Asymptotically Efficient Rank Invariant Test ProceduresRichard Peto; Julian PetoJournal of the Royal Statistical Society. Series A (General), Vol. 135, No. 2. (1972), pp. 185-207.Stable URL:http://links.jstor.org/sici?sici=0035-9238%281972%29135%3A2%3C185%3AAERITP%3E2.0.CO%3B2-QJournal of the Royal Statistical Society. Series A (General) is currently published by Royal Statistical Society.Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtainedprior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content inthe JSTOR archive only for your personal, non-commercial use.Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/journals/rss.html.Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academicjournals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers,and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community takeadvantage of advances in technology. For more information regarding JSTOR, please contact [email protected]://www.jstor.orgTue Nov 13 11:30:51 2007J. R. Statist. Soc. A, (1972), 135, Part 2, p. 185 Asymptotically Efficient Rank Invariant Test Procedures By RICHARDPETO AND JULIANPETO Radcliffe Infirmary, Institute of Psychiatry Oxford University University of London [Read before the ROYALSTATISTICAL on Wednesday, January 19th, 1972, the President SOCIETY Professor G. A. BARNARDin the Chair] Asymptotically efficient rank invariant test procedures for detecting differences between two groups of independent observations are derived. These are generalized to test between two groups of independent censored observations, to test between many groups of observations, and to test between groups after allowance for the effects of concomitant variables. One of these test procedures-the logrank-is particularly appropriate for comparing life tables, and can therefore be used in the analysis of clinical trials, industrial life-testing experiments and laboratory studies of animal carcinogenesis. It has greater local power than any other rank-invariant test procedure for detecting Lehmann-type differences between groups of independent observations subject to some right-censoring. The logrank test, although a rank test, can be presented in a format which exhibits the physical significance as well as the statistical significance of any important differences between groups of events. Keywords :LIFE TABLE; EXPERIMENTAL SURVIVAL CURVE ;EXPERIMENTAL CDF ;PRODUCT-LIMIT ESTIMATE; PERMUTATIONAL TEST ;RANK TEST ;TWO-GROUP TEST ;RIGHT CENSORING ; DEATH TIMES ;LOGRANK ;ASYMPTOTIC EFFICIENCY ;RANK INVARIANCE ;RELATIVE DEATH RATES;LEHMANN ALTERNATIVES ; CLINICAL TRIALS ; CENSORING ;WILCOXON RANK SUM TEST ;FAILURE TIMES. 1. INTRODUCTION THEprincipal advantage of any rank test, the absolute reliability of the significance level it generates whatever the distribution functions of the observations, is for many such tests offset by some loss of power. Various rank tests which are asymptotically efficient for particular distributions have been suggested during the last few years, and Hajek and Sidak (1967) have described a general method for the construction of such tests. Since any rank test is invariant under monotonic transformation of the data, distributions fall into disjoint "efficiency classes" within each of which the efficiency of any rank test is constant; the normal and lognormal distributions are evidently members of the same class, for example. Suppose z, (1 <i< N) are independent observations from the c.d.f. F(x, B,), where Oi --8, for 1 <i <n, Oi = 8, for n + 1<i <N, and the null hypothesis is H,: 8, = 8,. If a score is assigned to each observation, the scores being such that the group A sum of scores is an asymptotically efficient test statistic, the null hypothesis distribution of this sum may be derived permutationally. If each score is not calculated exactly from the observation values but is estimated from their ranks, the resulting rank test will be asymptotically efficient in the family F(x, 8); it will of course also be asympto- tically efficient in any other family in the same efficiency class as F(x, 8).186 RICHARDPETOAND JULIANPETO-Rank Invariant Test Procedures [Part 2, 2. SOME DEFINITIONS Let z be a real-valued random variable with c.d.f. F(x). (i) Let G(x) = 1-F(x). G is the survival curve of z. (ii) We shall be concerned only with either exact or (interval) censored observa- tions. For the latter, the only information recorded about the random variable is that it lies somewhere in the non-null interval (xl,x2). One observation of a random variable may thus generate either one or two data points according to whether the observation is exact or censored. If x, =co then the observation is right-censored with censoringpoint or value xl. In data consisting entirely of either exact or right-censored observations, the only data points other than CQ are the exact observation values and the censored observation values. (iii) Consider data consisting of observations of N random variables zi(1 <i< N). For these data, the experimental survival curve, H(x) is the survival curve under which the product of the likelihoods of the N observations is maximal. H has a dis-continuity at each exact observation value, since otherwise the likelihood of such observations would be infinitesimal. At each data point the values of H (or of the top and bottom of the step in H) are well defined? and these values are invariant under monotonic (rank-preserving) transformations of the data points. If there are no tied values and there is no censoring then the steps in Hare all of equal size and at the rth exact observation value H decreases from (N+ 1-r)/N to (N- r)/N. For partially right-censored data H(x) is the familiar life-table estimate