## Abstract

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 language | English |
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Pages (from-to) | 724-731 |

Number of pages | 8 |

Journal | Biometrika |

Volume | 106 |

Issue number | 3 |

DOIs | |

State | Published - 1 Sep 2019 |

Externally published | Yes |

### Bibliographical note

Funding Information:The first two authors contributed equally. We thank Micha Mandel and Richard Cook for helpful comments. We acknowledge funding from the U.S. National Institutes of Health.

Publisher Copyright:

© 2020 American Institute of Physics Inc.. All rights reserved.

## Keywords

- Constant-sum condition
- Kendall's tau
- Left truncation
- Right censoring
- Survival data

## ASJC Scopus subject areas

- Statistics and Probability
- Mathematics (all)
- Agricultural and Biological Sciences (miscellaneous)
- Agricultural and Biological Sciences (all)
- Statistics, Probability and Uncertainty
- Applied Mathematics