Sequential χ2and F tests and the related confidence intervals

Abstract

SUMMARY Sequential x2 and F tests are proposed for comparing more than two treatments. Extending earlier results of the author, one obtains approximations to the significance level and power of the tests, and approximate confidence intervals for the noncentrality parameter O., For each fixed 0 a confidence region is given for the angle c which the treatment effect vector in canonical coordinates makes with a fixed direction, and hence an overall confidence region is obtained for the pair (0, co). For comparing two treatments in clinical trials, Armitage (1975) has suggested a class of sequential tests which he calls repeated significance tests. These tests have been studied numerically by McPherson & Armitage (1971) and theoretically by Siegmund (1977, 1978), who also described a method for obtaining confidence intervals related to these tests and suggested a modification of the tests to increase the precision of the estimates; see also Siegmund & Gregory (1980). The goal of the present paper is to study analogous procedures for comparing more than two treatments. Simple examples involving three treatments might arise in comparing a drug at maximal dose, a drug at minimal dose and placebo, or drug A, drug B and drug A together with B. For theoretical analysis it is convenient to assume that observations are made sequentially on vectors Xn = (X1n, ..., Xrn)' (n = 1, 2, ...), where r denotes the number of treatments and Xij is the response of the jth patient assigned to treatment i. The results of this paper should be applicable to randomized schemes of allocating treatments which approximate this situation. The Xij are assumed to be independently and normally distributed with mean uij and common variance U2. The following discussion is restricted to the special case ,Usj = ,ui for all j. The theory can easily be modified to accommodate stratification on some concomitant variable, provided that the allocation of treatments is approximately balanced within strata. The one-way analysis of variance model with uij = ui for all j is customarily reparameterized in terms of ,u = r-1 2jp and ao = jug-,u, so that E(Xij) = Ju+oi with Z ox = 0. The log likelihood ratio statistic for.testing Ho: 1 = 0 against H1: Z a2 > 0 based on n observations is