
- van Lier, P.A.C., Muthén, B., van der Sar, R.M. & Crijnen, A.A.M. (2004).
Preventing disruptive behavior in elementary schoolchildren: Impact of a universal classroom-based intervention.
In Journal of Consulting and Clinical Psychology, 72, 467-478.
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Abstract
A population-based, randomized universal classroom intervention trial for the prevention of disruptive
behavior (i.e., attention-deficit/hyperactivity problems, oppositional defiant problems, and conduct
problems) is described. Impact on developmental trajectories in young elementary schoolchildren was
studied. Three trajectories were identified in children with high, intermediate, or low levels of
problems on all 3 disruptive behaviors at baseline. The intervention had a positive impact on the development
of all disruptive behavior problems in children with intermediate levels of these problems
at baseline. Effect sizes of mean difference at outcome were medium or small. In children with the
highest levels of disruptive behavior at baseline, a positive impact of the intervention was found
for conduct problems.
hide abstract
- Croudace, T.J., Jarvelin, M.R., Wadsworth, M.E. & Jones, P.B. (2003).
Developmental typology of trajectories to nighttime bladder control: Epidemiologic application of longitudinal
latent class analysis.
American Journal of Epidemiology, May 1;157(9):834-42.
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Abstract
The authors aimed to characterize developmental trajectories to nighttime continence by applying two
latent class models-longitudinal latent class analysis (LLCA) and latent class growth analysis (LCGA)-to
data on nighttime bed-wetting from a population-based birth cohort, the Medical Research Council
1946 National Survey of Health and Development cohort. Data on a binary outcome (wetting in the
past month vs. not wetting) were available for children at six ages (4, 6, 8, 9, 11, and 15 years)
assessed in 1950, 1952, 1954, 1955, 1957, and 1961. For 3,272 children with complete data (62.5%
of the cohort), results of sequential model comparisons (T classes vs. T + 1 classes) and chi-square
goodness-of-fit tests were evaluated using parametric bootstrapping. At least four trajectory classes
(LLCA and LCGA) were identified. Associations between class membership and the prevalence of
related measures were examined using a confirmatory latent class analysis approach. Inclusion of 1,483
children with partially incomplete data (n = 4,755; 90.9% of the cohort) enabled the authors
to refine trajectories further: normal development (prevalence = 84.0%); delayed acquisition of bladder
control ('transient' (8.7%) and 'persistent' (1.8%)), capturing primary enuresis; chronic bed-wetting
(2.6%), or experiencing night wetting until age 15 years; and a final trajectory (relapse =
2.9%) capturing secondary or onset enuresis. This empirically based, typologic approach to analysis
of extensive longitudinal data in a general population sample provides an alternative perspective
to that offered by traditional diagnostic criteria.
hide abstract
- Muthén, B. (2003).
Statistical and substantive checking in growth mixture modeling.
Psychological Methods, 8, 369-377.
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Abstract
This commentary discusses the Bauer and Curran (2003) investigation of growth mixture modeling. Single-class
modeling of non-normal outcomes is compared to modeling with multiple latent trajectory classes.
New statistical tests of multiple-class models are discussed. Principles for substantive investigation
of growth mixture model results are presented and illustrated by an example of high school
dropout predicted by low mathematics achievement development in grades 7 - 10.
hide abstract
- Muthén, B., Khoo, S.T., Francis, D. & Kim Boscardin, C. (2003).
Analysis of reading skills development from Kindergarten through first grade: An application of growth
mixture modeling to sequential processes.
Multilevel Modeling: Methodological Advances, Issues, and Applications. S.R. Reise & N. Duan (Eds). Mahaw, NJ: Lawrence Erlbaum Associates, pp.71-89.
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Abstract
Methods for investigating the influence of an early developmental process on a later process are discussed.
Conventional growth modeling is found inadequate but a growth mixture model is sufficiently
flexible. The growth mixture model allows for prediction of the later process using different trajectory
classes for the early process. The growth mixture model is applied to the study of progress in
reading skills among first-grade students.
hide abstract
- Oxford, M.L., Gilchrist, L.D., Morrison, D.M., Gillmore, M.R., Lohr M.J. & Lewis, S.M. (2003).
Alcohol use among adolescent mothers: Heterogeneity in growth curves.
Prevention Science, 4, 15-26.
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Abstract
"With a sample of adolescent mothers we examine patterns of alcohol use over a 10-year period of time.
Mixture modeling with MPLUS is used to identify latent trajectory classes based on alcohol consumption
over ten years. We found significant heterogeneity in alcohol use trajectories of adolescent mothers
during the transition from adolescence to adulthood as well as significant predictors and outcomes
that vary by latent class trajectory.
hide abstract
- Schaeffer, C.M., Petras, H., Ialongo, N., Poduska, J. & Kellam, S. (2003).
Modeling growth in boys aggressive behavior across elementary school: Links to later criminal involvement,
conduct disorder, and antisocial personality disorder.
Developmental Psychology, 39, 1020-1035.
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Abstract
Theoretical models of antisocial behavior development have proposed distinct pathways leading to criminal
activity. The present study used general growth mixture modeling (GGMM) to find empirical evidence
for these pathways within an epidemiologically-defined sample of 297 urban, primarily African-American
boys. Teacher-rated aggression, measured longitudinally from 1st-7th grades, was used to
define growth trajectories. Three distinct high-risk trajectories (chronic high, moderate, and increasing
aggression; 68% of boys) and one low-risk aggression trajectory (stable low aggression; 32%
of boys) were found. Boys with chronic high and increasing trajectories were at increased risk for
conduct disorder and juvenile arrest in adolescence, and antisocial personality disorder and adult
arrest in young adulthood. Boys with a moderate aggression trajectory were at risk for juvenile
and adult arrest. Concentration [NSI1] problems were highest among boys with a chronic high aggression
trajectory and also differentiated boys with increasing aggression from boys with stable low
aggression. Peer rejection was also higher among boys with chronic high aggression relative to the
low aggression group. The need for improved early identification of and interventions with boys with
distinct patterns of aggression is discussed.
hide abstract
- Muthén, B., Brown, C.H., Masyn, K., Jo, B., Khoo, S.T., Yang, C.C., Wang, C.P., Kellam, S., Carlin, J., & Liao, J. (2002).
General growth mixture modeling for randomized preventive interventions.
Biostatistics, 3, 459-475.
Mplus inputs and outputs used in this paper can be viewed and/or downloaded from the Examples
page.
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Abstract
This paper proposes growth mixture modeling to assess intervention effects in longitudinal randomized
trials. Growth mixture modeling represents unobserved heterogeneity among the subjects using a finite
mixture random effects model. The methodology allows one to examine the impact of an intervention
on subgroups characterized by different types of growth trajectories. Such modeling is informative
when examining effects on populations that contain individuals who have normative growth as well
as non-normative growth. The analysis identifies subgroup membership and allows theory-based modeling
of intervention effects in the different subgroups. An example is presented concerning a randomized
intervention in Baltimore public schools aimed at reducing aggressive classroom behavior, where
only students who were initially more aggressive showed benefits from the intervention.
hide abstract
- Muthén, B. (2001). Two-Part Growth Mixture Modeling.
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Abstract
This paper considers the analysis of repeated measures data. Conventional random
effects growth modeling in the tradition of Laird and Ware (1982) represents unobserved
heterogeneity among subjects in the form of random effects, i.e. continuous latent
variables. Growth mixture modeling (Muth¶en & Shedden, 1999; Muthen, 2001a, b;
Muthen, Brown, Masyn, Jo, Khoo, Yang, Wang, Kellam, Carlin, & Liao, 2000; Muthen
& Muthen, 1998-2001, Appendix 8) offers an important extension of conventional modeling
in that more general forms of unobserved heterogeneity can be captured using
categorical latent variables (latent classes). Growth mixture modeling as implemented
in the Mplus software (Muthen & Muthen, 1998-2001) allows for latent classes that
may have different shapes, antecedents, and consequences. A related longitudinal technique,
latent class growth analysis (Nagin, 1999), also studies unobserved heterogeneity
in the form of categorical latent variables. Growth mixture modeling, however, allows
categorical and continuous heterogeneity jointly, capturing potential further continuous
heterogeneity among individuals within the latent classes.
hide abstract
- Muthén, B. (2001).
Second-generation structural equation modeling with a combination of categorical and continuous latent
variables: New opportunities for latent class/latent growth modeling.
In Collins, L.M. & Sayer, A. (eds.), New Methods for the Analysis of Change (pp. 291-322). Washington, D.C.: APA.
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Abstract
New research provides an integration of categorical and continuous latent variable models. Given its
generality, it is fitting to describe the emerging methodology as second-generation SEM, where the
focus is on the generality of latent variable modeling (LVM). This LVM development promises to be extremely
beneficial to growth modeling. The aim of this paper is to briefly introduce new LVM analyses
in the form of General Growth Mixture Modeling (GGMM) and to show examples of the new analysis opportunities
for growth modeling that are opened up. Five different GGMM examples are given representing
five new types of growth analyses. The analyses are carried out by the new computer program Mplus
(Muthén & Muthén, 1998a). The presentation is non-technical in order to reach
applied researchers.
hide abstract
- Stoolmiller, M. (2001).
Synergistic interaction of child manageability problems and parent-discipline tactics in predicting future
growth in externalizing behavior for boys.
Developmental Psychology, 37, 814-825.
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Abstract
During early childhood for boys, manageability problems were hypothesized to disrupt parental discipline
practices. In turn, disrupted parental discipline practices were hypothesized to interact with
manageability problems during late childhood to predict change in antisocial behavior during the transition
from elementary to middle school. Results indicated that maternal retrospective perceptions
of unmanageability predicted observed maternal discipline practices, even when controlling for maternal
antisocial behavior and depressed mood and the disruptive and antisocial behavior of the boy.
Graphical analyses and latent class growth models indicated that temper tantrums interacted with maternal
discipline in predicting change in teacher ratings of antisocial behavior. The nature of the
interaction indicated that maternal discipline was a risk factor for growth in antisocial behavior
only for boys with high levels of tantrums.
hide abstract
- Muthén, B. & Muthén, L. (2000).
Integrating person-centered and variable-centered analyses: Growth mixture modeling with latent trajectory
classes.
Alcoholism: Clinical and Experimental Research, 24, 882-891.
Mplus inputs and outputs used in this paper can be viewed and/or downloaded from the Examples
page.
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Abstract
Background. Many alcohol research questions require methods that take a person-centered approach because
the interest is in finding heterogeneous groups of individuals such as those who are susceptible
to alcohol dependence and those who are not. A person-centered focus is also useful with longitudinal
data to represent heterogeneity in developmental trajectories. In alcohol, drug, and mental
health research the recognition of heterogeneity has led to theories of multiple developmental pathways.
Methods. This paper gives a brief overview of new methods that integrate variable- and
person-centered analyses. Methods discussed include latent class analysis, latent transition analysis,
latent class growth analysis, growth mixture modeling, and general growth mixture modeling. These
methods are presented in a general latent variable modeling framework that expands traditional
latent variable modeling by including not only continuous latent variables but also categorical latent
variables. Results. Four examples that use the NLSY data are presented to illustrate latent
class analysis, latent class growth analysis, growth mixture modeling, and general growth mixture
modeling. Latent class analysis of antisocial behavior found four classes. Four heavy drinking
trajectory classes were found. The relationship between the latent classes and their relationship
to background variables and consequences was studied. Conclusions. Person-centered and variable-centered
analyses have typically been seen as different activities that use different types of
models and software. This paper gives a brief overview of new methods that integrate variable- and
person-centered analyses. The general framework makes it possible to combine these models and to study
new models serving as a stimulus for asking research questions that have both person- and variable-centered
aspects."
hide abstract
- Muthén, B. & Shedden, K. (1999).
Finite mixture modeling with mixture outcomes using the EM algorithm.
Biometrics, 55, 463-469.
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Abstract
This paper discusses the analysis of an extended finite mixture model where the latent classes corresponding
to the mixture components for one set of observed variables influence a second set of observed
variables. The research is motivated by a repeated measurement study using a random coefficient
model to assess the influence of latent growth trajectory class membership on the probability of a binary
disease outcome. More generally, this model can be seen as a combination of latent class modeling
and conventional mixture modeling. The EM algorithm is used for estimation. As an illustration,
a random-coefficient growth model for the prediction of alcohol dependence from three latent classes
of heavy alcohol use trajectories among young adults is analyzed."
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- Seddig, D. (2023). Latent growth models for count outcomes: Specification, evaluation, and interpretation. Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2023.2175684
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- Feingold, A. (2022). Regression equivalent effect sizes for latent growth modeling and associated null hypothesis significance tests. Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2022.2139702
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- Asparouhov, T. & Muthén, B. (2021). Residual Structural Equation Models. Technical Report. Version 2. February 3, 2022.
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- Feingold, A. (2021). Effect of parameterization on statistical power and effect size estimation in latent growth modeling. Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2021.1878895
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- Zyphur, M.J., Allison, P.D., Tay, L. Voelkle, M.C., Preacher, K.J., Zhang, Z., Hamaker, E.L., Shamsollahi, A., Pierides, D.C., Koval, P. & Diener, E. (2020). From data to causes I: Building a general cross-lagged panel model (GCLM). Organizational Research Methods, 23(4), 651-687.
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- Zyphur, M.J., Allison, P.D., Tay, L. Voelkle, M.C., Preacher, K.J., Zhang, Z., Hamaker, E.L., Shamsollahi, A., Pierides, D.C., Koval, P. & Diener, E. (2020). From data to causes II: Comparing approaches to panel data analysis. Organizational Research Methods, 23(4), 688-716.
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- Feng, Y., Hancock, G.R, & Harring, J.R. (2019). Latent growth models with floors, ceilings, and random knots. Multivariate Behavioral Research, 54:5, 751-770. DOI: 10.1080/00273171.2019.1580556
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- Feingold, A. (2018). Time-varying effect sizes for quadratic growth models in multilevel and latent growth modeling. Structural Equation Modeling: A Multidisciplinary Journal. DOI: 10.1080/10705511.2018.1547110
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- Raykov, T., Marcoulides, G.A., Menold, N., Li, T., & Zhang, M. (2018). On examining intervention effects upon ability development using latent variable modeling. Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2018.1485494
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- Lee, T.K., Wickrama, K.A.S. & O’Neal, C.W. (2017): Application of Latent Growth Curve Analysis With Categorical Responses in Social Behavioral Research. Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2017.1375858
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- Grimm, K.J. & Liu, Y. (2016). Residual structures in growth models with ordinal outcomes. Structural Equation Modeling: A Multidisciplinary Journal, 23:3, 466-475, DOI: 10.1080/10705511.2015.1103192
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- Wang, C., Kohli, N., & Henn, L. (2015). A second-order longitudinal model for binary outcomes: Item response theory versus structural equation modeling. Structural Equation Modeling: A Multidisciplinary Journal. DOI: 10.1080/10705511.2015.1096744
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- Koppara, A., Wagner, M., Lange, C., Ernst, A., Wiese, B., Konig, H., Brettschneider, C., Riedel-Heller, S., Luppa, M., Weyerer, S., Werle, J., Bickel, H., Mosch, E., Pentzek, M., Fuchs, A., Wolfsgruber, S., Beauduccl, A., Scherer, M., Maier, W., & Jessen, F. (2015). Cognitive performance before and after the onset of subjective cognitive decline in old age. Alzheimer’s and Dementia: Diagnosis, Assessment and Disease Monitoring. 1-12. DOI: 10.1016/j.dadm.2015.02.005
view abstract
- Preacher, K. J., & Hancock, G. R. (2015). Meaningful aspects of change as novel random coefficients:
A general method for reparameterizing longitudinal models. Psychological Methods, 20(1), 84-101.
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- Peter, J., Scheef, L., Abdulkadir, A., Boecker, H., Heneka, M., Wagner, M., Koppara, A., Kloppel, S., & Jessen, F. (2014). Gray matter atrophy pattern in elderly with subjective memory impairment. Alzheimer's & Dementia, 10(1), 99-108. DOI: 10.1016/j.jalz.2013.05.1764
view abstract
- Feingold, A. (2014). Confidence interval estimation for standardized effect sizes in multilevel and latent growth modeling.
Journal of Consulting and Clinical Psychology, pre-print article.
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- Geiser, C., Bishop, J., Lockhart, G., Shiffman, S., & Grenard, J. (2013). Analyzing latent state-trait and multiple-indicator latent growth curve models as multilevel structural equation models.
Frontiers in Psychology, 4, 975. DOI: 10.3389/fpsyg.2013.00975
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Abstract
Latent state-trait (LST) and latent growth curve (LGC) models are frequently used in the analysis of longitudinal data.
Although it is well-known that standard single-indicator LGC models can be analyzed within either the structural equation
modeling (SEM) or multilevel (ML; hierarchical linear modeling) frameworks, few researchers realize that LST and
multivariate LGC models, which use multiple indicators at each time point, can also be specified as ML models. In the
present paper, we demonstrate that using the ML-SEM rather than the SL-SEM framework to estimate the parameters of these
models can be practical when the study involves (1) a large number of time points, (2) individually-varying times of
observation, (3) unequally spaced time intervals, and/or (4) incomplete data. Despite the practical advantages of the
ML-SEM approach under these circumstances, there are also some limitations that researchers should consider. We present
an application to an ecological momentary assessment study (N = 158 youths with an average of 23.49 observations of
positive mood per person) using the software Mplus (Muthén and Muthén, 1998–2012) and discuss advantages and disadvantages
of using the ML-SEM approach to estimate the parameters of LST and multiple-indicator LGC models.
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- Masyn, K., Petras, H. and Liu, W. (2013). Growth Curve Models with Categorical Outcomes. In Encyclopedia
of Criminology and Criminal Justice (pp. 2013-2025). Springer.
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- Kevin J. Grimm , Joel S. Steele , Nilam Ram & John R. Nesselroade (2013) Exploratory latent growth models in the structural equation
modeling framework. Structural Equation Modeling: A Multidisciplinary Journal, 20:4, 568-591, DOI: 10.1080/10705511.2013.824775
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Abstract
Latent growth modeling is often conducted using a confirmatory approach whereby specific struc-
tures of individual change (e.g., linear, quadratic, exponential, etc.) are fit to the observed
data, the best fitting model is chosen based on fit statistics and theoretical considerations, and
parameters from this model are interpreted. This confirmatory approach is appropriate when a
strong theory guides the model fitting process. However, this approach is often also used when
there is not a strong theory to guide the model fitting process, which might lead researchers to
misrepresent or miss key change characteristics present in their data. We discuss Tuckerized curves
(Tucker, 1958, 1966) as an exploratory way of modeling change processes based on principal components analysis and
propose an exploratory approach to latent growth modeling whereby minimal constraints are imposed
on the structure of within-person change. These methods are applied to longitudinal data on
cortisol response during a controlled experimental manipulation and height changes from early
childhood through adulthood collected from 2 different studies. We highlight the additional
insights gained, some of the benefits, limitations, and potential extensions of the exploratory
growth curve approach and suggest there is much to be gained from using such models to generate new
and potentially more precise theories about change and development.
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- Nidhi Kohli & Jeffrey R. Harring (2013): Modeling growth in latent variables using a piecewise function, Multivariate Behavioral Research, 48:3,370-397
DOI: 10.1080/00273171.2013.778191
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Abstract
Latent growth curve models with piecewise functions for continuous repeated
measures data have become increasingly popular and versatile tools for investigat- ing individual
behavior that exhibits distinct phases of development in observed variables. As an extension of
this framework, this research study considers a piecewise function for describing segmented
change of a latent construct over time where the latent construct is itself measured by multiple
indicators gathered at each measurement occasion. The time of transition from one phase to another
is not known a priori and thus is a parameter to be estimated. Utility of the model is highlighted
in 2 ways. First, a small Monte Carlo simulation is executed to show the ability of the model to
recover true (known) growth parameters, including the location of the point of transition (or
knot), under different manipulated condi- tions. Second, an empirical example using longitudinal
reading data is fitted via maximum likelihood and results discussed. Mplus (Version 6.1) code is
provided in Appendix C to aid in making this class of models accessible to practitioners.
hide abstract
- Malone, P. S., Northrup, T. F., Masyn, K. E., Lamis, D. A., & Lamont, A. E. (2012). Initiation and persistence of alcohol use in United States Black, Hispanic, and White male and female youth. Addictive Behaviors, 37, 299-305.
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Abstract
Background: The relation between early and frequent alcohol use and later difficulties is quite strong. However,
the degree that alcohol use persists, which is often a necessary cause for developing alcohol-related problems or
an alcohol use disorder, is not well studied, particularly with attention to race and gender. A novel statistical
approach, the Multi-facet Longitudinal Model, enables the concurrent study of age of initiation and persistence.
Methods: The models were applied to longitudinal data on youth alcohol use fromages 12 through 19, collected
in the (U.S.) National Longitudinal Survey of Youth 1997 cohort (N=8984).
Results: Results confirmed that Black adolescents initiate alcohol use at later ages than doWhite youth. Further,
after initiation, White adolescents were substantially more likely than Black adolescents to continue reporting
alcohol use in subsequent years. Hispanic teens showed an intermediate pattern. Gender differences were
more ambiguous, with a tendency for boys to be less likely to continue drinking after initiation than were girls.
Conclusions: Novel findings from the new analytic models suggest differential implications of early alcohol use
by race and gender. Early use of alcohol might be less consequential for males who initiate alcohol use early,
Black, and Hispanic youth than for their female and White counterparts.
hide abstract
- Geiser, C., Eid, M., Nussbeck, F.W., Courvoisier, D.S. & Cole, D.A. (2010).
Analyzing true change in longitudinal multitrait-multimethod studies: Application of a multimethod change
model to depression and anxiety in children.
Developmental Psychology, 46, 29-45.
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Abstract
"The authors show how structural equation modeling can be applied to analyze change in longitudinal
multitrait-multimethod
(MTMM) studies. For this purpose, an extension of latent difference models
(McArdle,
1988; Steyer, Eid, & Schwenkmezger, 1997) to multiple constructs and multiple methods is
presented.
The model allows investigators to separate true change from measurement error and to analyze
change
simultaneously for different methods. The authors also show how Campbell and Fiske’s (1959)
guidelines
for analyzing convergent and discriminant validity can be applied to the measurement of latent
change.
The practical application of the multimethod change model is illustrated in a reanalysis
of child
depression and anxiety scores (N 906 American children) that were assessed by self- and
parent reports
on three measurement occasions. The analyses revealed that (a) the convergent validity
of change was
low for both constructs and (b) sex was a significant predictor of self-reported, but
not of parent reported,
anxiety states. Finally, the authors discuss advantages and limitations and
compare the model with other
approaches for analyzing longitudinal MTMM data."
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- Morin, A.J.S., Maïano, C., Marsh, H.W., Janosz, M. & Nagengast, B. (2010). The longitudinal interplay of adolescents’ self-esteem and body image: A conditional autoregressive latent trajectory analysis.
Multivariate Behavioral Research, 46(2). DOI: 10.1080/00273171.2010.546731
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Abstract
"Self-esteem and body image are central to coping succsefully with the developmental challenges of
adolescence.
However, the current knowledge surrounding self-esteem and body image is fraught with
controversy.
This study attempts to clarify some of them by addressing three questions: (i) Are the
intra-individual
developmental trajectories of self-esteem and body image stable across adolescence?
(ii)
What is the direction of the relations between body image and self-esteem over time? (iii) What is
the
role of gender, ethnicity and pubertal development on those trajectories? This study relies on
Autoregressive
Latent Trajectory analyses based on data from a four-year, six-wave, prospective
longitudinal
study of 1001 adolescents. Self-esteem and body image levels remained high and stable
over
time, although body image levels also tended to increase slightly. The results show that levels of
self-esteem
were positively influenced by levels of body image. However, these effects remained
small
and most of the observed associations were cross-sectional. Finally, the effects of pubertal
development
on body image and self-esteem levels were mostly limited to non-Caucasian females
who appeared
to benefit from more advanced pubertal development. Conversely, Caucasian females
presented the lowest
self-esteem and body image levels of all, although for them more advanced
pubertal development levels
were associated with a slight rise in body image over time."
hide abstract
- Benner, A. & Graham, S. (2009).
The transition to high school as a developmental process among multiethnic urban youth.
Child Development, 80:2, 356–376.
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Abstract
"The high school transition was examined in an ethnically diverse, urban sample of 1,979 adolescents,
followed
from 7th to 10th grade (Mage = 14.6, SD = .37 in 7th grade). Twice annually, data were gathered
on adolescents’
perceptions of school climate, psychological functioning, and academic behaviors.
Piecewise growth
modeling results indicate that adolescents were doing well before the transition
but experienced transition
disruptions in psychological functioning and grades, and many continued
to struggle across high school. The
immediate experience of the transition appeared to be particularly
challenging for African American and
Latino students when the numerical representation of their ethnic
groups declined significantly from middle
to high school. Findings highlight the value of examining
the transition in a larger developmental context and
the importance of implementing transition
support."
hide abstract
- Grimm, K.J. & Ram, N. (2009).
Nonlinear growth models in Mplus and SAS.
Structural Equation Modeling, 16, 676-701.
Click here to view related
Mplus scripts.
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Abstract
"Nonlinear growth curves or growth curves that follow a specified nonlinear function in time enable
researchers
to model complex developmental patterns with parameters that are easily interpretable.
In
this article we describe how a variety of sigmoid curves can be fit using the Mplus structural
modeling
program and the nonlinear mixed-effects modeling procedure NLMIXED in SAS. Using
longitudinal achievement
data, collected as part of a study examining the effects of preschool
instruction on academic
gain, we illustrate the procedures for fitting growth models of logistic,
Gompertz, and Richards
functions. Brief notes regarding the practical benefits, limitations, and
choices faced in the fitting
and estimation of such models are included."
hide abstract
- Pettit, G.S., Keiley, M.K., Laird, R.D., Bates, J.E. & Dodge, K.A. (2007).
Predicting the developmental course of mother-reported monitoring across childhood and adolescence from
early proactive parenting, child temperament, and parents’ worries.
Journal of Family Psychology, 21, 206-217.
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Abstract
"Change in mothers’ reported monitoring and awareness of their children’s activities and
companions across
Grades 5, 6, 8, and 11 were examined with the use of latent factor growth
modeling. Proactive parenting
and resistant-to-control (RTC) child temperament assessed
prior to kindergarten, as well as
parents’ worries about their children’s behavior in Grades 5
and 8, were tested as factors associated
with change in monitoring over time. Higher proactive
parenting, lower RTC temperament, and the mounting
of a successful campaign to change
their children’s behavior were associated with higher monitoring
scores overall. Monitoring
levels decreased across time, but the rate of decline was steeper
among mothers with high
RTC children and slower among mothers who mounted a campaign and judged it
to be
effective. These findings shed light on factors contributing to continuity and change across
development
in a key domain of parenting."
hide abstract
- Clark, D., Birmaher, B., Axelson, D., Monk, K., Kalas, C., Ehmann, M., Bridge, J., Wood, S., Muthén, B., & Brent, D. (2005).
Fluoxetine for the treatment of childhood anxiety disorders: Open-label, long-term extension to a controlled
trial. Journal of the American Academy of Child & Adolescent Psychiatry, 44, 1263-1270.
