A Finite Mixture of Nonlinear Random Coefficient Models for Continuous Repeated Measures Data

Nonlinear random coefficient models (NRCMs) for continuous longitudinal data are often used for examining individual behaviors that display nonlinear patterns of development (or growth) over time in measured variables. As an extension of this model, this study considers the finite mixture of NRCMs that combine features of NRCMs with the idea of finite mixture (or latent class) models. The efficacy of this model is that it allows the integration of intrinsically nonlinear functions where the data come from a mixture of two or more unobserved subpopulations, thus allowing the simultaneous investigation of intra-individual (within-person) variability, inter-individual (between-person) variability, and subpopulation heterogeneity. Effectiveness of this model to work under real data analytic conditions was examined by executing a Monte Carlo simulation study. The simulation study was carried out using an R routine specifically developed for the purpose of this study. The R routine used maximum likelihood with the expectation–maximization algorithm. The design of the study mimicked the output obtained from running a two-class mixture model on task completion data.     

Nonlinear Structured Growth Mixture Models in Mplus and OpenMx

Growth mixture models (GMMs; B. O. Muthén & Muthén, 2000; B. O. Muthén & Shedden, 1999) are a combination of latent curve models (LCMs) and finite mixture models to examine the existence of latent classes that follow distinct developmental patterns. GMMs are often fit with linear, latent basis, multiphase, or polynomial change models because of their common use, flexibility in modeling many types of change patterns, the availability of statistical programs to fit such models, and the ease of programming. In this article, we present additional ways of modeling nonlinear change patterns with GMMs. Specifically, we show how LCMs that follow specific nonlinear functions can be extended to examine the presence of multiple latent classes using the Mplus and OpenMx computer programs. These models are fit to longitudinal reading data from the Early Childhood Longitudinal Study–Kindergarten Cohort to illustrate their use.     

Parsimonious Structural Equation Models for Repeated Measures Data,with Application to the Study of Consumer Preferences

Recent research reflects a growing awareness of the value of using structural equation models to analyze repeated measures data. However, such data, particularly in the presence of covariates, often lead to models that either fit the data poorly, are exceedingly general and hard to interpret, or are specified in a manner that is highly data dependent. This article introduces methods for developing parsimonious models for such data. The underlying technology uses reduced-rank representations of the variances, covariances and means of observed and latent variables. The value of this approach, which may be implemented using standard structural equation modeling software, is illustrated in an application study aimed at understanding heterogeneous consumer preferences. In this application, the parsimonious representations characterize systematic relationships among consumer demographics, attitudes and preferences that would otherwise be undetected. The result is a model that is parsimonious, illuminating, and fits the data well, while keeping data dependence to a minimum.     

Specifying Ability Growth Models Using a Multidimensional Item Response Model for Repeated Measures Categorical Ordinal Item Response Data

When categorical ordinal item response data are collected over multiple timepoints from a repeated measures design, an item response theory (IRT) modeling approach whose unit of analysis is an item response is suitable. This study proposes a few longitudinal IRT models and illustrates how a popular compensatory multidimensional IRT model can be utilized to formulate such longitudinal IRT models, which permits an investigation of ability growth at both individual and population levels. The equivalence of an existing multidimensional IRT model and those longitudinal IRT models is also elaborated so that one can make use of an existing multidimensional IRT model to implement the longitudinal IRT models.     

Regularized Finite Mixture Models for Probability Trajectories

Finite mixture models are widely used in the analysis of growth trajectory data to discover subgroups of individuals exhibiting similar patterns of behavior over time. In practice, trajectories are usually modeled as polynomials, which may fail to capture important features of the longitudinal pattern. Focusing on dichotomous response measures, we propose a likelihood penalization approach for parameter estimation that is able to capture a variety of nonlinear class mean trajectory shapes with higher precision than maximum likelihood estimates. We show how parameter estimation and inference for whether trajectories are time-invariant, linear time-varying, or nonlinear time-varying can be carried out for such models. To illustrate the method, we use simulation studies and data from a long-term longitudinal study of children at high risk for substance abuse. This work was supported in part by NIAAA grants R37 AA07065 and R01 AA12217 to RAZ.     

