Nonparametric estimation of the joint distribution of a survival time subject to interval censoring and a continuous mark variable

This article considers three nonparametric estimators of the joint distribution function for a survival time and a continuous mark variable when the survival time is interval censored and the mark variable may be missing for interval-censored observations. Finite and large sample properties are described for the nonparametric maximum likelihood estimator (NPMLE) as well as estimators based on midpoint imputation (MIDMLE) and coarsening the mark variable (CMLE). The estimators are compared using data from a simulation study and a recent phase III HIV vaccine efficacy trial where the survival time is the time from enrollment to infection and the mark variable is the genetic distance from the infecting HIV sequence to the HIV sequence in the vaccine. Theoretical and empirical evidence are presented indicating the NPMLE and MIDMLE are inconsistent. Conversely, the CMLE is shown to be consistent in general and thus is preferred.