Efficient Estimation in the Fine and Gray Model

Direct regression for the cumulative incidence function (CIF) has become increasingly popular since the Fine and Gray model was suggested (Fine and Gray) due to its more direct interpretation on the probability risk scale. We here consider estimation within the Fine and Gray model using the theory of semiparametric efficient estimation. We show that the Fine and Gray estimator is semiparametrically efficient in the case without censoring. In the case of right-censored data, however, we show that the Fine and Gray estimator is no longer semiparametrically efficient and derive the semiparametrically efficient estimator. This estimation approach involves complicated integral equations, and we therefore also derive a simpler estimator as an augmented version of the Fine and Gray estimator with respect to the censoring nuisance space. While the augmentation term involves the CIF of the competing risk, it also leads to a robustness property: the proposed estimators remain consistent even if one of the models for the censoring mechanism or the CIF of the competing risk are misspecified. We illustrate this robustness property using simulation studies, comparing the Fine–Gray estimator and its augmented version. When the competing cause has a high cumulative incidence we see a substantial gain in efficiency from adding the augmentation term with a very reasonable computation time. Supplementary materials for this article are available online.