WEIGHTED KAPLAN-MEIER TESTS FOR UMBRELLA ALTERNATIVES

WEI-CHUN LEE AND YUH-ING CHEN

Institute of Statistics, National Central University, Chung-Li 32054, Taiwan, R.O.C., e-mail: ychen@stat.ncu.edu.tw

(Received January 14, 2000; revised May 17, 2000)

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.

Key words and phrases: Asymptotic relative efficiency, Monte Carlo study, umbrella alternative, weighted Kaplan-Meier statistic, weighted logrank statistic.