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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

1
Department of Biostatistics, Harvard T. H. Chan School of Public Health, 655 Huntington Avenue, Boston, Massachusetts 02115, USA.
2
Department of Biostatistics and Epidemiology, University of Massachusetts, 715 N. Pleasant Street, Amherst, Massachusetts 01003, USA.

Abstract

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.

KEYWORDS:

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

PMID:
31427826
PMCID:
PMC6690171
[Available on 2020-09-01]
DOI:
10.1093/biomet/asz023

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