# Logit-Poisson Trick

Consider an analysis of discrete-choice logit, where the independent variables vary with the alternatives. Suppose that each individual is represented once in the sample, and that different individuals may have different choice sets (eg individual 1 may face alternatives 1,2,4, while individual 2 may face alternatives 3 and 4 only, etc).

Question: does the Poisson Trick work for this case? There seems to be no problem when everyone faces the same choice set, but the equivalence between logit (I'm using R's mlogit for this case) and the glm/Poisson analysis seems to break down (empirically) when different choice sets are involved.

Can anyone settle the question, one way or the other? If a demonstration could also be provided, I'd be very grateful.

• Could you please explain what you mean by the "Poisson Trick" and what it is intended to do here?
– whuber
Commented Dec 1, 2021 at 17:12
• (a) The Poisson Trick is a way of estimating multinomial logit models as glm/Poisson models, for cases (eg INLA) where a multinomial logit likelihood isn't directly available. Google Poisson Trick for more. (b) In answer to my own question: this was due to a specification error on my part. I estimated the MNL model without an intercept (to avoid choice-specific dummys), but forgot that I needed one in the Poisson formulation. With this added, mlogit and glm/poisson produce the same coeffs and std errors, as they should, even in the case of varying choice sets. Commented Dec 1, 2021 at 20:55
• I am not asking for my own edification. Please take my comment as a suggestion about how to make your question self-contained and recognizable to people qualified to answer it. It will improve the chances of receiving a clear, on-point answer.
– whuber
Commented Dec 1, 2021 at 21:44