Bootstrapping left truncated and right censored data

Survival data subject to left truncation and right censoring are encountered in many follow-up studies. One such situation is follow-up data collected under a cross-sectional sampling scheme. Efron (1981) described two different methods for bootstrapping right censored survival data, which he termed the "obvious" and the "simple" methods, and demonstrated that these two methods are equivalent. Using the non parametric estimate of the joint distribution of the truncation and censoring times and the nonparametric maximum likelihood estimate for the survival curve we generalize the "obvious" method of bootstrapping to the current data. A simulation study examining the large sample behavior of the extensions of both methods is presented. The methods are applied to obtain confidence bands for the nonparametric maximum likelihood estimate of the survival curve, confidence bands for the nonparametric maximum likelihood estimate of the truncation distribution, and confidence intervals for the proportion of truncated data. The simulation study shows that, for the specific non-trivial case illustrated, the two methods yield similar large sample results. However, the validity of the extension of the simple method, in general, remains unclear. The authors, therefore, recommend use of the obvious method. Real data applications are presented with AIDS Prevalent Cohort Data and the CDC Blood Transfusion Data.