Papers Using Special Mplus Features

References on this page are ordered by topic. References can also be viewed ordered by date.

Categorical Factor Analysis expand topic collapse topic

• Flora, D.B. & Curran P.J. (2004). An empirical evaluation of alternative methods of estimation for confirmatory factor analysis with ordinal data. Psychological Methods, 9, 466-491.
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  Abstract

  "Confirmatory factor analysis (CFA) is widely used for examining hypothesized relations among ordinal variables (e.g., Likert-type items). A theoretically appropriate method fits the CFA model to polychoric correlations using either weighted least squares (WLS) or robust WLS. Importantly, this approach assumes that a continuous, normal latent process determines each observed variable. The extent to which violations of this assumption undermine CFA estimation is not well-known. In this article, the authors empirically study this issue using a computer simulation study. The results suggest that estimation of polychoric correlations is robust to modest violations of underlying normality. Further, WLS performed adequately only at the largest sample size but led to substantial estimation difficulties with smaller samples. Finally, robust WLS performed well across all conditions."

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Complex Survey Data Analysis expand topic collapse topic

• Stapleton, L.M. (2008). Variance estimation using replication methods in structural equation modeling with complex sample data. Structural Equation Modeling, 15, 183-210.
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  "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."

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• Asparouhov, T. & Muthén, B. (2007). Testing for informative weights and weights trimming in multivariate modeling with survey data. Proceedings of the 2007 JSM meeting in Salt Lake City, Utah, Section on Survey Research Methods.
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  "Analyzing the informativeness of the sampling weights can lead to significant improvement in the precision of model estimation with survey data. A test for weights ignorability was proposed in Pfeffermann’s (1993). We propose a modification of this test which improves its performance for small and medium sample size problems. We also generalize the test to a test of equivalence between two different sets of sampling weights, which can be used to test the informativeness of individual weight components. We evaluate the performance of these techniques in simulation studies based on linear regression and multivariate factor analysis models. We also apply the test of equivalence to the problem of finding the optimal level of weight trimming and illustrate this approach with a practical example. We describe the implementation of these techniques in the software package Mplus."

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• Asparouhov, T. (2006). General multi-level modeling with sampling weights. Communications in Statistics: Theory and Methods, Volume 35, Number 3, 2006, pp. 439-460(22). An earlier version of this paper appeared as Mplus Web Notes: No. 8 with the title Weighting for unequal probability of selection in multilevel modeling. Refer to Mplus Web Notes: No. 8 for more details.
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  "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."

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• Asparouhov, T. & Muthen, B. (2006). Comparison of estimation methods for complex survey data analysis.
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  "Recently structural equation modeling software packages have implemented more accurate statistical methodology for analyzing complex survey data. The computational algorithms however vary across the packages and produce different results even for simple models. In this note we conduct simulation studies to compare the performance of the methods implemented in Mplus and LISREL. The Mplus algorithm produced more accurate results."

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• Asparouhov, T. & Muthen, B. (2006). Multilevel modeling of complex survey data. Proceedings of the Joint Statistical Meeting in Seattle, August 2006. ASA section on Survey Research Methods, 2718-2726.
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  "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."

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• Asparouhov, T. (2005). Sampling weights in latent variable modeling. Structural Equation Modeling, 12, 411-434.
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  "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."

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• Asparouhov, T. and Muthén, B. (2005). Multivariate statistical modeling with survey data. Proceedings of the Federal Committee on Statistical Methodology (FCSM) Research Conference.
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  "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."

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• Asparouhov, T. (2004). Stratification in multivariate modeling. Mplus Web Notes: No. 9. Click here for more details.
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  "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."

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Factor Mixture Analysis expand topic collapse topic

• Clark, S.L. (2010). Mixture modeling with behavioral data. Doctoral dissertation, University of California, Los Angeles.
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  "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."

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• Clark, S.L., Muthén, B., Kaprio, J., D’Onofrio, B.M., Viken, R., Rose, R.J., Smalley, S. L. (2009). Models and strategies for factor mixture analysis: Two examples concerning the structure underlying psychological disorders. Submitted for publication.
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  "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."

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• Kim, Y.K. & Muthén, B. (2009). Two-part factor mixture modeling:  Application to an aggressive behavior measurement instrument.  Structural Equation Modeling, 16, 602-624.
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  "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."

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• Lubke, G. & Muthén, B. (2007). Performance of factor mixture models as a function of model size, covariate effects, and class-specific parameters. Structural Equation Modeling, 14(1), 26–47.
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  "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."

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• Lubke, G., Muthén, B., Moilanen, I., McGough, J., Loo, S., Swanson, J., Yang, M., Taanila, A., Hurtig, T., Jarvelin, M. & Smalley, S. (2007). Subtypes versus severity differences in the Attention-Deficit/Hyperactivity disorder in the northern Finnish birth cohort. Journal of the American Academy of Child and Adolescent Psychiatry, 46, 1584-1593.
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  "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."

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• Lubke, G. & Neale, M. (2006). Distinguishing between latent classes and continuous factors: Resolution by maximum likelihood? Multivariate Behavioral Research, 41(4), 499–532.
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  "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."

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• Muthén, B. (2006). Should substance use disorders be considered as categorical or dimensional? Addiction, 101 (Suppl. 1), 6-16.
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  "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."

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• Muthén, B. & Asparouhov, T. (2006). Item response mixture modeling: Application to tobacco dependence criteria. Addictive Behaviors, 31, 1050-1066.
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  "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."

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• Lubke, G.H. & Muthén, B. (2005). Investigating population heterogeneity with factor mixture models. Psychological Methods, 10, 21-39.
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  "Sources of population heterogeneity may or may not be known. Factor mixture models can be used to explore unknown population heterogeneity while integrating known sources as covariates. Different ways to incorporate covariates are discussed in detail. Advantages of factor mixture modeling are described in comparison to other methods designed for data stemming from heterogenous populations. A step-by-step analysis of a subset of data from the Longitudinal Survey of American Youth (LSAY) illustrates how factor mixture models can be applied in an exploratory fashion to data collected at a single time point."

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General Mixture Modeling expand topic collapse topic

• Muthén, B. (2008). Latent variable hybrids:  Overview of old and new models. In Hancock, G. R., & Samuelsen, K. M. (Eds.), Advances in latent variable mixture models, pp. 1-24. Charlotte, NC: Information Age Publishing, Inc. Click here for information about the book.
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• Muthén, B. (2002). Beyond SEM: General latent variable modeling. Behaviormetrika, 29, 81-117.
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  "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."

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• Muthén, B. (2001). Latent variable mixture modeling. In G. A. Marcoulides & R. E. Schumacker (eds.), New Developments and Techniques in Structural Equation Modeling (pp. 1-33). Lawrence Erlbaum Associates. Mplus inputs and outputs used in this paper can be viewed and/or downloaded from the Examples page.
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  "This chapter gives an overview of latent variable modeling with both categorical and continuous latent variables. Conventional latent class, structural equation, and growth models are extended and integrated in a general modeling framework."

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• Muthén, B. (2001). Second-generation structural equation modeling with a combination of categorical and continuous latent variables: New opportunities for latent class/latent growth modeling. In Collins, L.M. & Sayer, A. (eds.), New Methods for the Analysis of Change (pp. 291-322). Washington, D.C.: APA. To receive a copy of the paper, contact the author and mention paper #82.
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  "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."

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Genetics Modeling expand topic collapse topic

• Kerner, B. & Muthén, B. (2009). Growth mixture modelling in families of the Framingham Heart Study. BMC Proceedings, 3, 1-5.
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  "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."

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• Bartels, M., Cacioppo, J.T., Hudziak, J.J., & Boomsma, D.I. (2008). Genetic and environmental contributions to stability in loneliness throughout childhood. American Journal of Medical Genetics Part B, 147B, 385-391.
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• D’Onofrio, B.M., Hulle, C.A., Waldman, I.D., Rodgers, J. L. Harden, K.P., Rathouz, P.J. & Lahey, B.B. (2008). Smoking during pregnancy and offspring externalizing problems: An exploration of genetic and environmental confounds. Development and Psychopathology, 20, 139-164. Mplus scripts can be obtained from the first author.
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  "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."

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• Rathouz, P.J., Hulle, C.A., Rodgers, J.L., Waldman, I.D., & Lahey, B.B. (2008). Specification, testing, and interpretation of gene-by-measured-environment interaction models in the presence of gene-environment correlations. Behavior Genetics, 38, 301-315.
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  "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."

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• Boomsma, D., Cacioppo, J., Muthén, B., Asparouhov, T. & Clark, S. (2007). Longitudinal genetic analysis for loneliness in Dutch twins. Twin Research and Human Genetics, 10, 267-273.
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  "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%."

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• Harden, K.P., Turkheimer, E. & Loehlin, J.C. (2006). Genotype by environment interaction in adolescents’ cognitive aptitude. Behavioral Genetics.
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  "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."

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• Muthén, B., Asparouhov, T. & Rebollo, I. (2006). Advances in behavioral genetics modeling using Mplus: Applications of factor mixture modeling to twin data. Twin Research and Human Genetics, 9, 313-324. Mplus inputs, outputs, and data are not yet available for this article.
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  "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."

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• Prescott, C.A. (2004). Using the Mplus computer program to estimate models for continuous and categorical data from twins. Behavior Genetics, 34, 17-40. Mplus inputs and outputs used in this paper can be viewed and/or downloaded from here.
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  "Historically, the focus of behavior genetic research was to obtain estimates of the sources of familial resemblance for a single phenotype. Current research strategies have moved beyond heritability estimates to the search for physiological and behavioral mechanisms by which genetic risk is translated into individual differences in behavior and disease liability. Such research questions often require multivariate designs and complex analytic models, including the analysis of continuous and categorical dependent variables within the same model. Recent advances in computer software for categorical data analysis have increased the tools available for researchers in behavior genetics. This paper describes how to use the Mplus software program (Muthén and Muthén, 1998, 2002) for the analysis of data obtained from twins. Example analyses include two- and five-group twin models for univariate and bivariate continuous and categorical variables. Data on alcoholism and age at first drink drawn from the Virginia Adult Twin Study of Psychiatric and Substance Use Disorders are used to illustrate how Mplus can be used to analyze multiple-category variables, recode and transform variables, select subgroups for analysis, handle subjects with incomplete data, include constraints to ensure non-negative loadings, include model covariates, model sex differences, and test alternative hypotheses about mediation of genetic risk by measured variables."

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Growth Mixture Modeling expand topic collapse topic

• Hunter, A. M., Muthén, B.O., Cook, I.A. & Leuchter, A. F. (2010). Antidepressant response trajectories and quantitative electroencephalography (QEEG) biomarkers in major depressive disorder. Journal of Psychiatric Research, 44, 90-98.
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  "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."

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• Muthén, B. & Brown, H. (2009). Estimating drug effects in the presence of placebo response: Causal inference using growth mixture modeling. Statistics in Medicine, 28, 3363-3385.
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  Abstract

  "Placebo-controlled randomized trials for antidepressants and other drugs often show a response for a sizeable percentage of the subjects in the placebo group. Potential placebo responders can be assumed to exist also in the drug treatment group, making it difficult to assess the drug effect. A key drug research focus should be to estimate the percentage of individuals among those who responded to the drug who would not have responded to the placebo (‘Drug Only Responders’). This paper investigates a finite mixture model approach to uncover percentages of up to four potential mixture components: Never Responders, Drug Only Responders, Placebo Only Responders, and Always Responders. Two examples are used to illustrate the modeling, a 12-week antidepressant trial with a continuous outcome (Hamilton D score) and a 7-week schizophrenia trial with a binary outcome (illness level). The approach is formulated in causal modeling terms using potential outcomes and principal stratification. Growth mixture modeling (GMM) with maximum-likelihood estimation is used to uncover the different mixture components. The results point to the limitations of the conventional approach of comparing marginal response rates for drug and placebo groups. It is useful to augment such reporting with the GMM-estimated prevalences for the four classes of subjects and the Drug Only Responder drug effect estimate. Copyright q 2009 John Wiley & Sons, Ltd."

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• Muthén, B., Brown, H., Leuchter, A. & Hunter A. (2009). General approaches to analysis of course: Applying growth mixture modeling to randomized trials of depression medication. Forthcoming in P.E. Shrout (ed.), Causality and Psychopathology: Finding the Determinants of Disorders and their Cures. Washington, DC: American Psychiatric Publishing.
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  Abstract

  "This chapter discusses the use of growth mixture modeling to assess treatment effects in clinical trials. The motivation is a study of depression medication in a double-blind placebo-controlled trial. Studies of this type typically show placebo response and placebo non response. Growth mixture modeling (GMM) is well suited for representing such heterogeneity among subjects in that it can identify di erent types of trajectory shapes. GMM can be seen as a combination of conventional mixed effects modeling and cluster analysis, also allowing prediction of class membership and estimation of each individual's 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."

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• Petras, H & Masyn, K. (2009). General growth mixture analysis with antecedents and consequences of change. To appear in Piquero, A. & Weisburd, D., Handbook of Quantitative Criminology.
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• Uher, R., Muthén, B., Souery, D., Mors, O., Jaracz, J., Placentino, A., Petrovic, A., Zobel, A., Henigsberg, N., Rietschel, M., Aitchison, K., Farmer, A. & McGuffin, P. (2009). Trajectories of change in depression severity during treatment with antidepressants. Psychological Medicine, published online October 29, 2009.
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  Abstract

  "Background: Response and remission defined by cut-off values on the last observed depression severity score are commonly used as outcome criteria in clinical trials, but ignore the time-course of symptomatic change and may lead to inefficient analyses. We explore alternative categorisation of outcome by naturally occurring trajectories of symptom change. Methods: Growth mixture models (GMM) were applied to repeated measurements of depression severity in 807 participants with major depression treated for 12 weeks with escitalopram or nortriptyline in the part-randomized Genome-based Therapeutic Drugs for Depression (GENDEP) study. Latent trajectory classes were validated as outcomes in drug efficacy comparison and pharmacogenetic analyses. Results: The final two-piece growth mixture model categorised participants into a majority (75%) following a gradual improvement trajectory and the remainder following a trajectory with rapid initial improvement. The rapid improvement trajectory was overrepresented among nortriptylinetreated participants and showed an antidepressant-specific pattern of pharmacogenetic associations. In contrast, conventional response and remission favoured escitalopram and produced chance results in pharmacogenetic analyses. Controlling for drop-out reduced drug differences on response and remission but did not affect latent trajectory results. Conclusions: Latent trajectory mixture models capture heterogeneity in the development of clinical response after the initiation of antidepressants and provide an outcome that is distinct from traditional endpoint measures. It differentiates between antidepressants with different modes of action and is robust against bias due to differential discontinuation."

