Estimation of the Survivorship Function Using the Cox-Proportional Hazard Model with Relaxed Tsiatis Assumptions
International Journal of Data Science and Analysis
Volume 4, Issue 6, December 2018, Pages: 106-111
Received: Oct. 10, 2018;
Accepted: Oct. 22, 2018;
Published: Jan. 25, 2019
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Valerie Atieno Odhiambo, Department of Statistics and Actuarial Sciences, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya
George Otieno Orwa, Department of Statistics and Actuarial Sciences, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya
Romanus Odhiambo, Department of Statistics and Actuarial Sciences, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya
Survival analysis is the primary statistical method of analysing time to event data. The most popular method for estimating the survivor function is the Cox-Proportional Hazard model. It assumes that the effect on the hazard function of a particular factor of interest remains unchanged throughout the observation. This is known as Proportional Hazards. Tsiatis assumed that the underlying hazard function is constant over distinct intervals. In the current study, no shape assumption is imposed other than that the hazard function is a smooth function with an arbitrary choice of a smoother. Such an approach involves the implementation of kernel-smoothing of the initial hazard estimate which have proved in studies to provide a trade-off between bias and variance. The cross-validation and plug-in bandwidth selectors are considered to determine the optimal bandwidth, h to be used as a smoothing parameter. Consequently, the survivorship function is estimated using the Cox-Proportional Hazards model. Proper application of the smoothing procedure is seen to improve the statistical performance of the resulting hazard rate estimator. No constraints are implored on the form of the underlying hazard proving to be less bias than Tsiatis’ method. This implies that the kernel smoothed survivorship function is more appropriate than the common standard techniques in survival analysis as it provides piecewise smooth estimates. Coverage probabilities of the estimate are then obtained which are found to be more accurate and closer to the nominal level compared to those estimated by Tsiatis.
Valerie Atieno Odhiambo,
George Otieno Orwa,
Estimation of the Survivorship Function Using the Cox-Proportional Hazard Model with Relaxed Tsiatis Assumptions, International Journal of Data Science and Analysis.
Vol. 4, No. 6,
2018, pp. 106-111.
Kaplan E. L. and Meier P., "Nonparametric Estimation from Incomplete Observations," Journal of the American Statistical Association, 1958.
Nadaraya E. A., "Some New Estimates for Distribution Functions," Theory of Probability and Its Applications, vol. 9, no. 3, pp. 497-500, 1964.
Breslow N., "Covariance Analysis of Censored Survival Data," Biometrics, vol. 30, pp. 89-99, 1974.
Tsiatis Anastasios A., "A Heuristic Estimate of the Asymptotic Variance of the Survival Probability in Cox's Regression Model," University of Wisconsin, 1978.
Rosenblatt John Rice and Murray, "Estimation of the Log Survivor Function and Hazard Function," Sankhya: The Indian Journal of Statistics, Series A (1961-2002), vol. 38, pp. 60-78, 1976.
Cox D. R., "Regression Models and Life-Tables," Journal of the Royal Statistical Society. Series D (The Statistician), 1972.
Anderson J. A. and Senthilselvan A., "Smooth Estimates for the Hazard Function," Journal of the Royal Statistical Society. Series B (Methodological), vol. 42, pp. 322--327, 1980.
Goodd I. J. and Gaskins R. A., "Nonparametric roughness penalties for probability densities," Biometrika, vol. 58, pp. 255-277, 1971.
Kalbfleisch J. D. and Prentice R. L., "Marginal likelihoods based on Cox's regression and life model," Biometrika, vol. 60, no. 2, pp. 267-278, 1973
Gijbels A. D., "Practical bandwidth selection in deconvolution kernel density estimation," Computational Statistics and Data Analysis, vol. 45, pp. 249 - 267, 2004
Lin D. Y., Fleming T. R. and Wei L. J., "Confidence Bands for Survival Curves Under the Proportional Hazards Model," Biometrika, vol. 81, no. 1, pp. 73-81, 1994.
Nair V. N., "Confidence Bands for Survival Functions With Censored Data: A Comparative Study," Technometrics, vol. 26, no. 3, pp. 265-275, 1984.
Bie O., Borgan O. and Liestol K., "Confidence Intervals and Confidence Bands for the Cumulative Hazard Rate Function and Their Small Sample Properties," Scandinavian Journal of Statistics, vol. 14, no. 3, pp. 221-233, 1987.
Breslow N. E., "Discussion of Professor Cox's paper," J Royal Stat Soc B, vol. 34, pp. 216-217, 1972.
Ramlau-Hansen H., "Smoothing Counting Process Intensities by Means of Kernel Functions," The Annals of Statistics, vol. 11, no. 2, pp. 453-466, 1983.
Tanner M. A. and Wong W. H., "The Estimation of the Hazard Function from Randomly Censored Data by the Kernel Method," The Annals of Statistics, vol. 11, pp. 989-993, 1983.
Tsiatis Anastasios A., "A Large Sample Study of Cox's Regression Model," The Annals of Statistics, vol. 9, pp. 93-108, 1981.