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.     

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).     

Inference on Two-Part Latent Variable Analysis Model With Multivariate Longitudinal Data

Semicontinuous variable analysis is a widely appreciated statistical method in such disciplines as social science, medicines, and economics. In detecting underlying structure and representing possible interrelationships, statistical analysis using a two-part model is appropriated. In this paper, we present a general extension of two-part model to the situation where the unobserved factors are included in the two parts to interpret external variability in semicontinuous variable. Auxiliary information on these factors is manifested by continuous responses via measurement model, while the interrelationships among factors are exploited through structural equation model. Moreover, under longitudinal setting, dynamic characteristics of responses between any two occasions are represented by transition model. Procedures for model fitting, parameter estimation, model selection and prediction are developed within the Bayesian paradigm. Markov Chains Monte Carlo method is used to implement posterior analysis. Empirical results including a simulation and a real example are used to illustrate the proposed methodology.     

On Latent Change Model Choice in Longitudinal Studies

An interval estimation procedure for proportion of explained observed variance in latent curve analysis is discussed, which can be used as an aid in the process of choosing between linear and nonlinear models. The method allows obtaining confidence intervals for the R ² indexes associated with repeatedly followed measures in longitudinal studies. In addition to facilitating evaluation of local model fit, the approach is helpful for purposes of differentiating between plausible models stipulating different patterns of change over time, and in particular in empirical situations characterized by large samples and high statistical power. The procedure is also applicable in cross-sectional studies, as well as with general structural equation models. The method is illustrated using data from a nationally representative study of older adults.     

Automated Latent Growth Curve Model Fitting: A Segmentation and Knot Selection Approach

Latent growth curve models are widely used in the social and behavioral sciences to study complex developmental patterns of change over time. The trajectories of these developmental patterns frequently exhibit distinct segments in the studied variables. Latent growth models with piecewise functions for repeated measurements of variables have become increasingly popular for modeling such developmental trajectories. A major problem with using piecewise models is determining the precise location of the point where the change in the process has occurred and uncovering the related number of segments. The purpose of this paper is to introduce an optimization procedure that can be used to determine both the segments and location of the knots in piecewise linear latent growth models. The procedure is illustrated using empirical data in order to detect the number of segments and change points. The results demonstrate the capabilities of the procedure for fitting latent growth curve models.     

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).     

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.     

Dyadic Curve-of-Factors Model: An Introduction and Illustration of a Model for Longitudinal Nonexchangeable Dyadic Data

The analysis of longitudinal data collected from nonexchangeable dyads presents a challenge for applied researchers for various reasons. This article introduces the dyadic curve-of-factors model (D–COFM), which extends the curve-of-factors model (COFM) proposed by McArdle (1988) for use with nonexchangeable dyadic data. The D–COFM overcomes problems with modeling composite scores across time and instead permits examination of the growth in latent constructs over time. The D–COFM also appropriately models the interdependency among nonexchangeable dyads. Different parameterizations of the D–COFM are illustrated and discussed using a real data set to aid applied researchers when analyzing dyadic longitudinal data.     

Specification Search for Identifying the Correct Mean Trajectory in Polynomial Latent Growth Models

This study investigated the optimal strategy for model specification search under the latent growth modeling (LGM) framework, specifically on searching for the correct polynomial mean or average growth model when there is no a priori hypothesized model in the absence of theory. In this simulation study, the effectiveness of different starting models on the search of the true mean growth model was investigated in terms of the mean and within-subject variance-covariance (V-C) structure model. The results showed that specifying the most complex (i.e., unstructured) within-subject V-C structure with the use of LRT, ΔAIC, and ΔBIC achieved the highest recovery rate (>85%) of the true mean trajectory. Implications of the findings and limitations are discussed.     

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.     

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.     

Testing Measurement Invariance Across Groups in Longitudinal Data: Multigroup Second-Order Latent Growth Model

In latent growth modeling, measurement invariance across groups has received little attention. Considering that a group difference is commonly of interest in social science, a Monte Carlo study explored the performance of multigroup second-order latent growth modeling (MSLGM) in testing measurement invariance. True positive and false positive rates in detecting noninvariance across groups in addition to bias estimates of major MSLGM parameters were investigated. Simulation results support the suitability of MSLGM for measurement invariance testing when either forward or iterative likelihood ratio procedure is applied.     

A Short Note on Interpretation in the Dual Change Score Model

The latent change score framework allows for estimating a variety of univariate trajectory models, such as the no change, linear change, exponential forms of change, as well as multivariate trajectory models that allow for coupling between two or more constructs. A particularly attractive feature of these models is that it is easy to decompose and interpret aspects of change. One particularly flexible model, the dual change score model, has two components of change: a proportional change component that depends on scores at the previous time point, and a constant change component that is additive. We demonstrate through simulation and an empirical example that in a correctly specified model, the correlation between the proportional change parameter and the mean of the constant change component can approach either ?1 or 1, thus complicating interpretation. We provide recommendations and code to aid researchers’ ability to diagnose this issue in their own data.     

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.     

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.     

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.     

A Unified Latent Growth Curve Model

Applying item response theory models to repeated observations has demonstrated great promise in developmental research. By allowing the researcher to take account of the characteristics of both item response and measurement error in longitudinal trajectory analysis, it improves the reliability and validity of latent growth curve analysis. This has enabled the study, to differentially weigh individual items and examine developmental stability and change over time, to propose a comprehensive modeling framework, combining a measurement model with a structural model. Despite a large number of components requiring attention, this study focuses on model formulation, evaluates the performance of the estimators of model parameters, incorporates prior knowledge from Bayesian analysis, and applies the model using an illustrative example. It is hoped that this fundamental study can demonstrate the breadth of this unified latent growth curve model.     

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.