Latent Growth Curve and Repeated Measures ANOVA Contrasts: What the Models are Telling You

The relationship between the latent growth curve and repeated measures ANOVA models is often misunderstood. Although a number of investigators have looked into the similarities and differences among these models, a cursory reading of the literature can give the impression that they are very different models. Here we show that each model represents a set of contrasts on the occasion means. We demonstrate that the fixed effects parameters of the estimated basis vector latent growth curve model are merely a transformation of the repeated measures ANOVA fixed effects parameters. We further show that differences in fit in models that estimate the same means structure can be due to the different error covariance structures implied by the model. We show these relationships both algebraically and through using data from a simulation.     

Multilevel Mixture Factor Models

Factor analysis is a statistical method for describing the associations among sets of observed variables in terms of a small number of underlying continuous latent variables. Various authors have proposed multilevel extensions of the factor model for the analysis of data sets with a hierarchical structure. These Multilevel Factor Models (MFMs) have in common that—as in multilevel regression analysis—variation at the higher level is modeled using continuous random effects. In this article, we present an alternative multilevel extension of factor analysis which we call the Multilevel Mixture Factor Model (MMFM). It is based on the assumption that higher level units belong to latent classes that differ in terms of the parameters of the factor model specified for the lower level units. We demonstrate the added value of MMFM compared with MFM, both from a theoretical and applied perspective, and we illustrate the complementarity of the two approaches with an empirical application on students' satisfaction with the University of Florence. The multilevel aspect of this application is that students are nested within study programs, which makes it possible to cluster these programs based on their differences in students' satisfaction.     

Mixture Models of Categorization

Many currently popular models of categorization are either strictly parametric (e.g., prototype models, decision bound models) or strictly nonparametric (e.g., exemplar models) (F. G. Ashby & L. A. Alfonso-Reese, 1995, Journal of Mathematical Psychology, 39, 216-233). In this article, a family of semiparametric classifiers is investigated where categories are represented by a finite mixture distribution. The advantage of these mixture models of categorization is that they contain several parametric models and nonparametric models as a special case. Specifically, it is shown that both decision bound models (F. G. Ashby & W. T. Maddox, 1992, Journal of Experimental Psychology: Human Perception and Performance, 16, 598-612; 1993, Journal of Mathematical Psychology, 37, 372-400) and the generalized context model (R. M. Nosofsky, 1986, Journal of Experimental Psychology: General, 115, 39-57) can be interpreted as two extreme cases of a common mixture model. Furthermore, many other (semiparametric) models of categorization can be derived from the same generic mixture framework. In this article, several examples are discussed and a parameter estimation procedure for fitting these models is outlined. To illustrate the approach, several specific models are fitted to a data set collected by S. C. McKinley and R. M. Nosofsky (1995, Journal of Experimental Psychology: Human Perception and Performance, 21, 128-148). The results suggest that semi-parametric models are a promising alternative for future model development.     

Multiple Comparisons For Repeated Measures Means

The paper shows how multiple comparison procedures for repeated measures means employing a pooled estimate of error variance must conform to the sphericity assumptions of the design in order to provide a valid test. Since it is highly unlikely that behavioral science data will satisfy this condition the paper presents a test statistic that, depending upon the design, will provide either an exact or robust test and is generalizable to designs containing any number of repeated factors. Finally, various critical values are enumerated to limit the joint level of significance at α.     

Understanding Linkages Among Mixture Models

The methodological literature on mixture modeling has rapidly expanded in the past 15 years, and mixture models are increasingly applied in practice. Nonetheless, this literature has historically been diffuse, with different notations, motivations, and parameterizations making mixture models appear disconnected. This pedagogical review facilitates an integrative understanding of mixture models. First, 5 prototypic mixture models are presented in a unified format with incremental complexity while highlighting their mutual reliance on familiar probability laws, common assumptions, and shared aspects of interpretation. Second, 2 recent extensions—hybrid mixtures and parallel-process mixtures—are discussed. Both relax a key assumption of classic mixture models but do so in different ways. Similarities in construction and interpretation among hybrid mixtures and among parallel-process mixtures are emphasized. Third, the combination of both extensions is motivated and illustrated by means of an example on oppositional defiant and depressive symptoms. By clarifying how existing mixture models relate and can be combined, this article bridges past and current developments and provides a foundation for understanding new developments.     

