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.

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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

http://psycnet.apa.org/books/10099/017.pdf

Best methods for the analysis of change: Recent advances, unanswered questions, future directions.

In this article, we describe a recent development in the analysis of attrition: using classification and regression trees (CART) and random forest methods to generate inverse sampling weights. These flexible machine learning techniques have the potential to capture complex nonlinear, interactive selection models, yet to our knowledge, their performance in the missing data analysis context has never been evaluated. To assess the potential benefits of these methods, we compare their performance with commonly employed multiple imputation and complete case techniques in 2 simulations. These initial results suggest that weights computed from pruned CART analyses performed well in terms of both bias and efficiency when compared with other methods. We discuss the implications of these findings for applied researchers.

Using Classification and Regression Trees (CART) and random forests to analyze attrition: Results from two simulations.

BackgroundEducational policy makers and test critics often assert that standardized test scores are strongly influenced by factors beyond individual differences in academic achievement such as fami...

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Race, Poverty and SAT Scores: Modeling the Influences of Family Income on Black and White High School Students' SAT Performance

We describe methods to fit structured latent growth curves to data from MZ and DZ twins. The well-known Gompertz, logistic and exponential curves may be written as a function of three components-asymptote, initial value, and rate of change. These components are allowed to vary and covary within individuals in a structured latent growth model . Such models are highly economical, requiring a small number of parameters to describe covariation across many occasions of measurement. We extend these methods to analyse longitudinal data from MZ and DZ twins and focus on the estimation of genetic and environmental variation and covariation in each of the asymptote, initial and rate of growth factors. For illustration, the models are fitted to longitudinal Bayley Infant Mental Development Scale data published by McArdle (1986). In these data, al l three components of growth appear strongly familial with the majority of variance associated with the shared environment; differences between the models were not great. Occasionspecific residual factors not associated with the curve components account for approximately 40% of variance of which a significant proportion is additive genetic. Though the growth curve model fit less well than some others, they make restrictive, falsifiable predictions about the mean, variance and twin covariance of other (not yet measured) occasions of measurement.

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Structured latent growth curves for twin data.

The purpose of this article is to describe a methodology for the evaluation of biometric genetic hypotheses in the context of a developmental model of growth and change. Linear structural equation models are described for longitudinal and twin data, including aspects of subject attrition and practice effects. These models are applied to 2 variables measured at several points in time in the New York Twin Study (Jarvik, Kallman, & Falek, 1962; Kallman, Feingold, & Bondy, 1951). The patterns of psychometric and biometric changes are different for the 2 intellectual variables. Substantive results are discussed in relation to gf/gc theory (Cattell, 1971; Horn, 1988), and some methodological limitations are emphasized.

John J. McArdle

Papers

Best methods for the analysis of change: Recent advances, unanswered questions, future directions.

Using Classification and Regression Trees (CART) and random forests to analyze attrition: Results from two simulations.

Race, Poverty and SAT Scores: Modeling the Influences of Family Income on Black and White High School Students' SAT Performance

Structured latent growth curves for twin data.

A contemporary method for developmental‐genetic analyses of age changes in intellectual abilities