Length-Biased Sampling With Right Censoring

When survival data arise from prevalent cases ascertained through a cross-sectional study, it is well known that the survivor function corresponding to these data is length biased and different from the survivor function derived from incident cases. Length-biased data have been treated both unconditionally and conditionally in the literature. In the latter case, where length bias is viewed as being induced by random left truncation of the survival times, the truncating distribution is assumed to be unknown. Conditioning on the observed truncation times hence causes very little loss of information. In many instances, however, it can be supposed that the truncating distribution is uniform, and it has been pointed out that under these circumstances, an unconditional analysis will be more informative. There are no results in the current literature that give the asymptotic properties of the unconditional nonparametric maximum likelihood estimator (NPMLE) of the unbiased survivor function in the presence of censoring. This article fills that gap by giving this NPMLE and its accompanying asymptotic properties when the data are purely length biased. An example of survival with dementia is presented in which the conditional and unconditional estimators are compared.