Comparison of Modern Methods for Analyzing Repeated Measures Data With Missing Values

Missing data are a pervasive problem in many psychological applications in the real world. In this article we study the impact of dropout on the operational characteristics of several approaches that can be easily implemented with commercially available software. These approaches include the covariance pattern model based on an unstructured covariance matrix (CPM-U) and the true covariance matrix (CPM-T), multiple imputation-based generalized estimating equations (MI-GEE), and weighted generalized estimating equations (WGEE). Under the missing at random mechanism, the MI-GEE approach was always robust. The CPM-T and CPM-U methods were also able to control the error rates provided that certain minimum sample size requirements were met, whereas the WGEE was more prone to inflated error rates. In contrast, under the missing not at random mechanism, all evaluated approaches were generally invalid. Our results also indicate that the CPM methods were more powerful than the MI-GEE and WGEE methods and their superiority was often substantial. Furthermore, we note that little or no power was sacrificed by using CPM-U method in place of CPM-T, although both methods have less power in situations where some participants have incomplete data. Some aspects of the CPM-U and MI-GEE methods are illustrated using real data from 2 previously published data sets. The first data set comes from a randomized study of AIDS patients with advanced immune suppression, the second from a cohort of patients with schizotypal personality disorder enrolled in a prevention program for psychosis.     

Consistent Partial Least Squares for Nonlinear Structural Equation Models

Partial Least Squares as applied to models with latent variables, measured indirectly by indicators, is well-known to be inconsistent. The linear compounds of indicators that PLS substitutes for the latent variables do not obey the equations that the latter satisfy. We propose simple, non-iterative corrections leading to consistent and asymptotically normal (CAN)-estimators for the loadings and for the correlations between the latent variables. Moreover, we show how to obtain CAN-estimators for the parameters of structural recursive systems of equations, containing linear and interaction terms, without the need to specify a particular joint distribution. If quadratic and higher order terms are included, the approach will produce CAN-estimators as well when predictor variables and error terms are jointly normal. We compare the adjusted PLS, denoted by PLSc, with Latent Moderated Structural Equations (LMS), using Monte Carlo studies and an empirical application.     

On Inclusion of Covariates for Class Enumeration of Growth Mixture Models

In this article, we directly question the common practice in growth mixture model (GMM) applications that exclusively rely on the fitting model without covariates for GMM class enumeration. We provide theoretical and simulation evidence to demonstrate that exclusion of covariates from GMM class enumeration could be problematic in many cases. Based on our findings, we provide recommendations for examining the class enumeration by the fitting model without covariates and discuss the potential of covariate inclusion as a remedy for the weakness of GMM class enumeration without including covariates. A real example on the development of children's cumulative exposure to risk factors for adolescent substance use is provided to illustrate our methodological developments.     

Simplified Estimation and Testing in Unbalanced Repeated Measures Designs

In this paper we propose a simple estimator for unbalanced repeated measures design models where each unit is observed at least once in each cell of the experimental design. The estimator does not require a model of the error covariance structure. Thus, circularity of the error covariance matrix and estimation of correlation parameters and variances are not necessary. Together with a weak assumption about the reason for the varying number of observations, the proposed estimator and its variance estimator are unbiased. As an alternative to confidence intervals based on the normality assumption, a bias-corrected and accelerated bootstrap technique is considered. We also propose the naive percentile bootstrap for Wald-type tests where the standard Wald test may break down when the number of observations is small relative to the number of parameters to be estimated. In a simulation study we illustrate the properties of the estimator and the bootstrap techniques to calculate confidence intervals and conduct hypothesis tests in small and large samples under normality and non-normality of the errors. The results imply that the simple estimator is only slightly less efficient than an estimator that correctly assumes a block structure of the error correlation matrix, a special case of which is an equi-correlation matrix. Application of the estimator and the bootstrap technique is illustrated using data from a task switch experiment based on an experimental within design with 32 cells and 33 participants.