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• Uher, R., Muthén, B., Souery, D., Mors, O., Jaracz, J., Placentino, A., Petrovic, A., Zobel, A., Henigsberg, N., Rietschel, M., Aitchison, K., Farmer, A. & McGuffin, P. (2009). Technical appendix: Methods and results of growth mixture modelling. Psychological Medicine, published online October 29, 2009.
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  Abstract

  "Recent advances in statistical modelling based on mixture extension of the latent growth model make it possible to categorise subjects based on temporal patterns of change with latent variable methods such as growth mixture modelling (GMM) that can provide unbiased estimates of trajectories of change in the presence of missing data (Muthen & 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."

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• Boscardin, C., Muthén, B., Francis, D. & Baker, E. (2008). Early identification of reading difficulties using heterogeneous developmental trajectories. Journal of Educational Psychology, 100, 192-208.
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  Abstract

  "Serious conceptual and procedural problems associated with current diagnostic methods call for alternative approaches to assessing and diagnosing students with reading problems. This study presents a new analytic model to improve the classification and prediction of children’s reading development. 411 children in kindergarten through 2nd grade were administered measures of phonological awareness, word recognition, and rapid naming skills. The application of growth mixture models provides a more dynamic view of the learning process and the correlates that affect the rate of reading development. Growth mixture modeling was used to examine the presence of heterogeneous developmental patterns and served to identify one group of students with distinct developmental patterns who are most at risk for reading difficulties. The results indicate that precursor reading skills such as phonological awareness and rapid naming are highly predictive of later reading development and that developmental profiles formed in kindergarten are directly associated with development in grades 1 and 2. Students identified as having reading-related difficulties in kindergarten exhibited slower development of reading skills in subsequent years of the study. Key words: Reading Development, Screening, Reading Skills, Achievement, Longitudinal Studies, Models"

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• Jung, T. & Wickrama, K.A.S. (2008). An introduction to latent class growth analysis and growth mixture modeling. Social and Personality Psychology Compass, 2, 302-317.
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  Abstract

  "In recent years, there has been a growing interest among researchers in the use of latent class and growth mixture modeling techniques for applications in the social and psychological sciences, in part due to advances in and availability of computer software designed for this purpose (e.g., Mplus and SAS Proc Traj). Latent growth modeling approaches, such as latent class growth analysis (LCGA) and growth mixture modeling (GMM), have been increasingly recognized for their usefulness for identifying homogeneous subpopulations within the larger heterogeneous population and for the identification of meaningful groups or classes of individuals. The purpose of this paper is to provide an overview of LCGA and GMM, compare the different techniques of latent growth modeling, discuss current debates and issues, and provide readers with a practical guide for conducting LCGA and GMM using the Mplus software."

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• Kreuter, F. & Muthen, B. (2008). Longitudinal modeling of population heterogeneity: Methodological challenges to the analysis of empirically derived criminal trajectory profiles. In Hancock, G. R., & Samuelsen, K. M. (Eds.), Advances in latent variable mixture models, pp. 53-75. Charlotte, NC: Information Age Publishing, Inc. Click here for information about the book.
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• Kreuter, F. & Muthen, B. (2008). Analyzing criminal trajectory profiles: Bridging multilevel and group-based approaches using growth mixture modeling. Journal of Quantitative Criminology, 24, 1-31.
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  Abstract

  "Over the last 25 years, a life-course/developmental perspective on criminal behavior has assumed increasing prominence in the criminological literature. This theoretical development has been accompanied by changes in the statistical models used to analyze criminological data. There are two main statistical modeling techniques currently used to model longitudinal data. These are growth curve models and latent class growth models, also known as grouped-based trajectory models. This paper is a contribution to the recent debate on the use of these two models. Using the well known Cambridge data on criminal conviction, this paper compares the two “classical” models – conventional growth curve model and group-based trajectory models in terms of their performance with these particular data. It also introduces two additional models that bridge the gap between conventional growth models and group-based trajectory models. Sensitivity and substantive conclusions are then discussed for the models with the best performance. The main goals of this paper are to broaden the set of tools available to criminologists in analyzing data from a life-course perspective, and to provide a concrete step-by-step illustration of such an analysis."

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• Muthén, B. & Asparouhov, T. (2008). Growth mixture modeling: Analysis with non-Gaussian random effects. In Fitzmaurice, G., Davidian, M., Verbeke, G. & Molenberghs, G. (eds.), Longitudinal Data Analysis, pp. 143-165. Boca Raton: Chapman & Hall/CRC Press.
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  Abstract

  "Growth mixture analysis represents unobserved heterogeneity among the subjects in their development using both random effects and finite mixtures. In particular, the mixture components allow different means of the random effects, although any parameter in the growth model can vary. This chapter gives an overview of examples motivating modeling with such trajectory classes. A general latent variable modeling framework is presented together with its maximum-likelihood estimation. Examples from criminology and education are analyzed. The choice of a normal or a non-parametric distribution for the random effects is discussed and investigated using a simulation study. Key words: Growth modeling, finite mixtures, latent variables, trajectory classes, maximum likelihood, non-parametric distribution. Complete address of first author: Graduate School of Education & Information Studies, Moore Hall, Box 951521, Los Angeles CA 90095-1521."

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• Tolvanen, A. (2008). Latent growth mixture modeling: A simulation study. Doctoral dissertation, Department of Mathematics, University of Jyvaskyla, Finland.
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  Abstract

  "Latent growth curve modeling (LGM) combined with the latent classes (LGMM) in the SEM context, is the method under investigation in this study. This dynamic way of analyzing longitudinal data takes an increasingly central position in the social sciences, e.g. in psychology. Despite twenty years development of the theory behind the LGM and LGMM, these are novel methods in analyzing data in practice. With limited sample size the functionality of the model is unknown. The aim of this dissertation was to examine the functionality of the linear LGM model with four repeated measurements, which is a typical case in longitudinal research. LGMM parameters were estimated using maximum likelihood estimation with robust standard errors (MLR). The effect of differences between latent classes in mean values of latent components with varying sample sizes is examined in this study. Other affecting factors examined are reliability of observed variables, number of repeated measures, model construct and additional measurement points. The functionality of LGMM was approached from three different viewpoints: 1) problems in estimation of model parameters expressed as number of failed estimations and as the number of negative variance estimates, 2) the ability of AIC, BIC and aBIC information criteria and VLMR, LMR and BLRT statistical tests to decide the number of latent classes, and 3) good parameter estimation, which was evaluated using four different criteria: MSE, proportion of bias in MSE, bias of standard error, and 95 % coverage. The results of Monte Carlo simulations suggest that from information criteria AIC, BIC aBIC and VLMR and LMR tests, BIC is most useful with small sample sizes ( ) and aBIC with large sample sizes ( ). The few results suggest that the BLRT test could be useful in any situation. More investigation is needed to further support the functionality of this test. The study reveals that the estimation of LGMM fails only in a few cases, and problems in estimation appear mainly in the negative variance estimates. The results of the simulations suggest that it is possible to identify the true two-latent classes when SMD is at least 2, in which case reliability of observed variables should be high and the sample size should be relatively large. In that case estimation produce good parameter estimates. When SMD is 4 or 5, the probability in identifying the right two-latent-class solution instead of the wrong one-class solution is greater than .70 with the smallest sample size (n=50) using BIC in models with high reliability. To achieve reliable results in estimation, the sample size should be greater than 50. n < 500 n ? 500 Key words: Latent growth mixture modeling, Monte Carlo simulation"

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• Muthen, B. (2006). The potential of growth mixture modeling. Commentary. Infant and Child Development, 15, 623-625.
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• Schaeffer, C.M., Petras, H., Ialongo, N., Masyn, K.E., Hubbard, S., Poduska, J., & Sheppard, K. (2006). A comparison of girl's and boy's aggressive-disruptive behavior trajectories across elementary school: Prediction to young adult antisocial outcomes. Journal of Consulting and Clinical Psychology, 74, 500-510.
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  Abstract

  "Multiple group analysis and general growth mixture modeling was used to determine whether aggressive– disruptive behavior trajectories during elementary school, and their association with young adulthood antisocial outcomes, vary by gender. Participants were assessed longitudinally beginning at age 6 as part of an evaluation of 2 school-based preventive programs. Two analogous trajectories were found for girls and boys: chronic high aggression– disruption (CHAD) and stable low aggression– disruption (LAD). A 3rd class of low moderate aggression– disruption (LMAD) for girls and increasing aggression– disruption (IAD) for boys also was found. Girls and boys in analogous CHAD classes did not differ in trajectory level and course, but girls in the CHAD and LAD classes had lower rates of antisocial outcomes than boys. Girls with the LMAD trajectory differed from boys with the IAD trajectory."

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• Greenbaum, P.E., Del Boca, F.K., Darkes, J., Wang, C. & Goldman, M.S. (2005). Variation in the drinking trajectories of freshman college students. Journal of Consulting and Clinical Psychology, 73, 229-238
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  Abstract

  "Recently, Del Boca, Darkes, Greenbaum, and Goldman (2004) examined temporal variations in drinking during the freshmen college year and the relationship of several risk factors to these variations. Here, using the same data, we investigate whether a single growth curve adequately characterizes the variability in individual drinking trajectories. Latent growth mixture modeling identified five drinking trajectory classes: Light-Stable, Light-Stable + High Holiday, Medium- Increasing, High-Decreasing, and Heavy-Stable. In multivariate predictor analyses, gender (i.e., more females) and lower alcohol expectancies distinguished the Light-Stable from other trajectories; only expectancies differentiated the High-Decreasing from the Heavy-Stable and Medium-Increasing classes. These findings move us closer to identification of individuals at risk for developing problematic trajectories and to development of interventions tailored to specific drinker classes."

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• Muthén, B. (2004). Latent variable analysis: Growth mixture modeling and related techniques for longitudinal data. In D. Kaplan (ed.), Handbook of quantitative methodology for the social sciences (pp. 345-368). Newbury Park, CA: Sage Publications.
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  Abstract

  "This chapter gives an overview of recent advances in latent variable analysis. Emphasis is placed on the strength of modeling obtained by using a flexible combination of continuous and categorical latent variables. To focus the discussion, analysis of longitudinal data using growth models will be considered. Continuous latent variables are common in growth modeling in the form of random effects that capture individual variation in development over time. The use of categorical latent variables in growth modeling is in contrast perhaps less familiar. The aim of this chapter is to show the usefulness of growth model extensions using categorical latent variables. Examples are drawn from research on achievement development and high school dropout and from research on delinquency development."

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• van Lier, P.A.C., Muthén, B., van der Sar, R.M. & Crijnen, A.A.M. (2004). Preventing disruptive behavior in elementary schoolchildren: Impact of a universal classroom-based intervention. In Journal of Consulting and Clinical Psychology, 72, 467-478.
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  Abstract

  "A population-based, randomized universal classroom intervention trial for the prevention of disruptive behavior (i.e., attention-deficit/hyperactivity problems, oppositional defiant problems, and conduct problems) is described. Impact on developmental trajectories in young elementary schoolchildren was studied. Three trajectories were identified in children with high, intermediate, or low levels of problems on all 3 disruptive behaviors at baseline. The intervention had a positive impact on the development of all disruptive behavior problems in children with intermediate levels of these problems at baseline. Effect sizes of mean difference at outcome were medium or small. In children with the highest levels of disruptive behavior at baseline, a positive impact of the intervention was found for conduct problems."

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• Croudace, T.J., Jarvelin, M.R., Wadsworth, M.E. & Jones, P.B. (2003). Developmental typology of trajectories to nighttime bladder control: Epidemiologic application of longitudinal latent class analysis. American Journal of Epidemiology, May 1;157(9):834-42. To request a copy of the paper, contact the first author.
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  Abstract

  "The authors aimed to characterize developmental trajectories to nighttime continence by applying two latent class models-longitudinal latent class analysis (LLCA) and latent class growth analysis (LCGA)-to data on nighttime bed-wetting from a population-based birth cohort, the Medical Research Council 1946 National Survey of Health and Development cohort. Data on a binary outcome (wetting in the past month vs. not wetting) were available for children at six ages (4, 6, 8, 9, 11, and 15 years) assessed in 1950, 1952, 1954, 1955, 1957, and 1961. For 3,272 children with complete data (62.5% of the cohort), results of sequential model comparisons (T classes vs. T + 1 classes) and chi-square goodness-of-fit tests were evaluated using parametric bootstrapping. At least four trajectory classes (LLCA and LCGA) were identified. Associations between class membership and the prevalence of related measures were examined using a confirmatory latent class analysis approach. Inclusion of 1,483 children with partially incomplete data (n = 4,755; 90.9% of the cohort) enabled the authors to refine trajectories further: normal development (prevalence = 84.0%); delayed acquisition of bladder control ('transient' (8.7%) and 'persistent' (1.8%)), capturing primary enuresis; chronic bed-wetting (2.6%), or experiencing night wetting until age 15 years; and a final trajectory (relapse = 2.9%) capturing secondary or onset enuresis. This empirically based, typologic approach to analysis of extensive longitudinal data in a general population sample provides an alternative perspective to that offered by traditional diagnostic criteria."

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• Muthén, B. (2003). Statistical and substantive checking in growth mixture modeling. Psychological Methods, 8, 369-377.
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  Abstract

  "This commentary discusses the Bauer and Curran (2003) investigation of growth mixture modeling. Single-class modeling of non-normal outcomes is compared to modeling with multiple latent trajectory classes. New statistical tests of multiple-class models are discussed. Principles for substantive investigation of growth mixture model results are presented and illustrated by an example of high school dropout predicted by low mathematics achievement development in grades 7 - 10."

