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On nonparametric likelihood ratio estimation of survival probabilities for censored data

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  • Li, Gang

Abstract

Thomas and Grunkemeier (1975) proposed a nonparametric likelihood ratio method for interval estimation of survival probabilities for randomly censored data. Their method always produces confidence intervals inside [0,1] and has a better performance than the normal approximation method based on the Kaplan-Meier estimate and Greenwood's formula. In this note we show that the likelihood ratio used by Thomas and Grunkemeier (1978) is a "genuine" nonparametric likelihood ratio. That is, it can be derived by considering the parameter space of all survival functions. This property is not shared by many existing empirical likelihood methods. We also note that this result is not a direct consequence of the fact proved by Kaplan and Meier (1958) that the maximum of the likelihood over the space of all survival functions is achieved in the subspace of all discrete survival functions supported on the observed uncensored lifetimes. Another objective of this note is to point out that the likelihood ratio approach can also be used to draw joint inferences on any finite number of probabilities, and test goodness of fit of a given survival function for censored data. A rigorous derivation for the limiting distribution of the likelihood ratio is also provided.

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  • Li, Gang, 1995. "On nonparametric likelihood ratio estimation of survival probabilities for censored data," Statistics & Probability Letters, Elsevier, vol. 25(2), pages 95-104, November.
    • Handle: RePEc:eee:stapro:v:25:y:1995:i:2:p:95-104
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    1. R.D. Gill, 1980. "Censoring and Stochastic Integrals," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 34(2), pages 124-124, June.
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    Cited by:

    1. Stuart Barber & Christopher Jennison, 1999. "Symmetric Tests and Confidence Intervals for Survival Probabilities and Quantiles of Censored Survival Data," Biometrics, The International Biometric Society, vol. 55(2), pages 430-436, June.
    2. Yang, Hanfang & Zhao, Yichuan, 2012. "Smoothed empirical likelihood for ROC curves with censored data," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 254-263.
    3. Junshan Shen & Shuyuan He, 2007. "Empirical likelihood for the difference of quantiles under censorship," Statistical Papers, Springer, vol. 48(3), pages 437-457, September.
    4. Shen, Junshan & Yuen, Kam Chuen & Liu, Chunling, 2016. "Empirical likelihood confidence regions for one- or two- samples with doubly censored data," Computational Statistics & Data Analysis, Elsevier, vol. 93(C), pages 285-293.
    5. Einmahl, J.H.J. & McKeague, I.W., 1999. "Confidence tubes for multiple quantile plots via empirical likelihood," Other publications TiSEM b64493f8-1c01-40fd-b16d-7, Tilburg University, School of Economics and Management.
    6. Yanqing Sun & Rajeshwari Sundaram & Yichuan Zhao, 2009. "Empirical Likelihood Inference for the Cox Model with Time‐dependent Coefficients via Local Partial Likelihood," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(3), pages 444-462, September.
    7. Pan, Xiao-Rong & Zhou, Mai, 2002. "Empirical Likelihood Ratio in Terms of Cumulative Hazard Function for Censored Data," Journal of Multivariate Analysis, Elsevier, vol. 80(1), pages 166-188, January.
    8. Zhou, Mai & Li, Gang, 2008. "Empirical likelihood analysis of the Buckley-James estimator," Journal of Multivariate Analysis, Elsevier, vol. 99(4), pages 649-664, April.
    9. Zhao, Meng & Zhao, Yichuan & McKeague, Ian W., 2015. "Empirical likelihood inference for the odds ratio of two survival functions under right censoring," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 304-312.
    10. Mai Zhou & Yifan Yang, 2015. "A recursive formula for the Kaplan–Meier estimator with mean constraints and its application to empirical likelihood," Computational Statistics, Springer, vol. 30(4), pages 1097-1109, December.
    11. Nubyra Ahmed & Sundarraman Subramanian, 2016. "Semiparametric simultaneous confidence bands for the difference of survival functions," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 22(4), pages 504-530, October.
    12. Sundarraman Subramanian, 2020. "Function-based hypothesis testing in censored two-sample location-scale models," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 26(1), pages 183-213, January.
    13. Li, Gang, 2003. "Nonparametric likelihood ratio goodness-of-fit tests for survival data," Journal of Multivariate Analysis, Elsevier, vol. 86(1), pages 166-182, July.
    14. Wang, Qihua & Shen, Junshan, 2008. "Estimation and confidence bands of a conditional survival function with censoring indicators missing at random," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 928-948, May.
    15. Qi-Hua Wang & Bing-Yi Jing, 2001. "Empirical Likelihood for a Class of Functionals of Survival Distribution with Censored Data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(3), pages 517-527, September.
    16. Bin Dong & David E. Matthews, 2012. "Empirical Likelihood for Cumulative Hazard Ratio Estimation with Covariate Adjustment," Biometrics, The International Biometric Society, vol. 68(2), pages 408-418, June.
    17. McKeague, Ian W. & Zhao, Yichuan, 2006. "Width-scaled confidence bands for survival functions," Statistics & Probability Letters, Elsevier, vol. 76(4), pages 327-339, February.
    18. Subramanian, Sundarraman, 2022. "Simultaneous confidence bands for survival functions from twice censorship," Statistics & Probability Letters, Elsevier, vol. 186(C).
    19. McKeague, Ian W. & Zhao, Yichuan, 2002. "Simultaneous confidence bands for ratios of survival functions via empirical likelihood," Statistics & Probability Letters, Elsevier, vol. 60(4), pages 405-415, December.
    20. Subramanian, Sundarraman, 2012. "Model-based likelihood ratio confidence intervals for survival functions," Statistics & Probability Letters, Elsevier, vol. 82(3), pages 626-635.
    21. Junshan Shen & Shuyuan He, 2008. "Empirical likelihood confidence intervals for hazard and density functions under right censorship," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(3), pages 575-589, September.
    22. Paul Blanche, 2020. "Confidence intervals for the cumulative incidence function via constrained NPMLE," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 26(1), pages 45-64, January.
    23. Zhao, Yichuan & Zhao, Meng, 2011. "Empirical likelihood for the contrast of two hazard functions with right censoring," Statistics & Probability Letters, Elsevier, vol. 81(3), pages 392-401, March.
    24. Subramanian, Sundarraman, 2016. "Bootstrap likelihood ratio confidence bands for survival functions under random censorship and its semiparametric extension," Journal of Multivariate Analysis, Elsevier, vol. 147(C), pages 58-81.
    25. Shen, Junshan & He, Shuyuan, 2006. "Empirical likelihood for the difference of two survival functions under right censorship," Statistics & Probability Letters, Elsevier, vol. 76(2), pages 169-181, January.

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