     

Detecting Social Desirability Bias Using Factor Mixture Models

Based on the conceptualization that social desirable bias (SDB) is a discrete event resulting from an interaction between a scale's items, the testing situation, and the respondent's latent trait on a social desirability factor, we present a method that makes use of factor mixture models to identify which examinees are most likely to provide biased responses, which items elicit the most socially desirable responses, and which external variables predict SDB. Problems associated with the common use of correlation coefficients based on scales' total scores to diagnose SDB and partial correlations to correct for SDB are discussed. The method is demonstrated with an analysis of SDB in the Attitude toward Interprofessional Service-Learning scale with a sample of students from health-related fields.     

Analyzing Job Analysis Data Using Mixture Rasch Models

An important piece of validity evidence to support the use of credentialing exams comes from performing a job analysis of the profession. One common job analysis method is the task inventory method, where people working in the field are surveyed using rating scales about the tasks thought necessary to safely and competently perform the job. This article describes how mixture Rasch models can be used to analyze these data, and how results from these analyses can help to identify whether different groups of people may be responding to job tasks differently. Three examples from different credentialing programs illustrate scenarios that can be found when applying mixture Rasch models to job analysis data. Discussion of what these results may imply for the development of credentialing exams and other analyses of job analysis data is provided.     

Effects of Covariance Heterogeneity on Three Procedures for Analyzing Multivariate Repeated Measures Designs

Empirical Type I error and power rates were estimated for (a) the doubly multivariate model, (b) the Welch-James multivariate solution developed by Keselman, Carriere and Lix (1993) using Johansen's results (1980), and for (c) the multivariate version of the modified Brown-Forsythe (1974) procedure. The performance of these procedures was investigated by testing within- blocks sources of variation in a multivariate split-plot design containing unequal covariance matrices. The results indicate that the doubly multivariate model did not provide effective Type I error control while the Welch-James procedure provided robust and powerful tests of the within-subjects main effect, however, this approach provided liberal tests of the interaction effect. The results also indicate that the modified Brown-Forsythe procedure provided robust tests of within-subjects main and interaction effects, especially when the design was balanced or when group sizes and covariance matrices were positively paired.     

Bayesian Inference for Growth Mixture Models with Latent Class Dependent Missing Data

Growth mixture models (GMMs) with nonignorable missing data have drawn increasing attention in research communities but have not been fully studied. The goal of this article is to propose and to evaluate a Bayesian method to estimate the GMMs with latent class dependent missing data. An extended GMM is first presented in which class probabilities depend on some observed explanatory variables and data missingness depends on both the explanatory variables and a latent class variable. A full Bayesian method is then proposed to estimate the model. Through the data augmentation method, conditional posterior distributions for all model parameters and missing data are obtained. A Gibbs sampling procedure is then used to generate Markov chains of model parameters for statistical inference. The application of the model and the method is first demonstrated through the analysis of mathematical ability growth data from the National Longitudinal Survey of Youth 1997 (Bureau of Labor Statistics, U.S. Department of Labor, 1997). A simulation study considering 3 main factors (the sample size, the class probability, and the missing data mechanism) is then conducted and the results show that the proposed Bayesian estimation approach performs very well under the studied conditions. Finally, some implications of this study, including the misspecified missingness mechanism, the sample size, the sensitivity of the model, the number of latent classes, the model comparison, and the future directions of the approach, are discussed.     

Nonlinear Growth Models as Measurement Models: A Second-Order Growth Curve Model for Measuring Potential

Recent methodological work has highlighted the promise of nonlinear growth models for addressing substantive questions in the behavioral sciences. In this article, we outline a second-order nonlinear growth model in order to measure a critical notion in development and education: potential. Here, potential is conceptualized as having three components—ability, capacity, and availability—where ability is the amount of skill a student is estimated to have at a given timepoint, capacity is the maximum amount of ability a student is predicted to be able to develop asymptotically, and availability is the difference between capacity and ability at any particular timepoint. We argue that single timepoint measures are typically insufficient for discerning information about potential, and we therefore describe a general framework that incorporates a growth model into the measurement model to capture these three components. Then, we provide an illustrative example using the public-use Early Childhood Longitudinal Study–Kindergarten data set using a Michaelis-Menten growth function (reparameterized from its common application in biochemistry) to demonstrate our proposed model as applied to measuring potential within an educational context. The advantage of this approach compared to currently utilized methods is discussed as are future directions and limitations.