Send to

Choose Destination
Biometrika. 2019 Sep;106(3):724-731. doi: 10.1093/biomet/asz023. Epub 2019 May 13.

Nonidentifiability in the presence of factorization for truncated data.

Author information

Department of Biostatistics, Harvard T. H. Chan School of Public Health, 655 Huntington Avenue, Boston, Massachusetts 02115, USA.
Department of Biostatistics and Epidemiology, University of Massachusetts, 715 N. Pleasant Street, Amherst, Massachusetts 01003, USA.


A time to event, [Formula: see text], is left-truncated by [Formula: see text] if [Formula: see text] can be observed only if [Formula: see text]. This often results in oversampling of large values of [Formula: see text], 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 [Formula: see text] and [Formula: see text] are quasi-independent. We derive a weaker factorization condition for the conditional distribution of [Formula: see text] given [Formula: see text] in the observable region that permits risk-set adjustment for estimation of the distribution of [Formula: see text], but not of the distribution of [Formula: see text]. Quasi-independence results when the analogous factorization condition for [Formula: see text] given [Formula: see text] holds also, in which case the distributions of [Formula: see text] and [Formula: see text] 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 [Formula: see text] or [Formula: see text] 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.


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

[Available on 2020-09-01]

Supplemental Content

Full text links

Icon for Silverchair Information Systems
Loading ...
Support Center