Alternatives to Rothman's approach for assessing synergism (or antagonism) in cohort studies

K.J. Rothman has explored in some detail the issue of assessing the potential presence of synergism (or antagonism) in data generated from either a cohort or a case-control study. Arguing that the "natural" scale for quantifying the joint effects of two or more factors acting in combination is the probability scale, he has proposed a procedure based on a ratio-type index for evaluating two-factor interaction in the presence of non-zero background effects. In this paper, the authors review the rationale underlying Rothman's approach for a cohort study. They then present what they maintain is a simpler and more appropriate test procedure (utilizing a linear contrast of the observed risks) for the additive approximation to his basic probabilistic model of "no interaction." A likelihood ratio test based on his original model is also proposed, as well as a closed form approximation to it. Finally, the assessment of interaction in cohort studies involving exposure factors measured at more than two levels is addressed.