Longitudinal confirmatory factor analysis: Common,time‐specific,item‐specific,and residual‐error components of variance

Following Cronbach (1970) and others, it is useful to decompose test score variation into common factor, time‐specific, item‐specific, and residual components. In the traditional approach to factor analysis, only two sources of variance can be estimated: common factor variance and a uniqueness term that confounds specific sources of variation and residual error. When the same items are measured on different occasions, however, it is possible to separate specific variance and residual error. Two approaches, the first‐order approach described by Raffalovich and Bohrnstedt (1987) and a second‐order approach based on Jöreskog and Sörbom (1989; Jöreskog, 1974) are considered initially. The two approaches, although based on different rationales, both suffer a similar weakness in that two of the four sources of variance are confounded. In the Raffalovich and Bohrnstedt approach, time‐specific variance is confounded with common factor variance that generalizes across items and time. In the second‐order approach based on Jöreskog and Sörbom, time‐specific variance is confounded with residual error. Here we demonstrate that by combining features from both approaches we can eliminate these weaknesses and estimate all four of Cronbach's sources of variance, and that this combined approach is easily generalized to a wide variety of applications.     

Software review

STATISTICA 5.0. StatSoft, 2325 East 13th Street, Tulsa, OK 74104, (918) 583–4149. $995 retail, academic site license—$2,000 for 10 copies. Requirements: 386 or better, 4 Meg RAM, Microsoft Windows 3.1 or 95.     

Binomial index of model fit: An elaboration

This article is an elaboration on the use of the binomial test of model fit value, which in this article will be referred to as the binomial index of model fit value, to gauge the degree that the data fit a path analytic or structural equation model. In addition, this article responds to the criticisms and comments made by Hsu (this issue), Drezner and Drezner (this issue), and Raykov and Penev (this issue) regarding the use of this approach to measuring the degree of model fit. We appreciate the comments provided by these authors. Their comments have assisted us in clarifying our reason for developing the binomial index of model fit procedure as well as our perception of its use.     

Unconstrained Structural Equation Models of Latent Interactions: Contrasting Residual- and Mean-Centered Approaches

Little, Bovaird and Widaman (2006) proposed an unconstrained approach with residual centering for estimating latent interaction effects as an alternative to the mean-centered approach proposed by Marsh, Wen, and Hau (2004, 2006). Little et al. also differed from Marsh et al. in the number of indicators used to infer the latent interaction factor and how they were represented, but this issue is separate from the mean versus residual centering distinction that was their primary focus. However, their implementation of the Marsh et al. mean-centered approach failed to incorporate the mean structure that Marsh et al. argued was necessary to obtain unbiased estimates. One might suppose that their new approach would suffer this same problem, an issue not addressed by Little et al. However, we demonstrate here why the Little et al. approach obviates this requirement that heretofore was thought to be necessary for all constrained, partially constrained, and unconstrained approaches. Both the Marsh et al. and Little et al. unconstrained approaches typically result in similar results and are much easier to implement than traditional constrained approaches. They differ primarily in that the Little et al. approach is a 2-step approach involving a potentially large number of separate analyses prior to estimating the structural equation model that apparently does not require the estimation of a mean structure, whereas the Marsh et al. approach is a 1-step approach that includes a mean structure.     

General Growth Mixture Analysis of Adolescents' Developmental Trajectories of Anxiety: The Impact of Untested Invariance Assumptions on Substantive Interpretations

Substantively, this study investigates potential heterogeneity in the developmental trajectories of anxiety in adolescence. Methodologically, this study demonstrates the usefulness of general growth mixture analysis (GGMA) in addressing these issues and illustrates the impact of untested invariance assumptions on substantive interpretations. This study relied on data from the Montreal Adolescent Depression Development Project (MADDP), a 4-year follow-up of more than 1,000 adolescents who completed the Beck Anxiety Inventory each year. GGMA models relying on different invariance assumptions were empirically compared. Each of these models converged on a 5-class solution, but yielded different substantive results. The model with class-varying variance–covariance matrices was retained as providing a better fit to the data. These results showed that although elevated levels of anxiety might fluctuate over time, they clearly do not represent a transient phenomenon. This model was then validated in relation to multiple predictors (mostly related to school violence) and outcomes (grade-point average, school dropout, depression, loneliness, and drug-related problems).     

