John J. McArdle

Latent Variable Modeling of Differences and Changes with Longitudinal Data

This review considers a common question in data analysis: What is the most useful way to analyze longitudinal repeated measures data? We discuss some contemporary forms of structural equation models (SEMs) based on the inclusion of latent variables. The specific goals of this review are to clarify basic SEM definitions, consider relations to classical models, focus on testable features of the new models, and provide recent references to more complete presentations. A broader goal is to illustrate why so many researchers are enthusiastic about the SEM approach to data analysis. We first outline some classic problems in longitudinal data analysis, consider definitions of differences and changes, and raise issues about measurement errors. We then present several classic SEMs based on the inclusion of invariant common factors and explain why these are so important. This leads to newer SEMs based on latent change scores, and we explain why these are useful.

Latent growth curves within developmental structural equation models.

This report uses structural equation modeling to combine traditional ideas from repeated-measures ANOVA with some traditional ideas from longitudinal factor analysis. A longitudinal model that includes correlations, variances, and means is described as a latent growth curve model (LGM). When merged with repeated-measures data, this technique permits the estimation of parameters representing both individual and group dynamics. The statistical basis of this model allows hypothesis testing of various developmental ideas, including models of alternative dynamic functions and models of the sources of individual differences in these functions. Aspects of these latent growth models are illustrated with a set of longitudinal WISC data from young children and by using the LISREL V computer program.

Dynamic but Structural Equation Modeling of Repeated Measures Data

The term “dynamic” is broadly defined as a pattern of change. Many scientists have searched for dynamics by calculating df/dt: the ratio of changes or differences d in a function f relative to changes in time t.This simple dynamic equation was used in the 16th and 17th century motion experiments of Galileo, in the 17th and 18th century gravitation experiments of Newton, and in the 19th century experiments of many physicists and chemists (see Morris, 1985). I also use this dynamic equation, but here I examine multivariate psychological change data using the 20th century developments of latent variable structural equation modeling.

Latent variable growth within behavior genetic models.

The purpose of this paper is to introduce one kind of latent-variable structural-equation model for multivariate longitudinal data which includes behavioral genetic components. A generic structural-equation model termedRAM (McArdle, J. J. and McDonald, R. P. (1984).Br. J. Math. Stat. Psychol.,37:239–251.) is used to define the univariate twin design, including both covariances and means. This model is extended to multivariate form using a latent-variable growth-curve model recently presented by W. Meredith and J. Tisak [(1984). “Tuckerizing” curves. Psychometric Society Annual Meetings]. The model presented herein further permits hypothesis testing of various biometric models of the sources of these individual differences in latent growth. Aspects of this model are illustrated using the LISREL algorithm [Jöreskog, K. G. and Sörbom, D. (1979).Advances in Factor Analysis and Structural Equation Models, Abt Books, Cambridge, Mass.] and longitudinal twin data on early childhood abilities [Wilson, R. S. (1983).Child Dev.54:298–316].

Modeling incomplete longitudinal and cross-sectional data using latent growth structural models

In this paper we describe some mathematical and statistical models for identifying and dealing with changes over age. We concentrate specifically on the use of a latent growth structural equation model approach to deal with issues of: (1) latent growth models of change, (2) differences in longitudinal and cross-sectional results, and (3) differences due to longitudinal attrition. This is a methodological paper using simulated data, but we base our models on practical and conceptual principles of modeling change in developmental psychology. Our results illustrate both benefits and limitations using structural models to analyze incomplete longitudinal data.

Alternative Structural Models for Multivariate Longitudinal Data Analysis

Structural equation models are presented as alternative models for examining longitudinal data. The models include (a) a cross-lagged regression model, (b) a factor model based on latent growth curves, and (c) a dynamic model based on latent difference scores. The illustrative data are on motivation and perceived competence of students during their first semester in high school. The 3 models yielded different results and such differences were discussed in terms of the conceptualization of change underlying each model. The last model was defended as the most reasonable for these data because it captured the dynamic interrelations between the examined constructs and, at the same time, identified potential growth in the variables.

Structural Factor Analysis Experiments with Incomplete Data

This article presents some benefits and limitations of structural equation models for multivariate experiments with incomplete data. Examples from studies of latent variable path models of cognitive performances illustrate analyses with four different kinds of incomplete data: (a) latent variables, (b) omitted variables, (c) randomly missing data, and (d) non- randomly missing data. Power based cost-benefit analyses for experimental design and planning are also presented. These incomplete data approaches are closely related to models used in classical experimental design, interbattery measurement analysis, longitudinal analyses, and behavioral genetic analyses. These structural equation methods for old experimental design problems indicate some new opportunities for future multivariate research.

Longitudinal Modeling of Developmental Changes in Psychological Research

In this article we provide a review of recent advances in longitudinal models for multivariate change. We first claim the need for dynamic modeling approaches as a way to evaluate psychological theories. We then describe one such approach, latent change score (LCS) models, and illustrate their utility with a summary of research findings in various areas of psychological science. We then highlight the most prominent features of LCS models. We conclude the article with suggestions for future research on multivariate models of change that can enhance our understanding of psychological science.

Age changes in processing speed as a leading indicator of cognitive aging.

Bivariate dual change score models were applied to longitudinal data from the Swedish Adoption/Twin Study of Aging to compare the dynamic predictions of 2-component theories of intelligence and the processing speed theory of cognitive aging. Data from up to 5 measurement occasions covering a 16-year period were available from 806 participants ranging in age from 50 to 88 years at the first measurement wave. Factors were generated to tap 4 general cognitive domains: verbal ability, spatial ability, memory, and processing speed. Model fitting indicated no dynamic relationship between verbal and spatial factors, providing no support for the hypothesis that age changes in fluid abilities drive age changes in crystallized abilities. The results suggest that, as predicted by the processing speed theory of cognitive aging, processing speed is a leading indicator of age changes in memory and spatial ability, but not verbal ability.

Longitudinal change in cognitive performance among individuals with mild cognitive impairment.

The authors used mixed-effects growth models to examine longitudinal change in neuropsychological performance over a 4-year period among 197 individuals who were either normal or had mild cognitive impairment (MCI) at baseline. At follow-up, the participants were divided into 4 groups: (a) controls: participants who were normal at both baseline and follow-up (n = 33), (b) stables: participants with MCI whose Clinical Dementia Rating-Sum of Boxes (CDR-SB) score did not differ between the first and last evaluations (n = 22), (c) decliners: participants with MCI whose CDR-SB score declined between the first and last evaluations (n = 95), and (d) converters: participants who received a clinical diagnosis of Alzheimers disease during the follow-up period (n = 47). Only the Episodic Memory factor showed a significantly greater rate of decline over the follow-up period among the converters. Two other factors were significantly lower in converters at baseline in comparison with other groups (the executive function factor and the general knowledge factor), but the rate of decline over time did not differ. Individuals with an APOE epsilon4 allele scored lower on the episodic memory and executive function factors at baseline.