Distinguishing Between Latent Classes and Continuous Factors with Categorical Outcomes: Class Invariance of Parameters of Factor Mixture Models

Factor mixture models are latent variable models with categorical and continuous latent variables that can be used as a model-based approach to clustering. A previous article covered the results of a simulation study showing that in the absence of model violations, it is usually possible to choose the correct model when fitting a series of models with different numbers of classes and factors within class. The response format in the first study was limited to normally distributed outcomes. This article has 2 main goals, first, to replicate parts of the first study with 5-point Likert scale and binary outcomes, and second, to address the issue of testing class invariance of thresholds and loadings. Testing for class invariance of parameters is important in the context of measurement invariance and when using mixture models to approximate nonnormal distributions. Results show that it is possible to discriminate between latent class models and factor models even if responses are categorical. Comparing models with and without class-specific parameters can lead to incorrectly accepting parameter invariance if the compared models differ substantially with respect to the number of estimated parameters. The simulation study is complemented with an illustration of a factor mixture analysis of 10 binary depression items obtained from a female subsample of the Virginia Twin Registry.     

The integration of continuous and discrete latent variable models: potential problems and promising opportunities

Structural equation mixture modeling (SEMM) integrates continuous and discrete latent variable models. Drawing on prior research on the relationships between continuous and discrete latent variable models, the authors identify 3 conditions that may lead to the estimation of spurious latent classes in SEMM: misspecification of the structural model, nonnormal continuous measures, and nonlinear relationships among observed and/or latent variables. When the objective of a SEMM analysis is the identification of latent classes, these conditions should be considered as alternative hypotheses and results should be interpreted cautiously. However, armed with greater knowledge about the estimation of SEMMs in practice, researchers can exploit the flexibility of the model to gain a fuller understanding of the phenomenon under study.     

Statistical and substantive checking in growth mixture modeling: comment on Bauer and Curran (2003)

This commentary discusses the D. J. Bauer and P. J. Curran (2003) investigation of growth mixture modeling. Single-class modeling of nonnormal outcomes is compared with 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.     

Handling Missing Covariates in Conditional Mixture Models Under Missing at Random Assumptions

Mixture modeling is a popular method that accounts for unobserved population heterogeneity using multiple latent classes that differ in response patterns. Psychologists use conditional mixture models to incorporate covariates into between-class and/or within-class regressions. Although psychologists often have missing covariate data, conditional mixtures are currently fit with a conditional likelihood, treating covariates as fixed and fully observed. Under this exogenous-x approach, missing covariates are handled primarily via listwise deletion. This sacrifices efficiency and does not allow missingness to depend on observed outcomes. Here we describe a modified joint likelihood approach that (a) allows inference about parameters of the exogenous-x conditional mixture even with nonnormal covariates, unlike a conventional multivariate mixture; (b) retains all cases under missing at random assumptions; (c) yields lower bias and higher efficiency than the exogenous-x approach under a variety of conditions with missing covariates; and (d) is straightforward to implement in available commercial software. The proposed approach is illustrated with an empirical analysis predicting membership in latent classes of conduct problems. Recommendations for practice are discussed.     

A realistic perspective on pattern representation in growth data: comment on Bauer and Curran (2003)

D. J. Bauer and P. J. Curran (2003) cautioned that results obtained from growth mixture models may sometimes be inaccurate. The problem they addressed occurs when a growth mixture model is applied to a single, general population of individuals but findings incorrectly support the conclusion that there are 2 subpopulations. In an artificial sampling experiment, they showed that this can occur when the variables in the population have a nonnormal distribution. A realistic perspective is that although a healthy skepticism to complex statistical results is appropriate, there are no true models to discover. Consequently, the issue of model misspecification is irrelevant in practical terms. The purpose of a mathematical model is to summarize data, to formalize the dynamics of a behavioral process, and to make predictions. All of this is scientifically valuable and can be accomplished with a carefully developed model, even though the model is false.     

Finite Mixtures for Simultaneously Modeling Differential Effects and Nonnormal Distributions

Regression mixture models have been increasingly applied in the social and behavioral sciences as a method for identifying differential effects of predictors on outcomes. Although the typical specification of this approach is sensitive to violations of distributional assumptions, alternative methods for capturing the number of differential effects have been shown to be robust. Yet, there is still a need to better describe differential effects that exist when using regression mixture models. This study tests a new approach that uses sets of classes (called differential effects sets) to simultaneously model differential effects and account for nonnormal error distributions. Monte Carlo simulations are used to examine the performance of the approach. The number of classes needed to represent departures from normality is shown to be dependent on the degree of skew. The use of differential effects sets reduced bias in parameter estimates. Applied analyses demonstrated the implementation of the approach for describing differential effects of parental health problems on adolescent body mass index using differential effects sets approach. Findings support the usefulness of the approach, which overcomes the limitations of previous approaches for handling nonnormal errors.     

