What to do when the configural invariance model has 0 degrees of freedom?

[Dijana Ostojić]

Hi all,

The configural invariance model has 0 degrees of freedom. Can I still proceed by fitting metric/scalar models and by using the delta-fit indices to judge invariance?

See below:

measurement <- '  
   F1 =~ X1 + X2 + X3

'

con_cfa <- cfa(measurement, data = df,
               group = "group")

weak_cfa <- cfa(measurement, data = df,
               group = "group",
               group.equal = c("loadings"))

strong_cfa <- cfa(measurement, data = df,
               group = "group"
               group.equal = c("loadings", "intercepts"))

lavTestLRT(con_cfa, weak_cfa, strong_cfa)

Chi-Squared Difference Test Df AIC BIC Chisq Chisq diff RMSEA Df diff Pr(>Chisq) con_cfa 0 369790 369941 0.000 weak_cfa 2 369796 369930 10.085 10.085 0.015843 2 0.006459 ** strong_cfa 4 369858 369976 76.078 65.993 0.044574 2 4.674e-15 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

fitMeasures(con_cfa, c("cfi","tli","rmsea","srmr"))

cfi tli rmsea srmr 1 1 0 0
fitMeasures(weak_cfa, c("cfi","tli","rmsea","srmr"))

cfi tli rmsea srmr 0.998 0.995 0.016 0.003
fitMeasures(strong_cfa, c("cfi","tli","rmsea","srmr"))

cfi tli rmsea srmr 0.984 0.977 0.033 0.011

[cff...@gmail.com]

I would guess you technically can do invariance testing even if the configural model has 0 df. Usually I advise against doing invariance testing (or trying to detect DIF) with too few items. I will let someone who's more familiar with the current software capabilities in lavaan to comment on whether there are better procedures than this traditional approach; it might depend also a bit on the purpose of doing such an analysis.

[cff...@gmail.com]

And what I partly mean by that is the use of "delta-fit indices" sounds like dated methodology to me.