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• Muthén, B., Khoo, S.T., Francis, D. & Kim Boscardin, C. (2003). Analysis of reading skills development from Kindergarten through first grade: An application of growth mixture modeling to sequential processes. Multilevel Modeling: Methodological Advances, Issues, and Applications. S.R. Reise & N. Duan (Eds). Mahaw, NJ: Lawrence Erlbaum Associates, pp.71-89.
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  Abstract

  "Methods for investigating the influence of an early developmental process on a later process are discussed. Conventional growth modeling is found inadequate but a growth mixture model is sufficiently flexible. The growth mixture model allows for prediction of the later process using different trajectory classes for the early process. The growth mixture model is applied to the study of progress in reading skills among first-grade students. "

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• Oxford, M.L., Gilchrist, L.D., Morrison, D.M., Gillmore, M.R., Lohr M.J. & Lewis, S.M. (2003). Alcohol use among adolescent mothers: Heterogeneity in growth curves. Prevention Science, 4, 15-26. To request a copy of the paper, contact the first author.
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  Abstract

  "With a sample of adolescent mothers we examine patterns of alcohol use over a 10-year period of time. Mixture modeling with MPLUS is used to identify latent trajectory classes based on alcohol consumption over ten years. We found significant heterogeneity in alcohol use trajectories of adolescent mothers during the transition from adolescence to adulthood as well as significant predictors and outcomes that vary by latent class trajectory."

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• Schaeffer, C.M., Petras, H., Ialongo, N., Poduska, J. & Kellam, S. (2003). Modeling growth in boys aggressive behavior across elementary school: Links to later criminal involvement, conduct disorder, and antisocial personality disorder. Developmental Psychology, 39, 1020-1035.
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  Abstract

  "Theoretical models of antisocial behavior development have proposed distinct pathways leading to criminal activity. The present study used general growth mixture modeling (GGMM) to find empirical evidence for these pathways within an epidemiologically-defined sample of 297 urban, primarily African-American boys. Teacher-rated aggression, measured longitudinally from 1st-7th grades, was used to define growth trajectories. Three distinct high-risk trajectories (chronic high, moderate, and increasing aggression; 68% of boys) and one low-risk aggression trajectory (stable low aggression; 32% of boys) were found. Boys with chronic high and increasing trajectories were at increased risk for conduct disorder and juvenile arrest in adolescence, and antisocial personality disorder and adult arrest in young adulthood. Boys with a moderate aggression trajectory were at risk for juvenile and adult arrest. Concentration [NSI1] problems were highest among boys with a chronic high aggression trajectory and also differentiated boys with increasing aggression from boys with stable low aggression. Peer rejection was also higher among boys with chronic high aggression relative to the low aggression group. The need for improved early identification of and interventions with boys with distinct patterns of aggression is discussed."

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• Muthén, B., Brown, C.H., Masyn, K., Jo, B., Khoo, S.T., Yang, C.C., Wang, C.P., Kellam, S., Carlin, J., & Liao, J. (2002). General growth mixture modeling for randomized preventive interventions. Biostatistics, 3, 459-475. Mplus inputs and outputs used in this paper can be viewed and/or downloaded from the Examples page.
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  Abstract

  "This paper proposes growth mixture modeling to assess intervention effects in longitudinal randomized trials. Growth mixture modeling represents unobserved heterogeneity among the subjects using a finite mixture random effects model. The methodology allows one to examine the impact of an intervention on subgroups characterized by different types of growth trajectories. Such modeling is informative when examining effects on populations that contain individuals who have normative growth as well as non-normative growth. The analysis identifies subgroup membership and allows theory-based modeling of intervention effects in the different subgroups. An example is presented concerning a randomized intervention in Baltimore public schools aimed at reducing aggressive classroom behavior, where only students who were initially more aggressive showed benefits from the intervention."

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• Stoolmiller, M. (2001). Synergistic interaction of child manageability problems and parent-discipline tactics in predicting future growth in externalizing behavior for boys. Developmental Psychology, 37, 814-825.
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  Abstract

  "During early childhood for boys, manageability problems were hypothesized to disrupt parental discipline practices. In turn, disrupted parental discipline practices were hypothesized to interact with manageability problems during late childhood to predict change in antisocial behavior during the transition from elementary to middle school. Results indicated that maternal retrospective perceptions of unmanageability predicted observed maternal discipline practices, even when controlling for maternal antisocial behavior and depressed mood and the disruptive and antisocial behavior of the boy. Graphical analyses and latent class growth models indicated that temper tantrums interacted with maternal discipline in predicting change in teacher ratings of antisocial behavior. The nature of the interaction indicated that maternal discipline was a risk factor for growth in antisocial behavior only for boys with high levels of tantrums."

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• Muthén, B. & Muthén, L. (2000). Integrating person-centered and variable-centered analyses: Growth mixture modeling with latent trajectory classes. Alcoholism: Clinical and Experimental Research, 24, 882-891. Mplus inputs and outputs used in this paper can be viewed and/or downloaded from the Examples page.
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  Abstract

  "Background. Many alcohol research questions require methods that take a person-centered approach because the interest is in finding heterogeneous groups of individuals such as those who are susceptible to alcohol dependence and those who are not. A person-centered focus is also useful with longitudinal data to represent heterogeneity in developmental trajectories. In alcohol, drug, and mental health research the recognition of heterogeneity has led to theories of multiple developmental pathways. Methods. This paper gives a brief overview of new methods that integrate variable- and person-centered analyses. Methods discussed include latent class analysis, latent transition analysis, latent class growth analysis, growth mixture modeling, and general growth mixture modeling. These methods are presented in a general latent variable modeling framework that expands traditional latent variable modeling by including not only continuous latent variables but also categorical latent variables.

  Results. Four examples that use the NLSY data are presented to illustrate latent class analysis, latent class growth analysis, growth mixture modeling, and general growth mixture modeling. Latent class analysis of antisocial behavior found four classes. Four heavy drinking trajectory classes were found. The relationship between the latent classes and their relationship to background variables and consequences was studied.

  Conclusions. Person-centered and variable-centered analyses have typically been seen as different activities that use different types of models and software. This paper gives a brief overview of new methods that integrate variable- and person-centered analyses. The general framework makes it possible to combine these models and to study new models serving as a stimulus for asking research questions that have both person- and variable-centered aspects."

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• Muthén, B. & Shedden, K. (1999). Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics, 55, 463-469.
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  Abstract

  "This paper discusses the analysis of an extended finite mixture model where the latent classes corresponding to the mixture components for one set of observed variables influence a second set of observed variables. The research is motivated by a repeated measurement study using a random coefficient model to assess the influence of latent growth trajectory class membership on the probability of a binary disease outcome. More generally, this model can be seen as a combination of latent class modeling and conventional mixture modeling. The EM algorithm is used for estimation. As an illustration, a random-coefficient growth model for the prediction of alcohol dependence from three latent classes of heavy alcohol use trajectories among young adults is analyzed."

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• Benner, A. & Graham, S. (2009). The transition to high school as a developmental process among multiethnic urban youth. Child Development, 80:2, 356–376.
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  Abstract

  "The high school transition was examined in an ethnically diverse, urban sample of 1,979 adolescents, followed from 7th to 10th grade (Mage = 14.6, SD = .37 in 7th grade). Twice annually, data were gathered on adolescents’ perceptions of school climate, psychological functioning, and academic behaviors. Piecewise growth modeling results indicate that adolescents were doing well before the transition but experienced transition disruptions in psychological functioning and grades, and many continued to struggle across high school. The immediate experience of the transition appeared to be particularly challenging for African American and Latino students when the numerical representation of their ethnic groups declined significantly from middle to high school. Findings highlight the value of examining the transition in a larger developmental context and the importance of implementing transition support."

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• Grimm, K.J. & Ram, N. (2009). Nonlinear growth models in Mplus and SAS. Structural Equation Modeling, 16, 676-701.
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  Abstract

  "Nonlinear growth curves or growth curves that follow a specified nonlinear function in time enable researchers to model complex developmental patterns with parameters that are easily interpretable. In this article we describe how a variety of sigmoid curves can be fit using the Mplus structural modeling program and the nonlinear mixed-effects modeling procedure NLMIXED in SAS. Using longitudinal achievement data, collected as part of a study examining the effects of preschool instruction on academic gain, we illustrate the procedures for fitting growth models of logistic, Gompertz, and Richards functions. Brief notes regarding the practical benefits, limitations, and choices faced in the fitting and estimation of such models are included."

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• Pettit, G.S., Keiley, M.K., Laird, R.D., Bates, J.E. & Dodge, K.A. (2007). Predicting the developmental course of mother-reported monitoring across childhood and adolescence from early proactive parenting, child temperament, and parents’ worries. Journal of Family Psychology, 21, 206-217.
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  Abstract

  "Change in mothers’ reported monitoring and awareness of their children’s activities and companions across Grades 5, 6, 8, and 11 were examined with the use of latent factor growth modeling. Proactive parenting and resistant-to-control (RTC) child temperament assessed prior to kindergarten, as well as parents’ worries about their children’s behavior in Grades 5 and 8, were tested as factors associated with change in monitoring over time. Higher proactive parenting, lower RTC temperament, and the mounting of a successful campaign to change their children’s behavior were associated with higher monitoring scores overall. Monitoring levels decreased across time, but the rate of decline was steeper among mothers with high RTC children and slower among mothers who mounted a campaign and judged it to be effective. These findings shed light on factors contributing to continuity and change across development in a key domain of parenting."

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• Clark, D., Birmaher, B., Axelson, D., Monk, K., Kalas, C., Ehmann, M., Bridge, J., Wood, S., Muthén, B., & Brent, D. (2005). Fluoxetine for the treatment of childhood anxiety disorders: Open-label, long-term extension to a controlled trial. Journal of the American Academy of Child & Adolescent Psychiatry, 44, 1263-1270.
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  Abstract

  "Objective: To assess the efficacy of fluoxetine for the long-term treatment of children and adolescents with anxiety disorders, including generalized anxiety disorder, separation anxiety disorder, and/or social phobia. Method: Children and adolescents (7–17 years old) with anxiety disorders were studied in open treatment for 1 year after they completed a randomized, controlled trial (RCT) comparing fluoxetine and placebo. The follow-up phase assessments included clinician, parent, and child ratings with measures of global severity, global improvement, and anxiety symptoms. Results: Subjects taking fluoxetine (n = 42) were compared with those taking no medication (n = 10) during follow-up on anxiety changes from the end of the RCT through the follow-up period. Statistical models included RCT assignment and follow-up psychological treatment. Excluded subjects took other medications (n = 4) or did not complete follow-up (n = 18). Compared with subjects taking no medication, subjects taking fluoxetine showed significantly superior follow-up outcomes on most measures, including clinician, parent, and child ratings. Conclusions: The results suggest that fluoxetine is clinically effective for the maintenance treatment of anxiety disorders in children and adolescents. A major limitation, however, was the lack of RCT methodology in the follow-up phase. RCTs are needed to determine the long-term risks and benefits of fluoxetine for this group. J. Am. Acad. Child Adolesc. Psychiatry, 2005;44(12):1263–1270. Key Words: anxiety disorders, fluoxetine, selective serotonin reuptake inhibitors."

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• Muthén, B. & Muthén, L. (2000). The development of heavy drinking and alcohol-related problems from ages 18 to 37 in a U.S. national sample. Journal of Studies on Alcohol, 61, 290-300.
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  Abstract

  "Objective. The purpose of this study is to add to the understanding of the development of heavy alcohol use and alcohol-related problems by examining data from the National Longitudinal Survey of Youth (NLSY), a general population sample that contains information on alcohol use for the ages 18-37. A key question in this study is how background characteristics of the individual influence this development and whether the influence of these background characteristics changes over time. Method. The data used in this study are a general population sample from the National Longitudinal Survey of Youth (NLSY). This study uses a multivariate outcome approach which focuses on individual variation in trajectories over age. The statistical analysis uses random coefficients in a latent variable framework. Across-age changes in the importance of the influence of background variables on the outcomes are modeled using varying centering points. Results. A key finding is that dropping out of high school has no effect on alcohol problems for individuals in their mid twenties, but is associated with significantly increased levels of alcohol problems for individuals in their mid thirties. In contrast, going on to college is associated with lower levels of heavy drinking when individuals reach their late twenties and their thirties. Strong gender and ethnicity effects seen in the twenties diminish when individuals reach their thirties. Conclusions. The trajectory analysis expands the knowledge of problematic alcohol development for individuals in their late twenties and thirties. The increasing detrimental effect of dropping out of high school up to the age 37 endpoint of the study raises questions about the effects of dropping out of high school later in life."

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IRT expand topic collapse topic

• Woods, C.M. (2009). Evaluation of MIMIC-Model Methods for DIF Testing With Comparison to Two-Group Analysis. Multivariate Behavioral Research, 44:1,1-27.
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  Abstract

  "Differential item functioning (DIF) occurs when an item on a test or questionnaire has different measurement properties for 1 group of people versus another, irrespective of mean differences on the construct. This study focuses on the use of multiple-indicator multiple-cause (MIMIC) structural equation models for DIF testing, parameterized as item response models. The accuracy of these methods, and the sample size requirements, are not well established. This study examines the accuracy of MIMIC methods for DIF testing when the focal group is small and compares results with those obtained using 2-group item response theory (IRT). Results support the utility of the MIMIC approach. With small focalgroup samples, tests of uniform DIF with binary or 5-category ordinal responses were more accurate with MIMIC models than 2-group IRT. Recommendations are offered for the application of MIMIC methods for DIF testing."

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• Dumenci, L. & Achenbach, T.M. (2008). Effects of estimation methods on making trait-level inferences from ordered categorical items for assessing psychopathology. Psychological Assessment, 20, 55-62.
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  Abstract

  "In assessments of attitudes, personality, and psychopathology, unidimensional scale scores are commonly obtained from Likert scale items to make inferences about individuals’ trait levels. This study approached the issue of how best to combine Likert scale items to estimate test scores from the practitioner’s perspective: Does it really matter which method is used to estimate a trait? Analyses of 3 data sets indicated that commonly used methods could be classified into 2 groups: methods that explicitly take account of the ordered categorical item distributions (i.e., partial credit and graded response models of item response theory, factor analysis using an asymptotically distribution-free estimator) and methods that do not distinguish Likert-type items from continuously distributed items (i.e., total score, principal component analysis, maximum-likelihood factor analysis). Differences in trait estimates were found to be trivial within each group. Yet the results suggested that inferences about individuals’ trait levels differ considerably between the 2 groups. One should therefore choose a method that explicitly takes account of item distributions in estimating unidimensional traits from ordered categorical response formats. Consequences of violating distributional assumptions were discussed."