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Abstract
"Objective: To assess the efficacy of fluoxetine for the long-term treatment of children and adolescents
with anxiety disorders,
including generalized anxiety disorder, separation anxiety disorder, and/or
social phobia. Method: Children and
adolescents (7–17 years old) with anxiety disorders were studied
in open treatment for 1 year after they completed a randomized,
controlled trial (RCT) comparing
fluoxetine and placebo. The follow-up phase assessments included clinician,
parent, and child ratings
with measures of global severity, global improvement, and anxiety symptoms. Results: Subjects
taking
fluoxetine (n = 42) were compared with those taking no medication (n = 10) during follow-up on anxiety
changes from
the end of the RCT through the follow-up period. Statistical models included RCT
assignment and follow-up psychological
treatment. Excluded subjects took other medications (n = 4)
or did not complete follow-up (n = 18). Compared with subjects
taking no medication, subjects taking
fluoxetine showed significantly superior follow-up outcomes on most measures, including
clinician,
parent, and child ratings. Conclusions: The results suggest that fluoxetine is clinically effective
for the
maintenance treatment of anxiety disorders in children and adolescents. A major limitation,
however, was the lack of RCT
methodology in the follow-up phase. RCTs are needed to determine the long-term
risks and benefits of fluoxetine for this
group. J. Am. Acad. Child Adolesc. Psychiatry, 2005;44(12):1263–1270.
Key Words: anxiety disorders, fluoxetine, selective
serotonin reuptake inhibitors."
hide abstract
- Muthén, B. & Muthén, L. (2000).
The development of heavy drinking and alcohol-related problems from ages 18 to 37 in a U.S. national
sample. Journal of Studies on Alcohol, 61, 290-300.
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Abstract
"Objective. The purpose of this study is to add to the understanding of the development of heavy alcohol
use and alcohol-related problems by examining data from the National Longitudinal Survey of Youth
(NLSY), a general population sample that contains information on alcohol use for the ages 18-37.
A key question in this study is how background characteristics of the individual influence this
development and whether the influence of these background characteristics changes over time. Method.
The data used in this study are a general population sample from the National Longitudinal
Survey of Youth (NLSY). This study uses a multivariate outcome approach which focuses on individual
variation in trajectories over age. The statistical analysis uses random coefficients in a latent
variable framework. Across-age changes in the importance of the influence of background variables
on the outcomes are modeled using varying centering points. Results. A key finding is that dropping
out of high school has no effect on alcohol problems for individuals in their mid twenties,
but is associated with significantly increased levels of alcohol problems for individuals in their
mid thirties. In contrast, going on to college is associated with lower levels of heavy drinking
when individuals reach their late twenties and their thirties. Strong gender and ethnicity effects
seen in the twenties diminish when individuals reach their thirties. Conclusions. The trajectory
analysis expands the knowledge of problematic alcohol development for individuals in their late
twenties and thirties. The increasing detrimental effect of dropping out of high school up to the
age 37 endpoint of the study raises questions about the effects of dropping out of high school later
in life."
hide abstract
- Muthén, B. & Curran, P. (1997).
General longitudinal modeling of individual differences in experimental designs: a latent variable framework
for analysis and power estimation.
Psychological Methods, 2, 371-402.
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Abstract
"The generality of latent variable modeling of individual difference in development over time is demonstrated
with a particular emphasis on randomized intervention studies. First, a brief overview is given
of biostatistical and psychometric approaches to repeated measures analysis. Second, the generality
of the psychometric approach is indicated by some nonstandard models. Third, a multiple-population
analysis approach is proposed for the estimation of treatment effects. The approach clearly describes
the treatment effect as development that differs from normative, control-group development.
This framework allows for interactions between treatment and initial status in their effects on development.
Finally, an approach for the estimation of power to detect treatment effects in this framework
is demonstrated. Illustrations of power calculations are carried out with artificial data, varying
the sample sizes, number of timepoints, and treatment effect sizes. Real data are used to illustrate
analysis strategies and power calculations. Further modeling extensions are discussed."
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- Asparouhov, T., & Muthén, B. (2020). IRT in Mplus. Version 4. Technical report. www.statmodel.com
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- Holtmann, J., Koch, T., Bohn, J., & Eid, M. (2020). Multimethod assessement of time-stable and time-variable interindividual differences: Introduction of a new multitrait-multimethod latent state-trait IRT model.
European Journal of Psychological Assessment. DOI: 10.1027/1015-5759/a000577
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- Montoya, A.K. & Jeon, M. (2019). MIMIC models for uniform and nonuniform DIF as moderated mediation models. Applied Psychological Measurement. DOI: 10.1177/0146621619835496
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- Muthén, B. & Asparouhov, T. (2016). Multi-Dimensional, Multi-Level, and Multi-Timepoint Item Response Modeling.
In van der Linden, W. J., Handbook of Item Response Theory. Volume One. Models, pp. 527-539. Boca Raton: CRC Press.
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Abstract
This chapter presents item response modeling techniques in the Mplus program (Muthén & Muthén,
2012) through the analysis of an example with three features common in behavioral science
applications: multiple latent variable dimensions, multilevel data, and multiple timepoints. The
dimensionality of a measurement instrument with categorical items is investigated using exploratory
factor analysis with bi-factor rotation. Variation across students and classrooms is investigated
using two-level exploratory and confirmatory bi-factor models. Change over grades is investigated
using a longitudinal two-level model. The analyses are carried out using weighted least-squares,
maximum-likelihood, and Bayesian analysis. The strengths of weighted least-squares and Bayes as a
complement to maximum-likelihood for this high-dimensional application are discussed. Mplus
scripts for all analyses are available at www.statmodel.com.
hide abstract
- Cho, S., Preacher, K.J., & Bottge, B.A. (2015). Detecting intervention effects in a cluster-randomized design using multilevel structural equation modeling for binary responses.
Applied Psychological Measurement, 39(8) 627–642. DOI: 10.1177/0146621615591094
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- Wall, M. M., Park, J. Y., & Moustaki, I. (2015). IRT modeling in the presence of zero-inflation with application to psychiatric disorder severity. Applied Psychological Measurement. DOI: 10.1177/0146621615588184
view abstract
- Huggins-Manley, A.C. & Algina, J. (2015). The partial credit model and generalized partial credit model as constrained nominal response models, with applications in Mplus. Structural Equation Modeling: A
Multidisciplinary Journal, DOI: 10.1080/10705511.2014.937374
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- Muthén, B. & Asparouhov T. (2014). IRT studies of many groups: The alignment method. Frontiers in Psychology, Volume 5, DOI: 10.3389/fpsyg.2014.00978
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Abstract
Asparouhov and Muth´en presented a new method for multiple-group confirmatory factor analysis (CFA), referred to as the alignment method. The alignment method can be used to estimate group-specific factor means and variances without requiring exact
measurement invariance. A strength of the method is the ability to conveniently estimate models for many groups, such as with comparisons of countries. This paper focuses on IRT applications of the alignment method. An empirical investigation is made of binary knowledge items administered
in two separate surveys of a set of countries. A Monte Carlo study is presented that shows how the quality of the alignment can be assessed.
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- Brown, A. & Maydeu-Olivares, A. (2013). How IRT can solve problems of ipsative data in forced-choice questionnaires. Psychological Methods, 18(1), 36-52. DOI: 10.1037/a0030641
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- Sawatzky, R., Ratner, P.A., Kopec, J.A., & Zumbo, B.D. (2011).
Latent variable mixture models: A promising approach for the validation of patient reported outcomes.
Quality of Life Research. DOI:10.1007/s11136-011-9976-6.
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Abstract
"Purpose
A fundamental assumption of patient-reported outcomes (PRO) measurement is that all individuals
interpret questions about their health status in a consistent manner, such that a measurement model
can be constructed that is equivalently applicable to all people in the target population. The
related assumption of sample homogeneity has been assessed in various ways, including the many approaches
to differential item functioning analysis.
Methods
This expository paper describes the use
of latent variable mixture modeling (LVMM), in conjunction with item response theory (IRT), to examine:
(a) whether a sample is homogeneous with respect to a unidimensional measurement model, (b) implications
of sample heterogeneity with respect to model-predicted scores (theta), and (c) sources of
sample heterogeneity. An example is provided using the 10 items of the Short-Form Health Status (SF-36®)
physical functioning subscale with data from the Canadian Community Health Survey (2003) (N =
7,030 adults in Manitoba).
Results
The sample was not homogeneous with respect to a unidimensional
measurement structure. Specification of three latent classes, to account for sample heterogeneity,
resulted in significantly improved model fit. The latent classes were partially explained by demographic
and health-related variables.
Conclusion
The illustrative analyses demonstrate the value
of LVMM in revealing the potential implications of sample heterogeneity in the measurement of PROs.
"
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- Reise, S. P., Moore, T. M., & Maydeu-Olivares, A. (2011). Target rotations and assessing the impact of model violations on the parameters of unidimensional item response theory models. Educational and Psychological Measurement, 71(4) 684–711. DOI: 10.1177/0013164410378690
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Abstract
"Peer Review Reise, Cook, and Moore (under review) proposed a “comparison modeling” approach to assess
the distortion in item parameter
estimates when a unidimensional item response theory (IRT) model is
imposed on multidimensional data. Central to their approach is
the comparison of item slope parameter
estimates from a unidimensional IRT model (a restricted model), with the item slope
parameter
estimates from the general factor in an exploratory bifactor IRT model (the unrestricted comparison
model). In turn,
these authors suggested that the unrestricted comparison bifactor model be derived
from a target factor rotation (Browne, 2001). The
goal of this study was to provide further empirical
support for the use of target rotations as a method for deriving a comparison
model. Specifically,
we conducted Monte Carlo analyses exploring: a) the use of the Schmid-Leiman (Schmid & Leiman, 1957)
orthogonalization
to specify a viable initial target matrix, and b) the recovery of true bifactor
pattern matrices using target rotations as
implemented in MPLUS (Asparouhov & Muthén, 2008).
Results suggest that to the degree that item response data conform to
independent cluster structure,
target rotations can be used productively to establish a plausible comparison model."
hide abstract
- Forero, C.G. & Maydeu-Olivares, A. (2009).
Estimation of IRT graded response models: Limited versus full information methods.
Psychological Methods, 14:3, 275–299.
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Abstract
"The performance of parameter estimates and standard errors in estimating F. Samejima’sThe performance
of parameter estimates and standard errors in estimating F. Samejima’s
graded response model was examined
across 324 conditions. Full information maximum
likelihood (FIML) was compared with a 3-stage
estimator for categorical item factor analysis
(CIFA) when the unweighted least squares method was
used in CIFA’s third stage. CIFA is
much faster in estimating multidimensional models, particularly
with correlated dimensions.
Overall, CIFA yields slightly more accurate parameter estimates, and FIML
yields slightly
more accurate standard errors. Yet, across most conditions, differences between methods
are
negligible. FIML is the best election in small sample sizes (200 observations). CIFA is the
best
election in larger samples (on computational grounds). Both methods failed in a number
of conditions,
most of which involved 200 observations, few indicators per dimension, highly
skewed items,
or low factor loadings. These conditions are to be avoided in applications."
hide abstract
- Woods, C.M. (2009).
Evaluation of MIMIC-model methods for DIF testing with comparison to two-group analysis.
Multivariate Behavioral Research, 44:1,1-27.
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Abstract
"Differential item functioning (DIF) occurs when an item on a test or questionnaire
has different measurement
properties for 1 group of people versus another,
irrespective of mean differences on the construct.
This study focuses on the use
of multiple-indicator multiple-cause (MIMIC) structural equation
models for DIF
testing, parameterized as item response models. The accuracy of these methods,
and
the sample size requirements, are not well established. This study examines
the accuracy of MIMIC methods
for DIF testing when the focal group is small
and compares results with those obtained using
2-group item response theory
(IRT). Results support the utility of the MIMIC approach. With small focalgroup
samples,
tests of uniform DIF with binary or 5-category ordinal responses
were more accurate
with MIMIC models than 2-group IRT. Recommendations are
offered for the application of MIMIC methods
for DIF testing."
hide abstract
- Dumenci, L. & Achenbach, T.M. (2008).
Effects of estimation methods on making trait-level inferences from ordered categorical items for assessing
psychopathology.
Psychological Assessment, 20, 55-62.
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Abstract
"In assessments of attitudes, personality, and psychopathology, unidimensional scale scores are commonly
obtained
from Likert scale items to make inferences about individuals’ trait levels. This study approached
the
issue of how best to combine Likert scale items to estimate test scores from the practitioner’s
perspective:
Does it really matter which method is used to estimate a trait? Analyses of 3
data sets
indicated that commonly used methods could be classified into 2 groups: methods that explicitly
take
account of the ordered categorical item distributions (i.e., partial credit and graded response
models of
item response theory, factor analysis using an asymptotically distribution-free estimator)
and methods
that do not distinguish Likert-type items from continuously distributed items (i.e.,
total score, principal
component analysis, maximum-likelihood factor analysis). Differences in
trait estimates were found to be
trivial within each group. Yet the results suggested that inferences
about individuals’ trait levels differ
considerably between the 2 groups. One should therefore choose
a method that explicitly takes account
of item distributions in estimating unidimensional traits
from ordered categorical response formats.
Consequences of violating distributional assumptions were
discussed."
hide abstract
- Zumbo, B.D. (2007).
Three generations of DIF analyses:Considering where it has been, where it is now, and where it is going.
Language Aseessment Quarterly, 4(2), 223–233.
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Abstract
"The purpose of this article is to reflect on the state of the theorizing and praxis of
DIF in general:
where it has been; where it is now; and where I think it is, and
should, be going. Along the way the
major trends in the differential item
functioning (DIF) literature are summarized and integrated
providing some organizing
principles that allow one to catalog and then contrast the various DIF
detection
methods and to shine a light on the future of DIF analyses. The three
generations of DIF are
introduced and described with an eye toward issues on the
horizon for DIF."
hide abstract
- MacIntosh, R. & Hashim, S. (2003).
Converting MIMIC model parameters to IRT parameters in DIF analysis.
Applied Psyhological Measurement, 27, 372-379.
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Abstract
"The purpose of this study is to document and compare two methods to estimate the statistical properties
of the converted Item Response Theory discrimination and difficulty parameters derived from MIMIC
model parameters. The delta method and Monte Carlo simulation provide similar variance estimates,
with differences attributed to rounding error. Discussed is the formulation of MIMIC models in Mplus
and how to obtain factor analytic estimates that are converted easily into IRT parameters. Also described
are the partial derivatives necessary to apply the delta method to estimate variances for the
converted parameters. Both item difficulty and discrimination parameters estimated from MIMIC parameters
were very close to the Multilog estimates. The variance estimates for most parameters were
similar, as well."
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- Weller, B.E., Bowen, N.K., & Faubert, S.J. (2020). Latent Class Analysis: A guide to best practice. Journal of Black Psychology. DOI: 10.1177/0095798420930932
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- Guertler, D., Moehring, A., Krause, K., Tomczyk, S., Freyer-Adam, J., Baumann, S., Bischof, G., Rumpf, H.J., Batra, A., Wurm, S., John, U., & Meyer, C. (2020). Latent alcohol use patterns and their link to depressive symptomatology in medical care patients. Addiction. DOI: 10.1111/add.15261
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- Evans, B. E., Kim, Y. & Hagquist, C. (2020). A latent class analysis of changes in adolescent substance use between 1988 and 2011 in Sweden: associations with sex and psychosomatic problems. Addiction. Vol. 115, 1932-1941. DOI: 10.1111/add.15040
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- Ferguson, S. L., G. Moore, E. W., & Hull, D. M. (2019). Finding latent groups in observed data: A primer on latent profile analysis in Mplus for applied researchers. International Journal of Behavioral Development. DOI: 10.1177/0165025419881721
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- Nylund-Gibson, K., Grimm, R., Masyn, K. (2019). Prediction from latent classes: A demonstration of different approaches to including distal outcomes in mixture models. Structural Equation Modeling: A Multidisciplinary Journal, 26:6, 967-985, DOI: 10.1080/10705511.2019.1590146
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Abstract
Including auxiliary variables such as antecedent and consequent variables in mixture models provides valuable insight in understanding the population heterogeneity embodied by a latent class variable. The model building process regarding how to include predictors/correlates and outcomes of the latent class variables into mixture models is an area of active research. As such, new methods of including these variables continue to emerge and best practices for the application of these methods in real data settings (including simple guidelines for choosing amongst them) are still not well established. This paper focuses on one type of auxiliary variable—distal outcomes—providing an overview of the methods currently available for estimating the effects of latent class membership on subsequent distal outcomes. We illustrate the recommended methods in the software packages Mplus and Latent Gold using a latent class model to capture population heterogeneity in students’ mathematics attitudes, linking latent class membership to two distal outcomes.
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- McLarnon, M.J. & O'Neill, T.A. (2018). Extensions of auxiliary variable approaches for the investigation of mediation, moderation, and conditional effects in mixture models. Organizational Research Methods. Vol. 21(4), 955-982. DOI: 10.1177/1094428118770731
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- Schmiege, S.J., Masyn, K.E. & Bryan, A.D. (2018). Confirmatory latent class analysis: Illustrations of empirically driven and theoretically driven model constraints. Organizational Research Methods. Vol. 21(4), 983-1001. DOI: 10.1177/1094428117747689
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- Nylund-Gibson, K., & Choi, A. Y. (2018). Ten frequently asked questions about latent class analysis. Translational Issues in Psychological Science, 4(4), 440–461. DOI: 10.1037/tps0000176
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Abstract
Latent class analysis (LCA) is a statistical method used to identify unobserved subgroups in a
population with a chosen set of indicators. Given the increasing popularity of LCA, our aim is to
equip psychological researchers with the theoretical and statistical fundamentals that we believe
will facilitate the application of LCA models in practice. In this article, we provide answers to
10 frequently asked questions about LCA. The questions included in this article were fielded from
our experience consulting with applied researchers interested in using LCA. The major topics
include a general introduction in the LCA; an overview of class enumeration (e.g., deciding on the
number of classes), including commonly used statistical fit indices; substantive interpretation of
LCA solutions; estimation of covariates and distal outcome relations to the latent class variable;
data requirements for LCA; software choices and considerations; distinctions and similarities among
LCA and related latent variable models; and extensions of the LCA model.
To illustrate the modeling ideas described in this article, we present an applied example using
LCA. Specifically, we use LCA to model individual differences in positive youth development among
college students and analyze demographic characteristics as covariates and a distal outcome of
overall life satisfaction. We also include key references that direct readers to more detailed and
technical discussions of these topics for which we provide an applied and introductory overview. We
conclude by mentioning future developments in research and
practice, including advanced cross-sectional and longitudinal extensions of LCA.
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- Nylund-Gibson, K. & Masyn, K. (2016). Covariates and mixture modeling: Results of a simulation study exploring the impact of misspecified effects on class enumeration. Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2016.1221313
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- Jiang, L., Chen, S., Zhang, B., Beals, J., Mitchell, C.M., Manson, S.M. & Roubideaux, Y. (2016). Longitudinal patterns of stages of change for exercise and lifestyle intervention outcomes: An application of latent class analysis with distal outcomes. Prevention Science, 17, 398-409. DOI 10.1007/s11121-015-0599-y
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- Raykov, T., Marcoulides, G.A., & Chang, C. (2016). Examining population heterogeneity in finite mixture settings using latent variable modeling. Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2015.1103193
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- Morin, A.J.S., Meyer, J.P., Creusier, J., Bietry, F. (2015). Multiple-group analysis of similarity in latent profile solutions. Organizational Research Methods, 19:2, 231-254, DOI: 10.1177/1094428115621148
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- Asparouhov, T. & Muthén, B. (2015). Residual associations in latent class and latent transition analysis. Structural Equation Modeling: A Multidisciplinary Journal, 22:2, 169-177, DOI: 10.1080/10705511.2014.935844
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Abstract
This paper explores a method for modeling associations among binary and
ordered categorical variables. The method has the advantage that maximum-
likelihood estimation can be used in multivariate models without numerical
integration because the observed data log-likelihood has an explicit form. The
association model is especially useful with mixture models to handle violations
of the local independence assumption. Applications to latent class and latent
transition analysis are presented.
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- Morgan, G. B. (2014). Mixed mode latent class analysis: An examination of fit index performance
for classification. Structural Equation Modeling: A Multidisciplinary Journal, DOI:
10.1080/10705511.2014.935751
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- Asparouhov, T. & Muthén, B. (2014) Auxiliary variables in mixture modeling: Three-step approaches using Mplus. Structural Equation Modeling: A Multidisciplinary Journal, 21:3, 329-341. The posted version corrects several typos in the published version. An earlier version of this paper was posted as web note 15.
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Abstract
This paper discusses alternatives to single-step mixture modeling. A 3-step method for latent
class predictor variables is studied in several different settings including latent class analysis, latent
transition analysis, and growth mixture modeling. It is explored under violations of its assumptions such as
with direct effects from predictors to latent class indicators. The 3-step method is also considered for distal
variables. The Lanza et al. (2013) method for distal variables is studied under several conditions including
violations of its assumptions. Standard errors are also developed for the Lanza method since these were not given
in Lanza et al. (2013).
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- Dziak, John J., Lanza, Stephanie T., & Tan, Xianming. (2014). Effect size, statistical power and sample size requirements for the bootstrap likelihood ratio test in latent class analysis. Structural Equation Modeling: A Multidisciplinary Journal, 21(4): 534–552. doi:10.1080/10705511.2014.919819.
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- Feingold, A., Tiberio, S.S., & Capaldi, D.M. (2013).
New approaches for examining associations with latent categorical variables: Applications to Substance
Abuse and Aggression. Psychology of Addictive Behaviors. DOI: 10.1037/a0031487
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Abstract
Assessments of substance use behaviors often include categorical variables that are frequently related to
other measures using logistic regression or chi-square analysis. When the categorical variable is latent
(e.g., extracted from a latent class analysis; LCA), classification of observations is often used to create
an observed nominal variable from the latent one for use in a subsequent analysis. However, recent simulation
studies have found that this classical three-step analysis championed by the pioneers of LCA produces
underestimates of the associations of latent classes with other variables. Two preferable but underused
alternatives for examining such linkages—each of which is most appropriate under certain conditions—are (a)
three-step analysis, which corrects the underestimation bias of the classical approach and (b) one-step
analysis. The purpose of this article is to dissuade researchers from conducting classical three-step
analysis and to promote the use of the two newer approaches that are described and compared. In addition, the
applications of these newer models—for use when the independent, the dependent, or both categorical variables
are latent—are illustrated through substantive analyses relating classes of substance abusers to classes of
intimate partner aggressors.
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- Masyn, K. E. (2013). Latent Class Analysis and Finite Mixture Modeling. In P. Nathan and T. Little (Eds.), The Oxford Handbook of Quantitative Methods (pp. 551-611). New York, NY. Oxford University Press.
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- Ibironke O., Koukounari A., Asaolu S., Moustaki I., & Shiff C. (2012) Validation of a new test for schistosoma haematobium based on detection of dra1 DNA fragments in urine: Evaluation through Latent Class Analysis.
PLoS Negl Trop Dis 6(1): e1464. DOI:10.1371/journal.pntd.0001464
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Abstract
Background: Diagnosis of urogenital schistosomiasis in chronically infected adults is challenging
but important, especially
because long term infection of the bladder and urinary tract can have dire consequences. We
evaluated three tests for viable infection: detection of parasite specific DNA Dra1 fragments,
haematuria and presence of parasite eggs for sensitivity
(Se) and specificity (Sp).
Methods: Over 400 urine specimens collected from adult volunteers in an endemic area in Western
Nigeria were assessed for haematuria then filtered in the field, the filter papers dried and later
examined for eggs and DNA. The results were stratified according to sex and age and subjected to
Latent Class analysis.
Conclusions: Presence of Dra1 in males (Se = 100%; Sp = 100%) exceeded haematuria (Se = 87.6%: Sp
= 34.7%) and detection of eggs (Se = 70.1%; Sp = 100%). In females presence of Dra1 was Se =
100%: Sp = 100%, exceeding haematuria (Se = 86.7%: Sp = 77.0%) and eggs (Se = 70.1%; Sp = 100%).
Dra1 became undetectable 2 weeks after praziquantel treatment. We
conclude detection of Dra1 fragment is a definitive test for the presence of Schistosoma
haematobium infection.
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- Asparouhov, T. & Muthén, B. (2011).
Using Bayesian priors for more flexible latent class analysis. In Proceedings of the 2011 Joint Statistical Meeting, Section on Government Statistics, pp 4979-4993.
Click here to view Mplus inputs, data, and outputs used in this
paper.
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Abstract
"Latent class analysis is based on the assumption that within each class the observed class indicator
variables
are independent of each other. We explore a new Bayesian approach that relaxes this
assumption
to an assumption of approximate independence. Instead of using a correlation matrix
with correlations
fixed to zero we use a correlation matrix where all correlations are estimated using
an informative
prior with mean zero but non-zero variance. This more flexible approach easily
accommodates LCA
model misspecifications and thus avoids spurious class formations that are
caused by the conditional
independence violations. Simulation studies and real data analysis are
conducted using Mplus."
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- Finch, W.H. & Bronk, K.C. (2011).
Conducting confirmatory latent class analysis using Mplus.
Structural Equation Modeling, 18, 132-151.
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Abstract
"Latent class analysis (LCA) is an increasingly popular tool that researchers can use to identify
latent
groups in the population underlying a sample of responses to categorical observed variables.
LCA
is most commonly used in an exploratory fashion whereby no parameters are specified a
priori. Although
this exploratory approach is reasonable when very little prior research has been
conducted in the
area under study, it can be very limiting when much is already known about the
variables and population.
Confirmatory latent class analysis (CLCA) provides researchers with a
tool for modeling and testing
specific hypotheses about response patterns in the observed variables.
CLCA is based on placing
specific constraints on the parameters to reflect these hypotheses. The
popular and easy-to-use latent
variable modeling software packageMplus can be used to conduct a
variety of CLCA types using these
parameter constraints. This article focuses on the basic principles
underlying the use of CLCA,
and the Mplus programming code necessary for carrying it out."
hide abstract
- Clark, S. & Muthén, B. (2009). Relating latent class analysis results to variables not included in the analysis.
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Abstract
"An important interest in mixture modeling is the investigation of what types of individuals belong to
each latent class by relating classes to covariates, concurrent outcomes and distal outcomes, also
known as auxiliary variables. This article presents results from real data examples and simulations
to show how various factors, such as the degree to which people are classified correctly into latent
classes and sample size, can impact the estimates and standard errors of auxiliary variable effects
and testing mean equality across classes. Based on the results of the examples and simulations, suggestions
are made about how to select auxiliary variables for a latent class analysis."
hide abstract
- Marsh, H.W., Lüdtke, O., Trautwein, U., & Morin, A.J.S. (2009).
Classical latent profile analysis of academic self-concept dimensions: Synergy of person- and variable-
centered approaches to theoretical models of self-concept.
Structural Equation Modeling, 16:2,191-225.
