## Friday, December 02, 2005

### ISIR blog live # 13 - factorial invariance method article for quantoids

Installment #13 of blogging "live" from the Internaltional Society for Intelligence Research (ISIR) 2005 conference in Albuquerque, NM.

Widaman. Factorial invariance and the representation of within-groups and between-groups differences: A reconsideration

Finally, a presentation for the quantoids in the audience (including Tim Keith and author of that hot new book on multiple regression--Multiple Regression and Beyond)! I've always made it a point to read most all of Keith Widaman's SEM/CFA writings.......both methodological and substantive.

Quick decision - this presentation is too complicated for me to track and summarize live (PPT slides with matrix algebra - my "matrix-algebra-as-a-second-language" abilities have decayed since graduate school). Below is the program abstact. Link to Widaman's web page is above. Link to article he is refuting is embedded in the abstract below.

Program Abstract

In a recent paper in Intelligence, Lubke, Dolan, Kelderman, and Mellenbergh (2003) argued that factorial invariance had implications for the study of within-group and between-group differences. Lubke et al. identified several levels of factorial invariance, the most restrictive being strict factorial invariance in which factor loadings, manifest variable intercepts, and unique factor variances are invariant across groups. Lubke et al. argued that a finding of strict factorial invariance implies, at a mathematical level, that the sources of within-group differences are identical to the sources of between-group differences. Several implications of this claim were drawn, such as the claim that a finding of high heritability of within-group differences (i.e., within-group differences are due primarily to differences in genetic endowment) implies that between-group differences are also due to genetic sources of variance. In 1974, Lewontin posed a thought experiment in which between-group differences were due to completely different sources than within-group differences. Lubke et al. argued that data consistent with the Lewontin thought experiment would lead to a rejection of strict factorial invariance if a model were fit to such data.

The core of the present presentation is to dispute the central conclusion offered by Lubke et al. The most important issue is the relation between levels of factorial invariance and the claim that strict invariance implies that between-group and within-group differences are due to the same underlying causes. First, the multiple-group confirmatory factor model will be described, together with increasing levels of factorial invariance (configural, weak, strong, and strict). Then, the central conclusion by Lubke et al. will be described as being due either to a linguistic error or to an error in conceiving the relation between parameters and sources of variance in hypothetical examples Lubke et al. considered. If strict factorial invariance holds, then the latent variables underlying the manifest variables are responsible for (a) within-group differences on the manifest variables, and (b) between-group differences in mean and variance on the manifest variables. Therefore, the latent variables, or factors, are mathematical entities that represent or embody all information about within- and between-group differences on the manifest variables. However, these latent factors can be combinations of genetic and environmental variance, and different combinations of genetic and environmental sources of variance can influence within- and between-group differences on the manifest variables.
Two simulated data sets designed to embody competing hypotheses for the Lewontin thought experiment will be described. In one data set, the sources of within-group differences are also responsible for between-group differences; in the second data set, the sources of within-group differences explain none of the between-group differences. Consideration of these data sets leads to the formulation of rules for study design that will enable a test of the hypothesis that sources of within-group differences are also responsible for between-group differences. The implications of these findings for research on intelligence are then explicated.

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