Abstract
It has been already noticed that the classical Greenwood formula may not be an unbiased estimator for the variance of the Kaplan–Meier Product Limit estimator (PLE). However, a rigorous proof for such a suggestion has not been available. In this paper, we investigate some small-sample properties of the PLE and show that the Greenwood formula strictly underestimates the variance of the PLE. Besides, some existing estimators for the variance of the PLE are also discussed.
Similar content being viewed by others
References
Breslow N.E., Crowley J. (1974). A large sample study of the life table and product limit estimates under random censorship. The Annals of Statistics 2, 437–453
Brookmeyer R., Crowley J. (1982). A confidence interval for the median survival time. Biometrics 38, 29–41
Chen Y.Y., Hollander M., Langberg N.A. (1982). Small-sample results for the Kaplan–Meier estimator. Journal of the American Statistical Association 77, 141–144
Guerts J.H.J. (1985). Some small-sample non-proportional hazard results for the Kaplan–Meier estimator. Statistica Neerlandica 39, 1–13
Guerts J.H.J. (1987). On the small-sample performance of Efron’s and Gill’s version of the product limit estimator under nonproportional hazards. Biometrics 43, 683–692
Fleming T.R., Harrington D.P. (1991). Counting processes and survival analysis. New York, Wiley
Greenwood M. (1926). The natural duration of cancer, reports on public health and medical subjects. Her Majesty’s Stationery Office 35, 1–39
Jennison C., Turnbull B.W. (1985). Repeated confidence interval for the median survival time. Biometrika 72, 619-625
Kalbfleisch J.D., Prentice R.L. (1981). The statistical analysis of failure time data. New York, Wiley
Kaplan E.L., Meier P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association 53, 457–481
Nelson W. (1969). Hazard plotting for incomplete failure data. Journal of Quality Technology 1, 27–52
Peto R., Pike M.C., Armitage P., et al. (1977). Design and analysis of randomized clinical trials requiring prolonged observation of each patient. II Analysis and Examples. British Journal of Cancer 35, 1–39
Phadia E.G., Shao P.Y.S. (1999). Exact moments of the product limit estimator. Statistics and Probability Letters 41, 277–286
Shen P.S. (2002). A note on the homogenetic estimate for the variance of the Kaplan–Meier estimator. Communications in Statistics—Theory and Methods 31, 1595–1603
Slud E.V., Byar D.P., Green S.B. (1984). A comparison of Reflected vs. Test- based confidence intervals for the median survival time, based on censored data. Biometrics 40, 587–600
Willett J.B., Singer J.D. (1993). Investigating onset, cessation, relapse and recovery: why you should, and how you can, use discrete-time survival analysis to examine event occurrence. Journal of Consulting and Clinical Psychology 61, 952–965
Zhao G.L. (1996). The homogenetic estimate for the variance of survival rate. Statistics in Medicine 15, 51–60
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Wang, J. Small-sample studies on right censored data with discrete failure times. AISM 60, 427–440 (2008). https://doi.org/10.1007/s10463-006-0087-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-006-0087-z