On selecting the better of two binomial populations

A new procedure for selecting the better of two binomial populations by using sequential analysis proposed. It is based on well known Play-the-Winner (PW) and Vector-at-a Time (VT) procedures and constructed to avoid some disadvantages of PW and VT rules.


Introduction
The paper is about selection the best of two treatments with a binomial response in view of a scientific rationale and human ethics. The goals are (i) to have a probability of at least P * of selecting the better treatment when it is sufficiently better and (ii) to minimize the number of people put on the poorer treatment or equivalently minimizing the expected number of failures that could have been avoided by using the better treatment throughout.
In [6] Play-the-Winner Rule (PWR) proposed. This is sequential method: trials are made one at a time on either treatment, a success implies that the next observation shall be drawn from the same treatment, while a failure causes a switch to the other treatment. There are many modifications of this rule. In [5] randomized PWR proposed, some modifications of this method are described in [3] and [1].
Sobel and Weiss compare PW and Vector-at a time rules (VT) [4]. VT is a rule with an equal allocation, both populations are sampled for every trial, so it does not fulfil our conditions. PWR has disadvantage that it admits the situation when only one population is observed.
Considering this a new procedure is proposed. We call it Mixed Rule (MR). This procedure is based on Play-the-Winner and Vector-at a time rules. We start the experiment with VT procedure and continue it until we get result "success and failure" or "failure and success". Further we apply only the treatment which gave result "success" until failure occurs. After that we return to applying VT rule. In this way at least one patient gets poorer treatment and more patients gets better treatment.
MR is compared with PW and VT procedures. Stopping rule is based on the difference of numbers of successes between two treatments [4] and the Sequential Probability Ratio Test [2] are used.

Analysis of the mixed rule
Following [4] let S(A) and S(B) be the numbers of A and B successes, = p − p 0, where p and p are probabilities of treatments A and B respectively.
We terminate the treatment when |S(A) − S(B)| = r, where integer r is chosen to be the smallest such that where P * and * are given by experimenter, CS denotes the event 'correct selection'.

We select treatment A when S(A)−S(B) = r and treatment B when S(A)−S(B) = −r.
Let N b be the number of trials in which treatment B is used. We say that treatment A is better ( > 0), so our purpose is to minimize E(N b ). CS is the choice A in our case, NT means 'next treatment'.
For the Mixed rule we denote We begin sampling from VT, hence P (CS) = B(0). Further where q = 1 − p and q = 1 − p , with the boundary conditions From (2) and (3) we obtain By setting P(CS) equal to P* we get quadratic in λ r equation. The solution of this gives The purpose is to select such r that (1) is satisfied for all p and p . Denote p 0 = p+p 2 . Then Using (4) and following [4] we get Now r is calculated from (6), (5)  Then we have recurrence relations: with boundary conditions U (r) = V (r) = W (−r) = 0. We find that Comparison is based on the expecting number of patients, which receive poorer treatment. Let P * = 0.95 and * = 0.2 according to [4]. We analyze situation when there is no a priori information about parameters. Mixed Rule is preferred under condition p > 0.5. This condition is satisfied in the most practical cases. In other cases Mixed Rule has not this advantage, but any of the procedures of this type can not give the best results for all cases. These results are shown in Table 1.

The sequential probability ratio test
This test is described in [2]. Let X = (X A , X B ) be the result of the trial. Suppose X has p.d.f. f (X, ·). Denote N v -the total number of vectors X. We wish to test H 0 : p = p = θ 0 against H 1 : p = p + = θ 1 . Define Z = ln f (X,θ 1 ) f (X,θ 0 ) . Then following Eq.(2.

4.3) from [2]
where A 1 = 1−β α , A 2 = β 1−α and α is the size of the type I error, β is the size of the type II error.
For the VT procedure X A (X B ) is 1 if the result is success and 0 otherwise and We get f (X, θ 1 ) from (9) with = 0. So Further the expected number of N v under H 1 is found from (8) and For the Mixed Rule X A (X B ) is of form (S...SF), suppose X A has i successes and X B has j successes, i, j = 0.1, 2, ..., X = (X A , X B ). The distribution of independent random variables X A and X B is geometric, so f (X, θ 0 ) = (p ) i+j (1 − p ) 2  These results are shown in Table 2, the entries in Table 3 were determined from simulations. In this case Mixed rule performs better than VT and similar to PWR with all p .