Advantage of the Gail-Simon Test (qualitative/ crossover interaction test) It is common to use a product of two variables to test whether an interaction is present. In regression analysis, for example, we can include both main effects and the interaction and see whether the beta of the interaction is significant (here for example).
However, Gail and Simon (1985) describe a method to test crossover interaction. Crossover (or qualitative) interactions are said to occur when treatment effects differ in direction for subsets. A non-crossover interaction arises when there is variation in the magnitude, but not in the direction, of treatment effects among subsets.
The common practice to test interaction can detect both, significant crossover and non-crossover interaction, while the approach of Gail and Simon is only significant in case of a crossover interaction. Doesn't this mean that we get a significant result with the usual approach if the Gail-Simon test is significant? If so, what is the advantage of the Gail-Simon test? Instead of using the Gail-Simon test I could run a "normal" interaction test and see what kind of interaction there is (crossover or not).
 A: I assume you are considering a single pre-specified interaction hypothesis, as these tests do not apply if the factors to test are "found" by inspecting relationships with $Y$.
You've described the situation well, and since confidence intervals (or better: Bayesian credible intervals) are often preferred over hypothesis tests, I would mainly compute an uncertainty interval for the interaction effect (which represents a double difference if both factors are dichotomous) and interpret that in most cases.  But that doesn't answer the question of reversal of an effect; it just estimates how different are the effects across levels of the interacting factor.  When there is reversal, the interaction effect will tend to be large, but it can also be large due to a major change in effect that does not make the effect change directions.  Hence there is a need for the Gail-Simon test.
The Bayesian approach would involve computing the posterior probability that the effects change sign when comparing two levels of an interacting factor.
When the interacting factor is not dichotomous, e.g. is continuous, one needs to model this more generally.  For example, if treatment and age interact and the age effect is not linear, one can see differently shaped age effects for the two treatments.  One can test nonlinear interactions by simultaneously testing all the interaction terms in a regression spline of age, for example.  I discuss this in my RMS book and course notes.
