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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  Abstract

  "We describe a multivariate, multilevel, pseudo maximum likelihood estimation method for multistage stratified cluster sampling designs, including finite population and unequal probability sampling. Multilevel models can be estimated with this method while incorporating the sampling design in the standard error computation. Design based adjustment of the likelihood ratio test (LRT) statistic is proposed. We also discuss multiple group and subpopulation analysis in this context. Simulation studies are conducted to evaluate the performance of the proposed estimator and test statistic. We also compare the estimators and the LRT adjustments implemented in Mplus and LISREL in simulation studies."

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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

• Kim, Y.K. & Muthén, B. (2007). Two-part factor mixture modeling:  Application to an aggressive behavior measurement instrument.  Accepted for publication in Structural Equation Modeling.
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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

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

  "Previous studies have documented that smoking during pregnancy (SDP) is associated with offspring externalizing problems, even when measured covariates were used to control for possible confounds. However, the association may be because of nonmeasured environmental and genetic factors that increase risk for offspring externalizing problems. The current project used the National Longitudinal Survey of Youth and their children, ages 4–10 years, to explore the relations between SDP and offspring conduct problems (CPs), oppositional defiant problems (ODPs), and attention-deficit/ hyperactivity problems (ADHPs) using methodological and statistical controls for confounds. When offspring were compared to their own siblings who differed in their exposure to prenatal nicotine, there was no effect of SDP on offspring CP and ODP. This suggests that SDP does not have a causal effect on offspring CP and ODP. There was a small association between SDP and ADHP, consistent with a causal effect of SDP, but the magnitude of the association was greatly reduced by methodological and statistical controls. Genetically informed analyses suggest that unmeasured environmental variables influencing both SDP and offspring externalizing behaviors account for the previously observed associations. That is, the current analyses imply that important unidentified environmental factors account for the association between SDP and offspring externalizing problems, not teratogenic effects of SDP."

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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

• 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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  "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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  "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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• Muthén, B., Brown, H., Leuchter, A. & Hunter A. (2008). 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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  "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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• 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 (in press). 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. For a copy of the paper, contact the first author and mention paper #85. 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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IRT expand topic collapse topic

• 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

• 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

• 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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• Kaplan, D. (2006). An overview of Markov chain methods for the study of stage-sequential developmental processes. Forthcoming in Developmental Psychology.
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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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• 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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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

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

• 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

• 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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• Muthén, B. & Asparouhov, T. (2008). Multilevel regression mixture analysis. Forthcoming in Journal of the Royal Statistical Society, Series A.
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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 mixture model. The two-level regression mixture analyses show that unobserved heterogeneity should not be presupposed to exist only on level 2 at the expense of level 1. Both the simulated and 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 used in conventional two-level regression analysis. Due to 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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Multilevel SEM expand topic collapse topic

• 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 specific