Four types of e¤ect modication - a classication based on directed acyclic graphs

By expressing the conditional causal risk di¤erence as a sum of products of stratum speci…c risk di¤erences and conditional probabilities, it is possible to give a classi…cation of the types of causal relationships that can give rise to e¤ect modi…cation on the risk di¤erence scale. Directed acyclic graphs make clear the necessary causal relationships for a particular variable to serve as an e¤ect modi…er for the causal risk di¤erence concerning two other variables. The directed acyclic graph causal framework thereby gives rise to a four-fold classi…cation for e¤ect modi…cation: direct e¤ect modi…cation, indirect e¤ect modi…cation, e¤ect modi…cation by proxy and e¤ect modi…cation by a common cause. Brief discussion is given to the case of multiple e¤ect modi…cation relationships and multiple e¤ect modi…ers as well as measures of e¤ect other than that of the causal risk di¤erence. Key Words: Causal inference; directed acyclic graphs; e¤ect modi…cation; interaction. Directed acyclic graphs have been used as causal diagrams in epidemiologic research for a variety of purposes. Directed acyclic graphs have been used to represent causal relations amongst variables; 3 they have been used extensively to determine the variables for which it is necessary to control for confounding in order to estimate causal e¤ects; 2;4 5 more recently they have been used by Hernán et al. to provide a classi…cation of the types of causal relationships that can give rise to selection bias. In this paper we follow their work by using directed acyclic graphs to provide a classi…cation of the types of causal relationships that can give rise to e¤ect modi…cation. Speci…cally, we consider what relationships an e¤ect modi…er variable may exhibit in relation to the variable constituting the cause and the variable constituting the e¤ect. Doing so yields a structural classi…cation of e¤ect modi…cation; the classi…cation is structural in that it makes reference to the structure of the causal directed acyclic graph. We …rst provide some discussion of the various measures of e¤ect used to assess e¤ect modi…cation. We will then focus on the causal risk di¤erence as a measure of e¤ect by which e¤ect modi…cation is assessed (though much of the discussion applies also to other measures of e¤ect as well) and we use directed acyclic graphs to provide a classi…cation of di¤erent types of e¤ect modi…cation. Extensions to conditioning on multiple items and to scales other than the risk di¤erence are discussed at the paper’s conclusion and in Appendix 1.