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.

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.

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.

"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."

"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."

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

"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."

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.

"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."

"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."

"'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."

"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."

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.

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

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.

"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."

"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."

"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."

"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. "

"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."

"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."

"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."

"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)."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

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.

"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."

"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."

"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."

"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."

"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."

"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%."

"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."

"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."

"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."

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.

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.

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.

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.

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.

"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. "

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.

"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."

"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.

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.

"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)."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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"

"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."

"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."

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

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.

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.

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.

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.

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.

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.

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.

"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.

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.

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.

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.

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.

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.

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."

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."

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.

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.

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.

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.

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

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.

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.

"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. "

"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."

"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."

"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."

"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."

"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."

"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."

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.

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.

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.

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).

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.

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.

"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."

"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."

"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."

"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."

"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."

"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."

"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."

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.

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.

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.

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.

"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."

"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."

"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."

"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."

"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."

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.

"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."

"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"

"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."

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.

"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. "

"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."

"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."

"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."

"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."

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.

"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."

"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."

"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"

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.

"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."

"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. "

"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. "

"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."

"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."

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.

"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."

"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."

"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."

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.

"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."

"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."

"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."

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.

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.

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.

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.

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.

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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. "

"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."

"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."

"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."

"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."

"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."

"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."

"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."

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.

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

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.

"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."

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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"

"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."

"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)."

"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."

"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. "

"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."

"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."

"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."

"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."

"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."

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.

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).

"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."

"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."

"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."

"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."

"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."

"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."

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.

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.

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.

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.

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.

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.

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.

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.

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.

"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."

"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."

"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."

"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."

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.

"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."

"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. "

"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."

"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."