One approach to consider is a multinomial logit model. If you are using factor analysis (or PCA) to combine items into scales, then the individual items are often treated as if the response pattern was cardinal (if these are response scales like terrible to terrific or strongly disagree to strongly agree). See, e.g., Wainer, H. (1976). Estimating coefficients in linear models: It don’t make no nevermind. Psychological Bulletin, 83, 213-217. https://doi.org/10.1037//0033-2909.83.2.213
If your response categories are numeric (e.g., none, 1 or 2, 3 to 5), consider using mid-ranges as estimates.