Latent Curve Models and Latent Change Score Models Estimated in R

When using multiple imputation in the analysis of incomplete data, a prominent guideline suggests that more than 10 imputed data values are seldom needed. This article calls into question the optimism of this guideline and illustrates that important quantities (e.g., p values, confidence interval half-widths, and estimated fractions of missing information) suffer from substantial imprecision with a small number of imputations. Substantively, a researcher can draw categorically different conclusions about null hypothesis rejection, estimation precision, and missing information in distinct multiple imputation runs for the same data and analysis with few imputations. This article explores the factors associated with this imprecision, demonstrates that precision improves by increasing the number of imputations, and provides practical guidelines for choosing a reasonable number of imputations to reduce imprecision for each of these quantities.     

Building Latent Class Growth Trees

Researchers use latent class growth (LCG) analysis to detect meaningful subpopulations that display different growth curves. However, especially when the number of classes required to obtain a good fit is large, interpretation of the encountered class-specific curves might not be straightforward. To overcome this problem, we propose an alternative way of performing LCG analysis, which we call LCG tree (LCGT) modeling. For this purpose, a recursive partitioning procedure similar to divisive hierarchical cluster analysis is used: Classes are split until a certain criterion indicates that the fit does not improve. The advantage of the LCGT approach compared to the standard LCG approach is that it gives a clear insight into how the latent classes are formed and how solutions with different numbers of classes relate. The practical use of the approach is illustrated using applications on drug use during adolescence and mood regulation during the day.     

The Latent Variable-Autoregressive Latent Trajectory Model: A General Framework for Longitudinal Data Analysis

In recent years, longitudinal data have become increasingly relevant in many applications, heightening interest in selecting the best longitudinal model to analyze them. Too often, traditional practice rather than substantive theory guides the specific model selected. This opens the possibility that alternative models might better correspond to the data. In this paper, we present a general longitudinal model that we call the Latent Variable-Autoregressive Latent Trajectory (LV-ALT) model that includes most other longitudinal models with continuous outcomes as special cases. It is capable of specializing to most models dictated by theory or prior research while having the capacity to compare them to alternative ones. If there is little guidance on the best model, the LV-ALT provides a way to determine the appropriate empirical match to the data. We present the model, discuss its identification and estimation, and illustrate how the LV-ALT reveals new things about a widely used empirical example.     

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.     

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.     

Using Design-Based Latent Growth Curve Modeling With Cluster-Level Predictor to Address Dependency

The authors compared the effects of using the true Multilevel Latent Growth Curve Model (MLGCM) with single-level regular and design-based Latent Growth Curve Models (LGCM) with or without the higher-level predictor on various criterion variables for multilevel longitudinal data. They found that random effect estimates were biased when the higher-level predictor was not included and that standard errors of the regression coefficients from the higher-level were underestimated when a regular LGCM was used. Nevertheless, random effect estimates, regression coefficients, and standard error estimates were consistent with those from the true MLGCM when the design-based LGCM included the higher-level predictor. They discussed implication for the study with empirical data illustration.     

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.     

Detecting Mixtures From Structural Model Differences Using Latent Variable Mixture Modeling: A Comparison of Relative Model Fit Statistics

The accuracy of structural model parameter estimates in latent variable mixture modeling was explored with a 3 (sample size) × 3 (exogenous latent mean difference) × 3 (endogenous latent mean difference) × 3 (correlation between factors) × 3 (mixture proportions) factorial design. In addition, the efficacy of several likelihood-based statistics (Akaike's Information Criterion [AIC], Bayesian Information Ctriterion [BIC], the sample-size adjusted BIC [ssBIC], the consistent AIC [CAIC], the Vuong-Lo-Mendell-Rubin adjusted likelihood ratio test [aVLMR]), classification-based statistics (CLC [classification likelihood information criterion], ICL-BIC [integrated classification likelihood], normalized entropy criterion [NEC], entropy), and distributional statistics (multivariate skew and kurtosis test) were examined to determine which statistics best recover the correct number of components. Results indicate that the structural parameters were recovered, but the model fit statistics were not exceedingly accurate. The ssBIC statistic was the most accurate statistic, and the CLC, ICL-BIC, and aVLMR showed limited utility. However, none of these statistics were accurate for small samples (n = 500).     