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• Zumbo, B.D. (2007). Three generations of DIF analyses:Considering where it has been, where it is now, and where it is going. Language Aseessment Quarterly, 4(2), 223–233.
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  Abstract

  "The purpose of this article is to reflect on the state of the theorizing and praxis of DIF in general: where it has been; where it is now; and where I think it is, and should, be going. Along the way the major trends in the differential item functioning (DIF) literature are summarized and integrated providing some organizing principles that allow one to catalog and then contrast the various DIF detection methods and to shine a light on the future of DIF analyses. The three generations of DIF are introduced and described with an eye toward issues on the horizon for DIF."

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• MacIntosh, R. & Hashim, S. (2003). Converting MIMIC model parameters to IRT parameters in DIF analysis. Applied Psyhological Measurement, 27, 372-379.
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  Abstract

  "The purpose of this study is to document and compare two methods to estimate the statistical properties of the converted Item Response Theory discrimination and difficulty parameters derived from MIMIC model parameters. The delta method and Monte Carlo simulation provide similar variance estimates, with differences attributed to rounding error. Discussed is the formulation of MIMIC models in Mplus and how to obtain factor analytic estimates that are converted easily into IRT parameters. Also described are the partial derivatives necessary to apply the delta method to estimate variances for the converted parameters. Both item difficulty and discrimination parameters estimated from MIMIC parameters were very close to the Multilog estimates. The variance estimates for most parameters were similar, as well."

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Latent Class Analysis expand topic collapse topic

• Clark, S. & Muthén, B. (2009). Relating latent class analysis results to variables not included in the analysis. Submitted for publication.
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  Abstract

  "An important interest in mixture modeling is the investigation of what types of individuals belong to each latent class by relating classes to covariates, concurrent outcomes and distal outcomes, also known as auxiliary variables. This article presents results from real data examples and simulations to show how various factors, such as the degree to which people are classified correctly into latent classes and sample size, can impact the estimates and standard errors of auxiliary variable effects and testing mean equality across classes. Based on the results of the examples and simulations, suggestions are made about how to select auxiliary variables for a latent class analysis."

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• Marsh, H.W., Lüdtke, O., Trautwein, U., & Morin, A.J.S. (2009). Classical latent profile analysis of academic self-concept dimensions: Synergy of person- and variable- centered approaches to theoretical models of self-concept. Structural Equation Modeling, 16:2,191-225.
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  Abstract

  "In this investigation, we used a classic latent profile analysis (LPA), a person-centered approach, to identify groups of students who had similar profiles for multiple dimensions of academic selfconcept (ASC) and related these LPA groups to a diverse set of correlates. Consistent with a priori predictions, we identified 5 LPA groups representing a combination of profile level (high vs. low overall ASC) and profile shape (math vs. verbal self-concepts) that complemented results based on a traditional variable-centered approach. Whereas LPA groups were substantially and logically related to the set of 10 correlates, much of the predictive power of individual ASC factors was lost in the formation of groups and the inclusion of the correlates into the LPA distorted the nature of the groups. LPA issues examined include distinctions between quantitative (level) and qualitative (shape) differences in LPA profiles, goodness of fit and the determination of the number of LPA groups, appropriateness of correlates as covariates or auxiliary variables, and alternative approaches to present and interpret the results."

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• Marsh, H.W., Lüdtke, O., Trautwein, U., & Morin, A.J.S. (2009). Classical latent profile analysis of academic self-concept dimensions: Synergy of person- and variable- centered approaches to theoretical models of self-concept. Structural Equation Modeling, 16:2,191-225.
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  Abstract

  "In this investigation, we used a classic latent profile analysis (LPA), a person-centered approach, to identify groups of students who had similar profiles for multiple dimensions of academic selfconcept (ASC) and related these LPA groups to a diverse set of correlates. Consistent with a priori predictions, we identified 5 LPA groups representing a combination of profile level (high vs. low overall ASC) and profile shape (math vs. verbal self-concepts) that complemented results based on a traditional variable-centered approach. Whereas LPA groups were substantially and logically related to the set of 10 correlates, much of the predictive power of individual ASC factors was lost in the formation of groups and the inclusion of the correlates into the LPA distorted the nature of the groups. LPA issues examined include distinctions between quantitative (level) and qualitative (shape) differences in LPA profiles, goodness of fit and the determination of the number of LPA groups, appropriateness of correlates as covariates or auxiliary variables, and alternative approaches to present and interpret the results."

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• Kreuter, F., Yan, T. & Tourangeau, R. (2008). Good item or bad – can latent class analysis tell?: The utility of latent class analysis for the evaluation of survey questions. Journal of the Royal Statistical Society, Series A, 171, 723-738.
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  Abstract

  "Latent class analysis has been used to model measurement error, to identify flawed survey questions and to estimate mode effects. Using data from a survey of University of Maryland alumni together with alumni records, we evaluate this technique to determine its usefulness for detecting bad questions in the survey context. Two sets of latent class analysis models are applied in this evaluation: latent class models with three indicators and latent class models with two indicators under different assumptions about prevalence and error rates. Our results indicated that the latent class analysis approach produced good qualitative results for the latent class models—the item that the model deemed the worst was the worst according to the true scores. However, the approach yielded weaker quantitative estimates of the error rates for a given item."

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• Nylund, K.L., Asparouhov, T., & Muthen, B. (2007). Deciding on the number of classes in latent class analysis and growth mixture modeling. A Monte Carlo simulation study. Structural Equation Modeling, 14, 535-569.
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  Abstract

  "Mixture modeling is a widely used modeling technique that is used to identify unobserved heterogeneity in a study population. The application of mixture models has allowed for a deeper understanding of many substantive areas. Despite their usefulness in practice, one unresolved issue in the application of mixture models is that there is not one commonly accepted statistical indicator of how many classes there are in a study population. This paper presents the results of a simulation study that looks at the performance of likelihood-based tests and the traditionally used Information Criterion (ICs) that are often used for determining the number of classes in mixture modeling. We look at the performance of these tests and indices for a series of Latent Class Analysis (LCA) and Growth Mixture Models (GMM) and evaluate their ability to correctly identify the number of classes in a given population. While the BIC was the best of the ICs, the bootstrap Likelihood Ratio Test (BLRT), which is now available in statistical software used for mixture modeling, proves to be a very consistent indicator of classes."

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• Geiser, C., Lehman, W., & Eid, M. (2006). Separating rotators from nonrotators in the Mental Rotation Test: A multigroup latent class analysis. Multivariate Behavioral Research, 41, 261-293.
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  Abstract

  "Items of mental rotation tests can not only be solved by mental rotation but also by other solution strategies. A multigroup latent class analysis of 24 items of the Mental Rotations Test (MRT) was conducted in a sample of 1,695 German pupils and students to find out how many solution strategies can be identified for the items of this test. The results showed that five subgroups (latent classes) can be distinguished. Although three of the subgroups differ mainly in the number of items reached, one class shows are very low performance. In another class, a special solution strategy is used. This strategy seems to involve analytic rather than mental rotation processes and is efficient only for a specialMRT item type, indicating that not allMRT items require a mental rotation approach. In addition, the multigroup analysis revealed significant sex differences with respect to the class assignment, confirming prior findings that on average male participants perform mental rotation tasks faster and better than female participants. Females were also overrepresented in the analytic strategy class. The results are discussed with respect to psychometric and substantive implications, and suggestions for the optimization of the MRT items are provided."

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Latent Transition Analysis expand topic collapse topic

• Kaplan, D. (2008). An overview of Markov chain methods for the study of stage-sequential developmental processes. Developmental Psychology, 44, 457-467.
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  Abstract

  "This article presents an overview of quantitative methodologies for the study of stage-sequential development based on extensions of Markov chain modeling. Four methods are presented that exemplify the flexibility of this approach: The manifest Markov model, the latent Markov model, latent transition analysis, and the mixture latent Markov model. A special case of the mixture latent Markov model, the so-called “mover-stayer"" model, is used in this study. Issues of model specification, estimation, and testing using the Mplus software environment are briefly discussed and the Mplus input syntax is provided. These four methods are applied to a single example of stage sequential development in reading competency in the early school years utilizing data from the Early Childhood Longitudinal Study – Kindergarten Cohort."

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• Bray, B.C. (2007). Examining gambling and substance use: Applications of advanced latent class modeling techniques for cross-sectional and longitudinal data. Doctoral dissertation, Pennsylvania State University.
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  Abstract

  "The purpose of the current project is to present three empirical studies that illustrate the application of advanced latent class modeling techniques for crosssectional and longitudinal data to research questions about gambling and substance use. The first empirical study used latent class analysis and conditional latent class analysis to identify and predict types of college-student gamblers using data from a large northeastern university. Four types of gamblers were identified for men and women: non-gamblers, cards and lotto players, cards and games of skill players, and multi-game players. There were substantial gender differences in the latent class membership probabilities: (1) men were most likely to be cards and lotto players whereas women were most likely to be non-gamblers; and (2) men were more likely than women to be cards and games of skill and multi-game players, and less likely to be non-gamblers. Significant predictors of gambling latent class membership included: school year, living in off-campus housing, Greek membership, and past-year alcohol use. There were substantial gender differences in the predictive effects of Greek membership and past-year alcohol use: (1) the effects of Greek membership were in different directions for men and women; and (2) pastyear alcohol use was more strongly related to gambling latent class membership for women. The second empirical study used latent class analysis to identify types of adolescent and young adult gamblers and used latent class analysis for repeated measures to identify types of drinking trajectories using data from the National Longitudiiii nal Study of Adolescent Health. Multivariable latent class modeling was used to examine the relation between gambling and drinking by linking specific types of gambling to specific types of drinking trajectories. Gambling and drinking were shown to be highly related: (1) consistent infrequent, light, or not intense drinkers were most likely to be non-gamblers; and (2) participants who were frequent, heavy, or intense drinkers at any time were most likely to gamble in all activities. Overall, drinking frequency appeared to be more predictive of gambling than was drinking quantity. The third empirical study used latent transition analysis to identify types of adolescent smokers and types of drinkers, and to describe smoking and drinking development over time using data from the National Longitudinal Survey of Youth 1997. Multiprocess modeling was used to examine the relation between smoking and drinking by modeling the development of smoking and the development of drinking simultaneously. Three types of smokers and three types of drinkers were identified: non-smokers, light smokers, heavy smokers, non-drinkers, light drinkers, and heavy drinkers. The majority of participants were non-smokers and nondrinkers. The behavior of non-smokers, non-drinkers, heavy smokers, and heavy drinkers was relatively stable across time whereas the behavior of light smokers and light drinkers was variable. Linking smoking and drinking showed that: (1) knowing type of smoking provided limited information about type of drinking; (2) transitioning from non-drinking to heavy drinking was progressively more likely for more serious types of smoking; (3) transitioning from heavy drinking to nondrinking was progressively less likely for more serious types of smoking; and (4) transitioning from light drinking to non-drinking was most likely for non-smokers whereas transitioning from light drinking to heavy drinking was most likely for heavy smokers. iv"

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• Nylund, K. (2007). Latent transition analysis: Modeling extensions and an application to peer victimization. Doctoral dissertation, University of California, Los Angeles.
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• Nylund, K.L., Muthén, B., Nishina, A., Bellmore, A. & Graham, S. (2006). Stability and instability of peer victimization during middle school: Using latent transition analysis with covariates, distal outcomes, and modeling extensions. Submitted for publication.
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  Abstract

  "This paper is an advanced application of latent transition analysis (LTA). Examining the peer victimization experiences of approximately 1300 urban, public school students across the 3 middle school years, we extend the conventional LTA model to simultaneously include time varying covariates with time varying effects, second-order effects, a mover-stayer variable, and distal outcomes. We present five key modeling steps that can be used in the application of the LTA model. The analyses yielded three victim classes based on victimization degree (victimized, sometimes victimized, nonvictimized). LTA indicated that when students transitioned between victimization classes, they were most likely to transition from a more victimized group into one of the less victimized groups. Further, results indicated that students who experience any sort of victimization, compared to those who do not, felt less safe at school, more socially anxious, and more depressed during certain middle school years. We also found that students who were chronically victimized in middle school reported more physical health problems and more social worries once in high school."

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Measurement Modeling expand topic collapse topic

• Raykov, T. (2009). Evaluation of convergent and discriminant validity with multitrait-multimethod correlations. Forthcoming in British Journal of Mathematical and Statistical Psychology.
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  Abstract

  "A procedure for evaluation of convergent and discriminant validity coefficients is outlined. The method yields interval estimates of these coefficients in a construct validation study conducted via the multitrait-multimethod approach, as well as permits examination of their population relationships. The procedure is readily employed in behavioral research using the increasingly popular latent variable modeling methodology. The described method is illustrated with a numerical example."

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• Raykov, T., Dimitrov, D.M. & Asparouhov, T. (2009). Evaluation of scale reliability with binary measures using latent variable modeling. Forthcoming in Structural Equation Modeling.
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  Abstract

  "A method for interval estimation of scale reliability with discrete data is outlined. The approach is applicable with multi-item instruments consisting of binary measures, and is developed within the latent variable modeling methodology. The procedure is useful for evaluation of consistency of single measures and of sum scores from item sets following the two-parameter logistic model or the one-parameter logistic model. An extension of the method is described for constructing confidence intervals of change in reliability due to instrument revision. The proposed procedure is illustrated with an example."

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• Raykov, T. (2008). “Alpha if item deleted”: A note on loss of criterion validity in scale development if maximizing coefficient alpha. British Journal of Mathematical and Statistical Psychology, 61, 275-285.
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  Abstract

  "This note is concerned with a validity-related limitation of the widely available and used index “alpha if item deleted” in the process of construction and development of multiplecomponent measuring instruments. Attention is drawn to the fact that this statistic can suggest dispensing with such scale components, whose removal leads to loss in criterion validity while maximising the popular coefficient alpha. As an alternative, a latent variable modelling approach is discussed that can be used for point and interval estimation of composite criterion validity (as well as reliability) after deletion of single components. The method can also be utilised to test conventional or minimum level hypotheses about associated population change in measurement quality indices."

