Abstract
It is well-known that the nonparametric maximum likelihood estimator (NPMLE) may severely under-estimate the survival function with left truncated data. Based on the Nelson estimator (for right censored data) and self-consistency we suggest a nonparametric estimator of the survival function, the iterative Nelson estimator (INE), for arbitrarily truncated and censored data, where only few nonparametric estimators are available. By simulation we show that the INE does well in overcoming the under-estimation of the survival function from the NPMLE for left-truncated and interval-censored data. An interesting application of the INE is as a diagnostic tool for other estimators, such as the monotone MLE or parametric MLEs. The methodology is illustrated by application to two real world problems: the Channing House and the Massachusetts Health Care Panel Study data sets.
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References
O. Aalen, “Nonparametric inference in connection with multiple decrement models,” Scand. J. Statisticsvol. 3 pp. 15–27, 1976.
O. Aalen, “Non-parametric estimation of partial transition probabilities in multiple decrement models,” Ann. Statist.vol. 6 pp. 534–545, 1978.
N. Breslow, “Discussion on Professor Cox's paper,” J. Roy. Statist. Soc. A vol. 34 pp. 2160–217, 1972.
R. Chappell, “Sampling design of multiwave studies with an application to the Massachusetts Health Care Panel Study,” Statist. in Medicinevol. 10 pp. 1945–1958, 1991.
T. R. Fleming and D. P. Harrington, “Nonparametric estimation of the survival distribution in censored data,” Commun. Statist.- Theor. Meth.vol. 20 pp. 2469–2486, 1984.
H. Frydman, “Anote on nonparametric estimation of the distribution function from interval-censored and truncated observations,” J. R. Stat. Soc. Ser. Bvol. 56 pp. 71–74, 1994.
J. Hyde, “Testing survival under right censoring and left truncation,” Biometrikavol. 64 pp. 225–230, 1977.
D. Lynden-Bell, “A method for allowing for known observational selection in small samples applied to 3CR quasars,” Mon. Nat. Royal Astr. Soc. vol. 155 pp. 95–118, 1971.
W. B. Nelson, “Hazard plotting for incomplete failure data,” J. Quality Technologyvol. 1 pp. 27–52, 1969.
W. Pan and R. Chappell, “Estimating survival curves with left truncated and interval censored data under monotone hazards.” Biostatistics Dept., Univ. of Wisconsin-Madison, Tech. Rept. 109, 1996. (To appear in Biometrics.)
W. Pan and R. Chappell, “Estimating survival curves with left truncated and interval censored data via the EMS algorithm,” To appear in Comm. in Statist.-Theory and Methods, 1998.
B. W. Silverman, M. C. Jones, J. D. Wilson and D. W. Nychka, “A smoothed EM approach to indirect estimation problems, with particular reference to stereology and emission tomography (with discussions),” J. R. Statist. Soc.B vol. 52 pp. 271–324, 1990.
T. M. Therneau, “A package for survival analysis in S,” available from StatLibat http://www.stat.cmu.edu, 1995.
W. Y. Tsai, “Estimation of the survival function with increasing failure rate based on left truncated and right censored data,” Biometrikavol. 75 pp. 319–324, 1988.
W. Y. Tsai, N. P. Jewell and M.-C. Wang, “A note on the product-limit estimator under right censoring and left truncation,” Biometrikavol. 74 pp. 883–886, 1987.
B. W. Turnbull, “The empirical distribution function with arbitrarily grouped, censored and truncated data,” J. Roy. Statist. Soc. B vol. 38 pp. 290–295, 1976.
M. Woodroofe, “Estimating a distribution function with truncated data (Corr: V15 p883),” Ann. Statist. vol. 13 163–177, 1985.
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Pan, W., Chappell, R. A Nonparametric Estimator of Survival Functions for Arbitrarily Truncated and Censored Data. Lifetime Data Anal 4, 187–202 (1998). https://doi.org/10.1023/A:1009637624440
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DOI: https://doi.org/10.1023/A:1009637624440