Mediation Analysis in a Latent Growth Curve Modeling Framework

This article presents several longitudinal mediation models in the framework of latent growth curve modeling and provides a detailed account of how such models can be constructed. Logical and statistical challenges that might arise when such analyses are conducted are also discussed. Specifically, we discuss how the initial status (intercept) and change (slope) of the putative mediator variable can be appropriately included in the causal chain between the independent and dependent variables in longitudinal mediation models. We further address whether the slope of the dependent variable should be controlled for the dependent variable's intercept to improve the conceptual relevance of the mediation models. The models proposed are illustrated by analyzing a longitudinal data set. We conclude that for certain research questions in developmental science, a multiple mediation model where the dependent variable's slope is controlled for its intercept can be considered an adequate analytical model. However, such models also show several limitations.     

Investigation of Mediational Processes Using Parallel Process Latent Growth Curve Modeling

This study investigated a method to evaluate mediational processes using latent growth curve modeling. The mediator and the outcome measured across multiple time points were viewed as 2 separate parallel processes. The mediational process was defined as the independent variable influencing the growth of the mediator, which, in turn, affected the growth of the outcome. To illustrate modeling procedures, empirical data from a longitudinal drug prevention program, Adolescents Training and Learning to Avoid Steroids, were used. The program effects on the growth of the mediator and the growth of the outcome were examined first in a 2-group structural equation model. The mediational process was then modeled and tested in a parallel process latent growth curve model by relating the prevention program condition, the growth rate factor of the mediator, and the growth rate factor of the outcome.     

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.     

Modeling Change in Effort Across a Low-Stakes Testing Session: A Latent Growth Curve Modeling Approach

ABSTRACT

We examined change in test-taking effort over the course of a three-hour, five test, low-stakes testing session. Latent growth modeling results indicated that change in test-taking effort was well-represented by a piecewise growth form, wherein effort increased from test 1 to test 4 and then decreased from test 4 to test 5. There was significant variability in effort for each of the five tests, which could be predicted from examinees’ conscientiousness, agreeableness, mastery approach goal orientation, and whether the examinee “skipped” or attended the initial testing session. The degree to which examinees perceived a particular test as important was related to effort for the difficult, cognitive test but not for less difficult, noncognitive tests. There was significant variability in the rates of change in effort, which could be predicted from examinees’ agreeableness. Interestingly, change in test-taking effort was not related to change in perceived test importance. Implications of these results for assessment practice and directions for future research 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.     

Accuracy of Estimates and Statistical Power for Testing Meditation in Latent Growth Curve Modeling

The latent growth curve modeling (LGCM) approach has been increasingly utilized to investigate longitudinal mediation. However, little is known about the accuracy of the estimates and statistical power when mediation is evaluated in the LGCM framework. A simulation study was conducted to address these issues under various conditions including sample size, effect size of mediated effect, number of measurement occasions, and R ₂ of measured variables. In general, the results showed that relatively large samples were needed to accurately estimate the mediated effects and to have adequate statistical power, when testing mediation in the LGCM framework. Guidelines for designing studies to examine longitudinal mediation and ways to improve the accuracy of the estimates and statistical power were discussed.     

Detecting Unobserved Heterogeneity in Latent Growth Curve Models

Growth mixture models combine latent growth curve models and finite mixture models to examine the existence of latent classes that follow distinct developmental patterns. Analyses based on these models are becoming quite common in social and behavioral science research because of recent advances in computing, the availability of specialized statistical programs, and the ease of programming. In this article, we show how mixture models can be fit to examine the presence of multiple latent classes by algorithmically grouping or clustering individuals who follow the same estimated growth trajectory based on an evaluation of individual case residuals. The approach is illustrated using empirical longitudinal data along with an easy to use computerized implementation.     

Evaluating Intercept-Slope Interactions in Latent Growth Modeling

The effects of misspecifying intercept-covariate interactions in a 4 time-point latent growth model were the focus of this investigation. The investigation was motivated by school growth studies in which students' entry-level skills may affect their rate of growth. We studied the latent interaction of intercept and a covariate in predicting growth with respect to 3 factors: sample size (100, 200, and 500), 4 levels of magnitude of interaction effect, and 3 correlation values between intercept and covariate (.3, .5, and .7). Correctly specified models were examined to determine power and Type I error rates, and misspecified models were examined to evaluate the effects on power, parameter estimation, bias, and fit indexes. Results showed that, under correctly specified models, power increased as expected with increasing sample size, larger magnitude of interaction, and larger intercept-covariate correlation. Under misspecification, omitting a non-null interaction results in significant change in the estimation of the direct effects of both covariate and intercept in both magnitude and direction, with results dependent on sign of parameter values for main effects and interaction. Including a spurious interaction does not affect estimation of direct effects of intercept and covariate but does increase standard errors. The primary problem in ignoring a non-null interaction lies in misinterpretation of the model, as interactions yield important insights into the nature of the processes being studied.     