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Mediational Modeling expand topic collapse topic

• MacKinnon, D.P., Lockwood, C.M., Brown, C.H., Wang, W., & Hoffman, J.M. (2007). The intermediate endpoint effect in logistic and probit regression.  Clinical Trials, 4, 499-513.
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  Abstract

  "Background An intermediate endpoint is hypothesized to be in the middle of the causal sequence relating an independent variable to a dependent variable. The intermediate variable is also called a surrogate or mediating variable and the corresponding effect is called the mediated, surrogate endpoint, or intermediate endpoint effect. Clinical studies are often designed to change an intermediate or surrogate endpoint and through this intermediate change influence the ultimate endpoint. In many intermediate endpoint clinical studies the dependent variable is binary, and logistic or probit regression is used. Purpose The purpose of this study is to describe a limitation of a widely used approach to assessing intermediate endpoint effects and to propose an alternative method, based on products of coefficients, that yields more accurate results. Methods The intermediate endpoint model for a binary outcome is described for a true binary outcome and for a dichotomization of a latent continuous outcome. Plots of true values and a simulation study are used to evaluate the different methods. Results Distorted estimates of the intermediate endpoint effect and incorrect conclusions can result from the application of widely used methods to assess the intermediate endpoint effect. The same problem occurs for the proportion of an effect explained by an intermediate endpoint, which has been suggested as a useful measure for identifying intermediate endpoints. A solution to this problem is given based on the relationship between latent variable modeling and logistic or probit regression. Limitations More complicated intermediate variable models are not addressed in the study, although the methods described in the article can be extended to these more complicated models. Conclusions Researchers are encouraged to use an intermediate endpoint method based on the product of regression coefficients. A common method based on difference in coefficient methods can lead to distorted conclusions regarding the intermediate effect. Clinical Trials 2007; 4: 499–513. http://ctj.sagepub.com"

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Missing Data Analysis expand topic collapse topic

• Muthén, B., Asparouhov, T., Hunter, A. & Leuchter, A. (2010). Growth modeling with non-ignorable dropout: Alternative analyses of the STAR*D antidepressant trial. Submitted for publication. Click here to view Mplus outputs used in this paper.
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  Abstract

  "This paper uses a general latent variable framework to study a series of models for non-ignorable missingness due to dropout. Both existing and new models are explored for pattern-mixture and selection type modeling. The missing data models are applied to longitudinal data from STAR*D, the largest antidepressant clinical trial in the U.S. to date. Despite the importance of this trial, STAR*D growth model analyses using non-ignorable missing data techniques have not been used until now. Using growth mixture modeling extended to handle non-ignorable missingness due to dropout, the STAR*D data are shown to feature distinct trajectory classes, including a low class corresponding to substantial improvement in depression and a minority class with a U-shaped curve corresponding to transient improvement. The trajectory class analysis provides a new way to assess drug efficiency. Software is available for sensitivity analysis using the extensive set of missing data models available in the general latent variable framework. "

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• Morgan-Lopez, A. A. & Fals-Steward, W. (2007). Analytic methods for modeling longitudinal data from rolling therapy groups with membership turnover. Journal of Consulting and Clinical Psychology, 75, 580-593.
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  Abstract

  "Interventions for a variety of emotional and behavioral problems are commonly delivered in the context of treatment groups, with many using rolling admission to sustain membership (i.e., admission, dropout and discharge from group is perpetual and ongoing). We present an overview of the analytic challenges inherent in rolling group data and outline commonly-used (but flawed) analytic and design approaches used to address (or sidestep) these issues. Moreover, we propose latent class pattern mixture modeling (LCPMM) as a statistically and conceptually defensible approach for modeling treatment data from rolling groups. The LCPMM approach is illustrated with rolling group data from a group-based alcoholism pilot treatment trial (N = 128). Different inferences were made with regard to treatment efficacy under LCPMM versus the commonly used standard group-clustered latent growth model (LGM); coupled with other preliminary findings in this area, inferences from LGMs may be overly liberal when applied to data from rolling groups. Continued work on data analytic difficulties in groups with membership turnover is critical for furthering the ecological validity of research on behavioral treatments."

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• Muthén, B., Jo, B. & Brown, H. (2003). Comment on the Barnard, Frangakis, Hill & Rubin article, Principal stratification approach to broken randomized experiments: A case study of school choice vouchers in New York City. Journal of the American Statistical Association, 98, 311-314. The Muthén et al. article can be downloaded from here. The Barnard et al. article can be found at http://biosun01.biostat.jhsph.edu/~cfrangak/papers/index.html. For background information and analyses using Mplus, see Mplus Web Note #5 and Jo (2002), Sensitivity of causal effects under ignorable and latent ignorable missing-data mechanisms, Draft. Contact the author. The Jo paper can be downloaded from here.
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Mixture (Latent Class) SEM expand topic collapse topic

• Van Horn, M. L., Jaki, T., Masyn, K., Ramey, S. L., Smith, J., & Antaramian, S. (2009). Assessing differential effects: Applying regression mixture models to identify variations in the influence of family resources on academic achievement. In press, Child Development.
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  Abstract

  "Developmental scientists frequently seek to understand effects of environmental contexts on development. Traditional analytic strategies assume similar environmental effects on all children, sometimes exploring possible moderating influences or exceptions (e.g. outliers) as a secondary step. These strategies are poorly matched to ecological models of human development which posit complex individual by environment interactions. An alternative conceptual framework is proposed that tests the hypothesis that the environment has differential (non-uniform) effects on children. A demonstration of the utility of this framework is provided by examining the effects of family resources on children’s academic outcomes in a multisite study (N=6305). Three distinctive groups of children were identified, including one group particularly resilient to influence of low levels of family resources. Predictors of group differences including parenting and child demographics are tested, the replicability of the results are examined, and findings are contrasted with those using traditional regression interaction effects. This approach is proposed as a partial solution to advance theories of the environment, social ecological systems research, and behavioral genetics in order to create well-tailored environments for children."

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• Guo, J., Wall, M. & Amemiya, Y. (2006). Latent class regression on latent factors. Biostatistics, 7, 145-163. This type of modeling can be done using ML techniques illustrated in the Mplus Version 3 User's Guide (first printed in March 2004), example 7.19. The authors emailed us and apologized for not seeing this Mplus capability earlier and not referencing it in the paper.
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  Abstract

  "In the research of public health, psychology, and social sciences, many research questions investigate the relationship between a categorical outcome variable and continuous predictor variables. The focus of this paper is to develop a model to build this relationship when both the categorical outcome and the predictor variables are latent (i.e. not observable directly). This model extends the latent class regression model so that it can include regression on latent predictors. Maximum likelihood estimation is used and two numerical methods for performing it are described: the Monte Carlo Expectation and Maximization algorithm (MCEM) and Gaussian quadrature followed by quasi-Newton algorithm. A simulation study is carried out to examine the behavior of the model under different scenarios. A data example involving adolescent health is used for demonstration where the latent classes of eating disorders risk are predicted by the latent factor body satisfaction."

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Multilevel Mixture Modeling expand topic collapse topic

• Henry, K. & Muthén, B. (2009). Multilevel latent class analysis: An application of adolescent smoking typologies with individual and contextual predictors. Forthcoming in Structural Equation Modeling.
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  Abstract

  "Latent Class Analysis (LCA) is a statistical method used to identify subtypes of related cases using a set of categorical and/or continuous observed variables. Traditional LCA assumes that observations are independent. However, multilevel data structures are common in social and behavioral research and alternative strategies are needed. In this paper, a new methodology, multilevel latent class analysis (MLCA), is described and an applied example is presented. Latent classes of cigarette smoking among 10,772 European American females in 9th grade who live in one of 206 rural communities across the U.S. are considered. A parametric and non-parametric approach for estimating a MLCA are presented and both individual and contextual predictors of the smoking typologies are assessed. Both latent class and indicator-specific random effects models are explored. The best model was comprised of three level 1 latent smoking classes (heavy smokers, moderate smokers, non-smokers), two random effects to account for variation in the probability of level 1 latent class membership across communities, and a random factor for the indicator-specific level 2 variances. Several covariates at the individual and contextual level were useful in predicting latent classes of cigarette smoking as well as the individual indicators of the latent class model. This paper will assist researchers in estimating similar models with their own data."

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• Muthén, B. & Asparouhov, T. (2009). Multilevel regression mixture analysis. Journal of the Royal Statistical Society, Series A, 172, 639-657.
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  Abstract

  "A two-level regression mixture model is discussed and contrasted with the conventional two-level regression model. Simulated and real data shed light on the modelling alternatives. The real data analyses investigate gender differences in mathematics achievement from the US National Education Longitudinal Survey.The two-level regression mixture analyses show that unobserved heterogeneity should not be presupposed to exist only at level 2 at the expense of level 1. Both the simulated and the real data analyses show that level 1 heterogeneity in the form of latent classes can be mistaken for level 2 heterogeneity in the form of the random effects that are used in conventional two-level regression analysis. Because of this, mixture models have an important role to play in multilevel regression analyses. Mixture models allow heterogeneity to be investigated more fully, more correctly attributing different portions of the heterogeneity to the different levels."

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• Asparouhov, T. & Muthen, B. (2008). Multilevel mixture models. In Hancock, G. R., & Samuelsen, K. M. (Eds.), Advances in latent variable mixture models, pp. 27-51. Charlotte, NC: Information Age Publishing, Inc. Click here for information about the book.
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• Van Horn, M.L., Fagan, A.A., Jaki, T., Brown, E.C., Hawkins, J.D., Arthur, M.W., Abbott, R.D., & Catalano, R. F. (2008). Using multilevel mixtures to evaluate intervention effects in group randomized trials. Multivariate Behavioral Research, 43(2), 289-326. PMC - In Process.
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  Abstract

  "There is evidence to suggest that the effects of behavioral interventions may be limited to specific types of individuals, but methods for evaluating such outcomes have not been fully developed. This study proposes the use of finite mixture models to evaluate whether interventions, and, specifically, group randomized trials, impact participants with certain characteristics or levels of problem behaviors. This study uses latent classes defined by clustering of individuals based on the targeted behaviors and illustrates the model by testing whether a preventive intervention aimed at reducing problem behaviors affects experimental users of illicit substances differently than problematic substance users or those individuals engaged in more serious problem behaviors. An illustrative example is used to demonstrate the identification of latent classes, specification of random effects in a multilevel mixture model, independent validation of latent classes, and the estimation of power for the proposed models to detect intervention effects. This study proposes specific steps for the estimation of multilevel mixture models and their power and suggests that this model can be applied more broadly to understand the effectiveness of interventions."

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Multilevel SEM expand topic collapse topic

• Preacher, K., Zyphur, M. & Zhang, Z. (2010). A general multilevel SEM framework for assessing multilevel mediation. Forthcoming in Psychological Methods.
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  Abstract

  "Several methods for testing mediation hypotheses with two-level nested data have been proposed using a multilevel modeling (MLM) paradigm. However, these MLM approaches do not accommodate mediation pathways with Level-2 outcomes and may produce conflated estimates of between- and within-level components of indirect effects. Moreover, these methods have each appeared in isolation, so a unified framework that integrates the existing methods, as well as new multilevel mediation models, is lacking. Here we show that a multilevel structural equation modeling (MSEM) paradigm can overcome these two limitations of mediation analysis using MLM. We present an integrative two-level MSEM mathematical framework that subsumes new and existing multilevel mediation approaches as special cases. We use several applied examples and accompanying software code to illustrate the flexibility of this framework and to show that different substantive conclusions can be drawn using MSEM versus MLM."

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• Gottfredson, N.C., Panter, A.T., Daye, C.E., Allen, W.F. & Wightman, L.F. (2009). The effects of educational diversity in a national samples of law students: Fitting multilevel latent variable models in data with categorical indicators. Multivariate Behavioral Research, 44, 305-331.
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  Abstract

  "Controversy surrounding the use of race-conscious admissions can be partially resolved with improved empirical knowledge of the effects of racial diversity in educational settings. We use a national sample of law students nested in 64 law schools to test the complex and largely untested theory regarding the effects of educational diversity on student outcomes. Social scientists who study these outcomes frequently encounter both latent variables and nested data within a single analysis. Yet, until recently, an appropriate modeling technique has been computationally infeasible, and consequently few applied researchers have estimated appropriate models to test their theories, sometimes limiting the scope of their research question. Our results, based on disaggregated multilevel structural equation models, show that racial diversity is related to a reduction in prejudiced attitudes and increased perceived exposure to diverse ideas and that these effects are mediated by more frequent interpersonal contact with diverse peers. These findings provide support for the idea that administrative manipulation of educational diversity may lead to improved student outcomes. Admitting a racially/ethnically diverse student body provides an educational experience that encourages increased exposure to diverse ideas and belief systems."

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• Marsh, H.W., Ludtke, O., Robitzsch, A., Trautwein, U., Asparouhov, T., Muthén, B., & Nagengast, B. (2009). Doubly-latent models of school contextual effects: Integrating multilevel and structural equation approaches to control measurement and sampling errors. Multivariate Behavioral Research, 44, 764-802.
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  Abstract

  "This article is a methodological-substantive synergy.Methodologically, we demonstrate latent-variable contextual models that integrate structural equation models (with multiple indicators) and multilevel models. These models simultaneously control for and unconfound measurement error due to sampling of items at the individual (L1) and group (L2) levels and sampling error due the sampling of persons in the aggregation of L1 characteristics to form L2 constructs. We consider a set 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 4 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 L1 latent interaction) are not significant.Although random-slopemodels show 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 set 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."

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• Muthén, B. & Asparouhov, T. (2009). Beyond multilevel regression modeling: Multilevel analysis in a general latent variable framework. To appear in The Handbook of Advanced Multilevel Analysis.  J. Hox & J.K Roberts (eds). Taylor and Francis.
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  Abstract

  "Multilevel modeling is often treated as if it concerns only regression analysis and growth modeling. Multilevel modeling, however, is relevant for nested data not only with regression and growth analysis but with all types of statistical analyses. This chapter has two aims. First, it shows that already in the traditional multilevel analysis areas of regression and growth there are several new modeling opportunities that should be considered. Second, it gives an overview with examples of multilevel modeling for path analysis, factor analysis, structural equation modeling, and growth mixture modeling. Examples include two extensions of two-level regression analysis with measurement error in the level 2 covariate and a level 1 mixture; two-level path analysis and structural equation modeling; two-level exploratory factor analysis of classroom misbehavior; two-level growth modeling using a two-part model for heavy drinking development; an unconventional approach to three-level growth modeling of math achievement; and multilevel latent class mediation of high school dropout using multilevel growth mixture modeling of math achievement development."

