Nonidentifiability in the presence of factorization for truncated data

A time to event, X , is left-truncated by T if X can be observed only if T < X . This often results in oversampling of large values of X , and necessitates adjustment of estimation procedures to avoid bias. Simple risk-set adjustments can be made to standard risk-set-based estimators to accommodate left truncation when T and X are quasi-independent. We derive a weaker factorization condition for the conditional distribution of T given X in the observable region that permits risk-set adjustment for estimation of the distribution of X , but not of the distribution of T. Quasi-independence results when the analogous factorization condition for X given T holds also, in which case the distributions of X and T are easily estimated. While we can test for factorization, if the test does not reject, we cannot identify which factorization condition holds, or whether quasi-independence holds. Hence we require an unverifiable assumption in order to estimate the distribution of X or T based on truncated data. This contrasts with the common understanding that truncation is different from censoring in requiring no unverifiable assumptions for estimation.We illustrate these concepts through a simulation of left-truncated and right-censored data.