Three classes of censored data rank tests: Strengths and weaknesses under censoring

SUMMARY Nonparametric tests are useful when definite parametric alternatives are not available. The behaviour under censoring of extensions of the tests is used to delineate three distinct classes of censored data rank tests: asymptotically efficient, approximately distribution-free and approximately unbiased. Section 2 gives qualitative comparisons; ? 3 gives some representative numerical comparisons. The Pitman efficiencies approximated by Lee, Desu & Gehan (1975) from simulations with 50 observations in each group are shown to agree with asymptotic calculations. 1. WEIGHTED LOG RANK TESTS It is assumed that two samples of randomly censored observations are available, that is, that the true lifetimes Yij have cumulative distribution functions F(y I Oj) for j = 1, ..., n(i); i = 1,2, that the censoring times Cij have cumulative distribution functions H(y I Of), and that the observable quantities are Xij = min (Yij, C) and Dij = [Xij = Yij], where [A] denotes the indicator function of the set A. It is assumed that 01 = 0, that F(t 1 0) is continuous in 0 and that the null hypothesis is 61 = 02 = 0. The 'true' value of 02 will be denoted by 0 and the functions F(t I 0) and H(t I 0) will be abbreviated to F and H. Survival distributions will be denoted by a bar, so that F(t) = 1- F(t). The hazard rates of F and F(t 10) will be denoted by A and AO, respectively. The test statistics studied are functions of: Di(t), the number of deaths observed before time t in the ith sample; Ri(t), the number at risk at time t in the ith sample; and Aj, an estimator of the cumulative hazard function for the ith sample. These functions are defined formally by n(i) n(i)