I am not sure how much benefit there would be for this over just running separate logistic regressions, and there would be the issue of the appropriate error structure. However, one of the statisticians said this.
"GENLOG (and LOGLINEAR) will fit models with multiple categorical dependents, but only with categorical predictors and "cell covariates" where a single value for a predictor for each cell is used.
The only option I can think of for a general model with multiple categorical responses other than ones that can be handled the way GENLOG does it is to use GENLIN or GENLINMIXED with REPEATED but not including the variable on REPEATED in the predictors."
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