{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T23:14:57Z","timestamp":1787008497877,"version":"build-2736575974"},"reference-count":11,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T00:00:00Z","timestamp":1557705600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,9,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>A time to event, $X$, is left-truncated by $T$ if $X$ can be observed only if $T&lt;X$. 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.<\/jats:p>","DOI":"10.1093\/biomet\/asz023","type":"journal-article","created":{"date-parts":[[2019,4,1]],"date-time":"2019-04-01T15:11:21Z","timestamp":1554131481000},"page":"724-731","source":"Crossref","is-referenced-by-count":5,"title":["Nonidentifiability in the presence of factorization for truncated data"],"prefix":"10.1093","volume":"106","author":[{"given":"B","family":"Vakulenko-Lagun","sequence":"first","affiliation":[{"name":"Department of Biostatistics, Harvard T. H. Chan School of Public Health, 655 Huntington Avenue, Boston, Massachusetts 02115, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J","family":"Qian","sequence":"additional","affiliation":[{"name":"Department of Biostatistics and Epidemiology, University of Massachusetts, 715 N. Pleasant Street, Amherst, Massachusetts 01003, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"S H","family":"Chiou","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, Harvard T. H. Chan School of Public Health, 655 Huntington Avenue, Boston, Massachusetts 02115, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R A","family":"Betensky","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, Harvard T. H. 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