Nonidentifiability in the presence of factorization for truncated data.

Bella Vakulenko-Lagun, J Qian, S H Chiou, R A Betensky

Research output: Contribution to journalArticle


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.
Original languageEnglish (US)
Pages (from-to)724-731
Number of pages8
Issue number3
StatePublished - Sep 1 2019



  • Censoring (Statistics)
  • Databases
  • Data
  • Constant-sum condition
  • Kendall's tau
  • Left truncation
  • Right censoring
  • Survival data

Cite this

Vakulenko-Lagun, B., Qian, J., Chiou, S. H., & Betensky, R. A. (2019). Nonidentifiability in the presence of factorization for truncated data. Biometrika, 106(3), 724-731.