Sample Size Requirements in Single- and Multiphase Growth Mixture Models: A Monte Carlo Simulation Study

Just as growth mixture models are useful with single-phase longitudinal data, multiphase growth mixture models can be used with multiple-phase longitudinal data. One of the practically important issues in single- and multiphase growth mixture models is the sample size requirements for accurate estimation. In a Monte Carlo simulation study, the sample sizes required for using these models are investigated under various theoretical and realistic conditions. In particular, the relationship between the sample size requirement and the number of indicator variables is examined, because the number of indicators can be relatively easily controlled by researchers in many multiphase data collection settings such as ecological momentary assessment. The findings not only provide tangible information about required sample sizes under various conditions to help researchers, but they also increase understanding of sample size requirements in single- and multiphase growth mixture models.     

Fitting Unstructured Finite Mixture Models in Longitudinal Design: A Recommendation for Model Selection and Estimation of the Number of Classes

In longitudinal design, investigating interindividual differences of intraindividual changes enables researchers to better understand the potential variety of development and growth. Although latent growth curve mixture models have been widely used, unstructured finite mixture models (uFMMs) are also useful as a preliminary tool and are expected to be more robust in identifying classes under the influence of possible model misspecifications, which are very common in actual practice. In this study, large-scale simulations were performed in which various normal uFMMs and nonnormal uFMMs were fit to evaluate their utility and the performance of each model selection procedure for estimating the number of classes in longitudinal designs. Results show that normal uFMMs assuming invariance of variance–covariance structures among classes perform better on average. Among model selection procedures, the Calinski–Harabasz statistic, which has a nonparametric nature, performed better on average than information criteria, including the Bayesian information criterion.     

Investigating Stage-Sequential Growth Mixture Models with Multiphase Longitudinal Data

This article investigates three types of stage-sequential growth mixture models in the structural equation modeling framework for the analysis of multiple-phase longitudinal data. These models can be important tools for situations in which a single-phase growth mixture model produces distorted results and can allow researchers to better understand population heterogeneity and growth over multiple phases. Through theoretical and empirical comparisons of the models, the authors discuss strategies with respect to model selection and interpreting outcomes. The unique attributes of each approach are illustrated using ecological momentary assessment data from a tobacco cessation study. Transitional discrepancy between phases as well as growth factors are examined to see whether they can give us useful information related to a distal outcome, abstinence at 6 months postquit. It is argued that these statistical models are powerful and flexible tools for the analysis of complex and detailed longitudinal data.     

Fitting Structural Equation Model Trees and Latent Growth Curve Mixture Models in Longitudinal Designs: The Influence of Model Misspecification

When conducting longitudinal research, the investigation of between-individual differences in patterns of within-individual change can provide important insights. In this article, we use simulation methods to investigate the performance of a model-based exploratory data mining technique—structural equation model trees (SEM trees; Brandmaier, Oertzen, McArdle, & Lindenberger, 2013)—as a tool for detecting population heterogeneity. We use a latent-change score model as a data generation model and manipulate the precision of the information provided by a covariate about the true latent profile as well as other factors, including sample size, under the possible influences of model misspecifications. Simulation results show that, compared with latent growth curve mixture models, SEM trees might be very sensitive to model misspecification in estimating the number of classes. This can be attributed to the lower statistical power in identifying classes, resulting from smaller differences of parameters prescribed by the template model between classes.     

Deciding on the Number of Classes in Latent Class Analysis and Growth Mixture Modeling: A Monte Carlo Simulation Study

Mixture modeling is a widely applied data analysis technique used to identify unobserved heterogeneity in a population. Despite mixture models' usefulness in practice, one unresolved issue in the application of mixture models is that there is not one commonly accepted statistical indicator for deciding on the number of classes in a study population. This article presents the results of a simulation study that examines the performance of likelihood-based tests and the traditionally used Information Criterion (ICs) used for determining the number of classes in mixture modeling. We look at the performance of these tests and indexes for 3 types of mixture models: latent class analysis (LCA), a factor mixture model (FMA), and a growth mixture models (GMM). We evaluate the ability of the tests and indexes to correctly identify the number of classes at three different sample sizes (n = 200, 500, 1,000). Whereas the Bayesian Information Criterion performed the best of the ICs, the bootstrap likelihood ratio test proved to be a very consistent indicator of classes across all of the models considered.     

