Estimation following sequential tests

Abstract

SUMMARY A method is given for obtaining confidence intervals following sequential tests. It involves defining an order relation among points on the stopping boundary and computing the probability of a deviation more extreme in this order relation than the observed one. Particular attention is given to the case of a normal mean with known or unknown variance. A comparison with the customary fixed sample size interval based on the same data is given. The purpose of this paper is to describe a method for obtaining confidence intervals following sequential tests. An order relation is defined among the points on the stopping boundary of the test. The confidence limits are determined by finding those values of the unknown parameter for which the probabilities of more extreme deviations in the order relation than the one observed have prescribed values. To facilitate understanding the proposed procedures, most of the paper is restricted to estimating the mean of a normal population with known variance following the class of sequential tests recommended by Armitage (1975) for clinical trials. The case of unknown variance is discussed briefly in ? 4. It is easy to see that the proposed method is valid more generally, although the probability calculations required to implement it depend on the specific parent distribution and stopping rule. A closely related method was proposed by Armitage (1958), who studied the case of binomial data numerically by enumeration of sample paths. Let xl, x2, ... be independent and normally distributed with unknown mean ,u and known variance cr2. Let sn =x1 + .. . + xn, and for given b > 0 consider the stopping rule