In non-linear models we can often present results on an additive
scale, by presenting marginal effect, or on a multiplicative scale,
by presenting odds ratios, or incidence-rate ratios, or hazard ratios.
Interpreting interactions on an additive scale is relatively complex
(see for example this
article by Edward Norton, Hua Wang, and Chunrong Ai). In the current
paper I illustrate how the interpretation of interactions is substantially
easier when interpreting the effects on a multiplicative scale, but
also shows that both types of effects answer subtly different questions,
meaning that there is an added value in having both tools in ones
toolbox.

A translation of the example that will work in Stata 10 or older.

A post on Statalist explaining why interaction effects in terms of odds ratios/hazard ratios/incidence rate ratios/etc. are so much easier than interaction effects in terms of marginal effects.

Another post on Statalist explaining why interaction effects in terms of odds ratios/hazard ratios/incidence rate ratios/etc. are so much easier than interaction effects in terms of marginal effects.

Example of a binary by binary interaction in a logit model with covariates.

Example of a continuous by continuous interaction in a logit model.

Example of a continuous by continuous interaction in a multinomial logit model

Example of a binary by continuous interaction in a negative binomial model

Example of a categorical by continuous interaction in a Cox regression model for survival data.

There is also an active debate on whether interaction effects in
non-linear models for binary dependent variables have any interpretation
at all. This article does not deal with that debate. I am less pessimistic
than others in this regard. The arguments for this optimism are set out in
this working paper.