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Abstract
"In this investigation, we used a classic latent profile analysis (LPA), a person-centered approach,
to
identify groups of students who had similar profiles for multiple dimensions of academic selfconcept
(ASC)
and related these LPA groups to a diverse set of correlates. Consistent with a priori
predictions,
we identified 5 LPA groups representing a combination of profile level (high vs. low
overall
ASC) and profile shape (math vs. verbal self-concepts) that complemented results based
on a traditional
variable-centered approach. Whereas LPA groups were substantially and logically
related to the
set of 10 correlates, much of the predictive power of individual ASC factors was lost
in the formation
of groups and the inclusion of the correlates into the LPA distorted the nature of
the groups. LPA
issues examined include distinctions between quantitative (level) and qualitative
(shape) differences
in LPA profiles, goodness of fit and the determination of the number of LPA
groups, appropriateness
of correlates as covariates or auxiliary variables, and alternative approaches
to present and interpret
the results."
hide abstract
- Kreuter, F., Yan, T. & Tourangeau, R. (2008).
Good item or bad – can latent class analysis tell?: The utility of latent class analysis for the evaluation
of survey questions.
Journal of the Royal Statistical Society, Series A, 171, 723-738.
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Abstract
"Latent class analysis has been used to model measurement error, to identify flawed
survey questions and
to estimate mode effects. Using data from a survey of University of Maryland
alumni together with
alumni records, we evaluate this technique to determine its usefulness
for detecting bad questions
in the survey context. Two sets of latent class analysis models are
applied in this evaluation: latent
class models with three indicators and latent class models with
two indicators under different assumptions
about prevalence and error rates. Our results indicated
that the latent class analysis approach
produced good qualitative results for the latent
class models—the item that the model deemed the
worst was the worst according to the true
scores. However, the approach yielded weaker quantitative
estimates of the error rates for a
given item."
hide abstract
- Nylund, K.L., Asparouhov, T., & Muthén, B. (2007).
Deciding on the number of classes in latent class analysis and growth mixture modeling. A Monte Carlo
simulation study.
Structural Equation Modeling, 14, 535-569.
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Abstract
"Mixture modeling is a widely used modeling technique that is used to identify unobserved heterogeneity
in a study population. The application of mixture models has allowed for a deeper understanding of
many substantive areas. Despite their usefulness in practice, one unresolved issue in the application
of mixture models is that there is not one commonly accepted statistical indicator of how many
classes there are in a study population. This paper presents the results of a simulation study that
looks at the performance of likelihood-based tests and the traditionally used Information Criterion
(ICs) that are often used for determining the number of classes in mixture modeling. We look at
the performance of these tests and indices for a series of Latent Class Analysis (LCA) and Growth
Mixture Models (GMM) and evaluate their ability to correctly identify the number of classes in a given
population. While the BIC was the best of the ICs, the bootstrap Likelihood Ratio Test (BLRT),
which is now available in statistical software used for mixture modeling, proves to be a very consistent
indicator of classes."
hide abstract
- Geiser, C., Lehman, W., & Eid, M. (2006).
Separating rotators from nonrotators in the Mental Rotation Test: A multigroup latent class analysis.
Multivariate Behavioral Research, 41, 261-293.
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Abstract
"Items of mental rotation tests can not only be solved by mental rotation but also by
other solution strategies.
A multigroup latent class analysis of 24 items of the Mental
Rotations Test (MRT) was conducted
in a sample of 1,695 German pupils and students
to find out how many solution strategies can
be identified for the items of this
test. The results showed that five subgroups (latent classes) can
be distinguished. Although
three of the subgroups differ mainly in the number of items reached, one
class
shows are very low performance. In another class, a special solution strategy is used.
This
strategy seems to involve analytic rather than mental rotation processes and is
efficient only for a
specialMRT item type, indicating that not allMRT items require a
mental rotation approach. In addition,
the multigroup analysis revealed significant
sex differences with respect to the class assignment,
confirming prior findings that on
average male participants perform mental rotation tasks faster
and better than female
participants. Females were also overrepresented in the analytic strategy class.
The results
are discussed with respect to psychometric and substantive implications, and
suggestions
for the optimization of the MRT items are provided."
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- Muthén, B. & Asparouhov, T. (2022). Latent transition analysis with random intercepts (RI-LTA). Psychological Methods, 27(1), 1–16. DOI: 10.1037/met0000370.
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Abstract
This paper demonstrates that the regular LTA model is unnecessarily restrictive and that an alternative model is readily available that typically fits the data much better, leads to better estimates of the transition probabilities, and extracts new information from the data. By allowing random intercept variation in the model, between-subject variation is separated from the within-subject latent class transitions over time allowing a clearer interpretation of the data. Analysis of four examples from the literature demonstrates the advantages of random intercept LTA. Model variations include Mover-Stayer analysis, multiple-group measurement invariance analysis, and analysis with covariates. Key words: Hidden Markov, mixtures, transition probabilities, latent traitstate, two-level LCA, measurement non-invariance, Mover-Stayer.
hide abstract
- Moore, S.A., Dowdy, E., Nylund-Gibson, K, & Furlong, M.J. (2019). A latent transition analysis of the longitudinal stability of dual-factor mental health in adolescence. Journal of School Psychology, 73, 56-73.
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- Choi, H.J. & Temple, J.R. (2016). Do gender and exposure to interparental violence moderate the stability of teen dating violence?: Latent transition analysis. Prevention Science, 17, 367-376. DOI 10.1007/s11121-015-0621-4
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- Nylund-Gibson, K., Grimm, R., Quirk, M., & Furlong, M. (2014): A latent transition mixture model using the three-step specification. Structural Equation Modeling: A Multidisciplinary Journal, 21, 439-454.
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- Koukounari, A., Donnelly, C.A., Moustaki, I., Tukahebwa, E.M., Kabatereine, N.B., et al. (2013). A latent markov modelling approach to the evaluation of
circulating cathodic antigen strips for schistosomiasis diagnosis pre- and post-praziquantel treatment in Uganda. PLoS Comput Biol 9(12): e1003402.
DOI:10.1371/journal.pcbi.1003402
Mplus scripts are available from the author.
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Abstract
Regular treatment with praziquantel (PZQ) is the strategy for human schistosomiasis control aiming
to prevent morbidity in later life. With the recent resolution on schistosomiasis elimination by the
65th World Health Assembly, appropriate diagnostic tools to inform interventions are keys to their success.
We present a discrete Markov chains modelling framework that deals with the longitudinal study design
and the measurement error in the diagnostic methods under study. A longitudinal detailed dataset from
Uganda, in which one or two doses of PZQ treatment were provided, was analyzed through Latent Markov
Models (LMMs). The aim was to evaluate the diagnostic accuracy of Circulating Cathodic Antigen
(CCA) and of double Kato-Katz (KK) faecal slides over three consecutive days for Schistosoma
mansoni infection simultaneously by age group at baseline and at two follow-up times post treatment.
Diagnostic test sensitivities and specificities and the true underlying infection prevalence over time
as well as the probabilities of transitions between infected and uninfected states are provided.
The estimated transition probability matrices provide parsimonious yet important insights into
the re-infection and cure rates in the two age groups. We show that the CCA diagnostic performance
remained constant after PZQ treatment and that this test was overall more sensitive but less specific
than single-day double KK for the diagnosis of S. mansoni infection. The probability of clearing infection
from baseline to 9 weeks was higher among those who received two PZQ doses compared to one PZQ dose for
both age groups, with much higher re-infection rates among children compared to adolescents and adults.
We recommend LMMs as a useful methodology for monitoring and evaluation and treatment decision research
as well as CCA for mapping surveys of S. mansoni infection, although additional diagnostic tools should
be incorporated in schistosomiasis elimination programs.
hide abstract
- Kam, C., Morin, A. J. S., Meyer, J. P., & Topolnytsky, A. (2016). Are commitment profiles stable and predictable? A latent transition analysis. Journal of Management, 42(6), 1462-1490. DOI: 10.1177/0149206313503010
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Abstract
Recent efforts have been made to identify and compare employees with profiles reflecting different
combinations of affective (AC), normative (NC), and continuance (CC) organizational commitment. To
date, the optimal profiles in terms of employee behavior and well-being have been found to be those
in which AC, NC and CC are all strong, or those where AC, or AC and NC, dominate. The poorest
outcomes are found for profiles where AC, NC and CC are all weak, or CC dominates. The primary goal
of the current study was to use Latent Profile Analysis (LPA) and Latent Transition Analysis (LTA)
to identify profile groups and examine changes in profile membership over an 8- month period in an
organization undergoing a strategic change. We also tested hypotheses concerning the relation
between perceived trustworthiness of management and employees’ commitment profile within and across
time. We found that commitment profiles have substantial temporal stability and that
trustworthiness positively predicts memberships in more desirable commitment profiles. There was
also some, albeit weak, evidence that changes in perceived trustworthiness were accompanied by
corresponding shifts in the commitment profile.
hide abstract
- Koukounari, A., Moustaki, I., Grassly, N.C., Blame, I. M., Basáñez, M., Gambhor, M., Mabey, D. C. W., Bailey, R. L., Burton, M. J., Solomon, A. W., & Donnelly, C. A. (2013).
Using a nonparametric multilevel latent markov model to evaluate diagnostics for trachoma. American Journal of Epidemiology. DOI: 10.1093/aje/kws345
Click here to view web materials associated with this paper.
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Abstract
In disease control or elimination programs, diagnostics are essential for assessing the impact of
interventions, refining treatment strategies, and minimizing the waste of scarce resources. Although
high-performance tests are desirable, increased accuracy is frequently accompanied by a
requirement for more elaborate infrastructure, which is often not feasible in the developing
world. These challenges are pertinent to mapping, impact monitoring, and surveillance in trachoma
elimination programs. To help inform rational design of diagnostics for trachoma elimina- tion, we
outline a nonparametric multilevel latent Markov modeling approach and apply it to 2 longitudinal
cohort studies of trachoma-endemic communities in Tanzania (2000–2002) and The Gambia
(2001–2002) to provide simultaneous inferences about the true population prevalence of
Chlamydia trachomatis infection and disease and the sensitivity, specificity, and predictive
values of 3 diagnostic tests for C. trachomatis infection. Estimates were obtained by using data
collected before and after mass azithromycin administration. Such estimates are par- ticularly
important for trachoma because of the absence of a true “gold standard” diagnostic test for C.
trachomatis. Estimated transition probabilities provide useful insights into key epidemiologic
questions about the persistence of disease and the clearance of infection as well as the
required frequency of surveillance in the postelimination
setting.
hide abstract
- Meeus, W., van de Schoot, R., Keijsers, L. & Branje, S. (2011).
Identity statuses as developmental trajectories. A five-wave longitudinal study in early to middle and
middle to late adolescents.
Journal of Youth and Adolescence.
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Abstract
"This study tested whether Marcia’s original
identity statuses of achievement, moratorium, early closure
(a
new label for foreclosure), and diffusion, can be considered
identity status trajectories. That
is, we examined
whether these statuses are distinct and relatively stable,
over-time configurations
of commitment strength, levels of
in-depth exploration of present commitments, and consideration
of
alternative commitments. The study examined
identity development in a five-wave study of 923 early-tomiddle
(49.3%
female) and 390 middle-to-late adolescents
(56.7% female), covering the ages of 12–20.
Using Latent
class growth analysis (LCGA), the authors found that
Marcia’s (1966) statuses are indeed
identity status trajectories.
Two kinds of moratorium were also found: the
classical moratorium
and searching moratorium. Support
was found for Waterman’s developmental hypothesis of the
identity
status model: the number of achievers was significantly
higher, and the number of diffusions lower,
in
middle-to-late adolescence than in early-to-middle adolescence.
Females were more often in the
advanced identity
status trajectories, and stable differences were found
between the trajectories in
psychosocial adjustment. Study
findings highlight that identity formation should be conceptualized
as
an over-time process."
hide abstract
- Meeus, W., van de Schoot, R., Klimstra, T. & Branje, S. (2011).
Personality types in adolescence: Change and stability and links with adjustment and relationships: A
five-wave longitudinal study in early-to-middle and middle-to-late adolescence.
Developmental Psychology, 47 (4), 1181–1195.
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Abstract
"We examined change and stability of the 3 personality types identified by Block and Block (1980) and
studied
their links with adjustment and relationships. We used data from a 5-wave study of 923
early-to-middle
and 390 middle-to-late adolescents, thereby covering the ages of 12–20 years. In Study
1,
systematic evidence for personality change was found, in that the number of overcontrollers and
undercontrollers
decreased, whereas the number of resilients increased. Undercontrol, in particular, was
found
to peak in early-to-middle adolescence. We also found substantial stability of personality types,
because
73.5% of the adolescents had the same personality type across the 5 waves. Personality
change
was mainly characterized by 2 transitions: overcontrol 3 resiliency and undercontrol 3 resiliency.
The
transitional analyses implied that the resilient type serves more often as the end point
of personality
development in adolescence than do overcontrol and undercontrol. Analyses of the personality
type
trajectories also revealed that the majority of adolescents who change personality type
across 5 years
made only 1 transition. Study 2 revealed systematic differences between resilients and
overcontrollers in
anxiety. Stable resilients were less anxious over time than were stable overcontrollers.
Further, change
from overcontrol to the resilient type was accompanied by decreases in anxiety,
whereas change from the
resilient type to overcontrol was accompanied by an increase in anxiety.
Similarly, systematic differences
between personality types were found in the formation of intimate
relationships."
hide abstract
- Petras, H., Masyn, K. & Ialongo, N. (2011). The developmental impact of two first grade preventive interventions on aggressive/disruptive behavior in childhood and adolescence: An application of latent transition growth mixture modeling.
Prevention Science, 12(3): 300–313. DOI: 10.1007/s11121-011-0216-7
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Abstract
"We examine the impact of two universal preventive interventions in first grade on the growth of aggressive/disruptive
behavior in grades 1-3 and 6-12 through the application of a latent transition growth
mixture model (LT-GMM). Both the classroom-centered and family-centered interventions were designed
to reduce the risk for later conduct problems by enhancing the child behavior management practices
of teachers and parents, respectively. We first modeled growth trajectories in each of the two time
periods with separate GMMs. We then associated latent trajectory classes of aggressive/disruptive
behavior across the two time periods using a transition model for the corresponding latent class variables.
Subsequently, we tested whether the interventions had direct effects on trajectory class membership
in grades 1-3 and 6-12. For males, both the classroom-centered and family-centered interventions
had significant direct effects on trajectory class membership in grades 6-12, whereas only the
classroom-centered intervention had a significant effect on class membership in grades 1-3. Significant
direct effects for females were confined to grades 1-3 for the classroom-centered intervention.
Further analyses revealed that both the classroom-centered and family-centered intervention males were
significantly more likely than control males to transition from the high trajectory class in grades
1-3 to a low class in grades 6-12. Effects for females in classroom-centered interventions went in
the hypothesized direction but did not reach significance."
hide abstract
- Cho, S., Cohen, A., Kim, S. & Bottge, B. (2010).
Latent transition analysis with a mixture item response theory measurement model.
Applied Psychological Measurement, 34(7), 483–504.
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Abstract
"A latent transition analysis (LTA) model was described with a mixture Rasch model (MRM) as
the measurement
model. Unlike the LTA, which was developed with a latent class measurement
model, the LTA-MRM
permits within-class variability on the latent variable, making it more useful
for measuring treatment
effects within latent classes. A simulation study indicated that model
recovery using the LTA-MRM
was good except for small sample size–short test conditions. A
real data application of a mathematics
intervention with middle school students indicated
that the LTA-MRM clearly detected the intervention
effect and also provided a means of helping
to better understand the effects compared to a standard
multiwave analysis of variance."
hide abstract
- Meeus, W., Van de Schoot, R., Keijsers, L., Schwartz, S. J. & Branje, S. (2010).
On the progression and stability of adolescent identity formation. A five-wave longitudinal study in
early-to-middle and middle-to-late adolescence.
Child Development, Volume 81, Number 5, Pages 1565–1581.
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Abstract
"This study examined identity development in a 5-wave study of 923 early-to-middle and 390 middle-to-late
adolescents
thereby covering the ages of 12–20. Systematic evidence for identity progression was
found: The
number of diffusions, moratoriums, and searching moratoriums (a newly obtained status) decreased,
whereas
the representation of the high-commitment statuses (2 variants of a [fore]closed identity:
‘‘early closure’’ and
‘‘closure,’’ and achievement) increased. We also found support for the
individual difference perspective: 63%
of the adolescents remained in the same identity status across
the 5 waves. Identity progression was characterized
by 7 transitions: diffusion fi moratorium, diffusion
fi early closure, moratorium fi closure,
moratorium fi achievement, searching moratorium fi
closure, searching moratorium fi achievement,
and early closure fi achievement."
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- Witkiewitz, K., Maisto, S. A., & Donovan, D. M. (2010). A comparison of methods for estimating change in drinking following alcohol treatment. Alcoholism: Clinical & Experimental Research. DOI: 10.1111/j.1530-0277.2010.01308.x
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Abstract
Background: The ultimate goal of alcohol treatment research is to develop interventions that help individuals reduce their alcohol use.
To determine whether a treatment is effective, researchers must then evaluate whether a particular treatment affects changes in drinking
behavior after treatment. Importantly, drinking following treatment tends to be highly variable between individuals and within individuals
across time.
Method: Using data from the COMBINE study (COMBINE Study Group, 2003), the current study compared 3 commonly used and novel methods for
analyzing changes in drinking over time: latent growth curve (LGC) analysis, growth mixture models, and latent Markov models. Specifically,
using self-reported drinking data from all participants (n = 1,383, 69% male), we were interested in examining how well the 3 estimated
models were able to explain observed changes in percent heavy drinking days during the 52 weeks following treatment.
Results: The results from all 3 models indicated that the majority of individuals were either abstinent or reported few heavy drinking
days during the 52-week follow-up and only a minority of individuals (10% or fewer) reported consistently frequent heavy drinking following
treatment. All 3 models provided a reasonably good fit to the observed data with the latent Markov models providing the closest fit. The
observed drinking trajectories evinced discontinuity, whereby individuals seem to transition between drinking and nondrinking across adjacent
follow-up assessment points. The LGC and growth mixture models both assumed continuous change and could not explain this discontinuity in the
observed drinking trajectories, whereas the latent Markov approach explicitly modeled transitions between drinking states.
Conclusions: The 3 models tested in the current study provided a unique look at the observed drinking among individuals who received
treatment for alcohol dependence. Latent Markov modeling may be a highly desirable methodology for gaining a better sense of transitions
between positive and negative drinking outcomes.
hide abstract
- Kaplan, D. (2008).
An overview of Markov chain methods for the study of stage-sequential developmental processes.
Developmental Psychology, 44, 457-467.
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Abstract
"This article presents an overview of quantitative methodologies for the study of stage-sequential development
based on extensions of Markov chain modeling. Four methods are presented that exemplify the
flexibility of this approach: The manifest Markov model, the latent Markov model, latent transition
analysis, and the mixture latent Markov model. A special case of the mixture latent Markov model, the
so-called “mover-stayer"" model, is used in this study. Issues of model specification, estimation,
and testing using the Mplus software environment are briefly discussed and the Mplus input syntax
is provided. These four methods are applied to a single example of stage sequential development in reading
competency in the early school years utilizing data from the Early Childhood Longitudinal Study
– Kindergarten Cohort."
hide abstract
- Bray, B.C. (2007).
Examining gambling and substance use: Applications of advanced latent class modeling techniques for
cross-sectional and longitudinal data.
Doctoral dissertation, Pennsylvania State University.
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Abstract
"The purpose of the current project is to present three empirical studies that illustrate the application of advanced latent class modeling techniques for crosssectional and longitudinal data to research questions about gambling and substance
use. The first empirical study used latent class analysis and conditional latent class analysis to identify and predict types of college-student gamblers using data from a large northeastern university. Four types of gamblers were identified for men and
women: non-gamblers, cards and lotto players, cards and games of skill players, and multi-game players. There were substantial gender differences in the latent class membership probabilities: (1) men were most likely to be cards and lotto
players whereas women were most likely to be non-gamblers; and (2) men were more likely than women to be cards and games of skill and multi-game players, and less likely to be non-gamblers. Significant predictors of gambling latent class
membership included: school year, living in off-campus housing, Greek membership, and past-year alcohol use. There were substantial gender differences in the predictive effects of Greek membership and past-year alcohol use: (1) the effects of
Greek membership were in different directions for men and women; and (2) pastyear alcohol use was more strongly related to gambling latent class membership for women. The second empirical study used latent class analysis to identify types of adolescent
and young adult gamblers and used latent class analysis for repeated measures to identify types of drinking trajectories using data from the National Longitudiiii nal Study of Adolescent Health. Multivariable latent class modeling was used to
examine the relation between gambling and drinking by linking specific types of gambling to specific types of drinking trajectories. Gambling and drinking were shown to be highly related: (1) consistent infrequent, light, or not intense drinkers were most likely to be non-gamblers; and (2)
participants who were frequent, heavy, or intense drinkers at any time were most likely to gamble in all activities. Overall, drinking frequency appeared to be more predictive of gambling than was drinking quantity. The third empirical study used latent transition analysis to identify types of
adolescent smokers and types of drinkers, and to describe smoking and drinking development over time using data from the National Longitudinal Survey of Youth 1997. Multiprocess modeling was used to examine the relation between smoking
and drinking by modeling the development of smoking and the development of drinking simultaneously. Three types of smokers and three types of drinkers were identified: non-smokers, light smokers, heavy smokers, non-drinkers, light drinkers, and heavy drinkers. The majority
of participants were non-smokers and nondrinkers. The behavior of non-smokers, non-drinkers, heavy smokers, and heavy drinkers was relatively stable across time whereas the behavior of light smokers and light drinkers was variable. Linking smoking and drinking showed that: (1)
knowing type of smoking provided limited information about type of drinking; (2) transitioning from non-drinking to heavy drinking was progressively more likely for more serious types of smoking; (3) transitioning from heavy drinking to nondrinking
was progressively less likely for more serious types of smoking; and (4) transitioning from light drinking to non-drinking was most likely for non-smokers whereas transitioning from light drinking to heavy drinking was most likely for heavy smokers. iv"
hide abstract
- Nylund, K. (2007).
Latent transition analysis: Modeling extensions and an application to peer victimization.
Doctoral dissertation, University of California, Los Angeles.
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- Nylund, K.L., Muthén, B., Nishina, A., Bellmore, A. & Graham, S. (2007). Stability and Instability of Peer Victimization during Middle School: Using Latent Transition Analysis with Covariates, Distal Outcomes, and Modeling Extensions.
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Abstract
"This paper is an advanced application of latent transition analysis (LTA). Examining the peer victimization
experiences of approximately 1300 urban, public school students across the 3 middle school years,
we extend the conventional LTA model to simultaneously include time varying covariates with time
varying effects, second-order effects, a mover-stayer variable, and distal outcomes. We present five
key modeling steps that can be used in the application of the LTA model. The analyses yielded three
victim classes based on victimization degree (victimized, sometimes victimized, nonvictimized). LTA
indicated that when students transitioned between victimization classes, they were most likely to
transition from a more victimized group into one of the less victimized groups. Further, results indicated
that students who experience any sort of victimization, compared to those who do not, felt less
safe at school, more socially anxious, and more depressed during certain middle school years. We
also found that students who were chronically victimized in middle school reported more physical health
problems and more social worries once in high school."
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- Asparouhov, T. & Muthén, B. (2023). Multiple group alignment for exploratory and structural equation models. Structural Equation Modeling: A Multidisciplinary Journal, 30(2), 169-191. DOI: 10.1080/10705511.2022.2127100
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- Billet, J., Meeusen, C. & Abts, K. (2021). The relationship between (sub)national identity, citizenship conceptions, and perceived ethnic threat in Flanders and Wallonia for the period 1995-2020: A measurement invariance testing strategy. Forthcoming in Frontiers in Political Science. DOI: 10.3389/fpos.2021.676551
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- Geiser, C., & Simmons, T. G. (in press). On the performance of multiple-indicator correlated traits-correlated (methods – 1) models. Psychological Test and Assessment Modeling
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Abstract
We examined the performance of two versions of the multiple-indicator correlated traits-correlated (methods – 1) [CT-C(M – 1)] model (Eid et al., 2008) in terms of convergence, improper solutions, parameter bias, standard error bias, and power to detect misspecified models. We also studied whether Yuan et al.’s (2015) correction procedure for the maximum likelihood chi-square model fit test yields accurate Type-I error rates and adequate power for these models. The models performed well except for underestimated standard errors for some parameters in specific small-sample conditions. Yuan et al.’s (2015) chi-square correction worked well for correctly specified models but showed limited power to detect misspecified models in small-sample, low-reliability conditions. We recommend that researchers using these models in smaller samples select highly reliable indicators.
hide abstract
- Raykov, T. & Marcoulides, G.A. (2021). On the pitfalls of estimating and using standardized reliability coefficients. Educational and Psychological Measurement, 81(4), 791-810. DOI: 10.1177/0013164420937345
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- Geiser, C., & Simmons, T. G. (2021). Do method effects generalize across traits (and what if they don’t)? Journal of Personality. DOI: 10.1111/jopy.12625
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- DeMars, C.E. (2019). Alignment as an alternative to anchor purification in DIF analyses. Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2019.1617151
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- Silva, B.C., & Littvay, L. (2019). Comparative research is harder than we thought: Regional differences in experts' understanding of electoral integrity questions. Political Analysis. DOI: 10.1017/pan.2019.24
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- Pendergast, L. L., von der Embse, N., Kilgus, S. P., & Eklund, K. R. (2017). Measurement equivalence: A non-technical primer on categorical multi-group confirmatory factor analysis in school psychology. Journal of School Psychology, 60, 65-82. DOI: 10.1016/j.jsp.2016.11.002
view abstract
- Lomazzi, V. (2017). Using alignment optimization to test the measurement invariance of gender role attitudes in 59 countries. methods, data, analyses, [S.l.], p. 27, ISSN 2190-4936.
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- Munck, I., Barber, C., & Torney-Purta, J. (2017). Measurement invariance in comparing attitudes toward immigrants among youth across Europe in 1999 and 2009: The alignment method applied to IEA CIVED and ICCS. Sociological Methods & Research, DOI: 10.1177/0049124117729691
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- Flake, J.K. & McCoach, D. B. (2017). An investigation of the alignment method with polytomous indicators under conditions of partial measurement invariance. Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2017.1374187
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- Wu, H. & Estabrook, R. (2016). Identification of confirmatory factor analysis models of different levels of invariance for ordered categorical outcomes. Psychometrika, 81:4, 1014-1045. DOI: 10.1007/s11336-016-9506-0
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contact first author
- Raykov, T. & Marcoulides, G.A. (2016). Scale reliability evaluation under multiple assumption violations. Structural Equation Modeling: A Multidisciplinary Journal, 23:2, 302-313, DOI: 10.1080/10705511.2014.938597
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- Koukounari, A., Pickles, A., Hill, J. & Sharp, H. (2015). Psychometric properties of the parent-infant caregiving touch scale. Frontiers in Psychology 6:1887. DOI: 10.3389/fpsyg.2015.01887
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- Schneider, S. (2016). Extracting response style bias from measures of positive and negative affect in aging research. The Journals of Gerontology: Series B. DOI: 10.1093/geronb/gbw103
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Abstract
"Objectives: This study investigated the role of response style biases in the assessment of positive
and negative affect in aging research; it addressed whether response styles (a) are associated with age-related changes in
cognitive abilities, (b) lead to distorted conclusions about age differences in affect, and (c)
reduce the convergent and predictive validity of affect measures in relation to health outcomes.