General mixture item response models with different item response structures: Exposition with an application to Likert scales

This article proposes a general mixture item response theory (IRT) framework that allows for classes of persons to differ with respect to the type of processes underlying the item responses. Through the use of mixture models, nonnested IRT models with different structures can be estimated for different classes, and class membership can be estimated for each person in the sample. If researchers are able to provide competing measurement models, this mixture IRT framework may help them deal with some violations of measurement invariance. To illustrate this approach, we consider a two-class mixture model, where a person’s responses to Likert-scale items containing a neutral middle category are either modeled using a generalized partial credit model, or through an IRTree model. In the first model, the middle category (“neither agree nor disagree”) is taken to be qualitatively similar to the other categories, and is taken to provide information about the person’s endorsement. In the second model, the middle category is taken to be qualitatively different and to reflect a nonresponse choice, which is modeled using an additional latent variable that captures a person’s willingness to respond. The mixture model is studied using simulation studies and is applied to an empirical example.     

Factors Affecting the Adequacy and Preferability of Semiparametric Groups-Based Approximations of Continuous Growth Trajectories

Psychologists have long been interested in characterizing individual differences in change over time. It is often plausible to assume that the distribution of these individual differences is continuous in nature, yet theory is seldom so specific as to designate its parametric form (e.g., normal). Semiparametric groups-based trajectory models (SPGMs) were thus developed to provide a discrete approximation for continuously distributed growth of unknown form. Previous research has demonstrated the adequacy of the approximation provided by SPGM but only under relatively narrow, theoretically optimal conditions. Under alternative conditions, which may be more common in practice (e.g., higher dimension random effects, smaller sample sizes), this study shows that approximation adequacy can suffer. Furthermore, this study also evaluates whether SPGM's discrete approximation is preferable to a parametric trajectory model that assumes normally distributed random effects when in fact the distribution is modestly nonnormal. The answer is shown to depend on distributional characteristics of both repeated measures (binary or continuous) and random effects (bimodal or skewed). Implications for practice are discussed in light of empirical examples on externalizing behavior.     

On growth curves and mixture models

The multilevel model of change and the latent growth model are flexible means to describe all sorts of population heterogeneity with respect to growth and development, including the presence of sub‐populations. The growth mixture model is a natural extension of these models. It comes at hand when information about sub‐populations is missing and researchers nevertheless want to retrieve developmental trajectories from sub‐populations. We argue that researchers have to make rather strong assumptions about the sub‐populations or latent trajectory classes in order to retrieve existing population differences. A simulated example is discussed, showing that a sample of repeated measures drawn from two sub‐populations easily leads to the mistaken inference of three sub‐populations, when assumptions are not met. The merits of methodological advises on this issue are discussed. It is concluded that growth mixture models should be used with understanding, and offer no free way to growth patterns in unknown sub‐populations. Copyright © 2006 John Wiley & Sons, Ltd.     

多阶段混合增长模型的影响因素：距离与形态

通过模拟研究, 考察潜类别距离和发展形态等因素对多阶段混合增长模型的模型选择和参数估计的影响：(1)潜类别距离越大, 模型选择和分类效果越好。(2)混合模型的选择, 应以一定样本量(至少200)为前提, 首先考虑BIC选出正确的分类模型, 再通过熵值、ARI等选择分类确定性较高的模型。(3)多阶段的发展形态对正确模型的选择和分类的确定性均有一定程度影响。(4)潜类别距离和样本量越大, 参数估计精度越高。(5)在判断分类准确性的指标中, ARI的选择更偏向于真实的模型。     

Observations on the Use of Growth Mixture Models in Psychological Research

Psychologists are applying growth mixture models at an increasing rate. This article argues that most of these applications are unlikely to reproduce the underlying taxonic structure of the population. At a more fundamental level, in many cases there is probably no taxonic structure to be found. Latent growth classes then categorically approximate the true continuum of individual differences in change. This approximation, although in some cases potentially useful, can also be problematic. The utility of growth mixture models for psychological science thus remains in doubt. Some ways in which these models might be more profitably used are suggested.     

