Tuesday, 10 May 2011

A proportional hazards regression model for the subdistribution with right-censored and left-truncated competing risks data

Xu Zhang, Mei-Jie Zhang and Jason Fine have a new paper in Statistics in Medicine. This covers the same ground as the paper by Geskus in Biometrics, in developing an approach to fitting the Fine-Gray proportional subdistribution hazard model for competing risks data with left truncated and right censored observations by using inverse probability weights (IPW). Bizarrely, the paper makes no reference at all to the Geskus paper. Presumably this is because the paper was first submitted in 2009 before Geskus's work was published (April 2010). However, it is strange that neither the authors nor the referees became aware of the work in the interim (i.e. acceptance of the paper wasn't until March 2011).

What is interesting is the differences between the approach taken in this paper compared to Geskus. The authors work on the basis that since X = min(T,C) is only observable if X > L, where T is the time of failure, L the time of left truncation and C the time of right censoring, the IPW should be calculated conditional on L < X. Zhang et al use a stabilised weight rather than the IPW to reduce the variability in the original weight. The weights they derive seem quite different to Geskus's as they depend on an estimate of overall survival, which will have to depend on the covariates if the semi-parametric model for the subdistribution hazard is to apply.
The authors suggest using Aalen additive hazard models for the overall survival (thus allowing for time varying covariate effects that can ensure the weights are consistent with the proportional subdistribution hazard model).

Zhang et al start from the general case where the truncation and censoring distributions depend on covariates (but are independent conditional on these covariates), though they only detail non-parametric estimates of the weights. Geskus argued that even if the censoring/truncation distribution depended on covariates that didn't imply it was necessary to include these covariates in the weightings.

Given these discrepancies it would be of interest to contrast and compare the two approaches to the same problem. If both approaches are effective, Geskus's seems preferable because the weights are much easier to calculate.

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