Four types of effect modification - A classification based on directed acyclic graphs

It is possible to classify the types of causal relationships that can give rise to effect modification on the risk difference scale by expressing the conditional causal risk-difference as a sum of products of stratum-specific risk differences and conditional probabilities. Directed acyclic graphs clarify the causal relationships necessary for a particular variable to serve as an effect modifier for the causal risk difference involving 2 other variables. The directed acyclic graph causal framework thereby gives rise to a 4-fold classification for effect modification: direct effect modification, indirect effect modification, effect modification by proxy and effect modification by a common cause. We briefly discuss the case of multiple effect modification relationships and multiple effect modifiers as well as measures of effect other than that of the causal risk difference.