On the law of Regular Minimality: Reply to Ennis

Ennis's critique touches on issues important for psychophysics, but the points he makes against the hypothesis that Regular Minimality is a basic property of sensory discrimination are not tenable.(1) Stimulus variability means that one and the same apparent stimulus value (as measured by experimenter) is a probabilistic mixture of true stimulus values. The notion of a true stimulus value is a logical necessity: variability and distribution presuppose the values that vary and are distributed (even if these values are represented by processes or sets rather than real numbers). Regular Minimality is formulated for true stimulus values. That a mixture of probabilities satisfying Regular Minimality does not satisfy this principle (unless it also satisfies Constant Self-Similarity) is an immediate consequence of my 2003 analysis. Stimulus variability can be controlled or estimated: the cases when observed violations of Regular Minimality can be accounted for by stimulus variability corroborate rather than falsify this principle. In this respect stimulus variability is no different from fatigue, perceptual learning, and other factors creating mixtures of discrimination probabilities in an experiment.(2) Could it be that well-behaved Thurstonian-type models are true models of discrimination but their parameters are so adjusted that the violations of Regular Minimality they lead to (due to my 2003 theorems) are too small to be detected experimentally? This is possible, but this amounts to admitting that Regular Minimality is a law after all, albeit only approximate: nothing in the logic of the Thurstonian-type representations per se prevents them from violating Regular Minimality grossly rather than slightly. Moreover, even very small violations predicted by a given class of Thurstonian-type models can be tested in specially designed experiments (perhaps under additional, independently testable assumptions). The results of one such experiment, in which observers were asked to alternately adjust to each other the values of stimuli in two observation areas, indicate that violations of Regular Minimality, if any, are far below limits of plausible interpretability.     

Evaluating the robustness of repeated measures analyses: The case of small sample sizes and nonnormal data

Repeated measures analyses of variance are the method of choice in many studies from experimental psychology and the neurosciences. Data from these fields are often characterized by small sample sizes, high numbers of factor levels of the within-subjects factor(s), and nonnormally distributed response variables such as response times. For a design with a single within-subjects factor, we investigated Type I error control in univariate tests with corrected degrees of freedom, the multivariate approach, and a mixed-model (multilevel) approach (SAS PROC MIXED) with Kenward–Roger’s adjusted degrees of freedom. We simulated multivariate normal and nonnormal distributions with varied population variance–covariance structures (spherical and nonspherical), sample sizes (N), and numbers of factor levels (K). For normally distributed data, as expected, the univariate approach with Huynh–Feldt correction controlled the Type I error rate with only very few exceptions, even if samples sizes as low as three were combined with high numbers of factor levels. The multivariate approach also controlled the Type I error rate, but it requires N ≥ K. PROC MIXED often showed acceptable control of the Type I error rate for normal data, but it also produced several liberal or conservative results. For nonnormal data, all of the procedures showed clear deviations from the nominal Type I error rate in many conditions, even for sample sizes greater than 50. Thus, none of these approaches can be considered robust if the response variable is nonnormally distributed. The results indicate that both the variance heterogeneity and covariance heterogeneity of the population covariance matrices affect the error rates.     

Index to Volume 112

For various nonnormal distributions, the power of the Student t test can be increased if continuous measures are transformed to ranks before the test is performed. The power of the test can also be increased almost as much and, even more for some distributions, if measures are replaced by dichotomous variables with the values 0 and 1, instead of ranks. Similarly, the power of a significance test of correlation can be increased if scores are transformed to ranks, that is, with the use of the Spearman rank correlation method. Power can also be increased almost as much and in some cases even more if dichotomous variables are introduced, that is, if the phi coefficient is used as a measure of correlation.     

The relationship between multilevel models and non‐parametric multilevel mixture models: Discrete approximation of intraclass correlation,random coefficient distributions,and residual heteroscedasticity

Multilevel data structures are common in the social sciences. Often, such nested data are analysed with multilevel models (MLMs) in which heterogeneity between clusters is modelled by continuously distributed random intercepts and/or slopes. Alternatively, the non‐parametric multilevel regression mixture model (NPMM) can accommodate the same nested data structures through discrete latent class variation. The purpose of this article is to delineate analytic relationships between NPMM and MLM parameters that are useful for understanding the indirect interpretation of the NPMM as a non‐parametric approximation of the MLM, with relaxed distributional assumptions. We define how seven standard and non‐standard MLM specifications can be indirectly approximated by particular NPMM specifications. We provide formulas showing how the NPMM can serve as an approximation of the MLM in terms of intraclass correlation, random coefficient means and (co)variances, heteroscedasticity of residuals at level 1, and heteroscedasticity of residuals at level 2. Further, we discuss how these relationships can be useful in practice. The specific relationships are illustrated with simulated graphical demonstrations, and direct and indirect interpretations of NPMM classes are contrasted. We provide an R function to aid in implementing and visualizing an indirect interpretation of NPMM classes. An empirical example is presented and future directions are discussed.     

Two-Step Estimation of Models Between Latent Classes and External Variables

We consider models which combine latent class measurement models for categorical latent variables with structural regression models for the relationships between the latent classes and observed explanatory and response variables. We propose a two-step method of estimating such models. In its first step, the measurement model is estimated alone, and in the second step the parameters of this measurement model are held fixed when the structural model is estimated. Simulation studies and applied examples suggest that the two-step method is an attractive alternative to existing one-step and three-step methods. We derive estimated standard errors for the two-step estimates of the structural model which account for the uncertainty from both steps of the estimation, and show how the method can be implemented in existing software for latent variable modelling.     

