Closed sequential t-tests

Abstract

Schneiderman & Armitage (1962) described a family of sequential procedures, called 'wedge plans', which could be used for testing the mean ,.t of a normal population with known variance CJ2. For tests of a null hypothesis Ho (that ,.t = 0) against two alternatives H1 (that It = ,t1) and H_1 (that It = -,tl), the boundaries favouring H1 and H_1 (i.e. for rejecting Ho) are the same as in the procedure of Sobel & Wald (1949), but the acceptance boundary (favouring Ho) is in general curved, and the boundaries are closedwith a maximum sample size N'. For one extreme member of the family (with N' = cx) the acceptance boundary coincides with that of the two-sided open sequential test of Wald (1947, Chapter 9); the other extreme is the 'restricted procedure' of Armitage (1957), where the acceptance boundary provides for termination at a fixed sample size N1 if neither rejection boundary has previously been crossed. These wedge plans, which derive their name from the shape of the acceptance boundary, thus provide a bridge between the open plans derived from Wald's theory and the restricted plans of Armitage (1957). The purpose of this note is to derive, from these results, some approximate procedures for use when o-2 is unknown. At one extreme one may use the open sequential t-test derived by Arnold & Goldberg, in the National Bureau of Standards (1951) tables, NBS-AMS-7. (These tables are unfortunately out of print; although some of the tabulated values can be derived from other tables (e.g. Resnikoff & Lieberman, 1957, and Slater, 1960), the tables in NBS-AMS-7 remain the most convenient source of reference for this test.) The other extreme, a restricted sequential t-test, has not previously been considered in any detail, and is discussed below. Between these extremes is a family of closed procedures, analogous to the family of wedge plans available when o-2 is known, and referred to above.