Statistical Methodology for Failure Time Data in the Presence of Truncation

We make several contributions to field of survival analysis when the failure time variable of interest is subject to various types of truncation. Our contributions are primarily focused on statistical methods for estimation of the distribution function of a failure time, and testing the association between a failure time and the truncation mechanism. Our first contribution solves the problem of how to estimate a failure time distribution in the presence of multiple right truncating (or left truncating) events, whereby truncation is both dependent and independent of the failure time. We derive consistent nonparametric estimators, as well as provide semi-parametric estimators with the intent of gained efficiency. We then extend this methodology to a double truncation setting where we relax the dependence between the failure time and the truncation times and then propose a consistent nonparametric estimator, as well as a more efficient semi-parametric estimator. Furthermore, we propose formal tests to test each of the dependence and independence models. By deriving tests of theses models, we further explore the idea of the testing various dependence models between the failure time and the truncation mechanism via conditional Kendall's tau. In the current literature there does not exist a consistent estimator of the conditional Kendall's tau when the failure time is right censored and dependent truncation exists. All of the current estimates for this parameter converge to a parameter that involves the censoring distribution. Therefore we propose two useful models of dependence for which we derive a consistent estimate for the a conditional Kendall's tau for dependent left truncated and right censored data. Ultimately these estimators prove to be useful as we develop an extension of the structural model used by Efron & Petrosian [8] to eliminate dependent truncation. The estimate of the conditional Kendall's tau enables us to find which value of the dependence parameter allows for independence between the failure time and the truncation time. This is done by choosing the value of the parameter that gives an estimate of the conditional Kendall's tau that is closest to 0.