A Comparison of Three-Step Approaches for Auxiliary Variables in Latent Class and Latent Profile Analysis

In this article, 3-step methods to include predictors and distal outcomes in commonly used mixture models are evaluated. Two Monte Carlo simulation studies were conducted to compare the pseudo class (PC), Vermunt’s (2010), and the Lanza, Tan, and Bray (LTB) 3-step approaches with respect to bias of parameter estimates in latent class analysis (LCA) and latent profile analysis (LPA) models with auxiliary variables. For coefficients of predictors of class membership, results indicated that Vermunt’s method yielded more accurate estimates for LCA and LPA compared to the PC method. With distal outcomes of latent classes and latent profiles, the LTB method produced the lowest relative bias of coefficient estimates and Type I error rates close to nominal levels.     

First- Versus Second-Order Latent Growth Curve Models: Some Insights From Latent State-Trait Theory

First-order latent growth curve models (FGMs) estimate change based on a single observed variable and are widely used in longitudinal research. Despite significant advantages, second-order latent growth curve models (SGMs), which use multiple indicators, are rarely used in practice, and not all aspects of these models are widely understood. In this article, our goal is to contribute to a better understanding of theoretical and practical differences between FGMs and SGMs. We define the latent variables in FGMs and SGMs explicitly on the basis of latent state–trait (LST) theory and discuss insights that arise from this approach. We show that FGMs imply a strict trait-like conception of the construct under study, whereas SGMs allow for both trait and state components. Based on a simulation study and empirical applications to the Center for Epidemiological Studies Depression Scale (Radloff, 1977 Radloff, L. S. 1977. The CES–D Scale: A self-report depression scale for research in the general population. Applied Psychological Measurement, 1: 385–401. [Crossref], [Web of Science ®] , [Google Scholar]) we illustrate that, as an important practical consequence, FGMs yield biased reliability estimates whenever constructs contain state components, whereas reliability estimates based on SGMs were found to be accurate. Implications of the state–trait distinction for the measurement of change via latent growth curve models are discussed.     

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).     

Relating Latent Change Score and Continuous Time Models

The primary goal of this article is to demonstrate the close relationship between 2 classes of dynamic models in psychological research: latent change score models and continuous time models. The secondary goal is to point out some differences. We begin with a brief review of both approaches, before demonstrating how the 2 methods are mathematically and conceptually related. It will be shown that most commonly used latent change score models are related to continuous time models by the difference equation approximation to the differential equation. One way in which the 2 approaches differ is the treatment of time. Whereas there are theoretical and practical restrictions regarding observation time points and intervals in latent change score models, no such limitations exist in continuous time models. We illustrate our arguments with three simulated data sets using a univariate and bivariate model with equal and unequal time intervals. As a by-product of this comparison, we discuss the use of phantom and definition variables to account for varying time intervals in latent change score models. We end with a reanalysis of the Bradway–McArdle longitudinal study on intellectual abilities (used before by McArdle & Hamagami, 2004) by means of the proportional change score model and the dual change score model in discrete and continuous time.     

A Note on Cluster Effects in Latent Class Analysis

This article examines the effects of clustering in latent class analysis. A comprehensive simulation study is conducted, which begins by specifying a true multilevel latent class model with varying within- and between-cluster sample sizes, varying latent class proportions, and varying intraclass correlations. These models are then estimated under the assumption of a single-level latent class model. The outcomes of interest are measures of bias in the Bayesian Information Criterion (BIC) and the entropy R ² statistic relative to accounting for the multilevel structure of the data. The results indicate that the size of the intraclass correlation as well as between- and within-cluster sizes are the most prominent factors in determining the amount of bias in these outcome measures, with increasing intraclass correlations combined with small between-cluster sizes resulting in increased bias. Bias is particularly noticeable in the BIC. In addition, there is evidence that class separation interacts with the size of the intraclass correlations and cluster sizes in producing bias in these measures.     

