Statistical Analysis of the Additive and Multiplicative Hypotheses of Multiple Exposure Synergy for Cohort and Case-Control Studies

Abstract

In epidemiological studies, where there are multiple causes of a particular disease, the issue arises as to whether the multiple causes have a synergistic relationship so that their combined effect is both greater than that of either activity alone, and greater than what one would expect by the sum of their individual risk contributions. Two hypotheses are frequently tested. The first hypothesis states that when the sources of disease act independently, the relative risk of disease, given exposure, is an additive relationship. Thus, the relative risk of dying from cause A adds to the relative risk of dying from cause B to determine the combined relative risk of dying when exposed to both A and B. A second hypothesis states that the relationship between disease and the two causal factors is multiplicative. In this case, the combined risk is the product of the individual risks. Of course synergism is itself a concept that is model dependent. For instance, a lack of synergism in a logit model of risk, as demonstrated by the statistical insignificance of an interaction term, leads to a multiplicative model of relative risk. Consider the following example. Suppose that the probability of dying from a disease depends on two factors, A and B. Let δ_A denote exposure to A, and δB denote exposure to B.Suppose further that the probability of dying is logistic and given by: P[D|δ_A,δ_B]=1/(1+e−(X_0β_0+δ_AX_Aβ_A+δ_BX_Bβ_B+δ_Aδ_BX_C)) where X_A, X_B, X_C are vectors of explanatory factor, and β are true but unknown coefficient vectors.

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