Conducting Confirmatory Latent Class Analysis Using Mplus

Latent class analysis (LCA) is an increasingly popular tool that researchers can use to identify latent groups in the population underlying a sample of responses to categorical observed variables. LCA is most commonly used in an exploratory fashion whereby no parameters are specified a priori. Although this exploratory approach is reasonable when very little prior research has been conducted in the area under study, it can be very limiting when much is already known about the variables and population. Confirmatory latent class analysis (CLCA) provides researchers with a tool for modeling and testing specific hypotheses about response patterns in the observed variables. CLCA is based on placing specific constraints on the parameters to reflect these hypotheses. The popular and easy-to-use latent variable modeling software package Mplus can be used to conduct a variety of CLCA types using these parameter constraints. This article focuses on the basic principles underlying the use of CLCA, and the Mplus programming code necessary for carrying it out.     

MplusAutomation: An R Package for Facilitating Large-Scale Latent Variable Analyses in Mplus

MplusAutomation is a package for R that facilitates complex latent variable analyses in Mplus involving comparisons among many models and parameters. More specifically, MplusAutomation provides tools to accomplish 3 objectives: to create and manage Mplus syntax for groups of related models; to automate the estimation of many models; and to extract, aggregate, and compare fit statistics, parameter estimates, and ancillary model outputs. We provide an introduction to the package using applied examples including a large-scale simulation study. By reducing the effort required for large-scale studies, a broad goal of MplusAutomation is to support methodological developments in structural equation modeling using Mplus.     

Structural Equation Models and Mixture Models With Continuous Nonnormal Skewed Distributions

In this article we describe a structural equation modeling (SEM) framework that allows nonnormal skewed distributions for the continuous observed and latent variables. This framework is based on the multivariate restricted skew t distribution. We demonstrate the advantages of skewed SEM over standard SEM modeling and challenge the notion that structural equation models should be based only on sample means and covariances. The skewed continuous distributions are also very useful in finite mixture modeling as they prevent the formation of spurious classes formed purely to compensate for deviations in the distributions from the standard bell curve distribution. This framework is implemented in Mplus Version 7.2.     

A Second-Order Conditionally Linear Mixed Effects Model With Observed and Latent Variable Covariates

A conditionally linear mixed effects model is an appropriate framework for investigating nonlinear change in a continuous latent variable that is repeatedly measured over time. The efficacy of the model is that it allows parameters that enter the specified nonlinear time-response function to be stochastic, whereas those parameters that enter in a nonlinear manner are common to all subjects. In this article we describe how a variant of the Michaelis-Menten (M-M) function can be fit within this modeling framework using Mplus 6.0. We demonstrate how observed and latent covariates can be incorporated to help explain individual differences in growth characteristics. Features of the model including an explication of key analytic decision points are illustrated using longitudinal reading data. To aid in making this class of models accessible, annotated Mplus code is provided.     

Application of Latent Growth Curve Analysis With Categorical Responses in Social Behavioral Research

Latent growth modeling allows social behavioral researchers to investigate within-person change and between-person differences in within-person change. Typically, conventional latent growth curve models are applied to continuous variables, where the residuals are assumed to be normally distributed, whereas categorical variables (i.e., binary and ordinal variables), which do not hold to normal distribution assumptions, have rarely been used. This article describes the latent growth curve model with categorical variables, and illustrates applications using Mplus software that are applicable to social behavioral research. The illustrations use marital instability data from the Iowa Youth and Family Project. We close with recommendations for the specification and parameterization of growth models that use both logit and probit link functions.     

Analyzing Different Types of Moderated Method Effects in Confirmatory Factor Models for Structurally Different Methods

An extension of two confirmatory factor models for multitrait-multimethod measurement designs with structurally different methods to the analysis of latent interaction effects is presented: the nonlinear latent difference (NL-LD) model and the nonlinear correlated trait–correlated method-minus-one (NL-CTC[M – 1]) model. Both models are compared with regard to (a) the psychometric definition of the latent variables, (b) the capabilities of explaining latent method effects, and (c) the analysis of latent interaction effects. Using the latent moderated structural equation approach, we show how moderated method effects can be examined in the NL-CTC(M – 1) model. This fine-grained analysis of method effects is not feasible using the classical NL-LD model. We propose an extended version of the NL-LD model, which recovers the results of the NL-CTC(M – 1) model. The different versions of the nonlinear multimethod models are illustrated using real data from a multirater study. Finally, the advantages and challenges of incorporating latent interaction effects in complex CFA–MTMM models are discussed.     

The Partial Credit Model and Generalized Partial Credit Model as Constrained Nominal Response Models,With Applications in Mplus

The purpose of this article is to demonstrate constraining the nominal response model in Mplus software to calibrate data under the partial credit model (PCM) and generalized partial credit model (GPCM). Currently, many researchers are uncertain if the PCM and GPCM can be estimated within Mplus. Through model constraint commands in Mplus, we demonstrate that both models can be estimated in recent versions of this software. We present an example of this approach with data from 522 respondents on a subset of items from the Math Self-Efficacy Scale (Betz & Hackett, 1983). It is demonstrated that the presented model code is a viable way of estimating the models in Mplus.     

