The identication of synergism in the su¢ cient-component cause framework

Various concepts of interaction are reconsidered in light of a su¢ cient-component cause framework. Conditions and statistical tests are derived for the presence of synergism within su¢ cient causes. The conditions derived are su¢ cient but not necessary for the presence of synergism. In the context of monotonic e¤ects, but not in general, the conditions which are derived are closely related but not identical to e¤ect modi…cation on the risk di¤erence scale. Some Key Words: Causal inference; e¤ect modi…cation; interaction; risk di¤erence; su¢ cientcomponent cause; synergism. The distinction between a biologic interaction or synergism and a statistical interaction has frequently been noted. 3 In the case of binary variables, concrete attempts have been made to articulate which types of counterfactual response patterns would constitute instances of interdependent e¤ects. 6 In what follows we reconsider the de…nition of causal interdependence and its relation to that of synergism in light of the su¢ cient-component cause framework. Consideration of this framework gives rise to a de…nition of "de…nite interdependence" which constitutes a su¢ cient but not necessary condition for the presence of synergism within the su¢ cient-component cause framework. We then derive various empirical conditions for the presence of synergism and provide a number of observations which illustrate the di¤erence between the concepts of de…nite interdependence and e¤ect modi…cation on the risk di¤erence scale. Although the material developed in this paper arguably has implications for applied data analysis, our principal aim here will be to extend theory: to consider various conceptual and mathematical relations between di¤erent notions of interaction. Synergism and Counterfactual Response Types Suppose that D and two of its causes, E1 and E2, are binary variables taking values 0 or 1. In the discussion that follows E1 and E2 are treated symmetrically so that E1 could be relabeled as E2 and E2 could be relabeled as E1. We assume a deterministic counterfactual framework. Let Dij(!) be the counterfactual value of D for individual ! if E1 were set to i and E2 were set to j. Robins has shown that standard statistical summaries of uncertainty due to sampling variability, such as p-values and con…dence intervals for proportions, have meaning in a deterministic model if and only if we regard (i) the n study subjects as having been randomly sampled from a large, perhaps hypothetical, source population of size N , such that n=N is very small and (ii) probability statements and expected values refer to proportions and averages in the source population. Because we plan to discuss statistical tests, we adopt (i) and (ii). For event E we will denote the complement of the event by E. The probability of an event E occurring, P (E = 1), we will frequently simply denote by P (E). If there were some individual ! for whom D10(!) = D01(!) = D00(!) = 0 but for whom D11(!) = 1 we might say that there was synergism between the e¤ect of E1 and E2 on D because in such a case there exists an individual for whom E1 or E2 alone is insu¢ cient for D but for whom E1 and E2 together yield D. There is thus joint action between E1 and E2 and so we might speak of synergism. Similarly if there were individuals for whom D11(!) = D01(!) = D00(!) = 0 and D10(!) = 1; or for whom D11(!) = D10(!) = D00(!) = 0 and D01(!) = 1; or for whom D11(!) = D01(!) = D10(!) = 0 and D00(!) = 1 we might again