A NOTE ON SELECTING THE BETTER BINOMIAL POPULATION

An inverse sampling procedure is proposed for the problem of selecting the better of two treatments when the responses are dichotomous. This procedure is particularly useful when it is desired to limit the number of failures during the decision making stage. The regret function of the procedure is derived and it is shown that this procedure has a minimax regret property when compared to a fixed sample procedure studied by Pradhan and Sathe [2]. Numerical evidence indicates that this procedure dominates the fixed sample procedure of Pradhan and Sathe over the entire parameter space.

ment, the one with more successes is chosen as the better treatment; (in case of a tie, one of the two treatments is selected as the better one at random with equal probability) and then that treatment is given to the remaining (N-2n) patients.
In this note we study the same problem but with an inverse sampling procedure.
We assume that N is an even integer, say N  2m, and the N patients are divided into m pairs.The two treatments are applied to each pair (assignment of the two pa- tients in any pair to the treatments being random) successively until k failures are observed with any one treatment.Then the other treatment is selected as the better treatment.If exactly k failures are observed on both the treatments simultaneously or if less than k failures are observed on both the treatments until all the N patients are treated, then one of the two treatments is selected as the better treat- ment at random with equal probability.The treatment selected as the better one is then applied to the remaining (if any) patients.
Several authors, e.g., Sobel and Weiss [3], have considered stopping rules based on the difference in the number of successes (and/or failures) with the two treatments.For @I and/or @2 small, (where @ denotes the probability of success i with the i th treatment) such procedures result in larger expected number of pa- tients treated and/or larger expected number of patients receiving the poorer treat- ment during the decision making stage.Our approach ensures that a decision is reached with at most 2k failures.In situations where an early (but correct) decl- sion is desired with minimal number of failures during the decision making stage, this approach would be better as it enables us to control that number of failures.
Our approach can be thought of as the "'truncated" version of the situation where the two treatments can be applied to an infinite number of patients.
In this article we study the particular case of k i.The more general case will be treated in a separate article.

Probability of Correct Selection
Let @ denote the probability of success and N the number of trials to the i i first failure with treatment T. (i 1 2) Without loss of generality, throughout 1 this article, we assume that @i > @2.Let the probability of correctly selecting T I as the better treatment be denoted by P P(@I, @2,m).Then we have r-I @2r-l(l-@l (1-@2)+ I m @2 m m r r-i r-i + I @i @2 r=l i1211 + (01 -02) {I-(01 @2)m}/(l-@I @2 )].

Expected Sample Size
Let S denote the number of patients treated before a decision as to the better r (i, j I, 2; i # j). (@ @2)m}/(1 @ @ 2Z @I @2 2 r=l (2.3) Observe that E(S) depends on @I and @2 only through (@i @2 and is increasing in (@i @2 ).

Regret Function
We define the regret function, R(@I' @2' m), as the expected number of failures in the 2m patients with our proposed decision procedure in excess of the expected number of failures if all the 2m patients had received the better treatment.

COMPARISON WITH CPS PROCEDURE (2.5)
We shall compare our procedure with the CPS procedure in terms of the respec- rive regret functions.The regret function obtained in Pradhan and Sathe [2] is: R'(O I, 02, m) (01 -02 where U is their probability of wrong selection (given by their equation (2.1)) and 2n (fixed) is the number of patients treated before a decision is reached.We thus see that (2.5) and (3.1) are similar in structure which facilitates their comparison.
Observe that for our procedure, the sample size, S, before a decision is reached is a random variable whereas in Pradhan and Sathe [2], the sample size 2n is fixed.
To make the comparison a just one, we set 2n E(S).In this situation, we see that R(O I, 02,m) _< R'(OI, 02,m) whenever P* _> I U.
In general it is not analytically easy to find the region in the (0 I, 02 space where P* _> i U inequality is satisfied.However, we can say something about the inequality as 01 i.
In the next theorem we show that our procedure has minimax regret when compared to the CPS procedure.
THEOREM 3.2: For 2n E(S) and @ we have
To prove this theorem we first need the following Lemma: LEMMA 3.1: For fixed (@I @2 and m, R(@I, @2,m) is a decreasing function of (@1@2).
To prove the lemma it suffices to show that g(x) (l-xm)/(1-x)2 m/(1-x) is de- creasing in x which can be seen to be the case by writing ).
i=I This proves the lemma.
The results for other values of m are almost identical and hence are not report- ed here.The effect of m is negligible since it appears in the expressions for P and E(S) (See (2.1) and (2.3))only as the power of (@i @2 ).
Thus, based on this numerical evidence, a much stronger statement than Theorem 3.1 can be made, namely that, our procedure dominates the CPS procedure over the entire (@I' @2 space.

ACKNOWLEDGEMENTS
The author thanks Mr. Michael Spevak for computational assistance and Professor Ajit Tamhane of Northwestern University for some useful discussions during the writing of this paper.
m and N 2 _> m.Using the tail probability representation of expectation we have m E(S) 21 Pr (S > 2r) r=l m 27.Pr (N _> r, N 2 _> )