I have a data set containing only categorical variables (both nominal and ordinal in nature). The dependent variable is also ordinal (with 4 categories). I was planning to run a categorical regression with optimal scaling instead of ordinal logistic regression aiming at obtaining a single beta coefficient for each independent variable (and also to account for the non-linearity of course). Because an overall comment is desired on whether the dependent variable is affected by a particular independent variable or not.
Now, to me, by theory a few of the variables seem to be related with each other. So, I am interested to check if multicollinearity exists and want to remove it to facilitate the regression. But I don't want to drop any variable because I have quite a few. The polychoric correlation matrix shows the highest pairwise correlation to be 0.69. Except from this and the other one, all others pairwise correlations are quite small.
As the variables are not continuous in my case, so how do I test the presence of multicollinearity in categorical regression and what is the remedy? How do I remove the effect of multicollinearity? I guess standardization will not help as these variables are categorical.
Expecting your kind advice on it.