Measurement invariance is usually tested using Multigroup Confirmatory Factor Analysis, which examines the change in the goodness-of-fit index (GFI) when cross-group constraints are imposed on a measurement model. Although many studies have examined the properties of GFI as indicators of overall model fit for single-group data, there have been none to date that examine how GFIs change when between-group constraints are added to a measurement model. The lack of a consensus about what constitutes significant GFI differences places limits on measurement invariance testing. We examine 20 GFIs based on the minimum fit function. A simulation under the two-group situation was used to examine changes in the GFIs (ΔGFIs) when invariance constraints were added. Based on the results, we recommend using Δcomparative fit index, ΔGamma hat, and ΔMcDonald's Noncentrality Index to evaluate measurement invariance. These three ΔGFIs are independent of both model complexity and sample size, and are not correlated with the o...

Evaluating Goodness-of-Fit Indexes for Testing Measurement Invariance

Statistical procedures for missing data have vastly improved, yet misconception and unsound practice still abound. The authors frame the missing-data problem, review methods, offer advice, and raise issues that remain unresolved. They clear up common misunderstandings regarding the missing at random (MAR) concept. They summarize the evidence against older procedures and, with few exceptions, discourage their use. They present, in both technical and practical language, 2 general approaches that come highly recommended: maximum likelihood (ML) and Bayesian multiple imputation (MI). Newer developments are discussed, including some for dealing with missing data that are not MAR. Although not yet in the mainstream, these procedures may eventually extend the ML and MI methods that currently represent the state of the art.

/pdf/missing-data-our-view-of-the-state-of-the-art-3xnwc88wmd.pdf

Missing data: Our view of the state of the art.

Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

Two Monte Carlo studies were conducted to examine the sensitivity of goodness of fit indexes to lack of measurement invariance at 3 commonly tested levels: factor loadings, intercepts, and residual variances. Standardized root mean square residual (SRMR) appears to be more sensitive to lack of invariance in factor loadings than in intercepts or residual variances. Comparative fit index (CFI) and root mean square error of approximation (RMSEA) appear to be equally sensitive to all 3 types of lack of invariance. The most intriguing finding is that changes in fit statistics are affected by the interaction between the pattern of invariance and the proportion of invariant items: when the pattern of lack of invariance is uniform, the relation is nonmonotonic, whereas when the pattern of lack of invariance is mixed, the relation is monotonic. Unequal sample sizes affect changes across all 3 levels of invariance: Changes are bigger when sample sizes are equal rather than when they are unequal. Cutoff points for t...

Sensitivity of Goodness of Fit Indexes to Lack of Measurement Invariance

The establishment of measurement invariance across groups is a logical prerequisite to conducting substantive cross-group comparisons (e.g., tests of group mean differences, invariance of structura...

A Review and Synthesis of the Measurement Invariance Literature: Suggestions, Practices, and Recommendations for Organizational Research

Latent growth curve techniques and longitudinal data are used to examine predictions from the theory of fluid and crystallized intelligence (Gf-Gc theory; J. L. Horn & R. B. Cattell, 1966, 1967). The data examined are from a sample (N approximately 1,200) measured on the Woodcock-Johnson Psycho-Educational Battery-Revised (WJ-R). The longitudinal structural equation models used are based on latent growth models of age using two-occasion "accelerated" data (e.g., J. J. McArdle & R. Q. Bell, 2000; J. J. McArdle & R. W. Woodcock, 1997). Nonlinear mixed-effects growth models based on a dual exponential rate yield a reasonable fit to all life span cognitive data. These results suggest that most broad cognitive functions fit a generalized curve that rises and falls. Novel multilevel models directly comparing growth curves show that broad fluid reasoning (Gf) and acculturated crystallized knowledge (Gc) have different growth patterns. In all comparisons, any model of cognitive age changes with only a single g factor yields an overly simplistic view of growth and change over age.

http://www.iapsych.com/wj3ewok/LinkedDocuments/McArdle2002.pdf

Comparative longitudinal structural analyses of the growth and decline of multiple intellectual abilities over the life span.

These latent difference score analyses were previously presented to the International Society for Behavioral Development, Bern, Switzerland, in July 1998, and at the Amer­ ican Psychological Association conference "New Methods for the of Change," Pennsylvania State University, in October 1998. This research was supported by Grants AG02695, AG04704, and AG07137 from the National Institute on Aging. These Na­ tional Longitudinal Survey of Youth (NLSY) data were selected by Patrick Curran of Duke University for a presentation on comparative longitudinal analyses at the for Research on Child Development in April 1997. All NLSY data used here are and the computer program sCripts used here are available from John ]. McArdle, so all analyses should be relatively easy to reproduce from the available files or from the original data. We thank our colleagues John R. Nesselroade, Paolo Ghisletta, and Patrick CUrran for their comments on drafts of this chapter.

Latent difference score structural models for linear dynamic analyses with incomplete longitudinal data.

A number of models for the analysis of moment structures, such as LISREL, have recently been shown to be capable of being given a particularly simple and economical representation, in terms of the Reticular Action Model (RAM). In contrast to previous treatments, a formal algebraic treatment is provided which shows that RAM directly incorporates many common structural models, including models describing the structure of means. It is also shown here that RAM treats coefficient matrices with patterned inverses simply and generally.

Some algebraic properties of the Reticular Action Model for moment structures.

The term “growth curve” is used to describe data where: (1) the same entities are repeatedly observed, (2) the same procedures of measurement and scaling of observations are used, and (3) the timing of the observations is known. Growth curves are now common in many areas of psychological research, and some of these are presented here. The term “growth curve analysis” denotes the processes of describing, testing hypotheses, and making scientific inferences about the growth and change patterns in a wide range of time-related phenomena. In this sense, growth curve analyses are a specific form of the larger set of developmental and longitudinal research methods, but the unique features of growth data permit unique kinds of analyses. Formal models for the analysis of growth curves which have been developed in many different substantive domains are described here in five sections: (1) An introduction to growth curves, (2) linear models of growth, (3) multiple groups in growth curve models, (4) aspects of dynamic theory for growth models, and (5) multiple variables in growth curve analyses. We conclude with a discussion of future issues raised by the current growth models.


Keywords:

growth curves;
latent growth models;
longitudinal data analysis;
mixed models;
multilevel models;
nonlinear models

Growth Curve Analysis in Contemporary Psychological Research

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.

John J. McArdle

Papers

Comparative longitudinal structural analyses of the growth and decline of multiple intellectual abilities over the life span.

Latent difference score structural models for linear dynamic analyses with incomplete longitudinal data.

Some algebraic properties of the Reticular Action Model for moment structures.

Growth Curve Analysis in Contemporary Psychological Research

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