In an analysis where the dependant variable
Y has 4 levels (say A, B, C, and D) and there are several independent variables (including important interaction terms), one could think of multiple ways to describe the data (in a frequentist approach).
First, one could (and probably should) fit a multinomial logistic regression, which would output only 3 OR (missing one for the reference level). These ORs are a bit tricky to interpret, especially when the reference level of
Y is chosen arbitrarily, as changing it will change the reported coefficients.
Second, one could make 4 binary dummy variables for each
Y level and fit 4 logistic models. This would output one OR for each level, representing the odd of encountering it or not. These ORs are much easier to read and interpret, for both statisticians and non-statisticians.
Given this (and hoping I didn't state too much nonsense), how is the second approach wrong? What should not be interpreted in such an approach? (this question is about the interpretation, not the maths which are clearly described in several places)
Human Context: with such a categorical dependant variable, I want to interpret the interaction term. No level would make sense as the reference level. Describing the variable would be easier with 4 ORs and it seems to fit the data pretty well (as I could witness using plots). I'm also afraid editors would not understand the OR from the multinominal model, which will be quite hard to explain to my non-statisticians colleagues anyway.