The Impact and Interpretation of Modeling Residual Noninvariance in Growth Mixture Models

Current practices for growth mixture modeling emphasize the importance of the proper parameterization and number of classes, but the impact of these decisions on latent class composition and the substantive implications has not been thoroughly addressed. Using measures of behavior from 575 middle school students, we compared the results of several multilevel growth mixture models. Results indicated a dramatic shift in class assignment as the models allowed class-varying parameters, with different substantive interpretations and resulting typologies. This research suggests that using variability as a criterion for class differences in a behavior typology can dramatically impact latent class membership. This study describes decisions and results from testing for noninvariance, with particular emphasis on how decisions about the nature of within-person variance can affect resulting subgroups and model parameters.     

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.     

Residual Structures in Growth Models With Ordinal Outcomes

Growth models allow for the study of within-person change and between-person differences in within-person change. Typically, these models are applied to continuous variables where the residuals are assumed to be normally distributed. With normally distributed residuals there are a variety of residual structures that can be imposed and tested, which have been shown to affect model fit and parameter estimation. This article concerns residual structures in growth models with binary and ordered categorical outcomes using the probit link function. Different residual structures and their appropriateness for growth data are discussed and their use is illustrated with longitudinal data collected as part of Head Start’s Family and Child Experiences Survey 1997 Cohort. We close with recommendations for the specification and parameterization of growth models that use the probit link.     

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.     

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.     

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.     

On the Application of the Three-Step Approach to Growth Mixture Models

This series of simulation studies evaluate, in the context of applied research settings, the impact of the parameterization of the covariance structure of the growth mixture model (GMM) on the regression coefficient and standard error estimates in the 3-step method. The results show that the 1-step approach performs better than the 3-step method across the simulation studies. However, the performance of the 3-step method depends slightly or importantly on the parameterization of the GGM from the first step, on the inclusion or not of the predictor at the first step of the analysis, on the population model, and on the type (i.e., logit vs. linear) and size of the regression coefficient estimates.     

Determining the Number of Latent Classes in Single- and Multiphase Growth Mixture Models

Stage-sequential (or multiphase) growth mixture models are useful for delineating potentially different growth processes across multiple phases over time and for determining whether latent subgroups exist within a population. These models are increasingly important as social behavioral scientists are interested in better understanding change processes across distinctively different phases, such as before and after an intervention. One of the less understood issues related to the use of growth mixture models is how to decide on the optimal number of latent classes. The performance of several traditionally used information criteria for determining the number of classes is examined through a Monte Carlo simulation study in single- and multiphase growth mixture models. For thorough examination, the simulation was carried out in 2 perspectives: the models and the factors. The simulation in terms of the models was carried out to see the overall performance of the information criteria within and across the models, whereas the simulation in terms of the factors was carried out to see the effect of each simulation factor on the performance of the information criteria holding the other factors constant. The findings not only support that sample size adjusted Bayesian Information Criterion would be a good choice under more realistic conditions, such as low class separation, smaller sample size, or missing data, but also increase understanding of the performance of information criteria in single- and multiphase growth mixture models.     

具有异方差的线性混合模型参数的谱分解估计

线性混合模型的未知参数分为两种,一种是固定效应,一种是方差分量对具有异方差的线性混合模型的固定效应和方差分量做谱分解估计,讨论它的有关统计性质.     

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.     

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.     

Nonlinear Growth Mixture Models With Fractional Polynomials: An Illustration With Early Childhood Mathematics Ability

Applications of growth mixture modeling have become widespread in the fields of medicine, public health, and the social sciences for modeling linear and nonlinear patterns of change in longitudinal data with presumed heterogeneity with respect to latent group membership. However, in contrast to linear approaches, there has been relatively less focus on methods for modeling nonlinear change. We introduce a nonlinear mixture modeling approach for estimating change trajectories that rely on the use of fractional polynomials within a growth mixture modeling framework. Fractional polynomials allow for more parsimonious and flexible models in comparison to conventional polynomial models. The procedures are illustrated through the use of math ability scores obtained from 499 children over a period of 3 years, with 4 measurement occasions. Techniques for identifying the best empirically derived growth mixture model solution are also described and illustrated by way of substantive example and a simulation.     

