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Nonparametric estimators of survival function under the mixed case interval-censored model with left truncation

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  • Pao-Sheng Shen

    (Tunghai University)

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.

Suggested Citation

  • Pao-Sheng Shen, 2020. "Nonparametric estimators of survival function under the mixed case interval-censored model with left truncation," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 26(3), pages 624-637, July.
    • Handle: RePEc:spr:lifeda:v:26:y:2020:i:3:d:10.1007_s10985-020-09493-2
    DOI: 10.1007/s10985-020-09493-2
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    References listed on IDEAS

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    1. Pan, Wei & Chappell, Rick & Kosorok, Michael R., 1998. "On consistency of the monotone MLE of survival for left truncated and interval-censored data," Statistics & Probability Letters, Elsevier, vol. 38(1), pages 49-57, May.
    2. Anton Schick & Qiqing Yu, 2000. "Consistency of the GMLE with Mixed Case Interval‐Censored Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(1), pages 45-55, March.
    3. Michael G. Hudgens, 2005. "On nonparametric maximum likelihood estimation with interval censoring and left truncation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(4), pages 573-587, September.
    4. Richard Peto, 1973. "Experimental Survival Curves for Interval‐Censored Data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 22(1), pages 86-91, March.
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    Cited by:

    1. Ouafae Benrabah & Feriel Bouhadjera & Elias Ould Saïd, 2022. "Local linear estimation of the regression function for twice censored data," Statistical Papers, Springer, vol. 63(2), pages 489-514, April.
    2. Pao-sheng Shen, 2022. "Nonparametric estimation for competing risks survival data subject to left truncation and interval censoring," Computational Statistics, Springer, vol. 37(1), pages 29-42, March.
    3. Prabhashi W. Withana Gamage & Christopher S. McMahan & Lianming Wang, 2023. "A flexible parametric approach for analyzing arbitrarily censored data that are potentially subject to left truncation under the proportional hazards model," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 29(1), pages 188-212, January.
    4. Pao-sheng Shen & Yingwei Peng & Hsin-Jen Chen & Chyong-Mei Chen, 2022. "Maximum likelihood estimation for length-biased and interval-censored data with a nonsusceptible fraction," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 28(1), pages 68-88, January.

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