An idea for multinomial logit model

Question

I am running a multinomial logistic regression. I have a sample size of 500.

I regressed the four categories/segments (A, B, C, D) on several independent variables including the intercept, setting A as the reference category, thus using the normal procedure. I understand that I have to interpret the estimated results for each segment B, C, and D in relation to the reference segment A. However, the respective interpretation is somehow hard, not intuitive, and partly confusing. It also makes it more difficult to understand the specifics of a group relative to the whole sample. However, that is what I would like to discuss.

Here I got an idea about analyzing the multinomial logit model differently and would like to know your opinions: I keep my four segments and add a fifths segment. This 5th segment (segment E) is the sum of all four segments. Thus, I duplicate the entire sample. I run a multinominal regression analysis on the (new) sample (N=1000 including the original A, B, C, and D categories as well as the extra category E) setting the category E as the reference group. In this case, I can interpret the obtained results for each of the segments A, B, C, D relative to the whole sample (category E). While this is very convenient/intuitive from an interpretation point of view I wonder whether there is anything wrong with that procedure from a methodological/econometrics point of view.

Let me stress again. Each observation in my model would occur twice, once in either group A, B, C or D and once in E. E is the aggregate of A, B, C and D. In the multinomial logistic regression I would use this aggregate as the reference category.

Answer 1

If you have nuisance parameters you may get overly confident estimates about them by duplicating your data. You can achieve exactly what you describe as your aim by parameterizing the model differently, i.e. by having an intercept and requiring that the group effects sum to zero ("sum to zero constraint").

Not all software will offer that to you, but if your software lets you specify your own log-likelihoods, you can always specify this yourself by having free parameters $\beta_1, \beta_2, \beta_3$ and specifying $\beta_4=-(\beta_1 + \beta_2 + \beta_3)$.

Answer 2

You can do the equivalent of this without inventing anything new. This is a question of reference coding vs. effect coding. The latter compares each level to the grand mean. Personally, I find reference coding (which your program already does, but you find confusing) much more intuitive. But, your opinion is your opinion.

In SAS PROC LOGISTIC, for example, you can do this with PARAM = EFFECT.. For R coding see this thread.