The Impact of Noninvariant Intercepts in Latent Means Models

Latent means methods such as multiple-indicator multiple-cause (MIMIC) and structured means modeling (SMM) allow researchers to determine whether or not a significant difference exists between groups' factor means. Strong invariance is typically recommended when interpreting latent mean differences. The extent of the impact of noninvariant intercepts on conclusions made when implementing both MIMIC and SMM methods was the main purpose of this study. The impact of intercept noninvariance on Type I error rates, power, and two model fit indices when using MIMIC and SMM approaches under various conditions were examined. Type I error and power were adversely affected by intercept noninvariance. Although the fit indices did not detect small misspecifications in the form of noninvariant intercepts, one did perform more optimally.     

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.     

Modeling Predictors of Latent Classes in Regression Mixture Models

The purpose of this study is to provide guidance on a process for including latent class predictors in regression mixture models. We first examine the performance of current practice for using the 1-step and 3-step approaches where the direct covariate effect on the outcome is omitted. None of the approaches show adequate estimates of model parameters. Given that Step 1 of the 3-step approach shows adequate results in class enumeration, we suggest using an alternative approach: (a) decide the number of latent classes without predictors of latent classes, and (b) bring the latent class predictors into the model with the inclusion of hypothesized direct covariate effects. Our simulations show that this approach leads to good estimates for all model parameters. The proposed approach is demonstrated by using empirical data to examine the differential effects of family resources on students’ academic achievement outcome. Implications of the study are discussed.     

General Growth Mixture Analysis of Adolescents' Developmental Trajectories of Anxiety: The Impact of Untested Invariance Assumptions on Substantive Interpretations

Substantively, this study investigates potential heterogeneity in the developmental trajectories of anxiety in adolescence. Methodologically, this study demonstrates the usefulness of general growth mixture analysis (GGMA) in addressing these issues and illustrates the impact of untested invariance assumptions on substantive interpretations. This study relied on data from the Montreal Adolescent Depression Development Project (MADDP), a 4-year follow-up of more than 1,000 adolescents who completed the Beck Anxiety Inventory each year. GGMA models relying on different invariance assumptions were empirically compared. Each of these models converged on a 5-class solution, but yielded different substantive results. The model with class-varying variance–covariance matrices was retained as providing a better fit to the data. These results showed that although elevated levels of anxiety might fluctuate over time, they clearly do not represent a transient phenomenon. This model was then validated in relation to multiple predictors (mostly related to school violence) and outcomes (grade-point average, school dropout, depression, loneliness, and drug-related problems).     

Growth Mixture Models Outperform Simpler Clustering Algorithms When Detecting Longitudinal Heterogeneity,Even With Small Sample Sizes

The expansion in the number of people entering higher education has resulted in a substantial increase in the proportion of students enrolling in nontraditional modes, such as part-time study. This study examined the question of whether part-time study curtails the development of the types of intellectual capabilities needed for a knowledge-based economy, because the students would have markedly less exposure to a stimulating campus environment than their full-time counterparts. Graduates from discrete full- and part-time programs from 1 university in Hong Kong completed a survey seeking perceptions of the nurturing of a range of capabilities, together with measures of teacher-student relationships and type of teaching experienced. Two hypotheses were tested by structural equation modeling: (a) the same mechanism for capability development operated for full- and part-time modes and (b) the principal element of the mechanism was the nature of teaching and the quality of teacher-student interaction. Hypothesis 1 was supported because configural invariance between hypothesized models for capability development between the 2 modes was found. Hypothesis 2 was also supported because the models showed that the principal influence on capability development came from teaching for understanding, through promoting active learning experiences and the degree and quality of teacher-student interaction.     

Residual Structures in Latent Growth Curve Modeling

This article presents relevant research on Bayesian methods and their major applications to modeling in an effort to lay out differences between the frequentist and Bayesian paradigms and to look at the practical implications of these differences. Before research is reviewed, basic tenets and methods of the Bayesian approach to modeling are presented and contrasted with basic estimation results from a frequentist perspective. It is argued that Bayesian methods have become a viable alternative to traditional maximum likelihood-based estimation techniques and may be the only solution for more complex psychometric data structures. Hence, neither the applied nor the theoretical measurement community can afford to neglect the exciting new possibilities that have opened up on the psychometric horizon.