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• Lüdtke, O., Marsh, H.W., Robitzsch, A., Trautwein, U., Asparouhov, T., & Muthén, B. (2008). The multilevel latent covariate model: A new, more reliable approach to group-level effects in contextual studies.  Psychological Methods, 13, 203-229.
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  Abstract

  "In multilevel modeling (MLM), group level (L2) characteristics are often measured by aggregating individual level (L1) characteristics within each group as a means of assessing contextual effects (e.g., group-average effects of SES, achievement, climate). Most previous applications have used a multilevel manifest covariate (MMC) approach, in which the observed (manifest) group mean is assumed to have no measurement error. This paper shows mathematically and with simulation results that this MMC approach can result in substantially biased estimates of contextual effects and can substantially underestimate the associated standard errors, depending on the number of L1 individuals in each of the groups, the number of groups, the intraclass correlation, the sampling ratio (the percentage of cases within each group sampled), and the nature of the data. To address this pervasive problem, we introduce a new multilevel latent covariate (MLC) approach that corrects for unreliability at L2 and results in unbiased estimates of L2 constructs under appropriate conditions. However, our simulation results also suggest that the contextual effects estimated in typical research situations (e.g., fewer than 100 groups) may be highly unreliable. Furthermore, under some circumstances when the sampling ratio approaches 100%, the MMC approach provides more accurate estimates. Based on three simulations and two real-data applications, we critically evaluate the MMC and MLC approaches and offer suggestions as to when researchers should most appropriately use one, the other, or a combination of both approaches. "

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• Asparouhov, T. & Muthén, B. (2007). Computationally efficient estimation of multilevel high-dimensional latent variable models. Proceedings of the 2007 JSM meeting in Salt Lake City, Utah, Section on Statistics in Epidemiology.
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  Abstract

  "Multilevel analysis often leads to modeling with multiple latent variables on several levels. While this is less of a problem with Gaussian observed variables, maximum-likelihood (ML) estimation with categorical outcomes presents computational problems due to multidimensional numerical integration. We describe a new method that compared to ML is both computationally efficient and has similar MSE. The method is an extension of the Muthen (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."

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• Grilli, L & Rampichini, C. (2007). Multilevel factor models for ordinal variables. Structural Equation Modeling, 14, 1-25.
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  Abstract

  "This article tackles several issues involved in specifying, fitting, and interpreting the results of multilevel factor models for ordinal variables. First, the problem of model specification and identification is addressed, outlining parameter interpretation. Special attention is devoted to the consequences on interpretation stemming from the usual choice of not decomposing the specificities into hierarchical components. Then a general strategy of analysis is outlined, highlighting the role of the exploratory steps. The theoretical topics are illustrated through an application to graduates' job satisfaction, where estimation is based on maximum likelihood via an Expectation-Maximization algorithm with adaptive quadrature."

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• Mehta, P. & Neale, M. (2005). People are variables too: Multilevel structural equations modeling. Psychological Methods, 10, 259-284. This paper draws on techniques illustrated in the Mplus Version 3 User's Guide (first printed in March 2004), example 9.10.
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  Abstract

  "The article uses confirmatory factor analysis (CFA) as a template to explain didactically multilevel structural equation models (ML-SEM) and to demonstrate the equivalence of general mixed-effects models and ML-SEM. An intuitively appealing graphical representation of complex ML-SEMs is introduced that succinctly describes the underlying model and its assumptions. The use of definition variables (i.e., observed variables used to fix model parameters to individual specific data values) is extended to the case of ML-SEMs for clustered data with random slopes. Empirical examples of multilevel CFA and ML-SEM with random slopes are provided along with scripts for fitting such models in SAS Proc Mixed, Mplus, and Mx. Methodological issues regarding estimation of complex ML-SEMs and the evaluation of model fit are discussed. Further potential applications of ML-SEMs are explored."

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• Yuan, K.H. & Hayashi, K. (2005). On Muthen's maximum likelihood for two-level covariance structure models. Psychometrika, 70, 147-167.
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  Abstract

  "Data in social and behavioral sciences are often hierarchically organized. Special statistical procedures that take into account the dependence of such observations have been developed. Among procedures for 2-level covariance structure analysis, Muth´en’s maximum likelihood (MUML) has the advantage of easier computation and faster convergence. When data are balanced, MUML is equivalent to the maximum likelihood procedure. Simulation results in the literature endorse the MUML procedure also for unbalanced data. This paper studies the analytical properties of the MUML procedure in general. The results indicate that the MUML procedure leads to correct model inference asymptotically when level-2 sample size goes to infinity and the coefficient of variation of the level-1 sample sizes goes to zero. The study clearly identifies the impact of level-1 and level-2 sample sizes on the standard errors and test statistic of the MUML procedure. Analytical results explain previous simulation results and will guide the design or data collection for the future applications of MUML."

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• Muthén, B., Khoo, S.T. & Gustafsson, J.E. (1997). Multilevel latent variable modeling in multiple populations. Unpublished technical report.
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  Abstract

  "Modeling is described for the simultaneous analysis of two-level data in several populations. A typical example is cluster sampling of students within schools, where schools of different types are represented, e.g. public and private schools. Multivariate measurements on each student are assumed to give rise to a latent variable model. Of interest is to study across-population differences and similarities with respect to the within- and between-group covariance matrices and with respect to the mean vector. The methodology is illustrated by a comparative analysis of achievement structures in Catholic and public schools. "

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• Muthén, B. (1994). Multilevel covariance structure analysis. In J. Hox & I. Kreft (eds.), Multilevel Modeling, a special issue of Sociological Methods & Research, 22, 376-398.
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Non-Compliance Modeling expand topic collapse topic

• Jo, B., Ginexi, E. & Ialongo, N. (2010). Handling missing data in randomized experiments with noncompliance. Forthcoming in Prevention Science.
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  Abstract

  "Treatment noncompliance and missing outcomes at posttreatment assessments are common problems in field experiments in naturalistic settings. Although the two complications often occur simultaneously, statistical methods that address both complications have not been routinely considred in data analysis practice in the prevention research field. This paper shows that identification and estimation of causal treatment effects considering both noncompliance and missing outcomes can be relatively easily conducted under various missing data assumptions. We review a few assumptions on missing data in the presence of noncompliance, including the latent ignorability proposed by Frangakis and Rubin (1999), and show how these assumptions can be used in the parametric CACE estimation framework. As an easy way of sensitivity analysis, we propose the use of alternative missing data assumptions, which will provide a range of causal effect estimates. In this way, we are less likely to settle with a possibly biased causal effect estimate based on a single assumption. We demonstrate how alternative missing data assumptions affect identification of causal effects, focusing on the complier average causal effect (CACE). The data from the Johns Hopkins School Intervention Study (Ialongo et al., 1999) will be used as an example."

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• Stormshak, E.A., Connell, A. & Dishion, T.J. (2009). An adaptive approach to family-centered intervention in schools: Linking intervention engagement to academic outcomes in middle and high school. Prevention Science, 10, 221-235.
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  Abstract

  "This study examined the impact of an adaptive approach to family intervention in public schools on academic outcomes from age 11 to 17. Students were randomly assigned to the three-session Family Check-Up (FCU), which is designed to motivate change in parenting practices by using an assessment-driven approach and strengths-based feedback. All services were voluntary, and approximately 25% of the families engaged in the FCU. Compared with matched controls, adolescents whose parents received the FCU maintained a satisfactory GPA into high school, and intervention engagement was associatedwith improved attendance. The highest-risk families were the most likely to engage in the family-centered intervention, suggesting the efficacy of integrating supportive services to families in the context of other schoolwide approaches to promote the success and achievement of vulnerable students."

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• Jo, B., Asparouhov, T. & Muthén, B. (2008). Intention-to-treat analysis in cluster randomized trials with noncompliance. Statistics in Medicine, 27, 5565-5577.
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  Abstract

  "In cluster randomized trials (CRT), individuals belonging to the same cluster are very likely to resemble one another, not only in terms of outcomes, but also in terms of treatment compliance behavior. Whereas the impact of resemblance in outcomes is well acknowledged, little attention has been given to the possible impact of resemblance in compliance behavior. This study defines compliance intraclass correlation as the level of resemblance in compliance behavior among individuals within clusters, and shows how compliance intraclass correlation can be a problem in evaluating intention-to-treat (ITT) effect in CRT. On the basis of Monte Carlo simulations, it is demonstrated that ignoring compliance information in analyzing data from CRT may result in substantially decreased power to detect ITT effect, mainly due to compliance intraclass correlation. As a way of avoiding additional loss of power to detect ITT effect in CRT accompanied by noncompliance, this study employs an estimation method, where ITT effect estimates are obtained based on compliance-type-specific treatment effect estimates. A multilevel mixture analysis using an ML-EM estimation method is used for this estimation."

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• Jo, B., Asparouhov, T., Muthén, B., Ialongo, N. & Brown, H. (2008). Cluster randomized trials with treatment noncompliance. Psychological Methods, 13, 1-18.
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  Abstract

  "Cluster randomized trials (CRT) have been widely used in field experiments treating a cluster (or group) of individuals as the unit of randomization. This study focuses particularly on situations where CRT are accompanied by a common complication in field experiments, namely treatment noncompliance. In CRT, compliance behavior may be related not only to individual characteristics of study participants, but also to the environment of clusters individuals belong to. Therefore, analyses ignoring the connection between compliance and CRT may not provide valid results. Although randomized field experiments often suffer from both noncompliance and clustering of the data, these features have been studied as separate rather than concurrent problems. On the basis of Monte Carlo simulations, this study demonstrates how CRT and noncompliance may affect statistical inferences and how these two complications can be accounted for simultaneously. In particular, the effect of randomized intervention on individuals who abide by the intervention assignment (complier average causal effect: CACE) will be the focus of the study. For estimation of intervention effects considering both noncompliance and CRT, an ML-EM estimation method is employed."

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• Dunn, G., Maracy, M., Dowrick, C., Ayuso-Mateos, J.L., Dalgard, O.S., Page, H., Lehtinen, V., Casey, P., Wilkinson, C., Vasquez-Barquero, J.L., & Wilkinson, G. (2003). Estimating psychological treatment effects from a randomized controlled trial with both non-compliance and loss to follow-up. British Journal of Psychiatry, 183, 323-331.
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  Abstract

  "Background: The Outcomes of Depression International Network (ODIN) trial evaluated the effects of two psychological interventions for the treatment of depression in primary care. Only about half of the patients in the treatment arm complied with the offer of treatment, prompting the question: 'what was the effect of treatment in those patients who actually received it?

  Aims: To illustrate the estimation of the effect of receipt of treatment in a randomised controlled trial subject to non-compliance and loss to follow-up.

  Method: We estimated the complier average causal effect (CACE) of treatment.

  Results: In the ODIN trial the effect of receipt of psychological intervention (an average of about 4 points on the Beck Depression Inventory) is about twice that of offering it.

  Conclusions: The statistical analysis of the results of a clinical trial subject to non-compliance to allocated treatment is now reasonably straightforward through estimation of a CACE and investigators should be encourage to present the results of analyses of the type as a routine component of a trial report."

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• Jo, B. & Muthén, B. (2003). Longitudinal studies with intervention and noncompliance: Estimation of causal effects in growth mixture modeling. In S.P. Reise & Duan, N. (eds.) Multilevel Modeling. Methodological Advances, Issues, and Applications (pp.112-139). Mahwah, New Jersey: Lawrence Erlbaum.
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• Jo, B. (2002). Statistical power in randomized intervention studies with noncompliance. Psychological Methods, 7, 178-193. Mplus inputs and outputs used in this paper can be viewed and/or downloaded from the Examples page.
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  Abstract

  "One of the common complications in randomized intervention trials is noncompliance of study participants. Noncompliance is an obstacle to fair statistical comparison and may lead to biased estimates of intervention effects. From a practical point of view, noncompliance is a major threat to obtaining power to detect intervention effects, because it is directly related to effect size and confidence interval. This study focuses on the exploration of cost-effective ways of optimizing intervention settings in the presence of noncompliance to reach a desirable level of statistical power. Given that compliance behavior of human participants is hard to control, it is demonstrated in the study that the quality of intervention effect estimates can be improved through more controllable factors such as appropriate statistical methods, pre-treatment covariates, and study design. This paper also provides power curves in various combinations of sample size and effect size as a guide for design of future studies."

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• Jo, B. (2002). Model misspecification sensitivity analysis in estimating causal effects of interventions with noncompliance. Statistics in Medicine, 21, 3161-3181.
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  Abstract

  "Randomized trials often face complications in assessing the effect of treatment because of study participants' noncompliance. If compliance type is observed in both the treatment and control conditions, the causal effect of treatment can be estimated for a targeted sub-population of interest based on compliance type. However, in practice, compliance type is not observed completely. Given this missing compliance information, the CACE (complier average causal effect) estimation approach provides a way to estimate differential effects of treatments by imposing the exclusion restriction for noncompliers. Under the exclusion restriction, the CACE approach estimates the effect of treatment assignment for compliers, but disallows the effect of treatment assignment for noncompliers. The exlusion restriction plays a key role in separating outcome distributions based on compliance type. However, the CACE estimate can be substantially biased if the assumption is violated. This study examines the bias mechanism in the estimation of CACE when the assumption of the exclusion restriction is violated. It is also examined how covariate information affects the sensitivity of the CACE estimate to violation of the exclusion restriction assumption."

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• Jo, B. (2002). Estimation of intervention effects with noncompliance: Alternative model specifications. Journal of Educational and Behavioral Statistics, 27, 385-409.
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  Abstract

  "This study examines alternative ways of specifying models in the CACE (complier average causal effect) estimation method, where the major interest is in estimating causal effects of treatments for compliers. A fundamental difficulty involved in the CACE estimation method is in dealing with missing compliance information among study participants. Given that, the assumption of the exclusion restriction plays a critical role in separating the distributions of compliers and noncompliers. If no pre-treatment covariates are available, assuming the exclusion restriction is unavoidable to obtain unique ML estimates in CACE models, although the assumption can be often unrealistic. One disadvantage of assuming the exclusion restriction is that the CACE estimate can be biased if the assumption is violated. Another disadvantage is that the assumption limits the flexibility of CACE modeling in practice. However, if pre-treatment covariates are available, more modeling options other than strictly forcing the exclusion restriction can be considered to establish identifiability of CACE models. This study explores modeling possibilities of CACE estimation within an ML-EM framework in the presence of covariate information."

