Abstract
The problem of testing for umbrella alternatives in a one-way layout with right-censored survival data is considered. Testing procedures based on the two-sample weighted Kaplan-Meier statistics suggested by Pepe and Fleming (1989, Biometrics, 45, 497–507; 1991, J. Roy. Statist. Soc. Ser. B, 53, 341–352) are suggested for both cases when the peak of the umbrella is known or unknown. The asymptotic relative efficiency of the weighted Kaplan-Meier test and the weighted logrank test proposed by Chen and Wolfe (2000, Statist. Sinica, 10, 595–612) is computed for the umbrella peak-known setting where the piecewise exponential survival distributions have the proportional or crossing hazards, or the related hazards differ at early or late times. Moreover, the results of a Monte Carlo study are presented to investigate the level and power performances of the umbrella tests. Finally, application of the proposed procedures to an appropriated data set is illustrated.
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References
Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference under Order Restrictions, Wiley, New York.
Chen, Y. I. and Wolfe, D. A. (1990). A study of distribution-free tests for umbrella alternatives, Biometrical J., 32, 47-57.
Chen, Y. I. and Wolfe, D. A. (2000). Umbrella tests for right-censored survival data, Statist. Sinica, 10, 595-612.
Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis, Wiley, New York.
Gill, R. D. (1980). Censoring and Stochastic Integrals, Mathematical Centre Tracts 124, Mathematisch Centrum, Amsterdam.
Homburger, F. and Treger, A. (1970). Transplantation technique for acceleration of carcinogenesis by benz[α]anthracene or 3,4,9,10-dibenzpyrene [benzo(rst)penta-phene], Journal of the National Cancer Institute, 44, 357-360.
Jonckheere, A. R. (1954). A distribution-free k-sample test against ordered alternatives, Biometrika, 41, 133-145.
Kaplan, E. L. and Meier, P. (1958). Nonparametric estimator from incomplete observations, J. Amer. Statist. Assoc., 53, 457-481.
Liu P. Y., Green, S., Wolf, M. and Crowley, J. (1993). Testing against ordered alternatives for censored survival data, J. Amer. Statist. Assoc., 88, 153-161.
Mack, M. A. and Wolfe, D. A. (1981). K-sample rank tests for umbrella alternatives, J. Amer. Statist. Assoc., 76, 175-181.
Pan, G. (1997). Confidence subset containing the unknown peaks of an umbrella ordering, J. Amer. Statist. Assoc., 92, 307-314.
Pepe, M. S. and Fleming, T. R. (1989). Weighted Kaplan-Meier statistics: A class of distance tests for censored survival data, Biometrics, 45, 497-507.
Pepe, M. S. and Fleming, T. R. (1991). Weighted Kaplan-Meier statistics: Large sample and optimality considerations, J. Roy. Statist. Soc. Ser. B, 53, 341-352.
Schervish, M. J. (1984). Multivariate normal probabilities with error bound, Applied Statistics, 33, 81-94.
Terpstra, T. J. (1952). The asymptotic normality and consistency of kendall's test against trend, when ties are present in one ranking, Indag. Math., 145, 327-333.
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Lee, WC., Chen, YI. Weighted Kaplan-Meier Tests for Umbrella Alternatives. Annals of the Institute of Statistical Mathematics 53, 835–852 (2001). https://doi.org/10.1023/A:1014673524005
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DOI: https://doi.org/10.1023/A:1014673524005