Measurement Invariance Testing Across Between-Level Latent Classes Using Multilevel Factor Mixture Modeling

This simulation study examines the efficacy of multilevel factor mixture modeling (ML FMM) for measurement invariance testing across unobserved groups when the groups are at the between level of multilevel data. To this end, latent classes are generated with class-specific item parameters (i.e., factor loading and intercept) across the between-level classes. The efficacy of ML FMM is evaluated in terms of class enumeration, class assignment, and the detection of noninvariance. Various classification criteria such as Akaike’s information criterion, Bayesian information criterion, and bootstrap likelihood ratio tests are examined for the correct enumeration of between-level latent classes. For the detection of measurement noninvariance, free and constrained baseline approaches are compared with respect to true positive and false positive rates. This study evidences the adequacy of ML FMM. However, its performance heavily depends on the simulation factors such as the classification criteria, sample size, and the magnitude of noninvariance. Practical guidelines for applied researchers are provided.     

Inferring Longitudinal Relationships Between Variables: Model Selection Between the Latent Change Score and Autoregressive Cross-Lagged Factor Models

This research focuses on the problem of model selection between the latent change score (LCS) model and the autoregressive cross-lagged (ARCL) model when the goal is to infer the longitudinal relationship between variables. We conducted a large-scale simulation study to (a) investigate the conditions under which these models return statistically (and substantively) different results concerning the presence of bivariate longitudinal relationships, and (b) ascertain the relative performance of an array of model selection procedures when such different results arise. The simulation results show that the primary sources of differences in parameter estimates across models are model parameters related to the slope factor scores in the LCS model (specifically, the correlation between the intercept factor and the slope factor scores) as well as the size of the data (specifically, the number of time points and sample size). Among several model selection procedures, correct selection rates were higher when using model fit indexes (i.e., comparative fit index, root mean square error of approximation) than when using a likelihood ratio test or any of several information criteria (i.e., Akaike’s information criterion, Bayesian information criterion, consistent AIC, and sample-size-adjusted BIC).     

The Impact of Various Class-Distinction Features on Model Selection in the Mixture Rasch Model

The purpose of the current study is to examine the performance of four information criteria (Akaike's information criterion [AIC], corrected AIC [AICC] Bayesian information criterion [BIC], sample-size adjusted BIC [SABIC]) for detecting the correct number of latent classes in the mixture Rasch model through simulations. The simulation study manipulated various class-distinction features (percentages of class-variant items, magnitudes, and patterns of item difficulty differences) and mixing proportions, assuming that a mixture Rasch model with two latent classes was the true model. Unlike previous studies that showed BIC's superiority to other indices, our findings from this study suggested that the four information criteria had differential performance depending on the percentage of class-variant items and the magnitude and pattern of item difficulty differences under a two-class structure. Furthermore, the present study revealed that AICC and SABIC generally performed as good as or better than their counterparts, AIC and BIC, respectively, for the class-class structure with a sample of 3,000.     

Detecting Appropriate Trajectories of Growth in Latent Growth Models: The Performance of Information-Based Criteria

Latent growth modeling (LGM) is a popular and flexible technique that may be used when data are collected across several different measurement occasions. Modeling the appropriate growth trajectory has important implications with respect to the accurate interpretation of parameter estimates of interest in a latent growth model that may impact educational policy decisions. A Monte Carlo simulation study was conducted to examine the accuracy of six information-based criteria (i.e., AIC, CAIC, AICC, BIC, nBIC, and HQIC) when selecting among various growth trajectories modeled using LGM under different sample size, number of time points, and growth trajectory scenarios. The accuracy of the information criteria generally improved as sample size increased. The cubic and linear growth models were distinguished most accurately by the information criteria. All of the nonlinear models were more easily distinguished as the number of time points increased. The comparative performance of the six information criteria was dependent upon the manipulated conditions. Implications of the findings are discussed.     

