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Conditional independence assumption of truncation and failure times conditioning on covariates is a fundamental and common assumption in the regression analysis of left-truncated and right-censored data. Testing for this assumption is essential to ensure the correct inference on the failure time, but this has often been overlooked in the literature. With consideration of challenges caused by left truncation and right censoring, tests for this conditional independence assumption are developed in which the generalized odds ratio derived from a Cox proportional hazards model on the failure time and the concept of Kendall's tau are combined. Except for the Cox proportional hazards model, no additional model assumptions are imposed, and the distributions of the truncation time and conditioning variables are unspecified. The asymptotic properties of the test statistic are established and an easy implementation for obtaining its distribution is developed. The performance of the proposed test has been evaluated through simulation studies and two real studies.

Inference for cause-specific hazards from competing risks data under interval censoring and possible left truncation has been understudied. Aiming at this target, a penalized likelihood approach for a Cox-type proportional cause-specific hazards model is developed, and the associated asymptotic theory is discussed. Monte Carlo simulations show that the approach performs very well for moderate sample sizes. An application to a longitudinal study of dementia illustrates the practical utility of the method. In the application, the age-specific hazards of AD, other dementia and death without dementia are estimated, and risk factors of all competing risks are studied.

While the currently available estimators for the conditional Kendall's tau measure of association between truncation and failure are valid for testing the null hypothesis of quasi-independence, they are biased when the null does not hold. This is because they converge to quantities that depend on the censoring distribution. The magnitude of the bias relative to the theoretical Kendall's tau measure of association between truncation and failure due to censoring has not been studied, and so its importance in real problems is not known. We quantify this bias in order to assess the practical usefulness of the estimators. Furthermore, we propose inverse probability weighted versions of the conditional Kendall's tau estimators to remove the effects of censoring and provide asymptotic results for the estimators. In simulations, we demonstrate the decrease in bias achieved by these inverse probability weighted estimators. We apply the estimators to the Channing House data set and an AIDS incubation data set.