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).