A Comparison of Label Switching Algorithms in the Context of Growth Mixture Models

Simulation studies involving mixture models inevitably aggregate parameter estimates and other output across numerous replications. A primary issue that arises in these methodological investigations is label switching. The current study compares several label switching corrections that are commonly used when dealing with mixture models. A growth mixture model is used in this simulation study, and the design crosses three manipulated variables—number of latent classes, latent class probabilities, and class separation, yielding a total of 18 conditions. Within each of these conditions, the accuracy of a priori identifiability constraints, a priori training of the algorithm, and four post hoc algorithms developed by Tueller et al.; Cho; Stephens; and Rodriguez and Walker are tested to determine their classification accuracy. Findings reveal that, of all a priori methods, training of the algorithm leads to the most accurate classification under all conditions. In a case where an a priori algorithm is not selected, Rodriguez and Walker’s algorithm is an excellent choice if interested specifically in aggregating class output without consideration as to whether the classes are accurately ordered. Using any of the post hoc algorithms tested yields improvement over baseline accuracy and is most effective under two-class models when class separation is high. This study found that if the class constraint algorithm was used a priori, it should be combined with a post hoc algorithm for accurate classification.     

The Impact of Ignoring Time Series Processes in Linear Growth Mixture Modeling

Growth mixture modeling (GMM) is a useful statistical method for longitudinal studies because it includes features of both latent growth modeling (LGM) and finite mixture modeling. This Monte Carlo simulation study explored the impact of ignoring 3 types of time series processes (i.e., AR(1), MA(1), and ARMA(1,1)) in GMM and manipulated the separation of the latent classes, the strength of the time series process, and whether the errors conformed to the time series process in 1 or 2 latent classes. The results showed that omitting time series processes resulted in more serious bias in parameter estimation as the distance between classes increased. However, when the class distances were small, ignoring time series processes contributed to the selection of the correct number of classes. When the GMM models correctly specified the time series process, only models with an AR(1) time series process produced unbiased parameter estimates in most conditions. It was also found that among design factors manipulated, the distance between classes prominently affected the identification of the number of classes and parameter estimation.     

Spurious Latent Classes in the Mixture Rasch Model

Mixture Rasch models have been used to study a number of psychometric issues such as goodness of fit, response strategy differences, strategy shifts, and multidimensionality. Although these models offer the potential for improving understanding of the latent variables being measured, under some conditions overextraction of latent classes may occur, potentially leading to misinterpretation of results. In this study, a mixture Rasch model was applied to data from a statewide test that was initially calibrated to conform to a 3‐parameter logistic (3PL) model. Results suggested how latent classes could be explained and also suggested that these latent classes might be due to applying a mixture Rasch model to 3PL data. To support this latter conjecture, a simulation study was presented to demonstrate how data generated to fit a one‐class 2‐parameter logistic (2PL) model required more than one class when fit with a mixture Rasch model.     

Predictions of Individual Change Recovered With Latent Class or Random Coefficient Growth Models

Popular longitudinal models allow for prediction of growth trajectories in alternative ways. In latent class growth models (LCGMs), person-level covariates predict membership in discrete latent classes that each holistically define an entire trajectory of change (e.g., a high-stable class vs. late-onset class vs. moderate-desisting class). In random coefficient growth models (RCGMs, also known as latent curve models), however, person-level covariates separately predict continuously distributed latent growth factors (e.g., an intercept vs. slope factor). This article first explains how complex and nonlinear interactions between predictors and time are recovered in different ways via LCGM versus RCGM specifications. Then a simulation comparison illustrates that, aside from some modest efficiency differences, such predictor relationships can be recovered approximately equally well by either model—regardless of which model generated the data. Our results also provide an empirical rationale for integrating findings about prediction of individual change across LCGMs and RCGMs in practice.     

Performance of Factor Mixture Models as a Function of Model Size,Covariate Effects,and Class-Specific Parameters

Abstract

Factor mixture models are designed for the analysis of multivariate data obtained from a population consisting of distinct latent classes. A common factor model is assumed to hold within each of the latent classes. Factor mixture modeling involves obtaining estimates of the model parameters, and may also be used to assign subjects to their most likely latent class. This simulation study investigates aspects of model performance such as parameter coverage and correct class membership assignment and focuses on covariate effects, model size, and class-specific versus class-invariant parameters. When fitting true models, parameter coverage is good for most parameters even for the smallest class separation investigated in this study (0.5 SD between 2 classes). The same holds for convergence rates. Correct class assignment is unsatisfactory for the small class separation without covariates, but improves dramatically with increasing separation, covariate effects, or both. Model performance is not influenced by the differences in model size investigated here. Class-specific parameters may improve some aspects of model performance but negatively affect other aspects.     

