Cause-specific cumulative incidence estimation and the Fine and Gray model under both left truncation and right censoring

Ronald Geskus has a new paper in Biometrics. This extends the Fine and Gray model, for regression of the subdistribution hazards for competing risks models, to the case of left truncated (and right censored) data. Essentially it is shown that the standard estimator of the cumulative incidence function (CIF) can be derived as a inverse-probability-weighting estimator. Thus the Fine-Gray model for left-truncated and right-censored data can be obtained as a weighted Cox-model where individuals who are censored or experience a competing risk at time s, have weights at time t>s given by where G(t) is the empirical CDF of the censoring distribution and H(t) is the empirical survivor distribution of the truncation distribution. This form appears to imply that independence is assumed between the censoring and truncation distributions. However, the equivalence of estimates of the CIF in the no-covariates case holds regardless of the relationship between censoring and truncation - only independence with failure times is required.