Method: A multidimensional item response theory model was used to extract response styles from
affect ratings provided by respondents to the psychosocial questionnaire (n = 6,295; aged 50–100
years) in the Health and Retirement Study (HRS). Results: The likelihood of extreme response styles
(disproportionate use of “not at all” and “very much” response catego- ries) increased
significantly with age, and this effect was mediated by age-related decreases in HRS cognitive test
scores. Removing response styles from affect measures did not alter age patterns in positive and
negative affect; however, it consist- ently enhanced the convergent validity (relationships with
concurrent depression and mental health problems) and predic- tive validity (prospective
relationships with hospital visits, physical illness onset) of the affect measures.
Discussion: The results support the importance of detecting and controlling response styles when
studying self-reported affect in aging research.
"
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- Geiser, C. & Lockhart, G. (2012).
A comparison of four approaches to account for method effects in latent state-trait analyses.
Psychological Methods. DOI: 10.1037/a0026977
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Abstract
"Latent state–trait (LST) analysis is frequently applied in psychological research to determine the degree
to
which observed scores reflect stable person-specific effects, effects of situations and/or person–
situation
interactions, and random measurement error. Most LST applications use multiple repeatedly
measured
observed variables as indicators of latent trait and latent state residual factors.
In practice, such
indicators often show shared indicator-specific (or method) variance over time. In
this article, the authors
compare 4 approaches to account for such method effects in LST models and
discuss the strengths and
weaknesses of each approach based on theoretical considerations, simulations,
and applications to actual
data sets. The simulation study revealed that the LST model with indicator-specific
traits (Eid, 1996) and
the LST model with M 1 correlated method factors (Eid, Schneider,
& Schwenkmezger, 1999)
performed well, whereas the model with M orthogonal method factors used
in the early work of Steyer,
Ferring, and Schmitt (1992) and the correlated uniqueness approach (Kenny,
1976) showed limitations
under conditions of either low or high method-specificity. Recommendations
for the choice of an
appropriate model are provided."
hide abstract
- Raykov, T. (2011). Evaluation of convergent and discriminant validity with multitrait-multimethod correlations. British Journal of Mathematical and Statistical Psychology, 64, 38–52. DOI:10.1348/000711009X478616
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Abstract
"A procedure for evaluation of convergent and discriminant validity coefficients is
outlined. The method
yields interval estimates of these coefficients in a construct
validation study conducted via the
multitrait-multimethod approach, as well as permits
examination of their population relationships.
The procedure is readily employed in
behavioral research using the increasingly popular latent variable
modeling methodology.
The described method is illustrated with a numerical example."
hide abstract
- Raykov, T., Dimitrov, D.M. & Asparouhov, T. (2010). Evaluation of scale reliability with binary measures using latent variable modeling. Structural Equation Modeling: A Multidisciplinary Journal, 17:2, 265-279, DOI: 10.1080/10705511003659417
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Abstract
"A method for interval estimation of scale reliability with discrete data is outlined.
The approach is
applicable with multi-item instruments consisting of binary measures, and is
developed within the latent
variable modeling methodology. The procedure is useful for
evaluation of consistency of single
measures and of sum scores from item sets following the
two-parameter logistic model or the one-parameter
logistic model. An extension of the
method is described for constructing confidence intervals
of change in reliability due to
instrument revision. The proposed procedure is illustrated with an
example."
hide abstract
- Raykov, T. (2008).
“Alpha if item deleted”: A note on loss of criterion validity in scale development if maximizing coefficient
alpha.
British Journal of Mathematical and Statistical Psychology, 61, 275-285.
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Abstract
"This note is concerned with a validity-related limitation of the widely available and used
index “alpha
if item deleted” in the process of construction and development of multiplecomponent
measuring instruments.
Attention is drawn to the fact that this statistic can suggest
dispensing with such scale
components, whose removal leads to loss in criterion validity while
maximising the popular coefficient
alpha. As an alternative, a latent variable modelling
approach is discussed that can be used for
point and interval estimation of composite criterion
validity (as well as reliability) after deletion
of single components. The method can also be
utilised to test conventional or minimum level hypotheses
about associated population change in
measurement quality indices."
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- Feingold, A., MacKinnon, D.P., & Capaldi, D.M. (2019). Mediation analysis with binary outcomes: Direct and indirect effects of pro-alcohol influences on alcohol use disorders. Addictive Behaviors, 93, 26-35.
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- McLarnon, M.J.W. & O'Neill, T.A. (2018). Extensions of auxiliary variable approaches for the investigation of mediation, moderation, and conditional effects in mixture models. Organizational Research Methods, 21(4), 955-982. DOI: 10.1177/1094428118770731
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- Goldsmith, K.A., Chalder, T., White, P.D., Sharpe, M., & Pickles, A. (2016). Measurement error, time lag, unmeasured confounding: Considerations for longitudinal estimation of the effect of a mediator in randomised clinical trials. Statistical Methods in Medical Research. DOI: 10.1177/0962280216666111
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- Koukounari, A., Stringaris, A., & Maughan, B. (2016). Pathways from maternal depression to young adult offspring depression: an exploratory longitudinal mediation analysis. International Journal of Methods in Psychiatric Research. DOI: 10.1002/mpr.1520
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- Nguyen, T.Q., Webb-Vargas, Y., Koning, I.K. & Stuart, E.A. (2016). Causal mediation analysis with a binary outcome and multiple continuous or ordinal mediators: Simulations and application to an alcohol intervention.
Structural Equation Modeling: A Multidisciplinary Journal, 23:3, 368-383 DOI: 10.1080/10705511.2015.1062730
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- Cheung, G.W. & Lau, R.S. (2015). Accuracy of parameter estimates and confidence intervals in moderated mediation models: A comparison of regression and latent moderated structural equations.
Organazational Research Methods. DOI: 10.1177/1094428115595869
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- Stride, C.B., Gardner, S.E., Catley, N. & Thomas, F. (2015). Mplus code for mediation, moderation and moderated mediation models.
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view Mplus examples/code
- De Stavola, B. L., Daniel, R. M., Ploubidis, G. B. & Micali, N. (2015). Mediation analysis with intermediate confounding: Structural equation modeling viewed through the causal inference lens.
American Journal of Epidemiology. DOI: 10.1093/aje/kwu239
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- Muthén, B. & Asparouhov T. (2015). Causal effects in mediation modeling: An introduction with applications to latent variables. Structural Equation Modeling: A Multidisciplinary Journal, 22(1), 12-23. DOI:10.1080/10705511.2014.935843
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Abstract
Causal inference in mediation analysis offers counterfactually-based causal
definitions of direct and indirect effects, drawing on research by Robins, Green- land, Pearl,
VanderWeele, Vansteelandt, Imai and others. This type of mediation effect estimation is little
known and seldom used among analysts using structural equation modeling (SEM). The aim of this
paper is to describe the new analysis opportunities in a way that is accessible to SEM analysts and
show examples of how to perform the analyses. An application is presented with an extension to a
latent mediator measured with multiple indicators.
hide abstract
- Wang, L., and Preacher, K. J. (2014).
Moderated mediation analysis using Bayesian methods.
Structural Equation Modeling: A Multidisciplinary Journal. 22(2), 249-263. DOI: 10.1080/10705511.2014.935256
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- Muthén, B. (2011).
Applications of causally defined direct and indirect effects in mediation analysis using SEM in Mplus.
Click here to view the Technical
appendix that goes with this paper and click here
for the Mplus input appendix. Click here
to view Mplus inputs, data, and outputs used in this paper.
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Abstract
"This paper summarizes some of the literature on causal effects in mediation analysis. It presents causally-defined
direct and indirect effects for continuous, binary, ordinal, nominal, and count variables.
The expansion to non-continuous mediators and outcomes offers a broader array of causal mediation
analyses than previously considered in SEM practice. A new result is the ability to handle mediation
by a nominal variable. Examples with a binary outcome and a binary, ordinal and nominal mediator
are given, using Mplus to compute the effects. The causal effects require strong assumptions even in
randomized designs, especially sequential ignorability, which is presumably often violated to some
extent due to mediator-outcome confounding. To study the effects of violating this assumption, it
is shown how a sensitivity analysis can be carried out using Mplus.
This can be used both in planning
a new study and in evaluating the results of an existing study."
hide abstract
- Emsley, R., Dunn, G. & White, I. (2010).
Mediation and moderation of treatment effects in randomised controlled trials of complex interventions.
Statistical Methods in Medical Research, 19, 237–270.
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Abstract
"Complex intervention trials should be able to answer both pragmatic and explanatory questions in order
to
test the theories motivating the intervention and help understand the underlying nature of the
clinical
problem being tested. Key to this is the estimation of direct effects of treatment and indirect
effects acting
through intermediate variables which are measured post-randomisation. Using psychological
treatment
trials as an example of complex interventions, we review statistical methods which
crucially evaluate
both direct and indirect effects in the presence of hidden confounding between
mediator and outcome.
We review the historical literature on mediation and moderation of treatment
effects. We introduce two
methods from within the existing causal inference literature, principal stratification
and structural mean
models, and demonstrate how these can be applied in a mediation context
before discussing approaches and
assumptions necessary for attaining identifiability of key parameters
of the basic causal model. Assuming
that there is modification by baseline covariates of the
effect of treatment (i.e. randomisation) on the mediator
(i.e. covariate by treatment interactions),
but no direct effect on the outcome of these treatment by covariate
interactions leads to the use
of instrumental variable methods. We describe how moderation can occur
through post-randomisation variables,
and extend the principal stratification approach to multiple group
methods with explanatory
models nested within the principal strata. We illustrate the new methodology
with motivating examples
of randomised trials from the mental health literature."
hide abstract
- MacKinnon, D.P., Lockwood, C.M., Brown, C.H., Wang, W., & Hoffman, J.M. (2007).
The intermediate endpoint effect in logistic and probit regression.
Clinical Trials, 4, 499-513.
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Abstract
"Background An intermediate endpoint is hypothesized to be in the middle of the causal sequence relating an independent variable to a dependent variable. The intermediate variable is also called a surrogate or mediating variable and the
corresponding effect is called the mediated, surrogate endpoint, or intermediate endpoint effect. Clinical studies are often designed to change an intermediate or surrogate endpoint and through this intermediate change influence the ultimate
endpoint. In many intermediate endpoint clinical studies the dependent variable is binary, and logistic or probit regression is used. Purpose The purpose of this study is to describe a limitation of a widely used approach to
assessing intermediate endpoint effects and to propose an alternative method, based on products of coefficients, that yields more accurate results. Methods The intermediate endpoint model for a binary outcome is described for a
true binary outcome and for a dichotomization of a latent continuous outcome. Plots of true values and a simulation study are used to evaluate the different methods. Results Distorted estimates of the intermediate endpoint effect and incorrect
conclusions can result from the application of widely used methods to assess the intermediate endpoint effect. The same problem occurs for the proportion of an effect explained by an intermediate endpoint, which has been suggested
as a useful measure for identifying intermediate endpoints. A solution to this problem is given based on the relationship between latent variable modeling and logistic or probit regression. Limitations More complicated intermediate variable models are not addressed in
the study, although the methods described in the article can be extended to these more complicated models. Conclusions Researchers are encouraged to use an intermediate endpoint method based on the product of regression coefficients. A common method based on
difference in coefficient methods can lead to distorted conclusions regarding the intermediate effect. Clinical Trials 2007; 4: 499–513. http://ctj.sagepub.com"
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- Raykov, T. & West, B. T. (2016). On enhancing plausibility of the missing at random assumption in incomplete data analyses via evaluation fo response-auxiliary variable correlations. Structural Equation Modeling: A Multidisciplinary Journal, 23(1) 45-53. DOI:10.1080/10705511.2014.937848
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- Muthén, B. (2015). General and specific factors in selection modeling. In M. Rosén et al. (eds.), Cognitive Abilities and Educational Outcomes, Methodology of Educational Measurement and Assessment (pp. 223-236). Switzerland, Springer International Publishing. DOI: 10.1007/978-3-319-43473-5_12
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Abstract
This chapter shows how analysis of data on selective subgroups can be
used to draw inference to the full, unselected group. This uses Pearson-Lawley
selection formulas which apply to not only regression analysis but also structural
equation modeling. The chapter shows the connection with maximum-likelihood
estimation with missing data assuming MAR versus using listwise deletion.
Applications are discussed of selection into military using the factor analysis
models for the variables used in the selection.
hide abstract
- Lugtig, P. (2014). Separating stayers, fast attriters, gradual attriters, and lurkers. Sociological Methods & Research,
43(4) 699-723. DOI: 10.1177/0049124113520305
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- McPherson, S., Barbosa-Leiker, C., Mamey, M. R., McDonell, M., Enders, C.K. & Roll, J. (2014). A ‘missing not at random’ (MNAR) and ‘missing at random’ (MAR) growth model comparison with a buprenorphine/naloxone clinical trial.
Addiction, 110(1), 51–58. DOI: 10.1111/add.12714
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- McPherson, S., Barbosa-Leiker, C., Burns, L. G., Howell, D. & Roll, J. (2012). Missing data in substance abuse treatment research: Current methods and modern approaches. Experimental and Clinical Psychopharmacology, 20(3), 243–250. DOI: 10.1037/a0027146
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- Power, R.A., Muthén, B.,Henigsberg, N., Mors, O., Placentino, A., Mendlewicz, J., Maier, W., McGuffin, P., Lewis, C.M., & Uher, R. (2012). Non-random dropout and the relative efficacy of escitalopram and nortriptyline in treating major depressive
disorder. Journal of Psychiatric Research. 46(10):1333-8. DOI: 10.1016/j.jpsychires.2012.06.014
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Abstract
"Most comparisons of the efficacy of antidepressants have relied on the assumption that missing data are
randomly distributed. Dropout rates differ between drugs, suggesting this assumption may not hold
true. This paper examines the effect of non-random dropout on a comparison of two antidepressant drugs,
escitalopram and nortriptyline, in the treatment of major depressive disorder. The GENDEP study
followed adult patients with major depressive disorder over 12 weeks of treatment, and the primary
analysis found no difference in efficacy of the two antidepressants under missing at random assumption.
By applying the recently developed Muthén-Roy model, we compared the relative efficacy of these
two antidepressants taking into account non-random distribution of missing outcomes (NMAR). Individuals
who dropped out of the study were those who were not responding to treatment. Based on the best
fitting NMAR model, it was found that escitalopram reduced symptom scores by an additional 1.4 points
on the Montgomery-Åsberg Depression Rating Scale (p ¼ 0.02), equivalent to 5% of baseline depression
severity, compared to nortriptyline. We conclude that association between dropout and worsening
symptoms led to an overestimate of the effectiveness of treatment, especially with nortriptyline, in
the primary analysis. These findings review the primary analysis of GENDEP and suggest that, when
non-random dropout is accounted for, escitalopram is more effective than nortriptyline in reducing
symptoms of major depression."
hide abstract
- Enders, C. (2011).
Missing not at random models for latent growth curve analyses.
Psychological Methods, 16, 1-16.
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show abstract
Abstract
"The past decade has seen a noticeable shift in missing data handling techniques that assume a missing
at
random (MAR) mechanism, where the propensity for missing data on an outcome is related to other
analysis
variables. Although MAR is often reasonable, there are situations where this assumption is
unlikely to hold,
leading to biased parameter estimates. One such example is a longitudinal study of
substance use where
participants with the highest frequency of use also have the highest likelihood
of attrition, even after
controlling for other correlates of missingness. There is a large body of
literature on missing not at random
(MNAR) analysis models for longitudinal data, particularly in the
field of biostatistics. Because these methods
allow for a relationship between the outcome variable
and the propensity for missing data, they require a
weaker assumption about the missing data mechanism.
This article describes 2 classic MNAR modeling
approaches for longitudinal data: the selection
model and the pattern mixture model. To date, these models
have been slow to migrate to the social
sciences, in part because they required complicated custom computer
programs. These models are now quite
easy to estimate in popular structural equation modeling programs,
particularly Mplus. The purpose
of this article is to describe these MNAR modeling frameworks and to
illustrate their application
on a real data set. Despite their potential advantages, MNAR-based analyses are not
without problems
and also rely on untestable assumptions. This article offers practical advice for implementing
and
choosing among different longitudinal models.
"
hide abstract
- Muthén, B., Asparouhov, T., Hunter, A. & Leuchter, A. (2011).
Growth modeling with non-ignorable dropout: Alternative analyses of the STAR*D antidepressant trial.
Psychological Methods, 16, 17-33.
Click here to view Mplus outputs used in this paper.
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Abstract
"This paper uses a general latent variable framework to study a series of models for non-ignorable missingness
due to dropout. Both existing and new models are explored for pattern-mixture and selection
type modeling. The missing data models are applied to longitudinal data from STAR*D, the largest
antidepressant clinical trial in the U.S. to date. Despite the importance of this trial, STAR*D growth
model analyses using non-ignorable missing data techniques have not been used until now. Using
growth mixture modeling extended to handle non-ignorable missingness due to dropout, the STAR*D data
are shown to feature distinct trajectory classes, including a low class corresponding to substantial
improvement in depression and a minority class with a U-shaped curve corresponding to transient improvement.
The trajectory class analysis provides a new way to assess drug efficiency. Software is
available for sensitivity analysis using the extensive set of missing data models available in the
general latent variable framework.
"
hide abstract
- Morgan-Lopez, A. A. & Fals-Steward, W. (2007). Analytic methods for modeling longitudinal data from rolling therapy groups with membership turnover. Journal of Consulting and Clinical Psychology, 75, 580-593.
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Abstract
"Interventions for a variety of emotional and behavioral problems are commonly delivered in the context
of treatment groups, with many using rolling admission to sustain membership (i.e., admission, dropout
and discharge from group is perpetual and ongoing). We present an overview of the analytic challenges
inherent in rolling group data and outline commonly-used (but flawed) analytic and design approaches
used to address (or sidestep) these issues. Moreover, we propose latent class pattern mixture
modeling (LCPMM) as a statistically and conceptually defensible approach for modeling treatment data
from rolling groups. The LCPMM approach is illustrated with rolling group data from a group-based
alcoholism pilot treatment trial (N = 128). Different inferences were made with regard to treatment
efficacy under LCPMM versus the commonly used standard group-clustered latent growth model (LGM); coupled
with other preliminary findings in this area, inferences from LGMs may be overly liberal when
applied to data from rolling groups. Continued work on data analytic difficulties in groups with membership
turnover is critical for furthering the ecological validity of research on behavioral treatments."
hide abstract
- Muthén, B., Jo, B. & Brown, H. (2003).
Comment on the Barnard, Frangakis, Hill & Rubin article, Principal stratification approach to broken
randomized experiments: A case study of school choice vouchers in New York City.
Journal of the American Statistical Association, 98, 311-314.
The Muthén et al. article can be downloaded from here.
The Barnard et al. article can be found at http://biosun01.biostat.jhsph.edu/~cfrangak/papers/index.html.
For background
information and analyses using Mplus, see Mplus Web Note
#5 and Jo (2002), Sensitivity of causal effects under ignorable
and latent ignorable missing-data mechanisms, Draft. Contact
the author. The Jo paper can be downloaded from here.
contact first author
- Muthén, B., Kaplan, D. & Hollis, M. (1987).
On structural equation modeling with data that are not missing completely at random.
Psychometrika, 52:3, 431-462.
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Abstract
"A general latent variable model is given which includes the specification of a missing data
mechanism.
This framework allows for an elucidating discussion of existing general multivariate
theory bearing
on maximum likelihood estimation with missing data. Here, missing completely at
random is not a prerequisite
for unbiased estimation in large samples, as when using the traditional
listwise or pairwise
present data approaches. The theory is connected with old and new
results in the area of selection
and factorial invariance. It is pointed out that in many applications,
maximum likelihood estimation
with missing data may be carried out by existing structural
equation modeling software, such as
LISREL and LISCOMP. Several sets of artifical data are
generated within the general model framework.
The proposed estimator is compared to the two
traditional ones and found superior."
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- Chénard-Poirier, Léandre-Alexis, Morin, A.J.S., & Boudrias, J.S.(2017). On the merits of coherent leadership empowerment behaviors: A mixture regression approach. Journal of Vocational Behavior. 103, 66-75 DOI: 10.1016/j.jvb.2017.08.003
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Abstract
This study aims to verify Lawler’s (1992, 2008) theoretical proposition that the
complementariness and coherence of leadership empowerment practices (LEB) need to be jointly
considered in order to adequately understand their relation with employees’ levels of behavioural
empowerment. Patterns of relations among three LEB (Delegation, Coaching, and Recognition), and
five indicators of behavioral empowerment were analyzed among a sample of 474 Canadian
employees. Lawler’s proposition was tested using a person-centered mixture regression approach. The
results revealed four distinct profiles of employees. At the profile level, results reveal that the joint
implementation of a similar level of LEB in a complementary manner relates to employees’ levels of
behavioral empowerment. However, within each profile, a lack of coherence in the levels of
implementation of the three LEB resulted in a more complex pattern of associations with employees’
levels of behavioral empowerment. Taken together, these results offer practical guidance to guide
supervisors in their utilization of LEB.
hide abstract
- Litson, K., Geiser, C., Burns, G. L., & Servera, M. (2016). Examining trait × method interactions using mixture distribution multitrait–multimethod models, Structural Equation Modeling: A Multidisciplinary Journal, DOI:
10.1080/10705511.2016.1238307
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- Jahanshahi, K. & Jin, Y. (2016) The built environment typologies in the UK and their influences on travel behaviour: new evidence through latent categorisation in structural equation modelling, Transportation Planning and Technology, 39:1, 59-77, DOI:10.1080/03081060.2015.1108083
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Abstract
"This paper uses a new latent categorisation approach (LCA) in structural equation modelling (SEM) to gain fresh insights into the in?uence of the built environment characteristics upon travel behaviour. So far as we are aware, this is the ?rst LCA-SEM application in this ?eld. We use all the main descriptors of the built environment in the UK National Travel Survey data in the analysis whilst accounting for the high correlations among the descriptors – this is achieved through de?ning a categorical rather than continuous latent variable for the built environment characteristics. This novel approach to de?ning a tangible typology of the built environment in the UK is capable of making the analytical results more cogent to formulating new, proactive land use planning and urban design measures as well as monitoring the outcomes of on-going planning and transport interventions. Since travel survey data are regularly collected across a large number of cities in the world, our approach helps to guide the design of future travel surveys for those cities in a way that enhances the analysis and monitoring of the impacts of planning and transport policies on travel choices."
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- O’Neill, T. A., McLarnon, M. J. W., Xiu, L., & Law, S. J. (2015). Core self-evaluations, perceptions of group potency, and job performance: The moderating role of individualism and collectivism cultural profiles. Journal of Occupational and Organizational Psychology. DOI: 10.1111/joop.12135
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contact first author.
Van Horn, M. L., Jaki, T., Masyn, K., Ramey, S. L., Smith, J. A., & Antaramian, S. (2009). Assessing differential effects: Applying regression mixture models to identify variations in the influence of family resources on academic achievement. Developmental Psychology, 45(5), 1298–1313. DOI: 10.1037/a0016427
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Abstract
"Developmental scientists frequently seek to understand effects of environmental contexts on development.
Traditional analytic strategies assume similar environmental effects on all children, sometimes
exploring possible moderating influences or exceptions (e.g. outliers) as a secondary step. These strategies
are poorly matched to ecological models of human development which posit complex individual
by environment interactions. An alternative conceptual framework is proposed that tests the hypothesis
that the environment has differential (non-uniform) effects on children. A demonstration of the
utility of this framework is provided by examining the effects of family resources on children’s academic
outcomes in a multisite study (N=6305). Three distinctive groups of children were identified,
including one group particularly resilient to influence of low levels of family resources. Predictors
of group differences including parenting and child demographics are tested, the replicability of the
results are examined, and findings are contrasted with those using traditional regression interaction
effects. This approach is proposed as a partial solution to advance theories of the environment,
social ecological systems research, and behavioral genetics in order to create well-tailored environments
for children."
hide abstract
- Guo, J., Wall, M. & Amemiya, Y. (2006).
Latent class regression on latent factors.
Biostatistics, 7, 145-163.
This type of modeling can be done using ML techniques illustrated in the Mplus Version 3 User's Guide
(first printed in March 2004), example 7.19. The authors emailed us and apologized for not seeing
this Mplus capability earlier and not referencing it in the paper.
show abstract
Abstract
"In the research of public health, psychology, and social sciences, many research questions investigate
the relationship between a categorical outcome variable and continuous predictor variables. The focus
of this paper is to develop a model to build this relationship when both the categorical outcome
and the predictor variables are latent (i.e. not observable directly). This model extends the latent
class regression model so that it can include regression on latent predictors. Maximum likelihood
estimation is used and two numerical methods for performing it are described: the Monte Carlo Expectation
and Maximization algorithm (MCEM) and Gaussian quadrature followed by quasi-Newton algorithm.
A simulation study is carried out to examine the behavior of the model under different scenarios.
A data example involving adolescent health is used for demonstration where the latent classes of
eating disorders risk are predicted by the latent factor body satisfaction."
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- Konold, T.R. & Sanders, E.A. (2023). The SEM reliability paradox in a Bayesian framework. Structural Equation Modeling: A Multidisciplinary Journal. DOI: 10.1080/10705511.2023.2220915
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- Sanders, E.A. & Konold, T.R. (2023). X matters too: How the blended slope problem manifests differently in unilevel vs. multilevel models. Methodology, 2023, Vol. 19(1), 1–23, DOI: 10.5964/meth.9925
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- Gillet, N., Morin, A.J.S., & Blais, A.-R. (in press). A multilevel person-centered perspective on the role of job demands and resources for employees' job engagement and burnout profiles. Group & Organization Management. Early view. DOI: 10.1177/10596011221100893.
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Abstract
The present study first examined the configurations, or profiles, taken by distinct global and specific facets of job engagement and burnout (by relying on a bifactor operationalization of these constructs) among a
nationally representative sample of Canadian Defence employees (n = 13,088; nested within 65 work units). The present study also adopted a multilevel perspective to investigate the role of job demands (work overload
and role ambiguity), as well as individual (psychological empowerment), workgroup (interpersonal justice), supervisor (transformational leadership), and organizational (organizational support) resources in the
prediction of profile membership. Latent profile analyses revealed five profiles of employees: BurnedOut/Disengaged (7.13%), Burned-Out/Involved (12.13%), Engaged (18.14%), Engaged/Exhausted (15.50%), and Normative
(47.10%). The highest levels of turnover intentions were observed in the BurnedOut/Disengaged profile, and the lowest in the Engaged profile. Employees’ perceptions of job demands and resources were also associated
with profile membership across both levels, although the effects of psychological empowerment were more pronounced than the effects of job demands and resources related to the workgroup, supervisor, and organization.
Individual level effects were also more pronounced than effects occurring at the work unit level, where shared perceptions of work overload and organizational support proved to be the key shared drivers of profile membership.
hide abstract
- Collie, R.J., Malmberg, L.-E., Martin, A.J., Sammons, P., & Morin, A.J.S. (2020). A multilevel person-centered examination of teachers’ workplace demands and resources: Links with work-related well-being. Frontiers in Psychology, 11, 626. DOI: 10.3389/fpsyg.2020.00626
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- Anne Mäkikangas, A. Tolvanen, A., Aunola, K., Feldt, T., Mauno1, S., & Kinnunen, U. (2018). Multilevel latent profile analysis with covariates: Identifying job characteristics profiles in hierarchical data as an example. Organizational Research Methods. DOI: 10.1177/1094428118760690
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- Henry, K. & Muthén, B. (2010). Multilevel latent class analysis: An application of adolescent smoking typologies with individual and contextual predictors. Structural Equation Modeling, 17, 193-215.
