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

Verbal memory is typically studied using immediate recall (IR) and delayed recall (DR) scores, although DR is dependent on IR capability. Separating these components may be useful for deciphering the genetic variation in age-related memory abilities. This study was conducted to (a) construct individual trajectories in IR and independent aspects of delayed recall, or residualized-DR (rDR), across older adulthood; and (b) identify genetic markers that contribute to four estimated phenotypes: IR and rDR levels and changes after age 60. A cognitively intact sample (N = 20,650 with 125,164 observations) was drawn from the U.S. Health and Retirement Study, a nationally representative study of adults aged 50 and older. Mixed effects regression models were constructed using repeated measures from data collected every two years (1996-2012) to estimate level at age 60 and change in memory post-60 in IR and rDR. Genome-wide association scans (GWAS) were conducted in the genotypic subsample (N = 7,486) using ~1.2 million single nucleotide polymorphisms (SNPs). One SNP (rs2075650) in TOMM40 associated with rDR level at the genome-wide level (p = 5.0x10-08), an effect that replicated in an independent sample from the English Longitudinal Study on Ageing (N = 6,898 with 41,328 observations). Meta-analysis of rDR level confirmed the association (p = 5.0x10-11) and identified two others in TOMM40 (rs71352238 p = 1.0x10-10; rs157582 p = 7.0x10-09), and one in APOE (rs769449 p = 3.1 x10-12). Meta-analysis of IR change identified associations with three of the same SNPs in TOMM40 (rs157582 p = 8.3x10-10; rs71352238 p = 1.9x10-09) and APOE (rs769449 p = 2.2x10-08). Conditional analyses indicate GWAS signals on rDR level were driven by APOE, whereas signals on IR change were driven by TOMM40. Additionally, we found that TOMM40 had effects independent of APOE e4 on both phenotypes. Findings from this first U.S. population-based GWAS study conducted on both age-related immediate and delayed verbal memory merit continued examination in other samples and additional measures of verbal memory.

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Genetic variants specific to aging-related verbal memory: Insights from GWASs in a population-based cohort.

/pdf/exploratory-data-mining-with-structural-equation-model-trees-2c5glm4jqf.pdf

Exploratory data mining with structural equation model trees

Background/Objective:
To develop a latent behavioral model by identifying and confirming the factor structure of health behaviors of people with spinal cord injury (SCI) and their relationships with biographic, injury, and educational characteristics.

A latent variable structural path model of health behaviors after spinal cord injury.

This article describes statistical research on the academic performance of student-athletes in college sports programs. We describe several statistical models used in the prediction of academic success defined by college persistence and graduation. Using longitudinal data on the academic performances of about 3,000 student-athletes in NCAA Division I collegiate sports programs, we formulate logit and multilevel logit statistical models for the prediction of graduation rates. These prediction models are based on academic, demographic, and athletic variables, and are used to account for differences in both the students and the colleges. These results show (1) moderate but significant relationships between precollege academic characteristics and college graduation, (2) small but significant differential validity of prediction between major student-athlete groups, (3) notable college-level variance in the average graduation rate, (4) small but significant within-college relationships between precolle...

Logit and Multilevel Logit Modeling of College Graduation for 1984-1985 Freshman Student-Athletes

In a recent article McArdle and Prescott (2005) showed how simultaneous estimation of the biometric parameters can be easily programmed using current mixed-effects modeling programs (e.g., SAS PROC MIXED). This article extends these concepts to deal with mixed-effect modeling of longitudinal twin data. The biometric basis of a polynomial growth curve model was used by Vandenberg and Falkner (1965) and this general class of longitudinal models was represented in structural equation form as a latent curve model by McArdle (1986). The new mixed-effects modeling approach presented here makes it easy to analyze longitudinal growth-decline models with biometric components based on standard maximum likelihood estimation and standard indices of goodness-of-fit (i.e., chi(2), df, epsilon(a)). The validity of this approach is first checked by the creation of simulated longitudinal twin data followed by numerical analysis using different computer programs (i.e., Mplus, Mx, MIXED, NLMIXED). The practical utility of this approach is examined through the application of these techniques to real longitudinal data from the Swedish Adoption/Twin Study of Aging (Pedersen et al., 2002). This approach generally allows researchers to explore the genetic and nongenetic basis of the latent status and latent changes in longitudinal scores in the absence of measurement error. These results show the mixed-effects approach easily accounts for complex patterns of incomplete longitudinal or twin pair data. The results also show this approach easily allows a variety of complex latent basis curves, such as the use of age-at-testing instead of wave-of-testing. Natural extensions of this mixed-effects longitudinal approach include more intensive studies of the available data, the analysis of categorical longitudinal data, and mixtures of latent growth-survival/frailty models.

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John J. McArdle

Papers

Genetic variants specific to aging-related verbal memory: Insights from GWASs in a population-based cohort.

Exploratory data mining with structural equation model trees

A latent variable structural path model of health behaviors after spinal cord injury.

Logit and Multilevel Logit Modeling of College Graduation for 1984-1985 Freshman Student-Athletes

Latent curve analyses of longitudinal twin data using a mixed-effects biometric approach.