{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T20:24:16Z","timestamp":1778790256726,"version":"3.51.4"},"reference-count":32,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T00:00:00Z","timestamp":1772582400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T00:00:00Z","timestamp":1772582400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Biometrical J"],"published-print":{"date-parts":[[2026,4]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>In order to estimate the proportion of \u201cimmune\u201d or \u201ccured\u201d subjects who will never experience failure, a sufficiently long follow\u2010up period is required. Several statistical tests have been proposed in the literature for assessing the assumption of sufficient follow\u2010up, meaning that the study duration is longer than the support of the survival times for the uncured subjects. These tests do not perform satisfactorily, especially in terms of Type I error. In addition, they are constructed based on the assumption that the survival time for the uncured subjects has a compact support, that is, the existence of a \u201ccure time.\u201d However, for practical purposes, the assumption of \u201ccure time\u201d is not realistic and the follow\u2010up would be considered sufficiently long if the probability for the event to happen after the end of the study is very small. Based on this observation, we formulate a more relaxed notion of \u201cpractically\u201d sufficient follow\u2010up characterized by the quantiles of the distribution and develop a novel nonparametric statistical test. The proposed method relies mainly on the assumption of a nonincreasing density function in the tail of the distribution. The test is then based on a shape\u2010constrained density estimator, such as the Grenander or the kernel\u2010smoothed Grenander estimator, and a bootstrap procedure is used for computation of the critical values. The performance of the test is investigated through an extensive simulation study, and the method is illustrated on breast cancer\u00a0data.<\/jats:p>","DOI":"10.1002\/bimj.70121","type":"journal-article","created":{"date-parts":[[2026,3,8]],"date-time":"2026-03-08T19:38:39Z","timestamp":1772998719000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Testing for Sufficient Follow\u2010Up in Survival Data With a Cure Fraction"],"prefix":"10.1002","volume":"68","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4929-9483","authenticated-orcid":false,"given":"Tsz Pang","family":"Yuen","sequence":"first","affiliation":[{"name":"Korteweg\u2010de Vries Institute for Mathematics University of Amsterdam Amsterdam The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eni","family":"Musta","sequence":"additional","affiliation":[{"name":"Korteweg\u2010de Vries Institute for Mathematics University of Amsterdam Amsterdam The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2026,3,4]]},"reference":[{"key":"e_1_2_9_2_1","doi-asserted-by":"publisher","DOI":"10.1146\/annurev-statistics-031017-100101"},{"key":"e_1_2_9_3_1","doi-asserted-by":"publisher","DOI":"10.1111\/biom.12999"},{"key":"e_1_2_9_4_1","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1069362738"},{"key":"e_1_2_9_5_1","doi-asserted-by":"publisher","DOI":"10.1214\/24-EJS2249"},{"key":"e_1_2_9_6_1","doi-asserted-by":"publisher","DOI":"10.1080\/10485252.2013.826356"},{"key":"e_1_2_9_7_1","doi-asserted-by":"publisher","DOI":"10.1080\/03461238.1956.10414944"},{"key":"e_1_2_9_8_1","doi-asserted-by":"publisher","DOI":"10.1111\/sjos.12294"},{"key":"e_1_2_9_9_1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139020893"},{"key":"e_1_2_9_10_1","doi-asserted-by":"publisher","DOI":"10.1111\/sjos.12730"},{"key":"e_1_2_9_11_1","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1958.10501452"},{"key":"e_1_2_9_12_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jspi.2007.03.032"},{"key":"e_1_2_9_13_1","doi-asserted-by":"publisher","DOI":"10.1214\/193940307000000202"},{"key":"e_1_2_9_14_1","doi-asserted-by":"publisher","DOI":"10.1214\/009053606000000100"},{"key":"e_1_2_9_15_1","doi-asserted-by":"publisher","DOI":"10.1201\/9781315112084-22"},{"key":"e_1_2_9_16_1","doi-asserted-by":"publisher","DOI":"10.1111\/stan.12101"},{"key":"e_1_2_9_17_1","doi-asserted-by":"publisher","DOI":"10.3150\/21-BEJ1417"},{"key":"e_1_2_9_18_1","doi-asserted-by":"publisher","DOI":"10.1002\/cjs.11771"},{"key":"e_1_2_9_19_1","doi-asserted-by":"publisher","DOI":"10.1214\/24-SS147"},{"key":"e_1_2_9_20_1","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/79.4.731"},{"key":"e_1_2_9_21_1","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1994.10476889"},{"key":"e_1_2_9_22_1","series-title":"Wiley Series in Probability and Statistics: Applied Probability and Statistics","volume-title":"Survival Analysis With Long\u2010Term Survivors","author":"Maller R.","year":"1996"},{"key":"e_1_2_9_23_1","doi-asserted-by":"publisher","DOI":"10.1201\/9780429032301"},{"key":"e_1_2_9_24_1","doi-asserted-by":"publisher","DOI":"10.1002\/sim.9610"},{"key":"e_1_2_9_25_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-7152(00)00063-8"},{"key":"e_1_2_9_26_1","doi-asserted-by":"publisher","DOI":"10.1186\/1471-2407-5-48"},{"key":"e_1_2_9_27_1","doi-asserted-by":"publisher","DOI":"10.1093\/humrep\/des428"},{"key":"e_1_2_9_28_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0140-6736(05)17947-1"},{"issue":"2","key":"e_1_2_9_29_1","first-page":"501","article-title":"A Penalized Maximum Likelihood Estimate of f(0+)$f(0+)$ When f$f$ is Non\u2010Increasing","volume":"3","author":"Woodroofe M.","year":"1993","journal-title":"Statistica Sinica"},{"key":"e_1_2_9_30_1","doi-asserted-by":"crossref","unstructured":"Xie P. M.Escobar\u2010Bach andI.Van Keilegom.2023. \u201cTesting for Sufficient Follow\u2010Up in Censored Survival Data by Using Extremes.\u201darXiv:2309.00868v1.","DOI":"10.1002\/bimj.202400033"},{"key":"e_1_2_9_31_1","doi-asserted-by":"publisher","DOI":"10.1016\/0167-7152(89)90113-2"},{"key":"e_1_2_9_32_1","doi-asserted-by":"crossref","unstructured":"Yuen T. P. E.Musta andI.Van Keilegom.2025. \u201cTesting for Sufficient Follow\u2010Up in Survival Data in Cure Models With Categorical Covariates.\u201darXiv:2505.13128.","DOI":"10.1002\/bimj.70121"},{"key":"e_1_2_9_33_1","doi-asserted-by":"publisher","DOI":"10.1080\/14697688.2018.1552791"}],"container-title":["Biometrical Journal"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/bimj.70121","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1002\/bimj.70121","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/bimj.70121","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T10:19:22Z","timestamp":1776766762000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/bimj.70121"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,4]]},"references-count":32,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["10.1002\/bimj.70121"],"URL":"https:\/\/doi.org\/10.1002\/bimj.70121","archive":["Portico"],"relation":{},"ISSN":["0323-3847","1521-4036"],"issn-type":[{"value":"0323-3847","type":"print"},{"value":"1521-4036","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,4]]},"assertion":[{"value":"2024-09-18","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-10-20","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-03-04","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e70121"}}