Comparing multinomial logistic models across groups

[Amanda Doi]

I have a multinomial logistic regression model - 1 DV with 3 categories and 5 IVs (4 continuous and 1 categorical). I would like to test whether my effects differ between different ethnic groups. Is it possible to do this without running the analysis for each group separately?

[Clyde Schechter]

Yes, by using interaction terms. See the excellent Richard Williams' https://www3.nd.edu/~rwilliam/stats2/l51.pdf for a full and clear explanation. The examples there do not include multinomial logistic regression, but the principles and methods are largely the same. The only real difference is that interpreting "effects" in multinomial logistic regression can be tricky (even when there are no interactions) because the marginal effect on outcome probability can actually be in the opposite direction from the regression coefficient! So you have to be very clear in advance what you mean by "effects" in this kind of model. But once you have settled on that, everything works just the same as it would with a simple linear regression.

[Amanda Doi]

Thank you for the response. I will read the paper you sent.

I read the explanations available at: https://www.statalist.org/forums/for...etation-method

I am always interested in comparison between the two categories (comparison category and reference category) and not the probability of belonging to one specific category, so I think I can rely on b coef. ?

Thank you very much once again.

[Clyde Schechter]

I am always interested in comparison between the two categories (comparison category and reference category) and not the probability of belonging to one specific category, so I think I can rely on b coef. ?

Well, there are two separate issues to consider.

Even in a simple logistic regression for a dichotomous outcome variable, there is some controversy about which is the more salient result: the regression coefficient or the marginal effect. In my own work, which is more oriented towards policy analysis and decision making, the marginal effect is the more important one because probabilities of outcomes are directly relevant to expected utility, and logistic regression coefficients are not. By contrast, those who are interested in the theory of causal modeling tend to advocate for the logistic regression coefficient as being more fundamental. In all honesty, I do not find their arguments persuasive even in the context of causal modeling, but perhaps I am a bit set in my ways. In any case, I think this whole issue is more or less a tempest in a teapot because the differences between these two approaches are very slight in practice and one rarely reaches different conclusions based on this. So let's leave this issue aside.

The issue that can bite hard is the disconnect between what the coefficients tell you and what the marginal effects tell you when you working with a polytomous outome in a multinomial logistic model. It can be entirely bewildering. And even if you are ignoring several of the outcomes and just looking at one particular outcome compared to the base category, you can still be drawn to opposite conclusions. Even if the regression coefficient of an effect is positive, in the multinomial context, it can still be true that a unit increase in that effect is associated with a decreased probability of the particular outcome. There will also be a decreased probability of the base case outcome in this scenario, and it will be true that the base case outcome probability decreases more than that of the particular outcome. But do this three times and your head will spin for hours, if not days. These models are very complicated; they have more moving parts than most people can keep track of. Even after you immerse yourself in your own project's results and finally have a full and deep understanding, if they are not very simple and straightforward, you will have a very hard time explaining them to anybody else.

My best advice is to bear in mind all of the arguments in the link that you cited. When you do have your results, make sure you understand them both in terms of the coefficients and also in terms of the marginal effects, regardless of which you ultimately decide is the more important result for your purposes. For I can almost guarantee you that when you write up or present your results from either perspective, you will draw questions about how it looks from the other perspective, questions you must be prepared to answer, and you must be able to explain apparent contradictions.

[Richard Williams]

I'm not reading this thread closely, but this handout may be useful if you want some aids to interpretation:

https://www3.nd.edu/~rwilliam/xsoc73994/Margins05.pdf