Nonparametric estimators of survival function under the mixed case interval-censored model with left truncation
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Nonparametric estimators of survival function under the mixed case interval-censored model with left truncation Pao-Sheng Shen1 Received: 3 December 2018 / Accepted: 3 January 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract It is well known that the nonparametric maximum likelihood estimator (NPMLE) can severely underestimate the survival probabilities at early times for left-truncated and interval-censored (LT-IC) data. For arbitrarily truncated and censored data, Pan and Chappel (JAMA Stat Probab Lett 38:49–57, 1998a, Biometrics 54:1053–1060, 1998b) proposed a nonparametric estimator of the survival function, called the iterative Nelson estimator (INE). Their simulation study showed that the INE performed well in overcoming the under-estimation of the survival function from the NPMLE for LT-IC data. In this article, we revisit the problem of inconsistency of the NPMLE. We point out that the inconsistency is caused by the likelihood function of the left-censored observations, where the left-truncated variables are used as the left endpoints of censoring intervals. This can lead to severe underestimation of the survival function if the NPMLE is obtained using Turnbull’s (JAMA 38:290–295, 1976) EM algorithm. To overcome this problem, we propose a modified maximum likelihood estimator (MMLE) based on a modified likelihood function, where the left endpoints of censoring intervals for left-censored observations are the maximum of left-truncated variables and the estimated left endpoint of the support of the left-censored times. Simulation studies show that the MMLE performs well for finite sample and outperforms both the INE and NPMLE. Keywords Left truncation · Interval censoring · NPMLE · EM algorithm
1 Introduction Left truncation often arises in epidemiology and individual follow-up studies. For example, under a prevalent cohort design (or the so-called cross-sectional sampling), an individual is selected only when he (or she) has already entered status 0 (e.g.,
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Pao-Sheng Shen [email protected] Department of Statistics, Tunghai University, Xitun District, Taichung 40704, Taiwan, ROC
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diagnosis of HIV infection or diabetes) sometime prior to calendar time τ0 and yet has not entered status 1 (e.g., development of AIDS or retinopathy). Hence, earlier onset of AIDS/retinopathy would then be a truncating force for the variable of interest. In addition to left-truncation, one is often unable to observe the true event time, but observe an interval. This situation is quite usual since the occurrence of a disease can only be observed at the time of a medical examination and the visiting sequences of patients are often random. Suppose that for individual i the calendar time of entering status 0 (denoted by Tsi ) can be accurately determined (e.g., HIV infection resulting from blood transfusion). The recruitment starts at τ0 and the follow-up is terminated at τe . For each individual i, let Ti∗ denote the time from Tsi to the calendar time of entering status 1. Let
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