Testing Inequality Constrained Hypotheses in SEM Models

Researchers often have expectations that can be expressed in the form of inequality constraints among the parameters of a structural equation model. It is currently not possible to test these so-called informative hypotheses in structural equation modeling software. We offer a solution to this problem using Mplus. The hypotheses are evaluated using plug-in p values with a calibrated alpha level. The method is introduced and its utility is illustrated by means of an example.     

An SEM Approach for the Evaluation of Intervention Effects Using Pre-Post-Post Designs

A directly applicable latent variable modeling procedure for classical item analysis is outlined. The method allows one to point and interval estimate item difficulty, item correlations, and item-total correlations for composites consisting of categorical items. The approach is readily employed in empirical research and as a by-product permits examining the latent structure of tentative versions of multiple-component measuring instruments. The discussed procedure is straightforwardly utilized with the increasingly popular latent variable modeling software Mplus, and is illustrated on a numerical example.     

Bayesian Evaluation of Inequality-Constrained Hypotheses in SEM Models Using Mplus

Researchers in the behavioral and social sciences often have expectations that can be expressed in the form of inequality constraints among the parameters of a structural equation model resulting in an informative hypothesis. The questions they would like an answer to are “Is the hypothesis Correct” or “Is the hypothesis incorrect?” We demonstrate a Bayesian approach to compare an inequality-constrained hypothesis with its complement in an SEM framework. The method is introduced and its utility is illustrated by means of an example. Furthermore, the influence of the specification of the prior distribution is examined. Finally, it is shown how the approach proposed can be implemented using Mplus.     

Data-Generating Mechanisms Versus Constructively Defined Latent Variables in Multitrait–Multimethod Analysis: A Comment on Castro-Schilo,Widaman, and Grimm (2013)

In a recent article, Castro-Schilo, Widaman, and Grimm (2013) compared different approaches for relating multitrait–multimethod (MTMM) data to external variables. Castro-Schilo et al. reported that estimated associations with external variables were in part biased when either the correlated traits–correlated uniqueness (CT-CU) or correlated traits–correlated (methods–1) [CT-C(M–1)] models were fit to data generated from the correlated traits–correlated methods (CT-CM) model, whereas the data-generating CT-CM model accurately reproduced these associations. Castro-Schilo et al. argued that the CT-CM model adequately represents the data-generating mechanism in MTMM studies, whereas the CT-CU and CT-C(M–1) models do not fully represent the MTMM structure. In this comment, we question whether the CT-CM model is more plausible as a data-generating model for MTMM data than the CT-C(M–1) model. We show that the CT-C(M–1) model can be formulated as a reparameterization of a basic MTMM true score model that leads to a meaningful and parsimonious representation of MTMM data. We advocate the use confirmatory factor analysis MTMM models in which latent trait, method, and error variables are explicitly and constructively defined based on psychometric theory.     

On the Estimation of Nonlinear Mixed-Effects Models and Latent Curve Models for Longitudinal Data

Nonlinear models are effective tools for the analysis of longitudinal data. These models provide a flexible means for describing data that follow complex forms of change. Exponential and logistic functions that include a parameter to represent an asymptote, for instance, are useful for describing responses that tend to level off with time. There are forms of nonlinear latent curve models and nonlinear mixed-effects model that are equivalent, and so given the same set of data, growth function, distributional assumptions, and method of estimation, the 2 models yield equivalent results. There are also forms that are strikingly different and can yield different interpretations for a given set of data. This article discusses cases in which nonlinear mixed-effects models and nonlinear latent curve models are equivalent and those in which they are different and clarifies the estimation needs of the different models. Examples based on empirical data help to illustrate these points.     

Moderated Mediation Analysis Using Bayesian Methods

Conventionally, moderated mediation analysis is conducted through adding relevant interaction terms into a mediation model of interest. In this study, we illustrate how to conduct moderated mediation analysis by directly modeling the relation between the indirect effect components including a and b and the moderators, to permit easier specification and interpretation of moderated mediation. With this idea, we introduce a general moderated mediation model that can be used to model many different moderated mediation scenarios including the scenarios described in Preacher, Rucker, and Hayes (2007). Then we discuss how to estimate and test the conditional indirect effects and to test whether a mediation effect is moderated using Bayesian approaches. How to implement the estimation in both BUGS and Mplus is also discussed. Performance of Bayesian methods is evaluated and compared to that of frequentist methods including maximum likelihood (ML) with 1st-order and 2nd-order delta method standard errors and mL with bootstrap (percentile or bias-corrected confidence intervals) via a simulation study. The results show that Bayesian methods with diffuse (vague) priors implemented in both BUGS and Mplus yielded unbiased estimates, higher power than the ML methods with delta method standard errors, and the ML method with bootstrap percentile confidence intervals, and comparable power to the ML method with bootstrap bias-corrected confidence intervals. We also illustrate the application of these methods with the real data example used in Preacher et al. (2007). Advantages and limitations of applying Bayesian methods to moderated mediation analysis are also discussed.     

