A General Framework for and New Normalization of Attributable Proportion

Abstract A unified theory is developed for attributable proportion (AP) and population attributable fraction (PAF) of joint effects, marginal effects or interaction among factors. We use a novel normalization with a range between –1 and 1 that gives the traditional definitions of AP or PAF when positive, but is different when they are negative. We also allow for an arbitrary number of factors, both those of primary interest and confounders, and quantify interaction as departure from a given model, such as a multiplicative, additive odds or disjunctive one. In particular, this makes it possible to compare different types of threeway or higher order interactions. Effect parameters are estimated on a linear or logit scale in order to find point estimates and confidence intervals for the various versions of AP and PAF, for prospective or retrospective studies. We investigate the accuracy of three confidence intervals; two of which use the delta method and a third bootstrapped interval. It is found that the delta method with logit type transformations, and the bootstrap, perform well for a wide range of models. The methodology is also applied to a multiple sclerosis (MS) data set, with smoking and two genetic variables as risk factors.