# Comparing regression coefficients across groups in multinomial logistic regression

Suppose that I fit a Bayesian multinomial logistic regression model where the dependent categorial variable indexes $x$ groups, and the predictors are the same across groups. I now have $x - 1$ sets of $p$ regression coefficients and want to compare the regression coefficients of one of the groups to the two others. One simple ad hoc way to do this is to compare the posterior distributions of the coefficients. But I'm sure there is a better way. Thanks.

It's not quite clear to me how you've set things up, but I have a feeling that you've bumped into this problem in which case the different sets of coefficients are not necessarily comparable at all without extra assumptions about the unobserved heterogeneity.

• Oh, snap. Yeah, that makes perfect sense. How would I standardize the dependent variable to make the coefficients more comparable? May 20, 2014 at 19:20
• So the magnitude is affected, but not the sign since sigma is constrained to the positive reals. So what if I compared the sign-function-transformed posterior distributions of the coefficients between groups. Then, for each coefficient, I could use a beta-Dirichlet process model to compute the posterior distribution of the probability that a pair of coefficients has the same sign, and then compare those distributions across pairs of regression coefficients to see whether the focal group is more like one group than another. May 20, 2014 at 19:33
• One more thing: I think that this would be equivalent to a fully integrated Bayesian model under the assumption that the beta-Dirichlet processes across all coefficients are conditionally independent of one another and of the other components of the multinomial logistic regression model. May 20, 2014 at 19:37