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

The t-test is a common statistical test of differences in means. Despite the fact that its extension, the paired t-test (t-testP), appears in most introductory statistics textbooks, it is less known that for repeated variables the t-testP is in fact a model of change that can be replicated within the Structural Equation Modeling (SEM) framework. We show how to perform the t-testP with latent change scores (LCS) models, which allow for direct testing of significance of mean changes, and moreover can explain inter-individual (and group) differences in changes over time.

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The paired t-test as a simple latent change score model

Previous analyses have identified a genetic contribution to the correlation between declines with age in processing speed and higher cognitive abilities. The goal of the current analysis was to apply the biometric dual change score model to consider the possibility of temporal dynamics underlying the genetic covariance between aging trajectories for processing speed and cognitive abilities. Longitudinal twin data from the Swedish Adoption/Twin Study of Aging, including 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 1st measurement wave. Factors were generated to tap 4 cognitive domains: verbal ability, spatial ability, memory, and processing speed. Model-fitting indicated that genetic variance for processing speed was a leading indicator of variation in age changes for spatial and memory ability, providing additional support for processing speed theories of cognitive aging.

/pdf/genetic-variance-in-processing-speed-drives-variation-in-w6rc351wpo.pdf

Genetic variance in processing speed drives variation in aging of spatial and memory abilities.

Five Steps in the Structural Factor Analysis of Longitudinal Data

This article explores new statistical methodologies for using longitudinal data in the early prediction of Alzheimer’s disease (AD). Specifically, the authors examine some new techniques that allow the joint or “shared” estimation of longitudinal components based on both duration (survival) and quantitative changes (growth curves). These new shared growth-survival parameter models may be used to characterize the declining functions that anticipate the onset of AD. The authors apply these models to data from the Kungsholmen Project, a longitudinal study of aging in Stockholm, Sweden. They examine age-based survival-frailty models for the onset of AD, latent growth-decline curve models for changes in cognition over age, and 3 alternative forms of models for the shared relationships of survival and early cognitive decline. The accuracy and reliability of this approach is considered for a better understanding of the developmental course of AD in these data, including the potential removal of biases due to sub...

Longitudinal models of growth and survival applied to the early detection of Alzheimer's disease

Classification and Regression Trees (CART), and their successors—bagging and random forests, are statistical learning tools that are receiving increasing attention. However, due to characteristics of censored data collection, standard CART algorithms are not immediately transferable to the context of survival analysis. Questions about the occurrence and timing of events arise throughout psychological and behavioral sciences, especially in longitudinal studies. The prediction power and other key features of tree-based methods are promising in studies where an event occurrence is the outcome of interest. This article reviews existing tree algorithms designed specifically for censored responses as well as recently developed survival ensemble methods, and introduces available computer software. Through simulations and a practical example, merits and limitations of these methods are discussed. Suggestions are provided for practical use.

John J. McArdle

Papers

The paired t-test as a simple latent change score model

Genetic variance in processing speed drives variation in aging of spatial and memory abilities.

Five Steps in the Structural Factor Analysis of Longitudinal Data

Longitudinal models of growth and survival applied to the early detection of Alzheimer's disease

Rationale and Applications of Survival Tree and Survival Ensemble Methods.