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Abstract
"Latent Class Analysis (LCA) is a statistical method used to identify subtypes of related cases using a set of categorical and/or continuous observed variables. Traditional LCA assumes that observations are independent. However, multilevel data structures are
common in social and behavioral research and alternative strategies are needed. In this paper, a new methodology, multilevel latent class analysis (MLCA), is described and an applied example is presented. Latent classes of cigarette smoking among
10,772 European American females in 9th grade who live in one of 206 rural communities across the U.S. are considered. A parametric and non-parametric approach for estimating a MLCA are presented and both individual and contextual predictors of the smoking
typologies are assessed. Both latent class and indicator-specific random effects models are explored. The best model was comprised of three level 1 latent smoking classes (heavy smokers, moderate smokers, non-smokers), two random effects
to account for variation in the probability of level 1 latent class membership across communities, and a random factor for the indicator-specific level 2 variances. Several covariates at the individual and contextual level were useful in predicting latent classes of cigarette
smoking as well as the individual indicators of the latent class model. This paper will assist researchers in estimating similar models with their own data."
hide abstract
- Muthén, B. & Asparouhov, T. (2009).
Multilevel regression mixture analysis.
Journal of the Royal Statistical Society, Series A, 172, 639-657.
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Abstract
"A two-level regression mixture model is discussed and contrasted with the conventional
two-level regression
model. Simulated and real data shed light on the modelling alternatives.
The real data analyses
investigate gender differences in mathematics achievement
from the US National Education Longitudinal
Survey.The two-level regression mixture analyses
show that unobserved heterogeneity should not
be presupposed to exist only at level 2 at the
expense of level 1. Both the simulated and the real data
analyses show that level 1 heterogeneity
in the form of latent classes can be mistaken for level
2 heterogeneity in the form of the
random effects that are used in conventional two-level regression
analysis. Because of this,
mixture models have an important role to play in multilevel regression
analyses. Mixture models
allow heterogeneity to be investigated more fully, more correctly attributing
different portions of
the heterogeneity to the different levels."
hide abstract
- Asparouhov, T. & Muthén, B. (2008).
Multilevel mixture models.
In Hancock, G. R., & Samuelsen, K. M. (Eds.), Advances in latent variable mixture models, pp. 27-51. Charlotte, NC: Information Age Publishing, Inc.
Click here for information about the book.
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contact second author
- Van Horn, M.L., Fagan, A.A., Jaki, T., Brown, E.C., Hawkins, J.D., Arthur, M.W., Abbott, R.D., & Catalano, R. F. (2008).
Using multilevel mixtures to evaluate intervention effects in group randomized trials.
Multivariate Behavioral Research, 43(2), 289-326. PMC - In Process.
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Abstract
"There is evidence to suggest that the effects of behavioral interventions may be
limited to specific types of individuals, but methods for evaluating such outcomes
have not been fully developed. This study proposes the use of finite mixture models
to evaluate whether interventions, and, specifically, group randomized trials, impact participants with certain characteristics or levels of problem behaviors.
This study uses latent classes defined by clustering of individuals based on the targeted
behaviors and illustrates the model by testing whether a preventive intervention
aimed at reducing problem behaviors affects experimental users of illicit substances differently than problematic substance
users or those individuals engaged in more serious problem behaviors. An illustrative example is used
to demonstrate the identification of latent classes, specification of random effects in a multilevel
mixture model, independent validation of latent classes, and the estimation of
power for the proposed models to detect intervention effects. This study proposes
specific steps for the estimation of multilevel mixture models and their power
and suggests that this model can be applied more broadly
to understand the effectiveness of interventions."
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- Morin, A.J.S., Gillet, N., Blais, A.-R., Comeau, C., & Houle, S.A. (2023, In Press). A multilevel perspective on the role of job demands, job resources, and need satisfaction employees' outcomes. Journal of Vocational Behavior. DOI: 10.1016/j.jvb.2023.103846.
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Abstract
This study investigates the mediator role of psychological need satisfaction for the effects of job demands and
resources on turnover intentions, psychological distress, and work-to-family conflict, simultaneously at the
employee and work unit levels. In doing so, we consider how need satisfaction, when considered at the work unit
level, creates a context likely to play an additional role in the prediction of these outcomes. These questions were
investigated using a combination of doubly latent multilevel confirmatory factor analyses and structural equation
models applied to responses provided by a large sample (N = 5,716 employees nested within 50 work units) of
Canadian Armed Forces/Department of National Defence personnel. The results supported the idea that work
environment effects on the outcomes considered in this study were mediated by psychological need satisfaction
at the individual and work unit levels and demonstrated that these associations were driven by global work
environment perceptions and global need satisfaction. Furthermore, need satisfaction was found to create a
context, at the work unit level, leading employees working in units including more highly satisfied co-workers to
present higher levels of turnover intentions but lower levels of work-to-family conflict than would be expected
based on their individual levels of need satisfaction.
hide abstract
- Asparouhov T, Muthén B. (2021). Robust Chi-Square in Extreme and Boundary Conditions: Comments on Jak et al. (2021). Psych, 3(3):542-551. DOI: 10.3390/psych3030035
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Abstract
In this article we describe a modification of the robust chi-square test of fit that yields more accurate type I error rates when the estimated model is at the boundary of the admissible space.
hide abstract
- McNeish, D. (2021). Specifying location-scale models for heterogeneous variances as multilevel SEMs. Organizational Research Methods, 24:3, 630-653. DOI: 10.1177/1094428120913083
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- Morin, A.J.S., Blais, A.-R., & Chénard-Poirier, L.A. (2021). Doubly latent multilevel procedures for organizational assessment and prediction. Journal of Business and Psychology. DOI: 10.1007/s10869-021-09736-5
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Abstract
Organizational research has a rich tradition of multilevel research anchored in a long-standing recognition that conclusions obtained at the individual level of analysis cannot be expected to generalize at other levels. Doubly latent multilevel methods, which provide the way to combine multilevel analyses and structural equation models into a single analytic framework, enrich this tradition. Yet, some critical technical considerations have yet to be systematically integrated in applied research. This article aims to introduce organizational researchers to the estimation of doubly latent multilevel models while taking into account: (a) the need to systematically assess the multilevel
measurement structure of the constructs included in one study; (b) the various types of measurement errors that can be controlled for (rather than simply estimated) as part of these models; (c) the importance of relying on a clear understanding of contextual versus climate constructs given their role in centering decisions. These issues are illustrated by an investigation of the multilevel relations between psychological empowerment, psychological health, and turnover intentions among a large sample (N = 5875 employees nested within 49 work units) of Canadian Defence employees.
hide abstract
- Hamaker, E. L., & Muthén, B. (2020). The fixed versus random effects debate and how it relates to centering in multilevel modeling. Psychological Methods, 25(3), 365–379. https://doi.org/10.1037/met0000239
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Abstract
In many disciplines researchers use longitudinal panel data to investigate the potentially causal relationship between two variables. However, the conventions and
concerns vary widely across disciplines. Here we focus on two concerns, that is: a) the concern about random effects versus xed effects, which is central in the
(micro-)econometrics/sociology literature; and b) the concern about grand mean versus group (or person) mean centering, which is central in the multilevel literature
associated with disciplines like psychology and educational sciences. We show that these two concerns are actually addressing the same underlying issue. We discuss
diverse modeling methods based on either multilevel regression modeling with the data in long format, or structural equation modeling with the data in wide format,
and compare these approaches with simulated data. We extend the multilevel model with random slopes and discuss the consequences of this. Subsequently, we provide
guidelines on how to choose between the diverse modeling options. We illustrate the use of these guidelines with an empirical example based on intensive longitudinal
data, in which we consider both a time-varying and a time-invariant covariate.
hide abstract
- Asparouhov, T., & Muthén, B. (2019). Latent variable centering of predictors and mediators in multilevel and time-series models. Structural Equation Modeling: A Multidisciplinary Journal, 26, 119-142. DOI: 10.1080/10705511.2018.1511375
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Abstract
We discuss different methods for centering a predictor or a mediator in multilevel models. We show how latent mean centering can be extended to models with random slopes. Implications are discussed for
estimating multilevel regression models with data missing on the predictors, estimating the contextual effect in multilevel, time-series and probit regression models, estimating the indirect effect in multilevel
mediation models, and estimating random tetrachoric autocorrelations for time-series models with categorical data.
hide abstract
- Diallo, T. M. O., & Lu, H. (2016): Consequences of misspecifying across-cluster time-specific residuals in multilevel latent growth curve models. Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2016.1247647
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- Kim, E.S., Dedrick, R.F., Cao, C. & Ferron, J.M. (2016). Multilevel factor analysis: Reporting guidelines and a review of reporting practices. Multivariate Behavioral Research, DOI: 10.1080/00273171.2016.1228042
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- Preacher, K.J., Zhang, Z. & Zyphur, M.J. (2016). Multilevel structural equation models for assessing moderation within and across levels of analysis. Psychological Methods. 21(2), 189-205. DOI: 10.1037/met0000052
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- Ryu, E. (2015). The role of centering for interaction of level 1 variables in multilevel structural equation models. Structural Equation Modeling: A Multidisciplinary Journal, 22:4, 617-630, DOI: 10.1080/10705511.2014.936491
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- Kim, E.S. & Cao, C. (2015). Testing group mean differences of latent variables in multilevel data using multiple-group multilevel CFA and multilevel MIMIC modeling. Multivariate Behavioral Research, 50:4, 436-456, DOI: 10.1080/00273171.2015.1021447
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- Cho, S., Preacher, K.J., & Bottge, B.A. (2015). Detecting intervention effects in a cluster-randomized design using multilevel structural equation modeling for binary responses. Applied Psychological Measurement. DOI: 10.1177/0146621615591094
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- Depaoli, S. & Clifton, J. P. (2015). A Bayesian approach to multilevel structural equation modeling with continuous and dichotomous outcomes. Structural Equation Modeling: A Multidisciplinary Journal, 22(3), 327-351. DOI: 10.1080/10705511.2014.937849
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- Schultze, M., Koch. T., & Eid, M. (2015). The effects of nonindependent rater
sets in multilevel–multitrait–multimethod models. Structural Equation Modeling: A Multidisciplinary
Journal, DOI: 10.1080/10705511.2014.937675
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- Dunn, C., Masyn, K.E., Jones, S.M., Subramanian, S.V., & Koenen, K.C. (2014). Measuring psychosocial environments using individual responses: an application of multilevel factor analysis to examining students in schools.
Prevention Science. DOI 10.1007/s11121-014-0523-x
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- Kelcey, B., McGinn, D. & Hill, H. (2014). Approximate measure invariance in cross-classified rater-mediated assessments. Frontiers in Medicine, 5:1469, 1-13. DOI: 10.3389/fpsyg.2014.01469
contact first author
- Morin, A.J.S., Marsh, H.W., Nagengast, B., & Scalas, L.F. (2014). Doubly latent multilevel analyses of classroom climate: An illustration.
The Journal of Experimental Education. DOI: 10.1080/00220973.2013.769412
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- Ryu, E. (2014). Model fit evaluation in multilevel structural equation models. Frontiers in Psychology, 5(81). DOI: 10.3389/fpsyg.2014.00081
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- Hox, J., van de Schoot, R. & Matthijsse, S. (2012).
How few countries will do? Comparative survey analysis from a Bayesian perspective.
Survey Research Methods, 6:2, 87-93.
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Abstract
"Meuleman and Billiet (2009) have carried out a simulation study aimed at the question how
many countries
are needed for accurate multilevel SEM estimation in comparative studies.
The authors concluded
that a sample of 50 to 100 countries is needed for accurate estimation.
Recently, Bayesian estimation
methods have been introduced in structural equation modeling
which should work well with much lower
sample sizes. The current study reanalyzes the
simulation of Meuleman and Billiet using Bayesian estimation
to find the lowest number of
countries needed when conducting multilevel SEM. The main result
of our simulations is that
a sample of about 20 countries is sufficient for accurate Bayesian estimation,
which makes
multilevel SEM practicable for the number of countries commonly available in large
scale
comparative surveys."
hide abstract
- Muthén, B. & Asparouhov, T. (2011).
Beyond multilevel regression modeling: Multilevel analysis in a general latent variable framework.
In J. Hox & J.K. Roberts (eds), Handbook of Advanced Multilevel Analysis, pp. 15-40. New York: Taylor and Francis.
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Abstract
"Multilevel modeling is often treated as if it concerns only regression
analysis and growth modeling.
Multilevel modeling, however, is relevant for nested data not only with regression and growth analysis
but with all types of statistical analyses. This chapter has two aims. First,
it shows that already in the traditional multilevel analysis areas of regression and growth there are several new modeling
opportunities that should be considered. Second, it gives an overview with examples of multilevel
modeling for path analysis, factor analysis, structural equation modeling, and growth mixture modeling.
Examples include two extensions of two-level regression analysis with measurement error in the
level 2 covariate and a level 1 mixture; two-level path analysis and structural equation modeling;
two-level exploratory factor analysis of classroom misbehavior; two-level growth modeling using a two-part
model for heavy drinking development; an unconventional approach to three-level growth modeling
of math achievement; and multilevel latent class mediation of high school dropout using multilevel
growth mixture modeling of math achievement development."
hide abstract
- Dedrick, R & Greenbaum, P. (2010).
Multilevel confirmatory factor analysis of a scale measuring interagency collaboration of children’s
mental health agencies.
Journal of Emotional and Behavioral Disorders, XX(X), 1-14.
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Abstract
"Multilevel confirmatory factor analysis was used to evaluate the factor structure underlying the 12-item,
three-factor Interagency Collaboration Activities Scale (ICAS) at the informant level and at the
agency level. Results from 378 professionals
(104 administrators, 201 service providers, and 73 case
managers) from 32 children’s mental health service agencies
supported a correlated three-factor model
at each level and indicated that the item loadings were not significantly (p <
.05) different across levels. Reliability estimates of the three factors (Financial and Physical Resource Activities,
Program Development and Evaluation Activities, and Collaborative Policy Activities) at the agency
level were .81, .60, and .72, respectively, whereas these estimates were .79, .82, and .85 at the individual
level. These multilevel results provide support
for the construct validity of the scores from
the ICAS. When the ICAS was examined in relation to Level 1 and Level
2 covariates, results showed that participants’ characteristics (i.e., age, job role, gender, educational level, and number of
months employed) and agency characteristics (i.e., state location and number of employees) were not
significantly (p > .05) related to levels of interagency collaboration."
hide abstract
- Preacher, K., Zyphur, M. & Zhang, Z. (2010).
A general multilevel SEM framework for assessing multilevel mediation.
Psychological Methods, 15, 209-233.
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Abstract
"Several methods for testing mediation hypotheses with two-level nested data have been proposed using
a multilevel modeling (MLM) paradigm. However, these MLM approaches do not accommodate mediation pathways
with Level-2 outcomes and may produce conflated estimates of between- and within-level components
of indirect effects. Moreover, these methods have each appeared in isolation, so a unified framework
that integrates the existing methods, as well as new multilevel mediation models, is lacking. Here
we show that a multilevel structural equation modeling (MSEM) paradigm can overcome these two limitations
of mediation analysis using MLM. We present an integrative two-level MSEM mathematical framework
that subsumes new and existing multilevel mediation approaches as special cases. We use several
applied examples and accompanying software code to illustrate the flexibility of this framework and
to show that different substantive conclusions can be drawn using MSEM versus MLM."
hide abstract
- Roesch, S. C. , Aldridge, A. A. , Stocking, S. N. , Villodas, F. , Leung, Q., Bartley, C. E. & Black, L. J. (2010).
Multilevel factor analysis and structural equation modeling of daily diary coping data: Modeling trait
and state variation.
Multivariate Behavioral Research, 45, 767-789.
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Abstract
"This study used multilevel modeling of daily diary data to model within-person
(state) and between-person
(trait) components of coping variables. This application
included the introduction of multilevel
factor analysis (MFA) and a compari-
son of the predictive ability of these trait/state factors. Daily
diary data were collected on a large .n D 366/ multiethnic sample over the course of 5 days.
Intraclass correlation coefficient for the derived factors suggested approximately
equal amounts of variability in coping usage at the state and trait levels. MFAs
showed that Problem-Focused Coping and
Social Support emerged as stable factors
at both the within-person and between-person levels. Other
factors (Minimization, Emotional Rumination, Avoidance, Distraction) were specific to the within-person
or between-person levels but not both. Multilevel structural equation modeling
(MSEM) showed that the prediction of daily positive and negative affect differed
as a function of outcome and level of coping factor. The Discussion section focuses
primarily on a conceptual and methodological understanding
of modeling state and trait coping using daily diary data with MFA and MSEM to examine covariation
among coping variables and predicting outcomes of interest."
hide abstract
- Gottfredson, N.C., Panter, A.T., Daye, C.E., Allen, W.F. & Wightman, L.F. (2009).
The effects of educational diversity in a national samples of law students: Fitting multilevel latent
variable models in data with categorical indicators.
Multivariate Behavioral Research, 44, 305-331.
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Abstract
"Controversy surrounding the use of race-conscious admissions can be partially
resolved with improved empirical knowledge of the effects of racial diversity in
educational settings. We use a national sample of law students nested in 64 law
schools to test the complex and largely untested theory regarding
the effects of educational diversity on student outcomes. Social scientists who study these outcomes
frequently encounter both latent variables and nested data within a single analysis.
Yet, until recently, an appropriate modeling technique has been computationally
infeasible, and consequently few applied researchers have estimated appropriate
models to test their theories, sometimes limiting the scope of their research question.
Our results, based on disaggregated multilevel structural equation
models, show that racial diversity is related to a reduction in prejudiced attitudes and increased
perceived exposure to diverse ideas and that these effects are mediated by more frequent interpersonal
contact with diverse peers. These findings provide support for the idea that administrative manipulation
of educational diversity may lead to improved student outcomes. Admitting a racially/ethnically
diverse student body provides an educational experience that encourages increased exposure to
diverse ideas and belief systems."
hide abstract
- Marsh, H.W., Ludtke, O., Robitzsch, A., Trautwein, U., Asparouhov, T., Muthén, B., & Nagengast, B. (2009).
Doubly-latent models of school contextual effects: Integrating multilevel and structural equation approaches
to control measurement and sampling errors.
Multivariate Behavioral Research, 44, 764-802.
Click here
to view the appendix that goes with this paper.
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Abstract
"This article is a methodological-substantive synergy. Methodologically, we demonstrate latent-variable contextual models that integrate structural equation models (with multiple indicators) and latent-variable multilevel models. These models simultaneously control for and unconfound unreliability due
to measurement error at the individual (L1) and group (L2) levels, and sampling error in the aggregation of L1-characteristics to form L2-constructs. We propose a 2x2 taxonomy of models that are latent or manifest in relation to sampling items (measurement error) and sampling of persons (sampling error),
and discuss when different models might be most useful. We demonstrate the flexibility of these four core models by extending them to include random slopes, latent (single-level or cross-level) interactions, and latent quadratic effects. Substantively we use these models to test the big-fish-little-pond
effect (BFLPE), showing that individual student levels of academic self-concept (L1-ASC) is positively associated with individual level achievement (L1-ACH) and negatively associated with school-average achievement (L2-ACH) – a finding with important policy implications for the way schools
are structured. Extending tests of the BFLPE in new directions, we show that the nonlinear effects of the L1-ACH (a latent quadratic effect) and the interaction between gender and L1-ACH (an L1 x L1 latent interaction) are not significant. Whilst random-slope models show there is no significant school-to-school
variation in relations between L1-ACH and L1-ASC, the negative effects of L2-ACH (the BFLPE) do vary somewhat with individual L1-ACH. We conclude with implications for diverse applications of the taxonomy of latent contextual models, including recommendations about their implementation,
effect size estimates (and confidence intervals) appropriate to multilevel models, and directions for further research in contextual effect analysis."
hide abstract
- Lüdtke, O., Marsh, H.W., Robitzsch, A., Trautwein, U., Asparouhov, T., & Muthén, B. (2008).
The multilevel latent covariate model: A new, more reliable approach to group-level effects in contextual
studies.
Psychological Methods, 13, 203-229.
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Abstract
"In multilevel modeling (MLM), group level (L2) characteristics are often measured by aggregating individual
level (L1) characteristics within each group as a means of assessing contextual effects (e.g.,
group-average effects of SES, achievement, climate). Most previous applications have used a multilevel
manifest covariate (MMC) approach, in which the observed (manifest) group mean is assumed to have
no measurement error. This paper shows mathematically and with simulation results that this MMC approach
can result in substantially biased estimates of contextual effects and can substantially underestimate
the associated standard errors, depending on the number of L1 individuals in each of the
groups, the number of groups, the intraclass correlation, the sampling ratio (the percentage of cases
within each group sampled), and the nature of the data. To address this pervasive problem, we introduce
a new multilevel latent covariate (MLC) approach that corrects for unreliability at L2 and results
in unbiased estimates of L2 constructs under appropriate conditions. However, our simulation results
also suggest that the contextual effects estimated in typical research situations (e.g., fewer
than 100 groups) may be highly unreliable. Furthermore, under some circumstances when the sampling
ratio approaches 100%, the MMC approach provides more accurate estimates. Based on three simulations
and two real-data applications, we critically evaluate the MMC and MLC approaches and offer suggestions
as to when researchers should most appropriately use one, the other, or a combination of both approaches.
"
hide abstract
- Asparouhov, T. & Muthén, B. (2007).
Computationally efficient estimation of multilevel high-dimensional latent variable models.
Proceedings of the 2007 JSM meeting in Salt Lake City, Utah, Section on Statistics in Epidemiology.
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Abstract
"Multilevel analysis often leads to modeling with multiple
latent variables on several levels. While this
is
less of a problem with Gaussian observed variables,
maximum-likelihood (ML) estimation with categorical
outcomes
presents computational problems due to multidimensional
numerical integration. We
describe a new
method that compared to ML is both computationally
efficient and has similar MSE. The
method is an extension
of the Muthén (1984) weighted least squares (WLS)
estimation method to multilevel
multivariate latent variable
models for any combination of categorical, censored,
and normal
observed variables. Using a new version of
the Mplus program, we compare MSE and the computational
time
for the ML and WLS estimators in a simulation
study."
hide abstract
- Grilli, L & Rampichini, C. (2007).
Multilevel factor models for ordinal variables.
Structural Equation Modeling, 14, 1-25.
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Abstract
"This article tackles several issues involved in specifying, fitting, and interpreting the results of
multilevel factor models for ordinal variables. First, the problem of model specification and identification
is addressed, outlining parameter interpretation. Special attention is devoted to the consequences
on interpretation stemming from the usual choice of not decomposing the specificities into hierarchical
components. Then a general strategy of analysis is outlined, highlighting the role of the
exploratory steps. The theoretical topics are illustrated through an application to graduates' job
satisfaction, where estimation is based on maximum likelihood via an Expectation-Maximization algorithm
with adaptive quadrature."
hide abstract
- Dyer, N.G., Hanges, P.J. & Hall, R.J. (2005).
Applying multilevel confirmatory factor analysis techniques to the study of leadership.
The Leadership Quarterly 16 (2005), 149–167.
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Abstract
"Statistical issues associated with multilevel data are becoming increasingly important to organizational
researchers.
This paper concentrates on the issue of assessing the factor structure of a construct
at aggregate levels
of analysis. Specifically, we describe a recently developed procedure for performing
multilevel confirmatory factor
analysis (MCFA) [Muthén, B.O. (1990). Mean and covariance structure
analysis of hierarchical data. Paper
presented at the Psychometric Society, Princeton, NJ; Muthén,
B.O. (1994). Multilevel covariance structure
analysis. Sociological Methods and Research, 22,
376–398], and provide an illustrative example of its application
to leadership data reflecting both
the organizational and societal level of analysis. Overall, the results of our
illustrative analysis
support the existence of a valid societal-level leadership construct, and show the potential of
this
multilevel confirmatory factor analysis procedure for leadership research and the field of I/O psychology
in
general."
hide abstract
- Mehta, P. & Neale, M. (2005).
People are variables too: Multilevel structural equations modeling.
Psychological Methods, 10, 259-284.
This paper draws on techniques illustrated in the Mplus Version 3 User's Guide (first printed in March
2004), example 9.10.
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Abstract
"The article uses confirmatory factor analysis (CFA) as a template to explain didactically multilevel
structural equation models (ML-SEM) and to demonstrate the equivalence of general mixed-effects models
and ML-SEM. An intuitively appealing graphical representation of complex ML-SEMs is introduced that
succinctly describes the underlying model and its assumptions. The use of definition variables (i.e.,
observed variables used to fix model parameters to individual specific data values) is extended
to the case of ML-SEMs for clustered data with random slopes. Empirical examples of multilevel CFA
and ML-SEM with random slopes are provided along with scripts for fitting such models in SAS Proc Mixed,
Mplus, and Mx. Methodological issues regarding estimation of complex ML-SEMs and the evaluation
of model fit are discussed. Further potential applications of ML-SEMs are explored."
hide abstract
- Yuan, K.H. & Hayashi, K. (2005).
On Muthén's maximum likelihood for two-level covariance structure models.
Psychometrika, 70, 147-167.
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Abstract
"Data in social and behavioral sciences are often hierarchically organized. Special statistical procedures
that take into account the dependence of such observations have been developed. Among procedures
for 2-level covariance structure analysis, Muth´en’s maximum likelihood (MUML) has the advantage of
easier computation and faster convergence. When data are balanced, MUML is equivalent to the maximum
likelihood procedure. Simulation results in the literature endorse the MUML procedure also for unbalanced
data. This paper studies the analytical properties of the MUML procedure in general. The results
indicate that the MUML procedure leads to correct model inference asymptotically when level-2
sample size goes to infinity and the coefficient of variation of the level-1 sample sizes goes to zero.
The study clearly identifies the impact of level-1 and level-2 sample sizes on the standard errors
and test statistic of the MUML procedure. Analytical results explain previous simulation results
and will guide the design or data collection for the future applications of MUML."
hide abstract
- Muthén, B., Khoo, S.T. & Gustafsson, J.E. (1997).
Multilevel latent variable modeling in multiple populations.
Unpublished technical report.
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Abstract
"Modeling is described for the simultaneous analysis of two-level data in several populations. A typical
example is cluster sampling of students within schools, where schools of different types are represented,
e.g. public and private schools. Multivariate measurements on each student are assumed to
give rise to a latent variable model. Of interest is to study across-population differences and similarities
with respect to the within- and between-group covariance matrices and with respect to the
mean vector. The methodology is illustrated by a comparative analysis of achievement structures in
Catholic and public schools."
hide abstract
- Muthén, B. & Satorra, A. (1995). Complex sample data in structural equation modeling. Sociological Methodology, 25, 267-316.
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Abstract
"Large-scale surveys using complex sample designs are frequently carried out by government agencies. The
statistical analysis technology available for such data is, however, limited in scope. This study
investigates and further develops statistical methods that could be used in software for the analysis
of data collected under complex sample designs. First, it identifies several recent methodological
lines of inquiry which taken together provide a powerful and general statistical basis for a complex
sample, structural equation modeling analysis. Second, it extends some of this research to new situations
of interest. A Monte Carlo study that empirically evaluates these techniques on simulated data
comparable to those in large-scale complex surveys demonstrates that they work well in practice.
