Isotonic estimation of survival under a misattribution of cause of death

Jinkyung Ha and Alexander Tsodikov have a new paper in Lifetime Data Analysis. This considers the problem of estimation of the cause specific hazard of death from a particular cause in the presence of competing risks and misattribution of cause of death. They assume they have right-censored data for which there is an associated cause of death, but that there is some known probability r(t) of misattributing the cause of death from a general cause to a specific cause (in this case pancreatic cancer) at time t.

The authors consider four estimators for the true underlying cause-specific hazards. Firstly they consider a naive estimator which obtains Nelson-Aalen estimates of the observed CSHs and transforms them to true hazards by solving the implied equations



This estimator is unbiased but has the drawback that there are negative increments to the cause-specific hazards.
The second approach is to apply a (constrained) NPMLE estimate for instance via an EM algorithm. The authors show that, unless the process is in discrete time (such that the number of failures at a specific time point increases as the sample size increases), this estimator is asymptotically biased.
The third and fourth approaches take the naive estimates and apply post-hoc algorithms to ensure monotonicity of the cumulative hazards, by using the maximum observed naive cumulative hazard up to time t (sup-estimator) or by applying the pool-adjacent-violators algorithm to the naive cumulative hazard. These estimators have the advantage of being both consistent and guaranteed to be monotonic.