A Comparison of Approaches for the Analysis of Interaction Effects Between Latent Variables Using Partial Least Squares Path Modeling

Although stratification in the workplace has received considerable attention in the literature on organizations, there is little systematic understanding about how individual and structural characteristics act to produce a stratified workplace. This study focuses on understanding the dynamics of workplace attainment by examining stratification within higher education administration. The purpose of the study is twofold—(a) to propose and test a model concerning how organizational and individual variables affect workplace attainment and (b) to demonstrate the application of structural equation modeling in estimating and testing this model. Results are discussed in terms of their theoretical significance and practical implications.     

Estimation of MIMIC Model Parameters with Multilevel Data

The purpose of this simulation study was to assess the performance of latent variable models that take into account the complex sampling mechanism that often underlies data used in educational, psychological, and other social science research. Analyses were conducted using the multiple indicator multiple cause (MIMIC) model, which is a flexible and effective tool for relating observed and latent variables. The data were simulated in a hierarchical framework (e.g., individuals nested in schools) so that a multilevel modeling approach would be appropriate. Analyses were conducted accounting for and not accounting for the nested data to determine the impact of ignoring such multilevel data structures in full structural equation models. Results highlight the differences in modeling results when the analytic strategy is congruent with the data structure and what occurs when this congruency is absent. Type I error rates and power for the standard and multilevel methods were similar for within-cluster variables and for the multilevel model with between-cluster variables. However, Type I error rates were inflated for the standard approach when modeling between-cluster variables.     

Multivariate Meta-Analysis as Structural Equation Models

Multivariate meta-analysis has become increasingly popular in the educational, social, and medical sciences. It is because the outcome measures in a meta-analysis can involve more than one effect size. This article proposes 2 mathematically equivalent models to implement multivariate meta-analysis in structural equation modeling (SEM). Specifically, this article shows how multivariate fixed-, random- and mixed-effects meta-analyses can be formulated as structural equation models. metaSEM (a free R package based on OpenMx) and Mplus are used to implement the proposed procedures. A real data set is used to illustrate the procedures. Formulating multivariate meta-analysis as structural equation models provides many new research opportunities for methodological development in both meta-analysis and SEM. Issues related to and extensions on the SEM-based meta-analysis are discussed.     

An Evaluation of Latent Growth Models for Propensity Score Matched Groups

Because random assignment is not possible in observational studies, estimates of treatment effects might be biased due to selection on observable and unobservable variables. To strengthen causal inference in longitudinal observational studies of multiple treatments, we present 4 latent growth models for propensity score matched groups, and evaluate their performance with a Monte Carlo simulation study. We found that the 4 models performed similarly with respect to model fit, bias of parameter estimates, Type I error, and power to test the treatment effect. To demonstrate a multigroup latent growth model with dummy treatment indicators, we estimated the effect of students changing schools during elementary school years on their reading and mathematics achievement, using data from the Early Childhood Longitudinal Study Kindergarten Cohort.     

Disentangling Shape from Level Effects in Person-Centered Analyses: An Illustration Based on University Teachers’ Multidimensional Profiles of Effectiveness

This study compares alternative ways of disentangling the effects of level (the tendency for a person to be high, medium, or low across all factors) and shape (the tendency for a person to have a distinct pattern of factors on which they are high, medium, or low) in profile analyses. This issue is particularly relevant to performance appraisals where it is often useful to identify specific strengths and weaknesses over and above a person global performance, but also to person-centered analyses more generally where the observation of qualitative (shape) differences between profiles is often used as justification for the added value of profiles. Substantively, this study illustrates these issues in the identification of profiles of teachers based on multidimensional students’ ratings of their effectiveness, using an archival data set of 31,951 class-average ratings based on the Students’ Evaluations of Educational Quality (SEEQ) instrument collected over a 13-year period. The results show the superiority of a factor mixture operationalization of teaching effectiveness in which a global effectiveness factor was used to control for unnecessary level effects in the profiles.