Due to the generality of the approaches, the methods cover not only continuous normal variables but
also continuous non-normal variables and dichotomous variables. Two methods designed to take into account
the complex sample structure were investigated in the Monte Carlo study. One method, termed aggregated
analysis, computes the usual parameter estimates but adjusts standard errors and goodness-of-fit
model testing. The other method, termed disaggregated analysis, includes a new set of parameters
reflecting the complex sample structure. Both of the methods worked very well. The conventional
method that ignores complex sampling worked poorly, supporting the need for development of special methods
for complex survey data."
hide abstract
- Muthén, B. (1994). Multilevel covariance structure analysis. In J. Hox & I. Kreft (eds.), Multilevel Modeling, a special issue of Sociological Methods & Research, 22, 376-398.
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- Muthén, B. (1990). Mean and covariance structure analysis of hierarchical data. Paper presented at the Psychometric Society meeting in Princeton, NJ, June 1990. UCLA Statistics Series 62.
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- Sobel, M. & Muthén, B. (2012). Compliance mixture modelling with a zero effect complier class and missing data. Biometrics, 68, 1037-1045. DOI: 10.1111/j.1541-0420.2012.01791.x
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Abstract
Randomized experiments are the gold standard for evaluating proposed treatments. The
intent to treat estimand (ITT) measures the effect of treatment assignment, but not the effect of treatment if
subjects take treatments to which they are not assigned. The desire to estimate the efficacy of the
treatment in this case has been the impetus for a substantial literature on compliance over the
last 15 years. In papers dealing with this issue, it is typically assumed there are different types
of subjects, for example, those who will follow treatment assignment (compliers), and those who
will always take a particular treatment irrespective of treatment assignment. The estimands of
primary interest are the complier proportion and the complier average treatment effect (CACE). To
estimate CACE, researchers have used various methods, for example, instrumental variables and
parametric mixture models, treating compliers as a single class. However, it is often
unreasonable to believe all compliers will be affected. This paper therefore treats compliers as a
mixture of two types, those belonging to a zero effect class, others to an effect class. Second, in
most experiments, some subjects drop out or simply do not report the value of the outcome variable,
and the failure to take into account missing data can lead to biased estimates of treatment
effects. Recent work on compliance in randomized experiments has addressed this issue by assuming
missing data are missing at random or latently ignorable. We extend this work to the case where
compliers are a mixture of types and also examine alternative types of non-ignorable
missing data assumptions.
hide abstract
- Jo, B., Ginexi, E. & Ialongo, N. (2010). Handling missing data in randomized experiments with noncompliance. Prevention Science, 11(4):384-96. DOI: 10.1007/s11121-010-0175-4
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Abstract
"Treatment noncompliance and missing outcomes at posttreatment assessments
are common problems in field
experiments in naturalistic settings. Although the two
complications often occur simultaneously, statistical
methods that address both complications
have not been routinely considred in data analysis
practice in the prevention
research field. This paper shows that identification and estimation of
causal treatment
effects considering both noncompliance and missing outcomes can be relatively easily
conducted
under various missing data assumptions. We review a few assumptions on
missing data in the
presence of noncompliance, including the latent ignorability proposed
by Frangakis and Rubin (1999),
and show how these assumptions can be used
in the parametric CACE estimation framework. As an easy
way of sensitivity analysis,
we propose the use of alternative missing data assumptions, which will
provide a range
of causal effect estimates. In this way, we are less likely to settle with a possibly
biased
causal effect estimate based on a single assumption. We demonstrate how alternative
missing
data assumptions affect identification of causal effects, focusing on the complier
average causal
effect (CACE). The data from the Johns Hopkins School Intervention
Study (Ialongo et al., 1999) will
be used as an example."
hide abstract
- Stormshak, E.A., Connell, A. & Dishion, T.J. (2009).
An adaptive approach to family-centered intervention in schools: Linking intervention engagement to academic
outcomes in middle and high school.
Prevention Science, 10, 221-235.
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Abstract
"This study examined the impact of an adaptive
approach to family intervention in public schools on
academic
outcomes from age 11 to 17. Students were
randomly assigned to the three-session Family Check-Up
(FCU),
which is designed to motivate change in parenting
practices by using an assessment-driven
approach and
strengths-based feedback. All services were voluntary, and
approximately 25% of the families
engaged in the FCU.
Compared with matched controls, adolescents whose
parents received the FCU
maintained a satisfactory GPA into
high school, and intervention engagement was associatedwith
improved
attendance. The highest-risk families were the most
likely to engage in the family-centered intervention,
suggesting
the efficacy of integrating supportive services to families
in the context of
other schoolwide approaches to promote the
success and achievement of vulnerable students."
hide abstract
- Jo, B., Asparouhov, T. & Muthén, B. (2008).
Intention-to-treat analysis in cluster randomized trials with noncompliance.
Statistics in Medicine, 27, 5565-5577.
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Abstract
"In cluster randomized trials (CRT), individuals belonging to the same cluster are
very likely to resemble
one another, not only in terms of outcomes, but also in terms of
treatment compliance behavior.
Whereas the impact of resemblance in outcomes is well
acknowledged, little attention has been given
to the possible impact of resemblance in
compliance behavior. This study defines compliance intraclass
correlation as the level
of resemblance in compliance behavior among individuals within clusters,
and shows
how compliance intraclass correlation can be a problem in evaluating intention-to-treat
(ITT)
effect in CRT. On the basis of Monte Carlo simulations, it is demonstrated that
ignoring compliance
information in analyzing data from CRT may result in substantially
decreased power to detect ITT
effect, mainly due to compliance intraclass correlation.
As a way of avoiding additional loss of
power to detect ITT effect in CRT accompanied
by noncompliance, this study employs an estimation method,
where ITT effect estimates
are obtained based on compliance-type-specific treatment effect estimates.
A multilevel
mixture analysis using an ML-EM estimation method is used for this estimation."
hide abstract
- Jo, B., Asparouhov, T., Muthén, B., Ialongo, N. & Brown, H. (2008).
Cluster randomized trials with treatment noncompliance.
Psychological Methods, 13, 1-18.
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Abstract
"Cluster randomized trials (CRT) have been widely used in field experiments
treating a cluster (or group)
of individuals as the unit of randomization. This
study focuses particularly on situations where
CRT are accompanied by a common
complication in field experiments, namely treatment noncompliance.
In
CRT, compliance behavior may be related not only to individual characteristics
of study participants,
but also to the environment of clusters individuals
belong to. Therefore, analyses ignoring the
connection between compliance and
CRT may not provide valid results. Although randomized field experiments
often
suffer from both noncompliance and clustering of the data, these features have
been studied
as separate rather than concurrent problems. On the basis of Monte
Carlo simulations, this study
demonstrates how CRT and noncompliance may
affect statistical inferences and how these two complications
can be accounted for
simultaneously. In particular, the effect of randomized intervention on
individuals
who abide by the intervention assignment (complier average causal effect: CACE)
will be
the focus of the study. For estimation of intervention effects considering
both noncompliance and CRT,
an ML-EM estimation method is employed."
hide abstract
- Dunn, G., Maracy, M., Dowrick, C., Ayuso-Mateos, J.L., Dalgard, O.S., Page, H., Lehtinen, V., Casey, P., Wilkinson, C., Vasquez-Barquero, J.L., & Wilkinson, G. (2003).
Estimating psychological treatment effects from a randomized controlled trial with both non-compliance
and loss to follow-up.
British Journal of Psychiatry, 183, 323-331.
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Abstract
"Background: The Outcomes of Depression International Network (ODIN) trial evaluated the effects of two
psychological interventions for the treatment of depression in primary care. Only about half of
the patients in the treatment arm complied with the offer of treatment, prompting the question: 'what
was the effect of treatment in those patients who actually received it? Aims: To illustrate the
estimation of the effect of receipt of treatment in a randomised controlled trial subject to non-compliance
and loss to follow-up. Method: We estimated the complier average causal effect (CACE)
of treatment. Results: In the ODIN trial the effect of receipt of psychological intervention (an
average of about 4 points on the Beck Depression Inventory) is about twice that of offering it. Conclusions:
The statistical analysis of the results of a clinical trial subject to non-compliance
to allocated treatment is now reasonably straightforward through estimation of a CACE and investigators
should be encourage to present the results of analyses of the type as a routine component of a
trial report."
hide abstract
- Jo, B. & Muthén, B. (2003).
Longitudinal studies with intervention and noncompliance: Estimation of causal effects in growth mixture
modeling.
In S.P. Reise & Duan, N. (eds.) Multilevel Modeling. Methodological Advances, Issues, and Applications (pp.112-139). Mahwah, New Jersey: Lawrence Erlbaum.
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- Jo, B. (2002).
Statistical power in randomized intervention studies with noncompliance.
Psychological Methods, 7, 178-193.
Mplus inputs and outputs used in this paper can be viewed and/or downloaded from the Examples
page.
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Abstract
"One of the common complications in randomized intervention trials is noncompliance of study participants.
Noncompliance is an obstacle to fair statistical comparison and may lead to biased estimates of
intervention effects. From a practical point of view, noncompliance is a major threat to obtaining
power to detect intervention effects, because it is directly related to effect size and confidence
interval. This study focuses on the exploration of cost-effective ways of optimizing intervention
settings in the presence of noncompliance to reach a desirable level of statistical power. Given
that compliance behavior of human participants is hard to control, it is demonstrated in the study that
the quality of intervention effect estimates can be improved through more controllable factors
such as appropriate statistical methods, pre-treatment covariates, and study design. This paper also
provides power curves in various combinations of sample size and effect size as a guide for design
of future studies."
hide abstract
- Jo, B. (2002).
Model misspecification sensitivity analysis in estimating causal effects of interventions with noncompliance.
Statistics in Medicine, 21, 3161-3181.
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show abstract
Abstract
"Randomized trials often face complications in assessing the effect of treatment because of study participants'
noncompliance. If compliance type is observed in both the treatment and control conditions,
the causal effect of treatment can be estimated for a targeted sub-population of interest based
on compliance type. However, in practice, compliance type is not observed completely. Given this
missing compliance information, the CACE (complier average causal effect) estimation approach provides
a way to estimate differential effects of treatments by imposing the exclusion restriction for
noncompliers. Under the exclusion restriction, the CACE approach estimates the effect of treatment
assignment for compliers, but disallows the effect of treatment assignment for noncompliers. The
exlusion restriction plays a key role in separating outcome distributions based on compliance type.
However, the CACE estimate can be substantially biased if the assumption is violated. This study examines
the bias mechanism in the estimation of CACE when the assumption of the exclusion restriction
is violated. It is also examined how covariate information affects the sensitivity of the CACE estimate
to violation of the exclusion restriction assumption."
hide abstract
- Jo, B. (2002).
Estimation of intervention effects with noncompliance: Alternative model specifications.
Journal of Educational and Behavioral Statistics, 27, 385-409.
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Abstract
"This study examines alternative ways of specifying models in the CACE (complier average causal effect)
estimation method, where the major interest is in estimating causal effects of treatments for compliers.
A fundamental difficulty involved in the CACE estimation method is in dealing with missing
compliance information among study participants. Given that, the assumption of the exclusion restriction
plays a critical role in separating the distributions of compliers and noncompliers. If no pre-treatment
covariates are available, assuming the exclusion restriction is unavoidable to obtain
unique ML estimates in CACE models, although the assumption can be often unrealistic. One disadvantage
of assuming the exclusion restriction is that the CACE estimate can be biased if the assumption
is violated. Another disadvantage is that the assumption limits the flexibility of CACE modeling in
practice. However, if pre-treatment covariates are available, more modeling options other than strictly
forcing the exclusion restriction can be considered to establish identifiability of CACE models.
This study explores modeling possibilities of CACE estimation within an ML-EM framework in the
presence of covariate information."
hide abstract
- Jo, B. & Muthén, B. (2001).
Modeling of intervention effects with noncompliance: A latent variable approach for randomized trials.
In G. Marcoulides & R.E. Schumacker (eds.) New Developments and Techniques in Structural Equation Modeling (pp. 57-87). Mahwah, New Jersey: Lawrence Erlbaum
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- Bauer, D.J. (2005).
The role of nonlinear factor-to-indicator relationships.
Psychological Methods, 10, 305-316.
This paper draws on techniques illustrated in the Mplus Version 3 User's Guide (first printed in March
2004), example 5.7.
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Abstract
"Measurement invariance is a necessary condition for the evaluation of factor mean differences over groups
or time. This article considers the potential problems that can arise for tests of measurement
invariance when the true factor-to-indicator relationship is nonlinear (quadratic) and invariant but
the linear factor model is nevertheless applied. The factor loadings and indicator intercepts of the
linear model will diverge across groups as the factor mean difference increases. Power analyses show
that even apparently small quadratic effects can result in rejection of measurement invariance at
moderate sample sizes when the factor mean difference is medium to large. Recommendations include the
identification of nonlinear relationships using diagnostic plots and consideration of newly developed
methods for fitting nonlinear factor models."
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- Pek, J., Sterba, S., Kok, B. & Bauer, D. (2009).
Estimating and visualizing nonlinear relations among latent variables: A semiparametric approach.
Multivariate Behavioral Research, 44, 407 - 436.
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Abstract
"The graphical presentation of any scientific finding enhances its description, in-
terpretation, and
evaluation. Research involving latent variables is no exception,
especially when potential nonlinear
effects are suspect. This article has multiple
aims. First, it provides a nontechnical overview of
a semiparametric approach to
modeling nonlinear relationships among latent variables using mixtures
of linear
structural equations. Second, it provides several examples showing how the method
works and
how it is implemented and interpreted in practical applications. In
particular, this article examines
the potentially nonlinear relationships between
positive and negative affect and cognitive processing.
Third, a recommended dis-
play format for illustrating latent bivariate relationships is demonstrated.
Finally,
the article describes an R package and an online utility that generate these displays
automatically."
hide abstract
- Bauer, D.J. (2005).
A semiparametric approach to modeling nonlinear relations among latent variables.
Structural Equation Modeling, 12, 513-534.
This paper draws on techniques illustrated in the Mplus Version 3 User's Guide (first printed in March
2004), example 7.26.
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- Brown, A. & Maydeu-Olivares, A. (2012).
Fitting a Thurstonian IRT model to forced-choice data using Mplus.
Behavioral Research Methods. DOI: 10.3758/s13428-012-0217-x.
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Abstract
"To counter response distortions associated with
the use of rating scales (a.k.a. Likert scales), items
can be
presented in a comparative fashion, so that respondents are
asked to rank the items within
blocks (forced-choice format).
However, classical scoring procedures for these
forced-choice designs
lead to ipsative data, which presents
psychometric challenges that are well described in the literature.
Recently,
Brown and Maydeu-Olivares (Educational
and Psychological Measurement 71: 460–502, 2011a)
introduced
a model based on Thurstone’s law of comparative
judgment, which overcomes the problems
of ipsative data.
Here, we provide a step-by-step tutorial for coding forcedchoice
responses, specifying
a Thurstonian item response
theory model that is appropriate for the design used, assessing
the
model’s fit, and scoring individuals on psychological
attributes. Estimation and scoring is performed
using Mplus,
and a very straightforward Excel macro is provided that
writes full Mplus input files
for any forced-choice design.
Armed with these tools, using a forced-choice design is now
as easy
as using ratings."
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- Brown, A. & Maydeu-Olivares, A. (2011).
Item response modeling of forced-choice questionnaires.
Educational and Psychological Measurement, 71:3, 460–502.
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Abstract
"Multidimensional forced-choice formats can significantly reduce the impact of numerous
response biases
typically associated with rating scales. However, if scored with
classical methodology, these questionnaires
produce ipsative data, which lead to distorted
scale relationships and make comparisons between
individuals problematic.
This research demonstrates how item response theory (IRT) modeling may
be
applied to overcome these problems. A multidimensional IRT model based on Thurstone’s
framework
for comparative data is introduced, which is suitable for use with
any forced-choice questionnaire
composed of items fitting the dominance response
model, with any number of measured traits, and any
block sizes (i.e., pairs, triplets,
quads, etc.). Thurstonian IRT models are normal ogive models with
structured factor
loadings, structured uniquenesses, and structured local dependencies. These models
can
be straightforwardly estimated using structural equation modeling (SEM) software
Mplus. A number
of simulation studies are performed to investigate how latent
traits are recovered under various
forced-choice designs and provide guidelines for
optimal questionnaire design. An empirical application
is given to illustrate how the
model may be applied in practice. It is concluded that when the
recommended design
guidelines are met, scores estimated from forced-choice questionnaires with the
proposed
methodology reproduce the latent traits well."
hide abstract
- Maydeu-Olivares, A. & Bockenholt, U. (2005).
Structural equation modeling of paired-comparison and ranking data.
Psychological Methods, 10, 285-304.
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Abstract
"L. L. Thurstone’s (1927) model provides a powerful framework for modeling individual differences in choice
behavior. An overview of Thurstonian models for comparative data is provided, including the classical
Case V and Case III models as well as more general choice models with unrestricted and factor-analytic
covariance structures. A flow chart summarizes the model selection process. The authors show
how to embed these models within a more familiar structural equation modeling (SEM) framework. The
different special cases of Thurstone’s model can be estimated with a popular SEM statistical package,
including factor analysis models for paired comparisons and rankings. Only minor modifications
are needed to accommodate both types of data. As a result, complex models for comparative judgments
can be both estimated and tested efficiently."
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-
Speyer, L.G., Zhu, X., Yang, Y., Ribeaud, D. & Eisner, M. (2024). On the Importance of Considering Concurrent Effects in Random-Intercept Cross-Lagged Panel Modelling: Example Analysis of Bullying and Internalising Problems. Multivariate Behavioral Research, DOI: 10.1080/00273171.2024.2428222
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- Muthén, B., Asparouhov, T. & Witkiewitz, K. (2024). Cross-lagged panel modeling with binary and ordinal outcomes. Advance online publication. DOI: 10.1037/met0000701
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Abstract
To date, cross-lagged panel modeling has been studied only for continuous outcomes. This paper presents methods that are suitable also when there are binary and ordinal outcomes. Modeling, testing, identification, and estimation are discussed. A two-part
ordinal model is proposed for ordinal variables with strong floor effects often seen in applications. An example considers the interaction between stress and alcohol use in an alcohol treatment study. Extensions to multiple-group analysis and modeling in the
presence of trends are discussed.
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- Tseng, M. (2024). Fitting Cross-Lagged Panel Models with the Residual Structural Equations Approach. Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2023.2296862
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- Muthén, B. & Asparouhov, T. (2024). Can cross-lagged panel modeling be relied on to establish cross-lagged effects? The case of contemporaneous and reciprocal effects. Psychological Methods. DOI: 10.1037/met0000661
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- Braun, L., Göllner, R., Rieger, S., Trautwein, U. and Spengler, M. (2021). How state and trait versions of self-esteem and depressive symptoms affect their interplay: A longitudinal experimental investigation Journal of Personality and Social Psychology: Personality Process and Individual Differences, 120(1), 206-255, DOI: 10.1037/pspp0000295
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- Shamsollah, A., Zyphur, M., & Ozkok, O. (2021). Long-run effects in dynamic systems: New tools for Cross-Lagged Panel Models. Organizational Research Methods. DOI: 10.1177/1094428121993228
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- Mulder, J.D. & Hamaker, E.L. (2020). Three extensions of the Random Intercept Cross-Lagged Panel Model. Structural Equation Modeling: A Multidisciplinary Journal. DOI: 10.1080/10705511.2020.1784738
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- Zyphur, M.J., Allison, P.D., Tay, L. Voelkle, M.C., Preacher, K.J., Zhang, Z., Hamaker, E.L., Shamsollahi, A., Pierides, D.C., Koval, P. & Diener, E. (2020). From data to causes I: Building a general cross-lagged panel model (GCLM). Organizational Research Methods, 23(4), 651-687.
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- Zyphur, M.J., Allison, P.D., Tay, L. Voelkle, M.C., Preacher, K.J., Zhang, Z., Hamaker, E.L., Shamsollahi, A., Pierides, D.C., Koval, P. & Diener, E. (2020). From data to causes II: Comparing approaches to panel data analysis. Organizational Research Methods, 23(4), 688-716.
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- Usami, S., Murayama, K., & Hamaker, E.L. (2019). A unified framework of longitudinal models to examine reciprocal relations. Psychological Methods, 24(4), 637-657. DOI: 10.1037/met0000210
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- Hamaker, E. L., Kuiper, R. M., & Grasman, R. P. P. P. (2015). A critique of the cross-lagged panel model. Psychological Methods, 20(1), 102-116. DOI: 10.1037/a0038889
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- Hox, J. & Lensvelt-Mulders, G. (2004).
Randomized response analysis in Mplus.
Structural Equation Modeling, 11, 615-620.
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Abstract
"This article describes a technique to analyze randomized response data using available structural equation
modeling (SEM) software. The randomized response technique was developed to obtain estimates
that are more valid when studying sensitive topics. The basic feature of all randomized response methods
is that the data are deliberately contaminated with error. This makes it difficult to relate
randomized responses to explanatory variables. In this tutorial, we present an approach to this problem,
in which the analysis of randomized response data is viewed as a latent class problem, with
different latent classes for the random and the truthful response. To illustrate this technique, an
example is presented using the program Mplus."
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- Eid, M., Geiser, C. & Koch, T. Structural Equation Modeling of Multiple Rater Data, New York, Guilford Press, 2024.
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supplemental webpage
- Tseng, M. (2024). Latent interaction effect in the CLPM model: A two-step multiple imputation analysis. Structural Equation Modeling: A Multidisciplinary Journal, 1–10. DOI: 10.1080/10705511.2024.2374349
view abstract
- Asparouhov, T. & Muthén, B. (2024). Penalized structural equation models. Structural Equation Modeling: A Multidisciplinary Journal, 31(3), 429–454. DOI: 10.1080/10705511.2023.2263913.
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Abstract
Penalized structural equation models (PSEM) is a new powerful estimation technique that can be used to tackle a variety of difficult structural estimation problems that can not be handled with previously developed methods. In this paper we describe the PSEM framework and illustrate the quality of the method with simulation studies. Maximum-likelihood and weighted least squares PSEM estimation is discussed for SEM models with continuous and categorical variables. We show that traditional EFA, multiple group alignment (MGA), and Bayesian SEM (BSEM) are examples of PSEM. The PSEM framework also extends standard SEM models with the possibility to structurally align various model parameters. Exploratory latent growth models, also referred to as Tuckerized curve models, can also be estimated in the PSEM framework and are illustrated here with simulation studies and an empirical example.
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- Asparouhov, T. & Muthén, B. (2023). Residual Structural Equation Models. Structural Equation Modeling: A Multidisciplinary Journal, 30, 1-31. DOI: 10.1080/10705511.2022.2074422
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- Nye, C. (2022). Reviewer Resources: Confirmatory Factor Analysis. Organizational Research Methods. DOI: 10.1177/10944281221120541
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- Pieters, C., Pieters, R. & Lemmens, A. (2022). Six methods for latent moderation analysis in marketing research: A comparison and guidelines. Journal of Marketing Research, DOI: 10.1177/00222437221077266.
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- Asparouhov, T. & Muthén, B. (2021). Bayesian estimation of single and multilevel models with latent variable interactions, Structural Equation Modeling: A Multidisciplinary Journal, 28:2, 314-328, DOI: 10.1080/10705511.2020.1761808
(*NOTE: Scripts refer to section numbers from the Mplus Web Note 23 version of this paper that are not present in the current version.)
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Abstract
In this article we discuss single and multilevel SEM models with latent variable interactions. We describe the Bayesian estimation for these models and show through simulation studies that the Bayesian
method outperforms other methods such as the maximum-likelihood method. We show that multilevel moderation models can easily be estimated with the Bayesian method.
hide abstract
- Feingold, A. (2019). New approaches for estimation of effect sizes and their con1dence intervals for treatment effects from randomized controlled trials. The Quantitative Methods for Psychology, 15:2, 96-111. DOI: 10.20982/tqmp.15.2.p096
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- Breitsohl, H. (2019). Beyond ANOVA: An Introduction to Structural Equation Models for Experimental Designs. Organizational Research Methods, 22(3), 649-677. DOI: 10.1177/1094428118754988
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- Zyphur, M. J., Voelkle, M. C., Tay, L., Allison, P. D., Preacher, K. J., Zhang, Z., Hamaker, E. L., Shamsollahi, A., Pierides, D. C., Koval, P., & Diener, E. (2019). From data to causes II: Comparing approaches to panel data analysis. Organizational Research Methods. DOI: 10.1177/1094428119847280
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Abstract
This paper compares a general cross-lagged model (GCLM) to other panel data methods based
on their coherence with a causal logic and pragmatic concerns regarding modeled dynamics and
hypothesis testing. We examine three ‘static’ models that do not incorporate temporal dynamics:
random- and fixed-effects models that estimate contemporaneous relationships; and latent curve
models. We then describe ‘dynamic’ models that incorporate temporal dynamics in the form of
lagged effects: cross-lagged models estimated in a structural equation model (SEM) or multilevel
model (MLM) framework; Arellano-Bond dynamic panel data methods; and autoregressive
latent trajectory models. We describe the implications of overlooking temporal dynamics in
static models and show how even popular cross-lagged models fail to control for stable factors
over time. We also show that Arellano-Bond and autoregressive latent trajectory models have
various shortcomings. By contrasting these approaches, we clarify the benefits and drawbacks of
common methods for modeling panel data, including the GCLM approach we propose. We
conclude with a discussion of issues regarding causal inference, including difficulties in
separating different types of time-invariant and time-varying effects over time.
hide abstract
- Zyphur, M. J., Allison, P. D., Tay, L., Voelkle, M. C., Preacher, K. J., Zhang, Z., Hamaker, E. L., Shamsollahi, A., Pierides, D. C., Koval, P., & Diener, E. (2019). From data to causes I: Building a general cross-lagged panel model (GCLM). Organizational Research Methods. DOI: 10.1177/1094428119847278
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Abstract
This is the first paper in a series of two that synthesizes, compares, and extends methods for
causal inference with longitudinal panel data in a structural equation modeling (SEM)
framework. Starting with a cross-lagged approach, this paper builds a General Cross-Lagged
Panel Model (GCLM) with parameters to account for stable factors while increasing the
range of dynamic processes that can be modeled. We illustrate the GCLM by examining the
relationship between national income and subjective well-being (SWB), showing how to
examine hypotheses about short-run (via Granger-Sims tests) versus long-run effects (via
impulse responses). When controlling for stable factors, we find no short-run or long-run
effects among these variables, showing national SWB to be relatively stable whereas income
is less so. Our second paper addresses the differences between the GCLM and other methods.