Incorporating Measurement Nonequivalence in a Cross-Study Latent Growth Curve Analysis

A large literature emphasizes the importance of testing for measurement equivalence in scales that may be used as observed variables in structural equation modeling applications. When the same construct is measured across more than one developmental period, as in a longitudinal study, it can be especially critical to establish measurement equivalence, or invariance, across the developmental periods. Similarly, when data from more than one study are combined into a single analysis, it is again important to assess measurement equivalence across the data sources. Yet, how to incorporate nonequivalence when it is discovered is not well described for applied researchers. Here, we present an item response theory approach that can be used to create scale scores from measures while explicitly accounting for nonequivalence. We demonstrate these methods in the context of a latent curve analysis in which data from two separate studies are combined to estimate a single longitudinal model spanning several developmental periods.     

Measurement Invariance Across Groups in Latent Growth Modeling

This Monte Carlo study investigated the impacts of measurement noninvariance across groups on major parameter estimates in latent growth modeling when researchers test group differences in initial status and latent growth. The average initial status and latent growth and the group effects on initial status and latent growth were investigated in terms of Type I error and bias. The location and magnitude of noninvariance across groups was related to the location and magnitude of bias and Type I error in the parameter estimates. That is, noninvariance in factor loadings and intercepts was associated with the Type I error inflation and bias in the parameter estimates of the slope factor (or latent growth) and the intercept factor (or initial status), respectively. As noninvariance became large, the degree of Type I error and bias also increased. On the other hand, a correctly specified second-order latent growth model yielded unbiased parameter estimates and correct statistical inferences. Other findings and implications on future studies were discussed.     

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.     

Number of Factors in Growth Curve Modeling

Basic growth curve models parameterize the mean and covariance structure of a set of repeated measures by latent factors that represent the polynomial influences of time. In practice it may be hard to choose the number of factors, i.e., the order of the polynomial. Simple calculations are proposed to estimate this order.     

Residual Structures in Growth Models With Ordinal Outcomes

Growth models allow for the study of within-person change and between-person differences in within-person change. Typically, these models are applied to continuous variables where the residuals are assumed to be normally distributed. With normally distributed residuals there are a variety of residual structures that can be imposed and tested, which have been shown to affect model fit and parameter estimation. This article concerns residual structures in growth models with binary and ordered categorical outcomes using the probit link function. Different residual structures and their appropriateness for growth data are discussed and their use is illustrated with longitudinal data collected as part of Head Start’s Family and Child Experiences Survey 1997 Cohort. We close with recommendations for the specification and parameterization of growth models that use the probit link.     

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.     

Latent Growth Modeling of Longitudinal Data: A Finite Growth Mixture Modeling Approach

Recent developments in finite mixture modeling allow for the identification of different developmental processes in distinct but unobserved subgroups within a population. The new approach, described within the general growth mixture modeling framework (Muthen, 2001, in press), extends conventional random coefficient growth models to incorporate a categorical latent trajectory variable representing latent classes or mixtures (i.e., the subgroups in the population whose membership must be inferred from the data). This article provides a didactic example of this new methodology with adolescent alcohol use data, which is shown to consist of a mixture of distinct subgroups, defined by unique growth trajectories and differing predictors and sequelae. The method is discussed as a useful tool for mapping hypotheses of development onto appropriate statistical models.     

The Effect of Sampling-Time Variation on Latent Growth Curve Models

Longitudinal data are often collected in waves in which a participant’s data can be collected at different times within each wave, resulting in sampling-time variation that is unaccounted for when waves are treated as single time points. Little research has been reported on the effects of this temporal imprecision on longitudinal growth-curve modeling. This article describes the results of a simulation study into the effect of sampling-time variation on parameter estimation, model fit, and model comparison with an empirical validation of the model fit and comparison results.     

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.     

Time-Varying Effect Sizes for Quadratic Growth Models in Multilevel and Latent Growth Modeling

Multilevel and latent growth modeling analysis (GMA) is often used to compare independent groups in linear random slopes of outcomes over time, particularly in randomized controlled trials. The unstandardized coefficient for the effect of group on the slope from a linear GMA can be transformed into a model-estimated effect size for the group difference at the end of a study. Because effect sizes vary nonlinearly in quadratic GMA, the effect size at the end of a study using quadratic GMA cannot be derived from a single coefficient, and cannot be used to estimate effect sizes at intermediate time points with backward extrapolation. This article formulates equations and associated input commands in Mplus for time-varying effect sizes for quadratic GMA. Illustrative analyses that produced these time-varying effect sizes were presented, and a Monte Carlo study found that bias in the effect sizes and their confidence intervals was ignorable.