A family of closed sequential procedures

Abstract

Armitage (1957) introduced a class of closed sequential procedures, called 'restricted', which have been proved useful in medical trials (Armitage, 1960) where the risk of a long sequential experiment is not easily borne. Restricted plans can be regarded as procedures for testing a null hypothesis, Ho, that the mean, It, of a random variate is zero. For two-sided alternatives there are three decisions, D1 (desirable when It > 0), D-1 (desirable when It < 0) and Do (desirable when It = 0). For one-sided alternatives, the choice is between two decisions, Do and, say, D1. In using restricted plans one sacrifices some of the reduction in average sample number achieved by the open plans of Wald (1947). For two-sided alternatives restricted plans have the same 'outer' boundaries leading to D1 and D-1 as when two Wald plans are used in conjunction (as proposed by Armitage (1947) and Sobel & Wald (1949)). Instead of the wedge-shaped 'inner' boundary of the Sobel-Wald plans, the restricted plans require a truncating boundary drawn after a certain number of observations, N1, the value of which is chosen to preserve certain specified type I and type II errors. It is the substitution of this boundary in place of the Sobel-Wald wedge that causes the increase in average sample number in the restricted plans. In this paper we describe a family of closed sequential plans for normal variates, which retain the same outer boundaries as the Sobel-Wald and the restricted plans. The limiting cases, at opposite extremes, of the family of plans considered here are (a) a restricted plan, and (b) an open plan similar to the Sobel-Wald plan. The new plans, which we call 'wedge' plans, have better average sample number characteristics than restricted plans. The wedge plans appear to have average sample number functions similar to those of the equivalent open plans. It is difficult to make a precise comparison of the average sample number functions because of the difficulty in finding comparable plans with exactly the same operating characteristic. There is, however, some evidence that the wedge plan may have rather smaller average sample numbers, for some values of Itt between Ito and #,, than the equivalent open plans. If this were so, the wedge plans would in this respect resemble the test procedures described by Anderson (1960), although the methods used here, and the resulting boundaries, are different from those of Anderson. In particular Anderson considers only two-decision procedures, whereas we consider also three-decision procedures. Results based on Monte Carlo trials for the twodecision problem ('one-sided') and the three-decision problem ('two-sided') are given in Tables 6-9. These are discussed in detail later.