Online supplemental materials offer an Excel file automating GCLM input for Mplus (with
an example also for Lavaan in R), and analyses using additional datasets and all program
input/output. We also offer an introductory GCLM presentation at
https://youtu.be/tHnnaRNPbXs. We conclude with a discussion of issues surrounding causal inference.
hide abstract
- Leite, W.L., Stapleton, L.M., & Bettini, E.F. (2018). Propensity score analysis of complex survey data with structural equation modeling: a tutorial with Mplus, Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2018.1522591
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- Asparouhov, T. & Muthén, B. (2019). Nesting and equivalence testing for Structural Equation Models. Structural Equation Modeling: A Multidisciplinary Journal, 26:2, 302-309, DOI: 10.1080/10705511.2018.1513795
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Abstract
In this article, we discuss the nesting and equivalence testing (NET) methodology developed
in Bentler and Satorra (2010). We describe how the methodology is implemented in Mplus for
the general structural equation model (SEM) model with continuous variables based on the
maximum-likelihood (ML) estimation as well as the general SEM model with categorical,
censored and continuous dependent variables based on the weighted least squares (WLS)
family of estimators. We use the NET methodology to address several model nesting questions
that arise in the bi-factor CFA model and the multiple group factor analysis model.
hide abstract
- Asparouhov, T. & Muthén, B. (2018). SRMR in Mplus. Technical Report. May 2, 2018.
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Abstract
In this note we describe the Mplus implementation of the SRMR (standardized
root mean squared residual) fit index for the models where the index
is computed. Starting with Mplus 8.1, changes have been implemented to
improve the quality of the fit index and to prevent failures. The index
is now computed for more models including SEM models with categorical
data estimated with the weighted least squares family of estimators:
wls/wlsm/wlsmv/ulsmv. We also discuss how the Mplus SRMR fit index
should be used in practice.
hide abstract
- Helm, J.L., Castro-Schilo, L., & Oravecz, Z. (2017). Bayesian versus maximum likelihood estimation of multitrait-multimethod confirmatory factor models. Structural Equation Modeling: A Multidisciplinary Journal, 24:1, 17-30. DOI: 10.1080/10705511.2016.1236261
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- Marsh, H. W., Guo, J., Nagengast, B., Parker, P. D., Asparouhov, T., Muthén, B., & Dicke, T. (2018). What to do when scalar invariance fails: The extended alignment method for multigroup factor analysis comparison of latent means across many groups. Psychological Methods, 23(3), 524-545. DOI: 10.1037/met0000113
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Abstract
Scalar invariance is an unachievable ideal that in practice can only be approximated; often using
potentially questionable approaches such as partial invariance based on a stepwise selection of
parameter estimates with large modification indices. Study 1 demonstrates an extension of the power
and flexibility of the alignment approach for comparing latent factor means in a large-scale
studies (30 OECD countries, 8 factors, 44 items and N = 249,840), for which scalar invariance is
typically not supported in the traditional confirmatory factor analysis approach to measurement
invariance (CFA-MI). Importantly, we introduce an alignment-within-CFA (AwC) approach, transforming
alignment from a largely exploratory tool into a confirmatory tool, and enabling analyses that
previously have not been possible with alignment (testing the invariance of uniquenesses and factor
variances/covariances; multiple-group MIMIC models; contrasts on latent means) and structural
equation models more generally. Specifically, it also allowed a comparison of gender difference in
a 30-country MIMIC AwC (i.e., a SEM with gender as a covariate) and a 60-group AwC CFA (i.e., 30
countries x 2 genders) analysis. Study 2, a simulation study following up issues raised in Study 1,
showed that latent means were more accurately estimated with alignment than with the scalar CFA-
MI, and particularly with partial invariance scalar models based on the heavily criticized stepwise
selection strategy. In summary, alignment augmented by AwC provides applied researchers from
diverse disciplines considerable flexibility to address substantively important issues when the
traditional CFA-MI scalar model does not fit the data.
hide abstract
- Muthén, B. & Asparouhov, T. (2016). Recent methods for the study of measurement invariance with many groups: Alignment and random effects.
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Abstract
This paper reviews and compares recently proposed factor analytic and item response theory (IRT) approaches to the study of invariance across groups. Two methods are
described and contrasted. The alignment method considers the groups as a fixed mode of variation, while the random-intercept, random-loading two-level method considers the groups as a random mode
of variation. Both maximum-likelihood and Bayesian analysis is applied. A survey of close to 50,000 subjects in 26 countries is used as an illustration. In addition, the two methods are studied by
Monte Carlo simulations. A list of considerations for choosing between the two methods is presented.
hide abstract
- Bauer, D. J. (2016). A more general model for testing measurement invariance and differential item functioning. Psychological Methods. DOI: 10.1037/met0000077
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- Raykov, T., Marcoulides, G. A. & Tong, B. (2015). Do two or more multicomponent instruments measure the same construct? Testing construct congruence using latent variable modeling.
Educational and Psychological Measurement. DOI: 10.1177/0013164415604705
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- Jahanshahi, K. Jin, Y. & Williams, I. (2015). Direct and indirect influences on employed adults’ travel in the UK: New insights from the National Travel Survey data 2002–2010.
Transportation Research Part A: Policy and Practice, 80, 288-306. DOI: 10.1016/j.tra.2015.08.007
view abstract
- Raykov, T., & Marcoulides, G. A. (2015). Scale reliability evaluation under multiple assumption
violations, Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2014.938597
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- Asparouhov, T. & Muthén B. (2015). Structural equation models and mixture models with continuous non-normal skewed distributions. Structural Equation Modeling: A Multidisciplinary Journal, DOI:
10.1080/10705511.2014.947375.
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Abstract
In this paper we describe a structural equation modeling framework that allows
non-normal skewed distributions for the continuous observed and latent variables. This framework
is based on the multivariate restricted skew t-distribution. We demonstrate the advantages of
skewed structural equation modeling over standard SEM modeling and challenge the notion that
structural equation models should be based only on sample means and covariances. The skewed
continuous distributions are also very useful in finite mixture modeling as they prevent the
formation of spurious classes formed purely to compensate for deviations in the distributions from
the standard bell curve distribution. This framework is implemented in Mplus Version 7.2.
hide abstract
- Kelava, A., Nagengast, B., Brandt, H. (2014). A nonlinear structural equation mixture modeling approach for
nonnormally distributed latent predictor variables. Structural Equation Modeling: A Multidisciplinary
Journal, 21:3, 468-481, DOI: 10.1080/10705511.2014.915379
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- Asparouhov T. & Muthén, B. (2014). Multiple-group factor analysis alignment. Structural Equation Modeling: A Multidisciplinary Journal, 21:4, 495-508. DOI: 10.1080/10705511.2014.919210
An earlier version of this paper was posted as web note 18.
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Abstract
This article presents a new method for multiple-group confirmatory factor analysis (CFA),
referred to as the alignment method. The alignment method can be used to estimate
group-specific factor means and variances without requiring exact measurement invariance. A
strength of the method is the ability to conveniently estimate models for many groups. The method
is a valuable alternative to the currently used multiple-group CFA methods for studying measurement
invariance that require multiple manual model adjustments guided by modifica- tion indexes.
Multiple-group CFA is not practical with many groups due to poor model fit of the scalar model and
too many large modification indexes. In contrast, the alignment method is based on the configural
model and essentially automates and greatly simplifies measurement invariance analysis. The method
also provides a detailed account of parameter invariance for
every model parameter in every group.
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- Morin, A.J.S., Moullec, G., Maïano, C., Layet, L., Just, J.L. & Ninot, G. (2011). Psychometric properties of the Center for Epidemiologic Studies Depression Scale (CES-D) in French clinical and non-clinical adults. Epidemiology and Public Health/Revue d’Épidémiologie et de Santé Publique, 59(5):327-40. DOI: 10.1016/j.respe.2011.03.061.
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Abstract
"Background: Previous research on the Center for Epidemiologic Studies Depression Scale (CESD) has five main limitations. First, no study provided evidence of the factorial equivalence of this instrument across samples of depressive and community participants. This is intriguing regarding
that the CES-D was specifically designed to identify clinical depression in epidemiological community adults. Second, only one study relied on systematic tests of measurement invariance as implemented within confirmatory factor analyses (CFA) and this study did not consider the higher
order depression structure, although it is the CES-D global scale score that is most often used in the context of epidemiological studies. It thus remains unknown whether the commonly recognized gender differences in depression could be related or not to measurement biases. Third,
few studies investigated the screening properties of the CES-D in non-English samples and their results have been inconsistent. Fourth, although the French version of the CES-D has previously been used in several studies, it has never been systematically validated among community and/or
depressed adults. Finally, very few studies took into account the ordered-categorical nature of the CES-D answer scale. The purpose of this study was thus to examine the construct validity (i.e. factorial, reliability; measurement invariance; latent mean invariance; convergent; screening
properties) of the CES-D in a French sample of depressed patients and community adults. Methods: A total sample of 469 participants, comprising 163 clinically depressed patients and 306 community adults, was involved in this study. The factorial validity and the measurement and
latent mean invariance of the CES-D, across gender and clinical status, were verified through CFAs based on ordered-categorical items. Correlation and receiver operator characteristic curves were also used to test the convergent validity and screening properties of the CES-D.
Results: The present results (i) provided support for the factor validity and reliability of a secondorder measurement model of depression based on the CES-D items; (ii) revealed the full measurement invariance of the first- and second-order measurement models across gender; (iii)
showed the partial strict measurement invariance (four uniquenesses had to be freely estimated, but the factor variances-covariances matrix also proved fully invariant) of the first-order factor model and the complete measurement invariance of the second-order model across patients and
community adults; (iv) revealed a lack of latent mean invariance across gender and across clinical and community subsamples (with women and patients reporting higher scores on all subscales and on the full scale); (v) confirmed the convergent validity of the CES-D with measures of
depression, self-esteem, anxiety and hopelessness; and (vi) demonstrated the efficacy of the screening properties of this instrument among clinical and non-clinical adults. Conclusion: This instrument may be useful for the assessment of depressive symptoms or for the
screening depressive disorders in the context of epidemiological studies targeting French patients and community men and women with a background similar to those from the present study."
hide abstract
- Schmitt, T. (2011). (2011).
Current methodological considerations in exploratory and confirmatory factor analysis.
Journal of Psychoeducational Assessment, 29, 304-321.
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Abstract
"Researchers must make numerous choices when conducting factor analyses, each of which can
have significant
ramifications on the model results. They must decide on an appropriate sample
size to achieve
accurate parameter estimates and adequate power, a factor model and estimation
method, a method for
determining the number of factors and evaluating model fit, and a rotation
criterion. Unfortunately,
researchers continue to use outdated methods in each of these
areas. The present article provides a
current overview of these areas in an effort to provide
researchers with up-to-date methods and considerations
in both exploratory and confirmatory
factor analysis. A demonstration was provided to illustrate
current approaches. Choosing
between confirmatory and exploratory methods is also discussed,
as researchers often make
incorrect assumptions about the application of each."
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- Bou, J.C. & Satorra, A. (2010).
A multigroup structural equation approach: A demonstration by testing variation of firm profitability
across EU samples.
Organizational Research Methods.
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Abstract
"We extend standard methodology for multigroup mean and covariance structure (MACS) analysis
to the case
where assessment of across-group variation of model parameters is the focus of the
study and the
data deviate from standard assumptions. The proposed methods are applied to analyze
an accounting profitability
database covering more than 100,000 firms in the 15 European Union
(EU) countries in the
period 1999 to 2003. A multivariate model with permanent and dynamic latent
components of profitability
is used to assess across-country variation of firm level profitability and
persistence. We show
that there are substantial differences among these countries, despite the partial
integration of their
economies. Estimation of supplementary parameters are proposed as a way
to characterize persistence
in each country, as well as across-group variation of model parameters.
This methodology is more
widely applicable in international organizational research."
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- Hayes, A.F. & Preacher, K.J. (2010).
Quantifying and testing indirect effects in simple mediation models when the constituent paths are nonlinear.
Multivariate Behavioral Research, 45, 627-660.
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Abstract
"Most treatments of indirect effects and mediation in the statistical methods liter-
ature and the corresponding
methods used by behavioral scientists have assumed
linear relationships between variables
in the causal system. Here we describe and
extend a method first introduced by Stolzenberg (1980) for
estimating indirect
effects in models of mediators and outcomes that are nonlinear functions but
linear
in their parameters. We introduce the concept of the instantaneous indirect effect
of X on Y
through M and illustrate its computation and describe a bootstrapping
procedure for inference.Mplus
code as well as SPSS and SAS macros are provided
to facilitate the adoption of this approach and ease
the computational burden on
the researcher."
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- Sass, D.A. & Schmitt, T.A. (2010).
A comparative investigation of rotation criteria within exploratory factor analysis.
Multivariate Behavioral Research, 45, 73-103.
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Abstract
"Exploratory factor analysis (EFA) is a commonly used statistical technique for
examining the relationships
between variables (e.g., items) and the factors (e.g.,
latent traits) they depict. There are several
decisions that must be made when using
EFA, with one of the more important being choice of the
rotation criterion. This selection
can be arduous given the numerous rotation criteria available and
the lack of
research/literature that compares their function and utility. Historically, researchers
have
chosen rotation criteria based on whether or not factors are correlated and
have failed to consider
other important aspects of their data. This study reviews
several rotation criteria, demonstrates
how they may perform with different factor
pattern structures, and highlights for researchers
subtle but important differences
between each rotation criterion. The choice of rotation criterion is
critical to ensure
researchers make informed decisions as to when different rotation criteria may
or
may not be appropriate. The results suggest that depending on the rotation
criterion selected and
the complexity of the factor pattern matrix, the interpretation
of the interfactor correlations and
factor pattern loadings can vary substantially.
Implications and future directions are discussed."
hide abstract
- van de Schoot, R., Hoijtink, H. & Dekovic, M. (2010). Testing inequality constrained hypotheses in SEM models. Structural Equation Modeling, 17, 443-463.
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Abstract
"Researchers often have expectations that can be expressed in the form of inequality constraints
among
the parameters of a structural equation model. It is currently not possible to test these
so-called
informative hypotheses in structural equation modeling software. We offer a solution to
this problem
using Mplus. The hypotheses are evaluated using plug-in p values with a calibrated
alpha level. The
method is introduced and its utility is illustrated by means of an example."
hide abstract
- Moosbrugger, H., Schermelleh-Engel, K., Kelava; A. & Klein, A. G (2009).
Testing multiple nonlinear effects in structural equation modeling: A comparison of alternative estimation
approaches. Invited Chapter in T. Teo & M. S. Khine (Eds.), Structural Equation Modelling in Educational Research: Concepts and Applications. Rotterdam, NL: Sense Publishers.
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- Tucker-Drob, E.M. (2009). Differentiation of cognitive abilities across the lifespan. Developmental Psychology, 45(4), 1097-1118. DOI: 10.1037/a0015864
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Abstract
"Existing representations of cognitive ability structure are exclusively based on linear patterns of
interrelations.
However, a number of developmental and cognitive theories predict that abilities
are
differentially related across ages (age differentiation-dedifferentiation) and across levels of
functioning
(ability differentiation). Nonlinear factor analytic models were applied to
multivariate cognitive
ability data from 6,273 individuals, ages 4 to 101 years, who were selected
to be nationally
representative of the United States population. Results consistently supported
ability differentiation,
but were less clear with respect to age differentiation-dedifferentiation.
Little evidence for age
modification of ability differentiation was found. These findings are
particularly informative about
the nature of individual differences in cognition, and the
developmental course of cognitive ability
level and structure."
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- Eid, M., Nussbeck, F., Geiser, C., Cole, D., Gollwitzer, M. & Lischetzke, T. (2008). Structural equation modeling of multitrait-multimethod data: Different models for different types of methods. Psychological Methods, 13, 230-253.
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- Geiser, C., Eid, M. & Nussbeck, F. W. (2008). On the meaning of the latent variables in the CT-C(M–1) model: A comment on Maydeu-Olivares & Coffman (2006). Psychological Methods, 13, 49-57.
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- Cheung, M.W.L. (2008).
A model for integrating fixed-, random-, and mixed-effects meta-analyses into structural equation modeling.
Psychological Methods, 13, 182–202.
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Abstract
"Meta-analysis and structural equation modeling (SEM) are two important statistical methods
in the behavioral,
social, and medical sciences. They are generally treated as two unrelated
topics in the literature.
The present paper proposes a model to integrate fixed-, random-, and
mixed-effects meta-analyses
into the SEM framework. By applying an appropriate
transformation on the data, studies in a meta-analysis
can be analyzed as subjects in a
structural equation model. This paper also highlights some
practical benefits of using the
SEM approach to conducting a meta-analysis. Specifically, the SEM
based meta-analysis can
be used to handle missing covariates, to quantify the heterogeneity of effect
sizes, and to
address the heterogeneity of effect sizes with mixture models. Examples are used to
illustrate
the equivalence between the conventional meta-analysis and the SEM based meta-analysis.
Future
directions on and issues related to the SEM based meta-analysis are discussed.
Keywords: Meta-analysis,
structural equation model, fixed-effects model, random-effects
model, mixed-effects model"
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- Maydeu-Olivares, A., & Coffman, D. L. (2006). Random intercept item factor analysis. Psychological Methods, 11, 344 –362.
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- Coenders, G., Batista-Foguet, J.M. & Saris, W. (2006).
Simple, efficient and distribution-free approach to interaction effects in complex structural equation
models.
Quality & Quantity, 42, 369-396.
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Abstract
"Structural equation models with mean structure and non-linear constraints are the
most frequent choice
for estimating interaction effects when measurement errors are present.
This article proposes eliminating
the mean structure and all the constraints but one, which
leads to a more easily handled model
that is more robust to non-normality and more general
as it can accommodate endogenous interactions
and thus indirect effects. Our approach
is compared to other approaches found in the literature with
a Monte Carlo simulation and
is found to be equally efficient under normality and less biased under
non-normality. An
empirical illustration is included."
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- Muthén, L.K. & Muthén, B.O. (2002).
How to use a Monte Carlo study to decide on sample size and determine power.
Structural Equation Modeling, 4, 599-620.
Mplus inputs and outputs used in this paper can be viewed and/or downloaded from the Examples
page.
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Abstract
"A common question asked by researchers is, 'What sample size do I need for my study?' Over the years,
several rules of thumb have been proposed. In reality there is no rule of thumb that applies to all
situations. The sample size needed for a study depends on many factors including the size of the
model, distribution of the variables, amount of missing data, reliability of the variables, and strength
of the relationships among the variables. The purpose of this paper is to demonstrate how substantive
researchers can use a Monte Carlo study to decide on sample size and determine power. Two
models ae used as examples, a confirmatory factor analysis (CFA) model and a growth model. The analyses
are carried out using the Mplus program (Muthén & Muthén, 1998)."
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- Yu, C.Y. (2002).
Evaluating cutoff criteria of model fit indices for latent variable models with binary and continuous
outcomes.
Doctoral dissertation, University of California, Los Angeles.
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Abstract
"The aims of this study are to first evaluate the performance of various model fit measures under different
model and data conditions, and secondly, to examine the adequacy of cutoff criteria for some model
fit measures. Model fit indices, along with some test statistics, are meant to assess model fit
in latent variable models. They are frequently applied to judge whether the model of interest is
a good fit to the data. Since Bentler and Bonett (1980) popularized the concept of model fit indices,
numerous studies have been done to propose new fit indices or to compare various fit indices. Most
of the studies, however, are limited to continuous outcomes and to measurement models, such as confirmatory
factor analysis models (CFA). The present study broadens the structure of models by including
the multiple causes and multiple indicators (MIMIC) and latent growth curve models. Moreover,
both binary and continuous outcomes are investigated in the CFA and MIMIC models. Weighted root-mean-square
residual (WRMR), a new fit index, is empirically evaluated and compared to the Tucker-Lewis
Index (TLI), the Comparative Fit Index (CFI), the root-mean-square error of approximation (RMSEA)
and the standardized root-mean square residual (SRMR). Few studies have investigated the adequacy of
cutoff criteria for fit indices. Thus study applied the method demonstrated in Hu and Bentler (1999)
to evaluate the adequacy of cutoff criteria for the fit indices. The adequacy of a conventional
probability level of 0.05 for chi-square to assess model fit is also investigated. With non-normal
continuous outcomes, the Satorra-Bentler rescaled chi-square (SB) is incorporated into the calculation
of TLI, CFI and RMSEA, and these SB-based fit measures are evaluated under various cutoff values.
An example of applying adequate cutoff values of overall fit indices is illustrated using the Holzinger
and Swineford data. Generally speaking, the use of SRMR with binary outcomes is not recommended.
A cutoff value close to 1.0 for WRMR is suitable under most conditions but is not recommended
for latent growth curve models with more time points. CFI performs relatively better than TLI and RMSEA,
and a cutoff value close to 0.96 for CFI has acceptable rejection rates across models when N is
greater than or equal to 250."
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- Muthén, B., du Toit, S.H.C., & Spisic, D. (1997).
Robust inference using weighted least squares and quadratic estimating equations in latent variable modeling
with categorical and continuous outcomes.
Unpublished technical report.
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Abstract
"This paper generalizes the robust weighted least-squares (WLS) approach of Muthén (1993) beyond
the binary factor analysis model to the general structural equation model considered in Muthén
(1984). A key feature in this generalization is the addition of covariates by which the means of
the outcome variables can vary across the individuals of the sample. The paper relates the robust
WLS approach to a generalized estimating equation (GEE) approach recently proposed by Melton and Liang
(1997) both with respect to statistical performance and computational speed. It is shown that except
for small sample sizes and strongly skewed distributions, the robust WLS approach performs statistically
almost as well as GEE, produces good standard error estimates, but gives considerably faster
computations. While in the Melton and Liang (1997) GEE context model testing is not straight-forward
and was not provided, robust chi-square model testing is easily obtained in the WLS approach.
As in Muthén (1984), the robust WLS approach is quite general in that it allows for a combination
of binary, ordered polytomous, and continuous outcome variables and allows for multiple-group
analysis.
"
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- Muthén, B. & Satorra, A. (1995).
Technical aspects of Muthén's LISCOMP approach to estimation of latent variable relations with
a comprehensive measurement model.
Psychometrika, 60, 489-503.
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Abstract
"Muthén (1984) formulated a general model and estimation procedure for structural equation modeling
with a mixture of dichotomous, ordered categorical, and continuous measures of latent variables.
A general three-stage procedure was developed to obtain estimates, standard errors, and a chi-square
measure of fit for a given structural model. While the last step uses generalized least-squares
estimation to fit a structural mode, the first two steps involve the computation of the statistics
used in this model fitting. A key component in the procedure was the development of a GLS weight matrix
corresponding to the asymptotic covariance matrix of the sample statistics computed in the first
two stages. This paper extends the description of the asymptotics involved and shows how the Muthén
formulas can be derived. The emphasis is placed on showing the asymptotic normality of the
estimates obtained in the first and second stage and the validity of the weight matrix used in the
GLS estimation of the third stage."
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- Muthén, B. (1989).
Tobit factor analysis.
British Journal of Mathematical and Statistical Psychology, 42, 241-250.
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Abstract
"A new approach is proposed for data which are skewed and have a sizeable
proportion of observation at
variable end points. Using a covariance
structure modelling lramework, the new approach assumes censored
multivariate
normal variables. Using bivariate information, this leads to the
use of 'tobit' correlations
in weighted least squares estimation. The
behaviour of the tobit approach is compared to
that of normal theory
estimation and ADF estimation."
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- Muthén, B. (1989).
Latent variable modeling in heterogeneous populations.
Psychometrika, 54:4, 557-585.
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Abstract
"Common applications of latent variable analysis fail to recognize that data may be obtained
from several
populations with different sets of parameter values. This article describes the
problem and gives
an overview of methodology that can address heterogeneity. Artificial examples
of mixtures are given,
where if the mixture is not recognized, strongly distorted results
occur. MIMIC structural modeling
is shown to be a useful method for detecting and describing
heterogeneity that cannot be handled
in regular multiple-group analysis. Other useful methods
instead take a random effects approach, describing
heterogeneity in terms of random parameter
variation across groups. These random effects models
connect with emerging methodology for
multilevel structural equation modeling of hierarchical data.
Examples are drawn from educational
achievement testing, psychopathology, and sociology of education.
Estimation is carded
out by the LISCOMP program."
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- Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49, 115-132.
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Abstract
"A structural equation model is proposed with a generalized measurement part, allowing for dichotomous
and ordered categorical variables (indicators) in addition to continuous ones. A
computationally feasible three-stage estimator is proposed for any combination of observed variable
types. This approach provides large-sample chi-square tests of fit and standard errors of
estimates for situations not previously covered. Two multiple-indicator modeling examples are
given. One is a simultaneous analysis of two groups with a structural equation model underlying
skewed Likert variables. The second is a longitudinal model with a structural model for multivariate
probit regressions."
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- Muthén, B. (1983).
Latent variable structural equation modeling with categorical data.
Journal of Econometrics, 22, 43-65.
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Abstract
"Structural equation modeling with latent variables is overviewed for situations involving a mixture of
dichotomous, ordered polytomous, and continuous indicators of latent variables.
Special emphasis is
placed on categorical variables. Models in psychometrics, econometrics and
biometrics are interrelated
via a general model due to Muthén. Limited information least
squares estimators and full information
estimation are discussed. An example is estimated with a
model for a four-wave longitudinal data
set, where dichotomous responses are related to each
other and a set of independent variables via
latent variables with a variance component
structure."
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- Muthén, B. (1979).
A structural Probit model with latent variables.
Journal of the American Statistical Association, 74, 807-811.
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Abstract
A model with dichotomous indicators of latent variables is developed. The latent variables are related to each other and to a set of exogenous variables in a system of structural relations. Identification and maximum likelihood estimation of the model are treated. A sociological
application is presented in which a theoretical construct (an attitude) is related to a set of background variables. The construct is not measured directly, but is indicated by the answers to a pair of questionairre statements.
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- Wheaton, B., Muthén, B., Alwin, D., & Summers, G. (1977). Assessing reliability and stability in panel models. In D. R. Heise (Ed.), Sociological Methodology 1977 (pp. 84-136). San Francisco: Jossey-Bass, Inc.
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- Asparouhov, T. & Muthén, B. (2018). Continuous - Time Survival Analysis in Mplus. Version 3. June 29, 2018.
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Abstract
Here we will describe the basic continuous time survival model implemented in Mplus and will provide some details on the basic modeling options that are available. Introduction to continuous time survival modeling can be found in Singer & Willett (2003), Hougaard (2000) or Klein & Moeschberger (1997). Survival analysis: techniques for. The survival models implemented in Mplus
includes many extensions of this basic model such as mixture survival models, survival models with random eects (frailty models), multilevel survival models, time varying covariate models, competing risk models, non-proportional hazard models etc. Describing the details of these models is beyond the scope of this document. In most cases however the material presented here applies to these extensions as well. More details on the models and algorithms implemented in Mplus can be found in Larsen (2004, 2005) and Asparouhov, Masyn & Muthén (2006). Practical applications of the Mplus methodology for continuous time survival modeling can be found in Muthén et al. (2009).
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- Raykov, T., Gorelick, P.B., Zajacova, A., & Marcoulides, G.A. (2017). On the potential of discrete time survival analysis using latent variable modeling: An application to the study of the vascular depression hypothesis. Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2017.1315305
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- Peugh, J. & Fan, X. (2016). Identifying unobserved hazard functions in discrete-time survival mixture analysis, Structural Equation Modeling: A Multidisciplinary Journal, DOI: 10.1080/10705511.2016.1242372
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- Using multivariate multilevel survival analysis to study reliability and change in hazard rates of emotions derived from parent-child dyadic social interaction. Tom Dishion and Jim Snyder (Eds.), Handbook of Coercion.
