Mike Johns writes:

Are you familiar with the work of Ai and Norton on interactions in logit/probit models? I’d be curious to hear your thoughts.

Ai, C.R. and Norton E.C. 2003. Interaction terms in logit and probit models. Economics Letters 80(1): 123-129.

A peer ref just cited this paper in reaction to a logistic model we tested and claimed that the “only” way to test an interaction in logit/probit regression is to use the cross derivative method of Ai & Norton. I’ve never heard of this issue or method. It leaves me wondering what the interaction term actually tests (something Ai & Norton don’t discuss) and why such an important discovery is not more widely known. Is this an issue that is of particular relevance to econometric analysis because they approach interactions from the difference-in-difference perspective?

Full disclosure, I’m coming from a social science/epi background. Thus, i’m not interested in the d-in-d estimator; I want to know if any variables modify the relationship between the focal IV and DV. The standard method of calculating and testing the interaction term seems perfectly reasonable for answering this question.

My reply: My quick answer is that with nonlinear functions, there is no single summary that tells the whole story. Different coefficients and different average predictive quantities are appropriate in different contexts. Some related material is here:
http://www.stat.columbia.edu/~gelman/research/published/ape17.pdf