The Impact of Multidimensionality on Extraction of Latent Classes in Mixture Rasch Models

This study investigates the effect of multidimensionality on extraction of latent classes in mixture Rasch models. In this study, two‐dimensional data were generated under varying conditions. The two‐dimensional data sets were analyzed with one‐ to five‐class mixture Rasch models. Results of the simulation study indicate the mixture Rasch model tended to extract more latent classes than the number of dimensions simulated, particularly when the multidimensional structure of the data was more complex. In addition, the number of extracted latent classes decreased as the dimensions were more highly correlated regardless of multidimensional structure. An analysis of the empirical multidimensional data also shows that the number of latent classes extracted by the mixture Rasch model is larger than the number of dimensions measured by the test.     

Detecting Unobserved Heterogeneity in Latent Growth Curve Models

Growth mixture models combine latent growth curve models and finite mixture models to examine the existence of latent classes that follow distinct developmental patterns. Analyses based on these models are becoming quite common in social and behavioral science research because of recent advances in computing, the availability of specialized statistical programs, and the ease of programming. In this article, we show how mixture models can be fit to examine the presence of multiple latent classes by algorithmically grouping or clustering individuals who follow the same estimated growth trajectory based on an evaluation of individual case residuals. The approach is illustrated using empirical longitudinal data along with an easy to use computerized implementation.     

A New Perspective on the Effects of Covariates in Mixture Models

In this simulation study, we explored the effect of introducing covariates to a growth mixture model when covariates were also generated by a mixture model. We varied the association between the latent classes underlying the growth trajectories and the covariates, the degree of separation between the latent classes underlying the covariates, the number of covariates included, and amount of missing data in the growth data. We found that adding covariates to the growth mixture model generally hurt class recovery except where the latent classes underlying the growth trajectories and the covariates were the same or very strongly associated, and there was a large degree of separation between the classes underlying the covariates. We found that when covariates were introduced, entropy might no longer be an accurate indicator of the distinctiveness of the growth trajectory classes.     

Estimation of Time-Unstructured Nonlinear Mixed-Effects Mixture Models

Change over time often takes on a nonlinear form. Furthermore, change patterns can be characterized by heterogeneity due to unobserved subpopulations. Nonlinear mixed-effects mixture models provide one way of addressing both of these issues. This study attempts to extend these models to accommodate time-unstructured data. We develop methods to fit these models in both the structural equation modeling framework as well as the Bayesian framework and evaluate their performance. Simulations show that the success of these methods is driven by the separation between latent classes. When classes are well separated, a sample of 200 is sufficient. Otherwise, a sample of 1,000 or more is required before parameters can be accurately recovered. Ignoring individually varying measurement occasions can also lead to substantial bias, particularly in the random-effects parameters. Finally, we demonstrate the application of these techniques to a data set involving the development of reading ability in children.     

The GRoLTS-Checklist: Guidelines for Reporting on Latent Trajectory Studies

Estimating models within the mixture model framework, like latent growth mixture modeling (LGMM) or latent class growth analysis (LCGA), involves making various decisions throughout the estimation process. This has led to a wide variety in how results of latent trajectory analysis are reported. To overcome this issue, using a 4-round Delphi study, we developed Guidelines for Reporting on Latent Trajectory Studies (GRoLTS). The purpose of GRoLTS is to present criteria that should be included when reporting the results of latent trajectory analysis across research fields. We have gone through a systematic process to identify key components that, according to a panel of experts, are necessary when reporting results for trajectory studies. We applied GRoLTS to 38 papers where LGMM or LCGA was used to study trajectories of posttraumatic stress after a traumatic event.     

Mixed Mode Latent Class Analysis: An Examination of Fit Index Performance for Classification

This Monte Carlo study examines the performance of fit indices commonly used by applied researchers interested in finite mixture modeling for the purposes of classification. Conditions for the simulation study were selected to reflect conditions found in applied educational and psychological research. The factors included in the investigation were metric level of indicators, sample size, and class prevalence. All models contained a combination of categorical and continuous indicators. All categorical indicators were dichotomous, and continuous indicators were normally distributed. The fit indices examined were Akaike’s information criterion, Bayesian information criterion (BIC), sample size-adjusted Bayesian information criterion (SSBIC), integrated classification likelihood criterion with Bayesian-type approximation, and Lo–Mendell–Rubin likelihood ratio test. Overall, SSBIC tended to identify the correct solution with higher frequency than other indices. BIC tended to identify the correct solution with higher frequency than the other indices in models with more continuous than categorical indicators, or when rare classes were absent.