How to compare results of factor analysis done on two groups? I have two groups, healthy and patients with schizophrenia. I used fa in psych package in R to compute the factor structure in each group. The factor structures appear different. How do I do a statistical test to tell if they are significantly different? Can I identify which factor is the most different?
 A: This test is best carried out using confirmatory factor analysis. I recommend the lavaan package, since you are using R. Check the author's tutorial for the how-to.
The idea is to fit the same model to each population, allowing different values of the parameters but the same structure. Then compare, using ANOVA (basically a likelihood ratio test), to a model that fits the two populations forcing corresponding coefficients to be the same.
There are several choices here: you might posit that the factor loadings are the same, but the error variances are different; or you might posit that the measurement models are the same but that intercepts differ ... so that schizophrenics obtain, overall, higher scores on the measure, but the covariance structure of the measure if the same. All of these options are available through lavaan. You need to give some thought to how you believe the models will differ between "normal" and schizophrenics.
If there is a significant difference between the models, you could look at how they differ. You might want to look at modification indices for the "equality" model and see which elements of the model are causing lack of fit.
