Interaction as Departure from Additivity in Case-Control Studies: A Cautionary Note

It has been argued that assessment of interaction should be based on departures from additive rates or risks. The corresponding fundamental interaction parameter cannot generally be estimated from case-control studies. Thus, surrogate measures of interaction based on relative risks from logistic models have been proposed, such as the relative excess risk due to interaction (RERI), the attributable proportion due to interaction (AP), and the synergy index (S). In practice, it is usually necessary to include covariates such as age and gender to control for confounding. The author uncovers two problems associated with surrogate interaction measures in this case: First, RERI and AP vary across strata defined by the covariates, whereas the fundamental interaction parameter is unvarying. S does not vary across strata, which suggests that it is the measure of choice. Second, a misspecification problem implies that measures based on logistic regression only approximate the true measures. This problem can be rectified by using a linear odds model, which also enables investigators to test whether the fundamental interaction parameter is zero. A simulation study reveals that coverage is much improved by using the linear odds model, but bias may be a concern regardless of whether logistic regression or the linear odds model is used.