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- Stoolmiller, M. and Snyder, J. (2013).
Embedding multilevel survival analysis of dyadic social interaction in structural equation models: Hazard rates as both outcomes and predictors.
Journal of Pediatric Psychology, DOI: 10.1093/jpepsy/jst076.
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Abstract
"Objective Demonstrate multivariate multilevel survival analysis within a larger structural equation model. Test the 3 hypotheses that when confronted
by a negative parent, child rates of angry, sad/fearful, and positive emotion will increase, decrease, and stay the same, respectively, for antisocial
compared with normal children. This same pattern will predict increases in future antisocial behavior. Methods Parent–child dyads were videotaped in
the fall of kindergarten in the laboratory and antisocial behavior ratings were obtained in the fall of kindergarten and third grade. Results
Kindergarten antisocial predicted less child sad/fear and child positive but did not predict child anger given parent negative. Less child positive and
more child neutral given parent negative predicted increases in third-grade antisocial behavior. Conclusions The model is a useful analytic tool for
studying rates of social behavior. Lack of positive affect or excess neutral affect may be a new risk factor for child antisocial behavior."
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- Yampolskaya, S., Armstrong, M. I., & King-Miller, T. (2011). Contextual and individual-level predictors of abused children’s reentry into out-of-home care: A multilevel mixture
survival analysis. Child Abuse & Neglect, 35, 670-679. doi:10.1016/j.chiabu.2011.05.005
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- Petras, H., Masyn, K., Buckley, J., Ialongo, N. & Kellam, S. (2010). Who is most at risk for school removal? A multilevel discrete-time survival analysis of individual and contextual-level influences.
Journal of Educational Psychology, 45(2): 171–191. DOI: 10.17105/SPR45-2.171-191
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Abstract
"The focus of this study was to prospectively investigate the effect of aggressive behavior and of classroom
behavioral context, as measured in fall of first grade on the timing of first school removal
across grades 1–7 in a sample of predominately urban minority youth from Baltimore, Maryland. Using
a multilevel discrete-time survival framework, we found that demographic characteristics of the students
as well as early individual and classroom level of aggression contribute to the onset of school
removal. Although early individual aggression was positively associated with the risk of school removal,
initially higher levels of classroom aggression corresponded to lower risk of school removal."
hide abstract
- Masyn, K. E. (2009).
Discrete-time survival factor mixture analysis for low-frequency recurrent event histories.
Research in Human Development, 6, 165-194.
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Abstract
"In this article, the latent class analysis framework for modeling single event
discrete-time survival
data is extended to low-frequency recurrent event histories.
A partial gap time model, parameterized
as a restricted factor mixture model, is
presented and illustrated using juvenile offending data.
This model accommodates
event-specific baseline hazard probabilities and covariate effects; event recurrences
within
a single time period; and accounts for within- and between-subject correlations
of
event times. This approach expands the family of latent variable survival
models in a way that allows
researchers to explicitly address questions about unobserved
heterogeneity in the timing of events
across the lifespan."
hide abstract
- Muthén, B., Asparouhov, T., Boye, M., Hackshaw, M. & Naegeli, A. (2009).
Applications of continuous-time survival in latent variable models for the analysis of oncology randomized
clinical trial data using Mplus. Technical Report.
Click here to view Mplus outputs used in this paper.
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- Masyn, K. E. (2008).
Modeling measurement error in event occurrence for single, non-recurring events in discrete-time survival
analysis.
In Hancock, G. R., & Samuelsen, K. M. (Eds.), Advances in latent variable mixture models, pp. 105-145. Charlotte, NC: Information Age Publishing, Inc.
Click here for information
about the book.
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- Asparouhov, T., Masyn, K. & Muthén, B. (2006).
Continuous time survival in latent variable models.
Proceedings of the Joint Statistical Meeting in Seattle, August 2006. ASA section on Biometrics, 180-187. Click here to download the files associated with this paper.
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Abstract
"We describe a general multivariate, multilevel framework for continuous time survival analysis that includes
joint
modeling of survival time variables and continuous and categorical observed and latent
variables. The proposed
framework is implemented in the Mplus software package. The survival time variables
are modeled with nonparametric
or parametric proportional hazard distributions and include
right censoring. The proposed modeling
framework includes finite mixtures of Cox regression models with
and without class-specific baseline hazards,
multilevel Cox regression models, and multilevel frailty
models. We illustrate the framework with several simulation
studies. Comparison is made with discrete
time survival models. We also investigate the effect of ties
on the proposed estimation method.
Simulation studies are conducted to compare the methods implemented in
Mplus with those implemented
in SAS."
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- Muthén, B. & Masyn, K. (2005).
Discrete-time survival mixture analysis. Journal of Educational and Behavioral Statistics, 30, 27-58.
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Abstract
"This article proposes a general latent variable approach to discrete-time survival analysis of nonrepeatable
events such as onset of drug use. It is shown how the survival analysis can be formulated as
a generalized latent class analysis of event history indicators. The latent class analysis can use
covariates and can be combined with the joint modeling of other outcomes such as repeated measures for
a related process. It is shown that conventional discrete-time survival analysis corresponds to a
single-class latent class analysis. Multiple-class extensions are proposed, including the special cases
of a class of long-term survivors and classes defined by outcomes related to survival. The estimation
uses a general latent variable framework, including both categorical and continuous latent variables
and incorporated in the Mplus program. Estimation is carried out using maximum likelihood via
the EM algorithm. Two examples serve as illustrations. The first example concerns recidivism after
incarceration in a randomized field experiment. The second example concerns school removal related
to the development of aggressive behavior in the classroom."
hide abstract
- Masyn, K. E. (2003).
Discrete-time survival mixture analysis for single and recurrent events using latent variables.
Doctoral dissertation, University of California, Los Angeles.
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Abstract
"Survival analysis refers to the general set of statistical methods developed specifically to model the
timing of events. This dissertation concerns a subset of those methods that deals with events measured
or occurring in discrete-time or grouped-time intervals. A method for modeling single event
discrete-time data utilizing a latent class regression (LCR) framework, originally presented by
Muthén and Masyn (2001), is further developed and detailed. It is shown that discrete-time
data can be represented as a set of binary event indicators and observed risk indicators that allow
estimation using a latent class regression specification under a missing-at-random assumption that
corresponds to the assumption of noninformative right-censoring. The modeling of the effects of
time-dependent and time-independent covariates with constant or time-varying effects is demonstrated
along with approaches to model testing. The LCR framework also allows for the modeling of unobserved
heterogeneity through finite mixture modeling, i.e., multiple latent classes. The problems of
ignoring unobserved heterogeneity and the challenges of discrete-time mixture model identification
and specification for single event data are discussed. The LCR model for single event data is extended
to recurrent event survival data with a focus on recurrent event processes with a low frequency
of recurrences. The gap time, counting process, and total time formulations in the continuous-time
setting are all reformulated for discrete-time and model specification and estimation is demonstrated
for all three. The proposed model accommodates event-specific baseline hazard probabilities
as well as event-specific covariate effects. The model also allows for multiple event occurrences
in a single time period for a single subject and accounts for within as well as between subject
correlation of event times though the same mixture modeling approach given for single event data.
All models are illustrated with data on the event times of domestic violence episodes perpetrated by
a sample of married men observed for 12 months after an alcohol treatment program. Opportunities
for future methodology developments for discrete-time models are discussed."
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- Muthén, B., Asparouhov, T. & Shiffman, S. (2024). Dynamic Structural Equation Modeling with Floor Effects. Submitted for publication.
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Abstract
Intensive longitudinal data analysis, commonly used in psychological studies, often
concerns outcomes that have strong floor effects, that is, a large percentage at its lowest
value. Ignoring a strong floor effect, using regular analysis with modeling assumptions
suitable for a continuous-normal outcome, is likely to give misleading results. This
paper suggests that two-part modeling may provide a solution. It can avoid potential
biasing effects due to ignoring the floor effect. It can also provide a more detailed
description of the relationships between the outcome and covariates allowing different
covariate effects for being at the floor or not and the value above the floor. A smoking
cessation example is analyzed to demonstrate available analysis techniques.
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- Muthén, B., Asparouhov, T. & Keijsers, L. (2024). Dynamic Structural Equation Modeling with Cycles. Submitted for publication.
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Abstract
Cyclical phenomena are commonly observed in many areas of repeated measurements, especially with intensive longitudinal data. A typical example is circadian
(24-hour) rhythm of physical measures such as blood pressure, heart rate, glucose
level, and alertness. This paper focuses on positive affect which is a common measure
in psychological studies and for which circadian rhythm has been observed but not
analyzed by modern statistical methods. The paper demonstrates that a large new
analysis arsenal is available for analysis of cyclical features in intensive longitudinal
data. This can help researchers extract more information from their data to get more
valid estimates of coupled processes and to get new theoretical insights into circadian
rhythms of mood. To assist in this effort, the analyses are based on general models
with a rich set of features while still being accessible without an unduly steep learning
curve. Scripts for the Mplus software are available for all the analyses presented.
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- Hamaker, E.L., Asparouhov, T, & Muthén, B. (2021). Dynamic structural equation modeling as a combination of time series modeling, multilevel modeling, and structural equation modeling. To be published as Chapter 31 in: The Handbook of Structural Equation Modeling (2nd edition); Rick H. Hoyle (Ed.); Publisher: Guilford Press.
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- Asparouhov, T. & Muthén, B. (2020). Comparison of models for the analysis of intensive longitudinal data. Structural Equation Modeling: A Multidisciplinary Journal, 27(2) 275-297, DOI: 10.1080/10705511.2019.1626733
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Abstract
In this note we discuss the differences between the residual dynamic structural equation models (RDSEM) and the dynamic structural equation models (DSEM). Both models are introduced in Asparouhov et al. (2018) for the purpose of adapting structural equation models for intensive longitudinal data. The DSEM model has been implemented in Mplus V8 while the RD- SEM model has been implemented in Mplus V8.1. Here we illustrate the differences between the models through several simulation studies. In addi- tion, we compare the models to the standard multilevel SEM model which ignores the autocorrelations in the data. We also compare the models to the multilevel longitudinal model, based on the REML estimation for linear mixed models. As will be shown below, the REML method partially ignores the autocorrelations in the data.
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- Mun, C.J., Suk, H.W., Davis, M.C., Karoly, P., Finan, P., Tennen, H., & Jensen, M.P. (2019). Investigating intraindividual pain variability: Methods, applications, issues, and directions. Pain. DOI: 10.1097/j.pain.0000000000001626
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- Lundgren, B. & Schultzberg, M. (2019). Application of the economic theory of self-control to model energyconservation behavioral change in households. Energy. DOI: 10.1016/j.energy.2019.05.217
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Abstract
Smart meters and in-house displays hold a promise of energy conservation for those who invest in such technology. Research has shown that households only have a limited interest in such technology and information is thus often neglected, with rather limited energy savings. Surprisingly few empirical investigations have a theoretical foundation that may explain what is going on from a behavioral perspective. In this study the economic theory of self-control is used to model energy-efficient behavior in middle-income households in Sweden. Our results show that different levels of energy-efficient behavior do not really have any impact on the actual consumption levels of electricity. Instead, different beliefs exist of being energy-efficient, but the households do not act accordingly. We recommend to policy makers that the payment time period should be changed to pre-paid electricity to stimulate the monitoring of bills and to introduce a gaming strategy to change incentives for energy conservation.
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- Öhrlund, I., Schultzberg, M. & Bartusch, C. (2019). Identifying and estimating the effects of a mandatory billing demand charge. Applied Energy, 237, 885-895. DOI: 10.1016/j.apenergy.2019.01.028
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- McNeish, D. & Hamaker, E.L. (2020). A primer on two-level dynamic structural equation models for intensive longitudinal data in Mplus. Psychological Methods, 25(5), 610–635. https://doi.org/10.1037/met0000250
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- Joly-Burra, E., Van der Linden, M. & Ghisletta, P. (2018). Intraindividual variability in inhibition and prospective memory in healthy older adults: Insights from response regularity and rapidity. Journal of Intelligence, 6(1), 13. DOI: 10.3390/jintelligence6010013
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- Schultzberg, M. & Muthén, B. (2018). Number of subjects and time points needed for multilevel time series analysis: A simulation study of dynamic structural equation modeling. Structural Equation Modeling: A Multidisciplinary Journal, 25:4, 495-515, DOI:10.1080/10705511.2017.1392862.
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Abstract
Dynamic Structural Equation modeling (DSEM) is a novel intensive longitudinal data (ILD) analysis framework. DSEM uses two-level modeling with time on level 1 and individuals on level 2. It models intra-individual changes over time and allows the parameters of these processes to vary across
individuals using random effects. DSEM merges time series, structural equation, multilevel, and time-varying effects models. Despite the well-known properties of these analysis areas by themselves, it is unclear how their sample size requirements and recommendations transfer to the DSEM
framework. ILD are sampled in two dimensions, across subjects and across repeated measures within subjects. This paper presents the results of a simulation study that examines the estimation quality of univariate two-level autoregressive models of order one, AR(1), using Bayesian analysis
in Mplus Version 8. Three features are varied in the simulations: complexity of the model, number of subjects, and number of time points per subject. The models cover empty random mean-only models and models using a random AR(1) mean, a random autoregressive coefficient, and a random residual
variance as mediators on level 2. The number of subjects and number of time points per subject are varied between 10 and 200, in various combinations. Special attention is given to the power and accuracy of the level 2 regression slopes. The results are summarized with sample size guidelines
for each model. Samples with many subjects and few time points are showed to perform substantially better than samples with few subjects and many time points.
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- Asparouhov, T., & Muthén, B. (2019). Latent variable centering of predictors and mediators in multilevel and time-series models. Structural Equation Modeling: A Multidisciplinary Journal26, 119-142. DOI: 10.1080/10705511.2018.1511375
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Abstract
We discuss different methods for centering a predictor or a mediator in multilevel models. We show how latent mean centering can be extended to models with random slopes. Implications are discussed for estimating multilevel regression models with data missing on the predictors,
estimating the contextual effect in multilevel, time-series and probit regression models, estimating the indirect effect in multilevel mediation models, and estimating random tetrachoric autocorrelations for time-series models with categorical data.
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- Asparouhov, T., Hamaker, E.L. & Muthen, B. (2018). Dynamic structural equation models. Structural Equation Modeling: A Multidisciplinary Journal, 25:3, 359-388, DOI: 10.1080/10705511.2017.1406803
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Abstract
This paper presents a dynamic structural equation model (DSEM), which can be used to study the evolution of observed and latent variables as well as the structural equation models over time. DSEM is suitable for analyzing intensive longitudinal data
(ILD) where observations from multiple individuals are collected at many points in time. The modeling framework encompasses previously published DSEM models and is a comprehensive attempt to combine time series modeling with structural equation modeling. DSEM is estimated with Bayesian
methods using the MCMC Gibbs sampler and the Metropolis-Hastings sampler. We provide a detailed description of the estimation algorithm as implemented in the Mplus software package. DSEM can be used for longitudinal analysis of any duration and with any number of observations across time.
Simulation studies are used to illustrate the framework and study the performance of the estimation method. Methods for evaluating model fit are also discussed. Continuous time modeling, uneven times of observations and subject-specific times of observations are discussed as well.
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- Hamaker, E.L., Asparouhov, T., Brose, A., Schmiedek, F. & Muthen, B. (2018). At the frontiers of modeling intensive longitudinal data: Dynamic structural equation models for the affective measurements from the COGITO study. Multivariate Behavioral Research, DOI: 10.1080/00273171.2018.1446819
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Abstract
Dynamic structural equation modeling (DSEM) is a newly emerging class of techniques by which we can model the dynamic patterns in intensive longitudinal data. When using DSEM to analyse the time series of multiple individuals, we
specify a time series model at the within-person level and allow for individual differences at the between-person level in the parameters that describe the dynamics. We use DSEM to analyze a?ective data from the COGITO study, which consists of two samples
of over one hundred individuals each who were measured for one hundred days each. We use composite scores of positive and negative affect and apply a multilevel vector autoregressive model to investigate individual di?erences
in means, autoregressions and cross-lagged effects. Then we extend the model with random residual variances, and finally we investigate whether the random effects mediate the e?ect of prior depression on later depression scores. We point out some
additional options, and discuss several unresolved issues.
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- Asparouhov, T., Hamaker, E.L. & Muthen, B. (2017). Dynamic Latent Class Analysis. Structural Equation Modeling: A Multidisciplinary Journal, 24:2, 257-269, DOI: 10.1080/10705511.2016.1253479
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Abstract
This article describes the general time-intensive longitudinal latent class modeling framework implemented in Mplus. For each individual a latent class variable is measured at each time point and the latent class changes across time follow a Markov process (i.e., a hidden or
latent Markov model), with subject-specific transition probabilities that are estimated as random effects. Such a model for single-subject data has been referred to as the regimeswitching state-space model. The latent class variable can be measured by continuous or
categorical indicators, under the local independence condition, or more generally by a classspecific structural equation model or a dynamic structural equation model. We discuss the Bayesian estimation based on Markov chain Monto Carlo, which allows modeling with
arbitrary long time series data and many random effects. The modeling framework is illustrated with several simulation studies.
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- Petras, H., Nieuwbeerta, P., & Piquero, A.R. (2009). Participation and frequency during criminal careers over the life span. Criminology, 48(2). DOI: 10.1111/j.1745-9125.2010.00197.x
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Abstract
"Recent advances and debates surrounding general/developmental and static/dynamic theories of crime can
be traced to the 1986 National Academy of Science Report on criminal careers and the discussion it
generated. A key point of contention lies in the interpretation of the age-crime curve. For Gottfredson
and Hirschi, the decline in the age-crime curve in early adulthood reflects decreasing individual
offending frequency (?) after the peak. Blumstein et al. claim that the decline in the aggregate
age-crime curve can also be attributable to the termination of criminal careers, and the average value
of ? could stay constant (or increase with age) for those offenders who remain active after that
peak. Using data from the Criminal Career and Life Course Study - including information on criminal
convictions over 60 years of almost 5,000 persons convicted in the Netherlands - and applying a Two-Part
Growth Model that explicitly distinguishes between participation and frequency - the paper assesses
the participation/frequency debate. Results suggest that the decline in the age-crime curve in
early adulthood reflects both decreasing individual offending participation and frequency after the
peak, that the probabilities of participation and frequency are significantly related at the individual
level, and that sex and marriage influence both participation and frequency."
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- Vazsonyi, A.T. & Keiley, M.K. (2007).
Normative developmental trajectories of aggressive behaviors in African American, American Indian, Asian
American, Caucasian, and Hispanic children and early adolescents.
Journal of Abnormal Child Psychology, 35, 1047-1062.
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Abstract
"The current 5-year accelerated longitudinal investigation
modeled the developmental trajectories of
aggressive
behaviors in 10,107 predominantly minority
(>70%; African American, American Indian, Asian
American,
and Hispanic) children and early adolescents (Kindergarten
through 8th grade, 49% female
youth) from lower to
lower–middle socioeconomic strata. Based on a two-part
latent growth model, findings
suggest that the probability
and frequency of aggressive behavior use decreases slightly
(linear)
through the elementary school years and then
increases as children move into middle school (quadratic).
Though
mean level differences were found across ethnic
and racial groups, socioeconomic strata,
and particularly by
sex at initial status, rates of change over time across all groups
were invariant.
Findings suggest that potential socialization
differences, if any, occur pre-Kindergarten in
all groups."
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- Witkiewitz, K., & Masyn, K. E. (2008). Drinking trajectories following an initial lapse. Psychology of Addictive Behaviors, 22(2), 157–167. DOI: 10.1037/0893-164X.22.2.157
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Abstract
"Relapse following alcohol treatment is a major problem for individuals who are alcohol dependent, yet
little is known about the course of drinking after the initial lapse. In the current study, discrete-time
survival analysis and latent growth mixture modeling were used to evaluate the time to first
lapse and the trajectories of post-lapse drinking in a sample of 563 individuals who received community
alcohol treatment. Results showed a decreasing risk of lapsing over time. After the initial lapse,
three trajectory subgroups provided a parsimonious representation of the heterogeneity in post-lapse
drinking frequency and quantity, with the majority of individuals reporting light, infrequent drinking.
Covariate analyses incorporating demographics, distal risk factors, time-to-first lapse, and
coping behavior as predictors of time-to-lapse and post-lapse drinking trajectories indicated alcohol
dependence and coping behavior were the strongest predictors of lapsing and post-lapse drinking behavior."
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- Brown, E.C., Catalano, C.B., Fleming, C.B., Haggerty, K.P. & Abbot, R.D. (2005).
Adolescent substance use outcomes in the Raising Healthy Children Project: A two-part latent growth curve
analysis.
Journal of Consulting and Clinical Psychology, 73, 699-710.
Mplus outputs used in this paper can be viewed and/or downloaded from the
Examples page.
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Abstract
"Raising Healthy Children (RHC) is a preventive intervention designed to promote positive youth development
by targeting developmentally appropriate risk and protective factors. This study tested the
efficacy of the RHC intervention on reducing alcohol, marijuana, and cigarette use during early to
middle adolescence. Ten public schools, comprising 959 students, were matched and assigned randomly
to either intervention or control conditions. A two-part latent growth modeling strategy was employed
to examine change in both use-vs.-nonuse and frequency-of-use outcomes. Results indicated significant
(p < .05) intervention effects in growth trajectories for frequency of alcohol and marijuana
use but not for use vs. nonuse. These findings provide support for preventive interventions that take
a social development perspective in targeting empirically supported risk and protective factors
and demonstrate the utility of two-part models in adolescent substance use research."
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- Muthén, B. (2001). Two-Part Growth Mixture Modeling.
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Abstract
This paper considers the analysis of repeated measures data. Conventional random
effects growth modeling in the tradition of Laird and Ware (1982) represents unobserved
heterogeneity among subjects in the form of random effects, i.e. continuous latent
variables. Growth mixture modeling (Muth¶en & Shedden, 1999; Muthen, 2001a, b;
Muthen, Brown, Masyn, Jo, Khoo, Yang, Wang, Kellam, Carlin, & Liao, 2000; Muthen
& Muthen, 1998-2001, Appendix 8) offers an important extension of conventional modeling
in that more general forms of unobserved heterogeneity can be captured using
categorical latent variables (latent classes). Growth mixture modeling as implemented
in the Mplus software (Muthen & Muthen, 1998-2001) allows for latent classes that
may have different shapes, antecedents, and consequences. A related longitudinal technique,
latent class growth analysis (Nagin, 1999), also studies unobserved heterogeneity
in the form of categorical latent variables. Growth mixture modeling, however, allows
categorical and continuous heterogeneity jointly, capturing potential further continuous
heterogeneity among individuals within the latent classes.
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- Breitsohl, H. (2019). Beyond ANOVA: An introduction to structural equation models for experimental designs. Organizational Research Methods, 22(3) 649-677. DOI: 10.1177/1094428118754988
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- Koukounari, A., Copas, A.J., & Pickles, A. (2019). A latent variable modelling approach for the pooled analysis of individual participant data on the association between depression and chlamydia infection in adolescence and young adulthood in the UK.
Journal of the Royal Statistical Society, 182, 101-130. DOI: 10.1111/rssa.12387
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- Raykov, T. & Marcoulides, G. A. (2015). Scale reliability evaluation with heterogeneous populations. Educational and Psychological Measurement, 75(1), 146-156. DOI: 10.1177/0013164414558587
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- Böckenholt, U. (2012, April 30). Modeling multiple response processes in judgment and
choice. Psychological Methods. Advance online publication. DOI: 10.1037/a0028111
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Abstract
"In this article, I show how item response models can be used to capture multiple response processes
in psychological applications. Intuitive and analytical responses, agree– disagree answers, response
refusals, socially desirable responding, differential item functioning, and choices among multiple
options are considered. In each of these cases, I show that the response processes can be measured
via pseudoitems derived from the observed responses. The estimation of these models via standard
software programs that allow for missing data is also discussed. The article concludes with two
detailed applications that illustrate the prevalence of multiple response processes."
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- Hatton, H., Donnellan, M.B., Masyn, K., Feldman, B.J., Larsen-Rife, D., & Conger, R.D. (2008).
Family and individual difference predictors of trait aspects of negative interpersonal behaviors during
emerging adulthood.
Journal of Family Psychology, 22, 448-455.
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Abstract
"A latent trait-state-occasion (TSO) model (D. A. Cole, N. C. Martin, & J. H. Steiger, 2005) was used
to isolate the trait and state components of negative interpersonal behaviors toward
a friend or romantic
partner during emerging adulthood. Results indicate that variance in negative interpersonal behaviors
was due to nearly equal portions of Trait and Occasion
factors. Variability in the trait aspects
of negative interpersonal behaviors was then predicted by theoretically relevant constructs. In
particular, mothers’ negative behaviors during adolescence,
adolescent core self-evaluations, negative
emotionality, and feelings of security in close relationships had independent effects in predicting
the enduring aspects of negative interpersonal behaviors. All told, these results indicate that
TSO models can be helpful tools for understanding the developmental antecedents of the trait-like aspects
of interpersonal processes.
"
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- Temme, D., Paulssen, M., & Dannewald, T. (2008).
Incorporating latent variables into discrete choice models – A simultaneous estimation approach using
SEM software.
BuR – Business Research, 1, 220-237.
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Abstract
"Integrated choice and latent variable (ICLV) models represent a promising new class of models which
merge
classic choice models with the structural equation approach (SEM) for latent variables. Despite
their
conceptual appeal, applications of ICLV models in marketing remain rare. We extend previous ICLV
applications
by first estimating a multinomial choice model and, second, by estimating hierarchical
relations
between latent variables. An empirical study on travel mode choice clearly demonstrates
the value of
ICLV models to enhance the understanding of choice processes. In addition to the usually
studied directly
observable variables such as travel time, we show how abstract motivations such
as power and hedonism
as well as attitudes such as a desire for flexibility impact on travel mode choice.
Furthermore, we show that
it is possible to estimate such a complex ICLV model with the widely
available structural equation modeling
package Mplus. This finding is likely to encourage more widespread
application of this appealing
model class in the marketing field."
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- Dagne, G.A., Howe, G.W., Brown, C.H., & Muthén, B. (2002).
Hierarchical modeling of sequential behavioral data: An empirical Bayesian approach.
Psychological Methods, 7, 262-280.
Mplus inputs and outputs used in this paper can be viewed and/or downloaded from the Examples
page.
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Abstract
"This paper reviews the common methods for measuring strength of contingency between two behaviors in
a behavioral sequence, the binomial z-score and the adjusted cell residual, and points out a number
of limitations with these approaches. It presents a new approach using log odds ratios and employing
empirical Bayes estimation in the context of hierarchical modeling, an approach not constrained by
these limitations. A series of hierarchical models is presented to test the stationarity of behavioral
sequences, the homogeneity of sequences across the sample of episodes, and whether covariates
can account for variation in sequences across the sample. These models are applied to observational
data taken from a study of the behavioral interactions of 254 couples, to illustrate their use."
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- Muthén, B. (1989).
The future of methodological training in educational psychology: The problem of teaching students to use new sophisticated
statistical techniques. In M. C. Wittrock, & F. Farley (Eds.), The Future of Educational Psychology ( pp. 181-189).
Hillsdale, NJ: Erlbaum Associates.
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