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• Jo, B. & Muthén, B. (2001). Modeling of intervention effects with noncompliance: A latent variable approach for randomized trials. In G. Marcoulides & R.E. Schumacker (eds.) New Developments and Techniques in Structural Equation Modeling (pp. 57-87). Mahwah, New Jersey: Lawrence Erlbaum
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Non-Linear Factor Analysis expand topic collapse topic

• Bauer, D.J. (2005). The role of nonlinear factor-to-indicator relationships. Psychological Methods, 10, 305-316. This paper draws on techniques illustrated in the Mplus Version 3 User's Guide (first printed in March 2004), example 5.7.
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  Abstract

  "Measurement invariance is a necessary condition for the evaluation of factor mean differences over groups or time. This article considers the potential problems that can arise for tests of measurement invariance when the true factor-to-indicator relationship is nonlinear (quadratic) and invariant but the linear factor model is nevertheless applied. The factor loadings and indicator intercepts of the linear model will diverge across groups as the factor mean difference increases. Power analyses show that even apparently small quadratic effects can result in rejection of measurement invariance at moderate sample sizes when the factor mean difference is medium to large. Recommendations include the identification of nonlinear relationships using diagnostic plots and consideration of newly developed methods for fitting nonlinear factor models."

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Non-Linear SEM with a Semi-Parametric Approach expand topic collapse topic

• Pek, J., Sterba, S., Kok, B. & Bauer, D. (2009). Estimating and visualizing nonlinear relations among latent variables: A semiparametric approach. Multivariate Behavioral Research, 44, 407 - 436.
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  Abstract

  "The graphical presentation of any scientific finding enhances its description, in- terpretation, and evaluation. Research involving latent variables is no exception, especially when potential nonlinear effects are suspect. This article has multiple aims. First, it provides a nontechnical overview of a semiparametric approach to modeling nonlinear relationships among latent variables using mixtures of linear structural equations. Second, it provides several examples showing how the method works and how it is implemented and interpreted in practical applications. In particular, this article examines the potentially nonlinear relationships between positive and negative affect and cognitive processing. Third, a recommended dis- play format for illustrating latent bivariate relationships is demonstrated. Finally, the article describes an R package and an online utility that generate these displays automatically."

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• Bauer, D.J. (2005). A semiparametric approach to modeling nonlinear relations among latent variables. Structural Equation Modeling, 12, 513-534. This paper draws on techniques illustrated in the Mplus Version 3 User's Guide (first printed in March 2004), example 7.26.
  
  

Paired-Comparison and Ranking Data SEM expand topic collapse topic

• Maydeu-Olivares, A. & Bockenholt, U. (2005). Structural equation modeling of paired-comparison and ranking data. Psychological Methods, 10, 285-304.
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  Abstract

  "L. L. Thurstone’s (1927) model provides a powerful framework for modeling individual differences in choice behavior. An overview of Thurstonian models for comparative data is provided, including the classical Case V and Case III models as well as more general choice models with unrestricted and factor-analytic covariance structures. A flow chart summarizes the model selection process. The authors show how to embed these models within a more familiar structural equation modeling (SEM) framework. The different special cases of Thurstone’s model can be estimated with a popular SEM statistical package, including factor analysis models for paired comparisons and rankings. Only minor modifications are needed to accommodate both types of data. As a result, complex models for comparative judgments can be both estimated and tested efficiently."

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Randomized Response Modeling of Sensitive Questions Using Latent Classes expand topic collapse topic

• Hox, J. & Lensvelt-Mulders, G. (2004). Randomized response analysis in Mplus. Structural Equation Modeling, 11, 615-620.
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  Abstract

  "This article describes a technique to analyze randomized response data using available structural equation modeling (SEM) software. The randomized response technique was developed to obtain estimates that are more valid when studying sensitive topics. The basic feature of all randomized response methods is that the data are deliberately contaminated with error. This makes it difficult to relate randomized responses to explanatory variables. In this tutorial, we present an approach to this problem, in which the analysis of randomized response data is viewed as a latent class problem, with different latent classes for the random and the truthful response. To illustrate this technique, an example is presented using the program Mplus."

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Structural Equation Modeling expand topic collapse topic

• Marsh, H. W., Lüdtke, O., Muthén, B., Asparouhov, T., Morin, A. J. S., Trautwein, U. & Nagengast, B. (2010). A new look at the big-five factor structure through exploratory structural equation modeling. Forthcoming in Psychological Assessment.
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  Abstract

  "NEO instruments are widely used to assess big-five personality factors, but confirmatory factor analyses (CFA) conducted at the item-level do not support their a priori structure, due in part, to the overly restrictive CFA assumptions. We demonstrate that exploratory structural equation modeling (ESEM), an integration of CFA and EFA, overcomes these problems with responses (n= 3390) to the 60-item NEO-FFI: (a) ESEM fits the data better and results in substantially more differentiated (less correlated) factors than CFA; (b) tests of gender invariance with the 13-model ESEM taxonomy of full measurement invariance of factor loadings, factor variance-covariances, item uniquenesses, correlated uniquenesses, item intercepts, differential item functioning, and latent means show that women score higher on all NEO big-five factors; (c) longitudinal analyses support measurement invariance over time and the maturity principle (decreases in Neuroticism, increases in Agreeableness, Openness and Conscientiousness). Based on ESEM, we addressed substantively important questions with broad applicability to personality research that could not be appropriately addressed with either traditional EFA or CFA approaches. "

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• Asparouhov, T. & Muthén, B. (2009). Exploratory structural equation modeling. Structural Equation Modeling, 16, 397-438.
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  Abstract

  "Exploratory factor analysis (EFA) is a frequently used multivariate analysis technique in statistics. Jennrich and Sampson (1966) solved a significant EFA factor loading matrix rotation problem by deriving the direct Quartimin rotation. Jennrich was also the first to develop standard errors for rotated solutions, although these have still not made their way into most statistical software programs. This is perhaps because Jennrich’s achievements were partly overshadowed by the subsequent development of confirmatory factor analysis (CFA) by Jöreskog (1969). The strict requirement of zero cross-loadings in CFA, however, often does not fit the data well and has led to a tendency to rely on extensive model modification to find a well-fitting model. In such cases, searching for a well-fitting measurement model may be better carried out by EFA (Browne, 2001). Furthermore, misspecification of zero loadings usually leads to distorted factors with overestimated factor correlations and subsequent distorted structural relations. This article describes an EFA-SEM (ESEM) approach, where in addition to or instead of a CFA measurement model, an EFA measurement model with rotations can be used in a structural equation model. The ESEM approach has recently been implemented in the Mplus program. ESEM gives access to all the usual SEM parameters and the loading rotation gives a transformation of structural coefficients as well. Standard errors and overall tests of model fit are obtained. Geomin and Target rotations are discussed. Examples of ESEM models include multiple-group EFA with measurement and structural invariance testing, test–retest (longitudinal) EFA, EFA with covariates and direct effects, and EFA with correlated residuals. Testing strategies with sequences of EFA and CFA models are discussed. Simulated and real data are used to illustrate the points."

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• Marsh, H.W., Muthén, B., Asparouhov, A., Lüdtke, O., Robitzsch, A., Morin, A.J.S., & Trautwein, U. (2009). Exploratory structural equation modeling, integrating CFA and EFA: Application to students’ evaluations of university teaching. Structural Equation Modeling, 16, 439-476.
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  Abstract

  "This study is a methodological-substantive synergy, demonstrating the power and flexibility of exploratory structural equation modeling (ESEM) methods that integrate confirmatory and exploratory factor analyses (CFA and EFA), as applied to substantively important questions based on multidimentional students’ evaluations of university teaching (SETs). For these data, there is a well established ESEM structure but typical CFA models do not fit the data and substantially inflate correlations among the nine SET factors (median rs D .34 for ESEM, .72 for CFA) in a way that undermines discriminant validity and usefulness as diagnostic feedback. A 13-model taxonomy of ESEM measurement invariance is proposed, showing complete invariance (factor loadings, factor correlations, item uniquenesses, item intercepts, latent means) over multiple groups based on the SETs collected in the first and second halves of a 13-year period. Fully latent ESEM growth models that unconfounded measurement error from communality showed almost no linear or quadratic effects over this 13-year period. Latent multiple indicators multiple causes models showed that relations with background variables (workload/difficulty, class size, prior subject interest, expected grades) were small in size and varied systematically for different ESEM SET factors, supporting their discriminant validity and a construct validity interpretation of the relations. A new approach to higher order ESEM was demonstrated, but was not fully appropriate for these data. Based on ESEM methodology, substantively important questions were addressed that could not be appropriately addressed with a traditional CFA approach."

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• Moosbrugger, H., Schermelleh-Engel, K., Kelava; A. & Klein, A. G (2009). Testing multiple nonlinear effects in structural equation modeling: A comparison of alternative estimation approaches. Invited Chapter in T. Teo & M. S. Khine (Eds.), Structural Equation Modelling in Educational Research: Concepts and Applications. Rotterdam, NL: Sense Publishers.
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• Tucker-Drob, E.M. (2009). Differentiation of cognitive abilities across the lifespan. Accepted for publication in Developmental Psychology.
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  Abstract

  "Existing representations of cognitive ability structure are exclusively based on linear patterns of interrelations. However, a number of developmental and cognitive theories predict that abilities are differentially related across ages (age differentiation-dedifferentiation) and across levels of functioning (ability differentiation). Nonlinear factor analytic models were applied to multivariate cognitive ability data from 6,273 individuals, ages 4 to 101 years, who were selected to be nationally representative of the United States population. Results consistently supported ability differentiation, but were less clear with respect to age differentiation-dedifferentiation. Little evidence for age modification of ability differentiation was found. These findings are particularly informative about the nature of individual differences in cognition, and the developmental course of cognitive ability level and structure."

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• Cheung, M.W.L. (2008). A model for integrating fixed-, random-, and mixed-effects meta-analyses into structural equation modeling. Psychological Methods, 13, 182–202.
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  Abstract

  "Meta-analysis and structural equation modeling (SEM) are two important statistical methods in the behavioral, social, and medical sciences. They are generally treated as two unrelated topics in the literature. The present paper proposes a model to integrate fixed-, random-, and mixed-effects meta-analyses into the SEM framework. By applying an appropriate transformation on the data, studies in a meta-analysis can be analyzed as subjects in a structural equation model. This paper also highlights some practical benefits of using the SEM approach to conducting a meta-analysis. Specifically, the SEM based meta-analysis can be used to handle missing covariates, to quantify the heterogeneity of effect sizes, and to address the heterogeneity of effect sizes with mixture models. Examples are used to illustrate the equivalence between the conventional meta-analysis and the SEM based meta-analysis. Future directions on and issues related to the SEM based meta-analysis are discussed. Keywords: Meta-analysis, structural equation model, fixed-effects model, random-effects model, mixed-effects model"

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• Coenders, G., Batista-Foguet, J.M. & Saris, W. (2006). Simple, efficient and distribution-free approach to interaction effects in complex structural equation models. Quality & Quantity, 42, 369-396.
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  Abstract

  "Structural equation models with mean structure and non-linear constraints are the most frequent choice for estimating interaction effects when measurement errors are present. This article proposes eliminating the mean structure and all the constraints but one, which leads to a more easily handled model that is more robust to non-normality and more general as it can accommodate endogenous interactions and thus indirect effects. Our approach is compared to other approaches found in the literature with a Monte Carlo simulation and is found to be equally efficient under normality and less biased under non-normality. An empirical illustration is included."

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• Muthén, B., du Toit, S.H.C. & Spisic, D. (1997). Robust inference using weighted least squares and quadratic estimating equations in latent variable modeling with categorical and continuous outcomes. Unpublished technical report.
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  Abstract

  "This paper generalizes the robust weighted least-squares (WLS) approach of Muthén (1993) beyond the binary factor analysis model to the general structural equation model considered in Muthén (1984). A key feature in this generalization is the addition of covariates by which the means of the outcome variables can vary across the individuals of the sample. The paper relates the robust WLS approach to a generalized estimating equation (GEE) approach recently proposed by Melton and Liang (1997) both with respect to statistical performance and computational speed. It is shown that except for small sample sizes and strongly skewed distributions, the robust WLS approach performs statistically almost as well as GEE, produces good standard error estimates, but gives considerably faster computations. While in the Melton and Liang (1997) GEE context model testing is not straight-forward and was not provided, robust chi-square model testing is easily obtained in the WLS approach. As in Muthén (1984), the robust WLS approach is quite general in that it allows for a combination of binary, ordered polytomous, and continuous outcome variables and allows for multiple-group analysis. "

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• Muthén, B. & Satorra, A. (1995). Technical aspects of Muthén's LISCOMP approach to estimation of latent variable relations with a comprehensive measurement model. Psychometrika, 60, 489-503.
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  Abstract

  "Muthén (1984) formulated a general model and estimation procedure for structural equation modeling with a mixture of dichotomous, ordered categorical, and continuous measures of latent variables. A general three-stage procedure was developed to obtain estimates, standard errors, and a chi-square measure of fit for a given structural model. While the last step uses generalized least-squares estimation to fit a structural mode, the first two steps involve the computation of the statistics used in this model fitting. A key component in the procedure was the development of a GLS weight matrix corresponding to the asymptotic covariance matrix of the sample statistics computed in the first two stages. This paper extends the description of the asymptotics involved and shows how the Muthén formulas can be derived. The emphasis is placed on showing the asymptotic normality of the estimates obtained in the first and second stage and the validity of the weight matrix used in the GLS estimation of the third stage."

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• Muthén, B. (1989). Tobit factor analysis. British Journal of Mathematical and Statistical Psychology, 42, 241-250.
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  Abstract

  "A new approach is proposed for data which are skewed and have a sizeable proportion of observation at variable end points. Using a covariance structure modelling lramework, the new approach assumes censored multivariate normal variables. Using bivariate information, this leads to the use of 'tobit' correlations in weighted least squares estimation. The behaviour of the tobit approach is compared to that of normal theory estimation and ADF estimation."

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Survival Analysis expand topic collapse topic

• Masyn, K. E. (2009). Discrete-time survival factor mixture analysis for low-frequency recurrent event histories. Research in Human Development, 6, 165-194.
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  Abstract

  "In this article, the latent class analysis framework for modeling single event discrete-time survival data is extended to low-frequency recurrent event histories. A partial gap time model, parameterized as a restricted factor mixture model, is presented and illustrated using juvenile offending data. This model accommodates event-specific baseline hazard probabilities and covariate effects; event recurrences within a single time period; and accounts for within- and between-subject correlations of event times. This approach expands the family of latent variable survival models in a way that allows researchers to explicitly address questions about unobserved heterogeneity in the timing of events across the lifespan."