Alternative Approaches to Addressing Non-Normal Distributions in the Application of IRT Models to Personality Measures

It is generally assumed that the latent trait is normally distributed in the population when estimating logistic item response theory (IRT) model parameters. This assumption requires that the latent trait be fully continuous and the population homogenous (i.e., not a mixture). When this normality assumption is violated, models are misspecified, and item and person parameter estimates are inaccurate. When normality cannot be assumed, it might be appropriate to consider alternative modeling approaches: (a) a zero-inflated mixture, (b) a log-logistic, (c) a Ramsay curve, or (d) a heteroskedastic-skew model. The first 2 models were developed to address modeling problems associated with so-called quasi-continuous or unipolar constructs, which apply only to a subset of the population, or are meaningful at one end of the continuum only. The second 2 models were developed to address non-normal latent trait distributions and violations of homogeneity of error variance, respectively. To introduce these alternative IRT models and illustrate their strengths and weaknesses, we performed real data application comparing results to those from a graded response model. We review both statistical and theoretical challenges in applying these models and choosing among them. Future applications of these and other alternative models (e.g., unfolding, diffusion) are needed to advance understanding about model choice in particular situations.     

Robustness of Parameter Estimation to Assumptions of Normality in the Multidimensional Graded Response Model

A central assumption that is implicit in estimating item parameters in item response theory (IRT) models is the normality of the latent trait distribution, whereas a similar assumption made in categorical confirmatory factor analysis (CCFA) models is the multivariate normality of the latent response variables. Violation of the normality assumption can lead to biased parameter estimates. Although previous studies have focused primarily on unidimensional IRT models, this study extended the literature by considering a multidimensional IRT model for polytomous responses, namely the multidimensional graded response model. Moreover, this study is one of few studies that specifically compared the performance of full-information maximum likelihood (FIML) estimation versus robust weighted least squares (WLS) estimation when the normality assumption is violated. The research also manipulated the number of nonnormal latent trait dimensions. Results showed that FIML consistently outperformed WLS when there were one or multiple skewed latent trait distributions. More interestingly, the bias of the discrimination parameters was non-ignorable only when the corresponding factor was skewed. Having other skewed factors did not further exacerbate the bias, whereas biases of boundary parameters increased as more nonnormal factors were added. The item parameter standard errors recovered well with both estimation algorithms regardless of the number of nonnormal dimensions.     

Structural Equation Modeling with Small Samples: Test Statistics

Structural equation modeling is a well-known technique for studying relationships among multivariate data. In practice, high dimensional nonnormal data with small to medium sample sizes are very common, and large sample theory, on which almost all modeling statistics are based, cannot be invoked for model evaluation with test statistics. The most natural method for nonnormal data, the asymptotically distribution free procedure, is not defined when the sample size is less than the number of nonduplicated elements in the sample covariance. Since normal theory maximum likelihood estimation remains defined for intermediate to small sample size, it may be invoked but with the probable consequence of distorted performance in model evaluation. This article studies the small sample behavior of several test statistics that are based on maximum likelihood estimator, but are designed to perform better with nonnormal data. We aim to identify statistics that work reasonably well for a range of small sample sizes and distribution conditions. Monte Carlo results indicate that Yuan and Bentler's recently proposed F-statistic performs satisfactorily.     

Growth modeling with nonignorable dropout: alternative analyses of the STAR*D antidepressant trial

This article uses a general latent variable framework to study a series of models for nonignorable missingness due to dropout. Nonignorable missing data modeling acknowledges that missingness may depend not only on covariates and observed outcomes at previous time points as with the standard missing at random assumption, but also on latent variables such as values that would have been observed (missing outcomes), developmental trends (growth factors), and qualitatively different types of development (latent trajectory classes). These alternative predictors of missing data can be explored in a general latent variable framework with the Mplus program. A flexible new model uses an extended pattern-mixture approach where missingness is a function of latent dropout classes in combination with growth mixture modeling. A new selection model not only allows an influence of the outcomes on missingness but allows this influence to vary across classes. Model selection is discussed. The missing data models are applied to longitudinal data from the Sequenced Treatment Alternatives to Relieve Depression (STAR*D) study, the largest antidepressant clinical trial in the United States to date. Despite the importance of this trial, STAR*D growth model analyses using nonignorable missing data techniques have not been explored until now. The STAR*D data are shown to feature distinct trajectory classes, including a low class corresponding to substantial improvement in depression, a minority class with a U-shaped curve corresponding to transient improvement, and a high class corresponding to no improvement. The analyses provide a new way to assess drug efficiency in the presence of dropout.