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

In prior research we have shown how linear structural equation models and computer programs (e.g., LISREL) may be simply and directly used to provide alternatives for the traditional biometric twin design. We use structural equations and path models to define biometric group differences, we write traditional common-factor models in the same way, and then we take a detailed look at some alternative multivariate and biometric models. We contrast the biometricfactors covariance structure approach used by Loehlin and Vandenberg (1968), Martin and Eaves (1977), and others with the psychometric-factors approach used by McArdle et al. (1980) and others. We use the multivariate primary mental abilities data on monozygotic (MZ) and dizygotic (DZ) twins from Loehlin and Vandenberg (1968) to detail fundamental differences in model specification and results. We extend both multivariate biometric approaches using exploratory and confirmatory multiple-factor models. These comparisons show that each alternative multivariate methodology has useful features for empirical applications.

Alternative common factor models for multivariate biometric analyses.

The authors analyzed longitudinal data from a cognitive training experiment--Advanced Cognitive Training for Independent and Vital Elderly--using several alternative contemporary statistical models to test dynamic hypotheses based on latent change scores. The analyses focused on pretest and posttest data for only the group who received Reasoning training compared with the No-Contact (control) group. The initial structural equation modeling (SEM) path model isolated several training effects and an important source of transfer of training, Near-->Far, but this transfer was not increased due to training. The subsequent models, which accounted for pretest differences and latent changes, implied that only the Near measurements were influenced by training, and the change transfer was small. Introduction of common factors for both Near and Far measurements showed the factor patterns were unaffected by training or time and suggested training was a broader effect than in any single variable. The bivariate analysis of common factors did not appear to alter the previous results. Addition of demographic covariates and latent mixture analysis of the trained group led to further results. The uses of contemporary SEMs with experimental data are discussed.

A latent change score analysis of a randomized clinical trial in reasoning training.

In this paper, we model gender differences in cognitive ability in China using a new sample of middle-aged and older Chinese respondents. Modeled after the American Health and Retirement Study (HRS), the CHARLS Pilot survey respondents are 45 years and older in two quite distinct provinces-Zhejiang, a high-growth industrialized province on the East Coast, and Gansu, a largely agricultural and poor province in the West-in a sense new and old China. Our cognition measures proxy for two different dimensions of adult cognition-episodic memory and intact mental status. On both measures, Chinese women score much lower than do Chinese men, a gender difference that grows among older Chinese cohorts. We relate both these cognition scores to schooling, urban residence, family and community levels of economic resources, and height. We find that cognition is more closely related to mean community resources than to family resources, especially for women, suggesting that in traditional poor Chinese communities there are strong economic incentives to favor boys at the expense of girls. We also find that these gender differences in cognitive ability have been steadily decreasing across birth cohorts as the economy of China grew rapidly. Among cohorts of young adults in China, there is no longer any gender disparity in cognitive ability. This parallels the situation in the United States where cognition scores of adult women actually exceed those of adult men.

Gender Differences in Cognition among Older Adults in China

tempts, Journal of the American Academy of Child and Adolescent Psychiatry, 29, 221-235 (1990). 4. P.D. Trautman, M.J. Rotheram-Borus, S.A. Dopkins, and N. Lewin, Psychiatric diag noses in minority female adolescent suicide at tempters, Journal of the American Academy of Child and Adolescent Psychiatry, 30, 617-622 (1991). 5. M.J. Rotheram-Borus, P.D. Trautman, S.A. Dopkins, and P.E. Shout, Cognitive style and pleasant activities among female adolescent suicide attempters, Journal of Consulting and Clin ical Psychology, 58, 554-561 (1990). 6. A. Spirito, L. Brown, J. Overholser, and G. Fritz, Attempted suicide in adolescence: A review and critique of the literature, Clinical Psychology Review, 9, 335-363 (1989). 7. T.M. Achenbach, Manual for the Youth Self Report and 1991 Profile (University of Vermont, Department of Psychiatry, Burlington, 1991); T.M. Achenbach, Manual for the Child Behavior Checklist, 1991 Profile (University of Vermont, Department o? Psychiatry, Burlington, 1991); M. Kovacs and A.T. Beck, An empirical-clinical ap proach toward a definition of childhood depres sion, in Depression in Childhood: Diagnosis, Treat ment and Conceptual Models, J.G. Schulterbrandt and A. Raskin, Eds. (Raven Press, New York, 1977); D.H. Olsen, J. Portner, and Y. Lavee, FACES-III (University of Minnesota, Department of Family Social Science, St. Paul, 1985); M.E.P. Seligman, C Peterson, N.J. Kaslow, R.L. Tanen baum, L.B. Alloy, and L.Y. Abramson, Attribu tional style and depressive symptoms among children, Journal of Abnormal Psychology, 93, 235 238 (1984). 8. M.B. Summerville, M.F. Abbate, A.M. Sie gel, J. Serravezza, and N.J. Kaslow, Psychopa thology in urban female minority adolescents with suicide attempts, Journal of the American Academy of Child and Adolescent Psychiatry, 31, 663 668 (1992). 9. M.B. Summerville, N.J. Kaslow, M.F. Ab ba te, and S. Cronan, Psychopathology, family functioning, and cognitive style in urban African American adolescents with suicide attempts, Jour nal of Abnormal Child Psychology, 22, 221-235 (1994). 10. K.J. Doepke, A.D. Dingle, and M.B. Summerville, Follow-up and outpatient treatment in African American youth following a suicide attempt, manuscript submitted for publication (1995). 11. M.N. Wilson and T.F.J. Toison, Family support in the black community, Journal of Clinical Child Psychology, 19, 347-355 (1990). 12. J.D. Rodick, S.W. Hengeller, and C.L. Hanson, An evaluation of the Family Adaptabil ity and Cohesion Evaluation Scales and the Cir cumplex model, Journal of Abnormal Child Psychol ogy, 14, 77-87 (1986).

https://www.jstor.org/stable/info/20182379

John J. McArdle

Papers

Alternative common factor models for multivariate biometric analyses.

A latent change score analysis of a randomized clinical trial in reasoning training.

Gender Differences in Cognition among Older Adults in China

Current Directions in Structural Factor Analysis

Perspectives on Mathematical/Statistical Model Building (MASMOB) in research on aging.