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.     

A Nonlinear Dynamic Latent Class Structural Equation Model

In this article, we propose a nonlinear dynamic latent class structural equation modeling (NDLC-SEM). It can be used to examine intra-individual processes of observed or latent variables. These processes are decomposed into parts which include individual- and time-specific components. Unobserved heterogeneity of the intra-individual processes are modeled via a latent Markov process that can be predicted by individual- and time-specific variables as random effects. We discuss examples of sub-models which are special cases of the more general NDLC-SEM framework. Furthermore, we provide empirical examples and illustrate how to estimate this model in a Bayesian framework. Finally, we discuss essential properties of the proposed framework, give recommendations for applications, and highlight some general problems in the estimation of parameters in comprehensive frameworks for intensive longitudinal data.     

Latent Class Analysis for Multiple Discrete Latent Variables: A Study on the Association Between Violent Behavior and Drug-Using Behaviors

This article proposes a new type of latent class analysis, joint latent class analysis (JLCA), which provides a set of principles for the systematic identification of the subsets of joint patterns for multiple discrete latent variables. Inferences about the parameters are obtained by a hybrid method of expectation-maximization and Newton–Raphson algorithms. We apply JLCA in an investigation of adolescent violent behavior and drug-using behaviors. The data are from 4,957 male high-school students who participated in the Youth Risk Behavior Surveillance System in 2015. The JLCA approach identifies the different joint patterns of 4 latent variables: violent behavior, alcohol consumption, tobacco cigarette smoking, and other drug use. The JLCA uncovers 4 common violent behaviors and 3 representative behavioral patterns for each of 3 other latent variables. In addition, the JLCA supports 3 common joint classes, representing the most probable simultaneous patterns for being violent and being a drug user among adolescent males.     

Propensity Score Analysis of Complex Survey Data with Structural Equation Modeling: A Tutorial with Mplus

Propensity score (PS) analysis aims to reduce bias in treatment effect estimates obtained from observational studies, which may occur due to non-random differences between treated and untreated groups with respect to covariates related to the outcome. We demonstrate how to use structural equation modeling (SEM) for PS analysis to remove selection bias due to latent covariates and estimate treatment effects on latent outcomes. Following the discussion of the design and analysis stages of PS analysis with SEM, an example is presented which uses the Mplus software to analyze data from the 1999 School and Staffing Survey (SASS) and 2000 Teacher Follow-up Survey (TFS) to estimate the effects teacher’s participation in a network of teachers on the teacher’s perception of workload manageability.     

Modeling Unobserved Heterogeneity Using Latent Profile Analysis: A Monte Carlo Simulation

Latent profile analysis (LPA) has become a popular statistical method for modeling unobserved population heterogeneity in cross-sectionally sampled data, but very few empirical studies have examined the question of how well enumeration indexes accurately identify the correct number of latent profiles present. This Monte Carlo simulation study examined the ability of several classes of enumeration indexes to correctly identify the number of latent population profiles present under 3 different research design conditions: sample size, the number of observed variables used for LPA, and the separation distance among the latent profiles measured in Mahalanobis D units. Results showed that, for the homogeneous population (i.e., the population has k = 1 latent profile) conditions, many of the enumeration indexes used in LPA were able to correctly identify the single latent profile if variances and covariances were freely estimated. However, for a heterogeneous population (i.e., the population has k = 3 distinct latent profiles), the correct identification rate for the enumeration indexes in the k = 3 latent profile conditions was typically very low. These results are compared with the previous cross-sectional mixture modeling studies, and the limitations of this study, as well as future cross-sectional mixture modeling and enumeration index research possibilities, are discussed.     

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.     

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.