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  • SEM with group comparison: Fit indices using "ginvariant(all)" differ from fit indices when ignoring groups. Why?

    Dear Statalisters,

    In his (excelent!) book Discovering Structural Equation Modelling Using Stata, Alan Acock teaches us how to test for measurement invariance in Stata.

    Using data from the National Longitudinal Survey of Youth (1997), three items about depression are linked to one latent factor and four items about government responsibility are linked to a second latent factor. In Figure 4.2 of Section 5.3 he shows us the solution when women and men are combined into one group using both the "group(female)" and the "ginvariant(all)" options:
    Code:
    net from http://www.stata-press.com/data/dsemus/
    net get dsemus
    use multgrp_cfa, clear
    
    sem (Depress -> x1 x2 x3) ///
        (Gov_Resp -> x4 x9 x10 x12), ///
        standardized group(female) ginvariant(all)
    estat gof, stats(all)
    The latter command gives us the fit indices.
    Comparing the SEM model to the situation when ignoring the grouping variable, using
    Code:
    sem (Depress -> x1 x2 x3) ///
        (Gov_Resp -> x4 x9 x10 x12), ///
        standardized 
    estat gof, stats(all)
    gives us exactly the same model (= as expected: hurray!!), but - except for the Coefficient of determination - very different fit indices (= very unexpected: huh?!).

    Why?

    Kind regards, Adriaan Hoogendoorn
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