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• Masyn, K. E. (2008). Modeling measurement error in event occurrence for single, non-recurring events in discrete-time survival analysis. In Hancock, G. R., & Samuelsen, K. M. (Eds.), Advances in latent variable mixture models, pp. 105-145. Charlotte, NC: Information Age Publishing, Inc. Click here for information about the book.
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• Asparouhov, T., Masyn, K. & Muthen, B. (2006). Continuous time survival in latent variable models. Proceedings of the Joint Statistical Meeting in Seattle, August 2006. ASA section on Biometrics, 180-187.
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  Abstract

  "We describe a general multivariate, multilevel framework for continuous time survival analysis that includes joint modeling of survival time variables and continuous and categorical observed and latent variables. The proposed framework is implemented in the Mplus software package. The survival time variables are modeled with nonparametric or parametric proportional hazard distributions and include right censoring. The proposed modeling framework includes finite mixtures of Cox regression models with and without class-specific baseline hazards, multilevel Cox regression models, and multilevel frailty models. We illustrate the framework with several simulation studies. Comparison is made with discrete time survival models. We also investigate the effect of ties on the proposed estimation method. Simulation studies are conducted to compare the methods implemented in Mplus with those implemented in SAS."

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• Muthén, B. & Masyn, K. (2005). Discrete-time survival mixture analysis. Journal of Educational and Behavioral Statistics, 30, 27-58.
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  Abstract

  "This article proposes a general latent variable approach to discrete-time survival analysis of nonrepeatable events such as onset of drug use. It is shown how the survival analysis can be formulated as a generalized latent class analysis of event history indicators. The latent class analysis can use covariates and can be combined with the joint modeling of other outcomes such as repeated measures for a related process. It is shown that conventional discrete-time survival analysis corresponds to a single-class latent class analysis. Multiple-class extensions are proposed, including the special cases of a class of long-term survivors and classes defined by outcomes related to survival. The estimation uses a general latent variable framework, including both categorical and continuous latent variables and incorporated in the Mplus program. Estimation is carried out using maximum likelihood via the EM algorithm. Two examples serve as illustrations. The first example concerns recidivism after incarceration in a randomized field experiment. The second example concerns school removal related to the development of aggressive behavior in the classroom."

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• Masyn, K. E. (2003). Discrete-time survival mixture analysis for single and recurrent events using latent variables. Doctoral dissertation, University of California, Los Angeles.
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  Abstract

  "Survival analysis refers to the general set of statistical methods developed specifically to model the timing of events. This dissertation concerns a subset of those methods that deals with events measured or occurring in discrete-time or grouped-time intervals. A method for modeling single event discrete-time data utilizing a latent class regression (LCR) framework, originally presented by Muthén and Masyn (2001), is further developed and detailed. It is shown that discrete-time data can be represented as a set of binary event indicators and observed risk indicators that allow estimation using a latent class regression specification under a missing-at-random assumption that corresponds to the assumption of noninformative right-censoring. The modeling of the effects of time-dependent and time-independent covariates with constant or time-varying effects is demonstrated along with approaches to model testing. The LCR framework also allows for the modeling of unobserved heterogeneity through finite mixture modeling, i.e., multiple latent classes. The problems of ignoring unobserved heterogeneity and the challenges of discrete-time mixture model identification and specification for single event data are discussed. The LCR model for single event data is extended to recurrent event survival data with a focus on recurrent event processes with a low frequency of recurrences. The gap time, counting process, and total time formulations in the continuous-time setting are all reformulated for discrete-time and model specification and estimation is demonstrated for all three. The proposed model accommodates event-specific baseline hazard probabilities as well as event-specific covariate effects. The model also allows for multiple event occurrences in a single time period for a single subject and accounts for within as well as between subject correlation of event times though the same mixture modeling approach given for single event data. All models are illustrated with data on the event times of domestic violence episodes perpetrated by a sample of married men observed for 12 months after an alcohol treatment program. Opportunities for future methodology developments for discrete-time models are discussed."

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Two-Part Growth Modeling with a Preponderance of Zeros expand topic collapse topic

• Petras, H., Nieuwbeerta, P., & Piquero, A.R. (2009). Participation and frequency during criminal careers over the life span. Accepted for publication in Criminology.
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  Abstract

  "Recent advances and debates surrounding general/developmental and static/dynamic theories of crime can be traced to the 1986 National Academy of Science Report on criminal careers and the discussion it generated. A key point of contention lies in the interpretation of the age-crime curve. For Gottfredson and Hirschi, the decline in the age-crime curve in early adulthood reflects decreasing individual offending frequency (?) after the peak. Blumstein et al. claim that the decline in the aggregate age-crime curve can also be attributable to the termination of criminal careers, and the average value of ? could stay constant (or increase with age) for those offenders who remain active after that peak. Using data from the Criminal Career and Life Course Study - including information on criminal convictions over 60 years of almost 5,000 persons convicted in the Netherlands - and applying a Two-Part Growth Model that explicitly distinguishes between participation and frequency - the paper assesses the participation/frequency debate. Results suggest that the decline in the age-crime curve in early adulthood reflects both decreasing individual offending participation and frequency after the peak, that the probabilities of participation and frequency are significantly related at the individual level, and that sex and marriage influence both participation and frequency."

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• Vazsonyi, A.T. & Keiley, M.K. (2007). Normative developmental trajectories of aggressive behaviors in African American, American Indian, Asian American, Caucasian, and Hispanic children and early adolescents. Journal of Abnormal Child Psychology, 35, 1047-1062.
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  Abstract

  "The current 5-year accelerated longitudinal investigation modeled the developmental trajectories of aggressive behaviors in 10,107 predominantly minority (>70%; African American, American Indian, Asian American, and Hispanic) children and early adolescents (Kindergarten through 8th grade, 49% female youth) from lower to lower–middle socioeconomic strata. Based on a two-part latent growth model, findings suggest that the probability and frequency of aggressive behavior use decreases slightly (linear) through the elementary school years and then increases as children move into middle school (quadratic). Though mean level differences were found across ethnic and racial groups, socioeconomic strata, and particularly by sex at initial status, rates of change over time across all groups were invariant. Findings suggest that potential socialization differences, if any, occur pre-Kindergarten in all groups."

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• Witkiewitz, K. & Masyn, K. (2007). Drinking trajectories following an initial lapse.  In press, Psychology of Addictive Behaviors.
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  Abstract

  "Relapse following alcohol treatment is a major problem for individuals who are alcohol dependent, yet little is known about the course of drinking after the initial lapse. In the current study, discrete-time survival analysis and latent growth mixture modeling were used to evaluate the time to first lapse and the trajectories of post-lapse drinking in a sample of 563 individuals who received community alcohol treatment. Results showed a decreasing risk of lapsing over time. After the initial lapse, three trajectory subgroups provided a parsimonious representation of the heterogeneity in post-lapse drinking frequency and quantity, with the majority of individuals reporting light, infrequent drinking. Covariate analyses incorporating demographics, distal risk factors, time-to-first lapse, and coping behavior as predictors of time-to-lapse and post-lapse drinking trajectories indicated alcohol dependence and coping behavior were the strongest predictors of lapsing and post-lapse drinking behavior."

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• Brown, E.C., Catalano, C.B., Fleming, C.B., Haggerty, K.P. & Abbot, R.D. (2005). Adolescent substance use outcomes in the Raising Healthy Children Project: A two-part latent growth curve analysis. Journal of Consulting and Clinical Psychology, 73, 699-710. Mplus outputs used in this paper can be viewed and/or downloaded from the Examples page.
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  Abstract

  "Raising Healthy Children (RHC) is a preventive intervention designed to promote positive youth development by targeting developmentally appropriate risk and protective factors. This study tested the efficacy of the RHC intervention on reducing alcohol, marijuana, and cigarette use during early to middle adolescence. Ten public schools, comprising 959 students, were matched and assigned randomly to either intervention or control conditions. A two-part latent growth modeling strategy was employed to examine change in both use-vs.-nonuse and frequency-of-use outcomes. Results indicated significant (p < .05) intervention effects in growth trajectories for frequency of alcohol and marijuana use but not for use vs. nonuse. These findings provide support for preventive interventions that take a social development perspective in targeting empirically supported risk and protective factors and demonstrate the utility of two-part models in adolescent substance use research."

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Miscellaneous expand topic collapse topic

• Hatton, H., Donnellan, M.B., Masyn, K., Feldman, B.J., Larsen-Rife, D., & Conger, R.D. (2008). Family and individual difference predictors of trait aspects of negative interpersonal behaviors during emerging adulthood. Journal of Family Psychology, 22, 448-455.
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  Abstract

  "A latent trait-state-occasion (TSO) model (D. A. Cole, N. C. Martin, & J. H. Steiger, 2005) was used to isolate the trait and state components of negative interpersonal behaviors toward a friend or romantic partner during emerging adulthood. Results indicate that variance in negative interpersonal behaviors was due to nearly equal portions of Trait and Occasion factors. Variability in the trait aspects of negative interpersonal behaviors was then predicted by theoretically relevant constructs. In particular, mothers’ negative behaviors during adolescence, adolescent core self-evaluations, negative emotionality, and feelings of security in close relationships had independent effects in predicting the enduring aspects of negative interpersonal behaviors. All told, these results indicate that TSO models can be helpful tools for understanding the developmental antecedents of the trait-like aspects of interpersonal processes. "

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• Temme, D., Paulssen, M., & Dannewald, T. (2008). Incorporating latent variables into discrete choice models – A simultaneous estimation approach using SEM software. BuR – Business Research, 1, 220-237.
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  Abstract

  "Integrated choice and latent variable (ICLV) models represent a promising new class of models which merge classic choice models with the structural equation approach (SEM) for latent variables. Despite their conceptual appeal, applications of ICLV models in marketing remain rare. We extend previous ICLV applications by first estimating a multinomial choice model and, second, by estimating hierarchical relations between latent variables. An empirical study on travel mode choice clearly demonstrates the value of ICLV models to enhance the understanding of choice processes. In addition to the usually studied directly observable variables such as travel time, we show how abstract motivations such as power and hedonism as well as attitudes such as a desire for flexibility impact on travel mode choice. Furthermore, we show that it is possible to estimate such a complex ICLV model with the widely available structural equation modeling package Mplus. This finding is likely to encourage more widespread application of this appealing model class in the marketing field."

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• Dagne, G.A., Howe, G.W., Brown, C.H., & Muthén, B. (2002). Hierarchical modeling of sequential behavioral data: An empirical Bayesian approach. Psychological Methods, 7, 262-280. Mplus inputs and outputs used in this paper can be viewed and/or downloaded from the Examples page.
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  Abstract

  "This paper reviews the common methods for measuring strength of contingency between two behaviors in a behavioral sequence, the binomial z-score and the adjusted cell residual, and points out a number of limitations with these approaches. It presents a new approach using log odds ratios and employing empirical Bayes estimation in the context of hierarchical modeling, an approach not constrained by these limitations. A series of hierarchical models is presented to test the stationarity of behavioral sequences, the homogeneity of sequences across the sample of episodes, and whether covariates can account for variation in sequences across the sample. These models are applied to observational data taken from a study of the behavioral interactions of 254 couples, to illustrate their use."

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• Muthén, L.K. & Muthén, B.O. (2002). How to use a Monte Carlo study to decide on sample size and determine power. Structural Equation Modeling, 4, 599-620. Mplus inputs and outputs used in this paper can be viewed and/or downloaded from the Examples page.
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  Abstract

  "A common question asked by researchers is, 'What sample size do I need for my study?' Over the years, several rules of thumb have been proposed. In reality there is no rule of thumb that applies to all situations. The sample size needed for a study depends on many factors including the size of the model, distribution of the variables, amount of missing data, reliability of the variables, and strength of the relationships among the variables. The purpose of this paper is to demonstrate how substantive researchers can use a Monte Carlo study to decide on sample size and determine power. Two models ae used as examples, a confirmatory factor analysis (CFA) model and a growth model. The analyses are carried out using the Mplus program (Muthén & Muthén, 1998)."

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• Yu, C.Y. (2002). Evaluating cutoff criteria of model fit indices for latent variable models with binary and continuous outcomes. Doctoral dissertation, University of California, Los Angeles.
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  Abstract

  "The aims of this study are to first evaluate the performance of various model fit measures under different model and data conditions, and secondly, to examine the adequacy of cutoff criteria for some model fit measures. Model fit indices, along with some test statistics, are meant to assess model fit in latent variable models. They are frequently applied to judge whether the model of interest is a good fit to the data. Since Bentler and Bonett (1980) popularized the concept of model fit indices, numerous studies have been done to propose new fit indices or to compare various fit indices. Most of the studies, however, are limited to continuous outcomes and to measurement models, such as confirmatory factor analysis models (CFA). The present study broadens the structure of models by including the multiple causes and multiple indicators (MIMIC) and latent growth curve models. Moreover, both binary and continuous outcomes are investigated in the CFA and MIMIC models. Weighted root-mean-square residual (WRMR), a new fit index, is empirically evaluated and compared to the Tucker-Lewis Index (TLI), the Comparative Fit Index (CFI), the root-mean-square error of approximation (RMSEA) and the standardized root-mean square residual (SRMR). Few studies have investigated the adequacy of cutoff criteria for fit indices. Thus study applied the method demonstrated in Hu and Bentler (1999) to evaluate the adequacy of cutoff criteria for the fit indices. The adequacy of a conventional probability level of 0.05 for chi-square to assess model fit is also investigated. With non-normal continuous outcomes, the Satorra-Bentler rescaled chi-square (SB) is incorporated into the calculation of TLI, CFI and RMSEA, and these SB-based fit measures are evaluated under various cutoff values. An example of applying adequate cutoff values of overall fit indices is illustrated using the Holzinger and Swineford data. Generally speaking, the use of SRMR with binary outcomes is not recommended. A cutoff value close to 1.0 for WRMR is suitable under most conditions but is not recommended for latent growth curve models with more time points. CFI performs relatively better than TLI and RMSEA, and a cutoff value close to 0.96 for CFI has acceptable rejection rates across models when N is greater than or equal to 250."

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