Sunday, 20 November 2011

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

$\begin{matrix} d\hat{\Lambda}^{Obs}_{1}(t) &=& r(t)d\Lambda_{2}(t) + d\Lambda_{1}(t)\\ d\hat{\Lambda}^{Obs}_{2}(t) &=& (1-r(t))d\Lambda_{2}(t)\\ \end{matrix}$

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.