Truncation occurs in cohort studies with complex sampling schemes. When truncation is ignored or incorrectly assumed to be independent of the event time in the observable region, bias can result. We derive completely nonparametric bounds for the survivor function under truncation and censoring; these extend prior nonparametric bounds derived in the absence of truncation. We also define a hazard ratio function that links the unobservable region in which event time is less than truncation time, to the observable region in which event time is greater than truncation time, under dependent truncation. When this function can be bounded, and the probability of truncation is known approximately, it yields narrower bounds than the purely nonparametric bounds. Importantly, our approach targets the true marginal survivor function over its entire support, and is not restricted to the observable region, unlike alternative estimators. We evaluate the methods in simulations and in clinical applications.
Keywords: Biased sampling; Cross-ratio function; Kendall’s tau; Peterson-type bounds; Quasi-independence; product-form estimator.