Attributable fractions for sufficient cause interactions.

A number of results concerning attributable fractions for sufficient cause interactions are given. Results are given both for etiologic fractions (i.e. the proportion of the disease due to a particular sufficient cause) and for excess fractions (i.e. the proportion of disease that could be eliminated by removing a particular sufficient cause). Results are given both with and without assumptions of monotonicity. Under monotonicity assumptions, exact formulas can be given for the excess fraction. When etiologic fractions are of interest or when monotonicity assumptions do not hold for excess fractions then only lower bounds can be given. The interpretation of the results in this paper and in a proposal by Hoffmann et al. (2006) are discussed and compared. A method is described to estimate the lower bounds on attributable fractions using marginal structural models. Identification is discussed in settings in which time-dependent confounding may be present.