Fitting Nonlinear Latent Growth Curve Models With Individually Varying Time Points

Individual growth trajectories of psychological phenomena are often theorized to be nonlinear. Additionally, individuals’ measurement schedules might be unique. In a structural equation framework, latent growth curve model (LGM) applications typically have either (a) modeled nonlinearity assuming some degree of balance in measurement schedules, or (b) accommodated truly individually varying time points, assuming linear growth. This article describes how to fit 4 popular nonlinear LGMs (polynomial, shape-factor, piecewise, and structured latent curve) with truly individually varying time points, via a definition variable approach. The extension is straightforward for certain nonlinear LGMs (e.g., polynomial and structured latent curve) but in the case of shape-factor LGMs requires a reexpression of the model, and in the case of piecewise LGMs requires introduction of a general framework for imparting piecewise structure, along with tools for its automation. All 4 nonlinear LGMs with individually varying time scores are demonstrated using an empirical example on infant weight, and software syntax is provided. The discussion highlights some advantages of modeling nonlinear growth within structural equation versus multilevel frameworks, when time scores individually vary.     

A Bayesian Approach for Estimating Multilevel Latent Contextual Models

In many applications of multilevel modeling, group-level (L2) variables for assessing group-level effects are generated by aggregating variables from a lower level (L1). However, the observed group mean might not be a reliable measure of the unobserved true group mean. In this article, we propose a Bayesian approach for estimating a multilevel latent contextual model that corrects for measurement error and sampling error (i.e., sampling only a small number of L1 units from a L2 unit) when estimating group-level effects of aggregated L1 variables. Two simulation studies were conducted to compare the Bayesian approach with the maximum likelihood approach implemented in Mplus. The Bayesian approach showed fewer estimation problems (e.g., inadmissible solutions) and more accurate estimates of the group-level effect than the maximum likelihood approach under problematic conditions (i.e., small number of groups, predictor variable with a small intraclass correlation). An application from educational psychology is used to illustrate the different estimation approaches.     

On Fitting Nonlinear Latent Curve Models to Multiple Variables Measured Longitudinally

This article shows how nonlinear latent curve models may be fitted for simultaneous analysis of multiple variables measured longitudinally using Mx statistical software. Longitudinal studies often involve observation of several variables across time with interest in the associations between change characteristics of different variables measured within individuals. Other applications involve repeated measures for distinguishable individuals nested within small groups, such as families, with interest in the associations between change characteristics in variables for individuals within groups. This article shows how Mx can be used to carry out analysis of multiple variables measured over time where at least one variable is described by a function that includes one or more parameters that enter the model nonlinearly. An example is provided.     

Bias-Adjusted Three-Step Latent Markov Modeling With Covariates

Latent Markov models with covariates can be estimated via 1-step maximum likelihood. However, this 1-step approach has various disadvantages, such as that the inclusion of covariates in the model might alter the formation of the latent states and that parameter estimation could become infeasible with large numbers of time points, responses, and covariates. This is why researchers typically prefer performing the analysis in a stepwise manner; that is, they first construct the measurement model, then obtain the latent state classifications, and subsequently study the relationship between covariates and latent state memberships. However, such a stepwise approach yields downward-biased estimates of the covariate effects on initial state and transition probabilities. This article, shows how to overcome this problem using a generalization of the bias-corrected 3-step estimation method proposed for latent class analysis (Asparouhov & Muthén, 2014; Bolck, Croon, & Hagenaars, 2004; Vermunt, 2010). We give a formal derivation of the generalization to latent Markov models and discuss how it can be used with many time points by incorporating it into a Baum–Welch type of expectation-maximization algorithm. We evaluate the method through a simulation study and illustrate it using an application on household financial portfolio change. Our study shows that the proposed correction method yields unbiased parameter estimates and accurate standard errors, except for situations with very poorly separated classes and a small sample.     

On the Correspondence between the Latent Growth Curve and Latent Change Score Models

There has been a great deal of work in the literature on the equivalence between the mixed-effects modeling and structural equation modeling (SEM) frameworks in specifying growth models (Willett &; Sayer, 1994). However, there has been little work on the correspondence between the latent growth curve model (LGM) and the latent change score model (see Grimm, Zhang, Hamagami, &; Mazzocco, 2013 Grimm, K. J., Zhang, Z., Hamagami, F., &; Mazzocco, M. M. (2013). Modeling nonlinear change via latent change and latent acceleration frameworks: Examining velocity and acceleration of growth trajectories. Multivariate Behavioral Research, 48, 117–143.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). We demonstrate that four popular variants of the latent change score model – the no change, constant change, proportional change, and dual change models – have LGM equivalents. We provide equations that allow the translation of parameters from one approach to the other and vice versa. We then illustrate this equivalence using mathematics achievement data from the National Longitudinal Survey of Youth.