Nonparametric estimation of the cumulative intensities in an interval censored competing risks model

Halina Frydman and Jun Liu have a new paper in Lifetime Data Analysis. This concerns non-parametric estimation for competing risks models under interval censoring. The problem of estimating the cumulative incidence functions (or sub-distribution functions) under interval censoring has been considered by Hudgens et al (2001) and involves an extension of the NPMLE for standard survival data under interval censoring.

The resulting estimates of the cumulative incidence functions are only defined up to increments on intervals. Moreover, the intervals by which the CIFs are defined are not the same for each competing risk. This causes problems if one wants to convert the CIFs into estimates of the cumulative cause-specific hazards. Frydman and Liu propose estimating the cumulative cause-specific hazards by first constraining the NPMLEs of the CIFs to have the same intervals of support (NB: this is just a sub-set of the set of all NPMLEs involving sub-dividing the intervals) and adopting a convention to distribute the increment within the resulting sub-intervals (they assume an equal distribution across sub-interval).

In addition they show that an ad-hoc estimator of the cumulative hazards based on the convention that the support of each interval of the NPMLE for each CIF occurs at its midpoint leads to biased results. They also show that their estimator has standard N^0.5 convergence when the support of the observation time distribution is discrete and finite.