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2 J. R. Statist. Soc. B (1980), 42, No. 1, pp Selecting the Better Bernoulli Treatment Using a Matched Samples Design By AJIT C. TAMHANE Northwestern University, Evanston, Illinois [Received June Final revision February 1979] SUMMARY The problem of selecting the better Bernoulli treatment using a matched samples design is considered in the framework of the indifference-zone approach. A singlestage procedure is proposed and its properties are studied. Tables of sample sizes for implementing the proposed procedure are given. A comparison is made with an independent samples design and the associated Sobel-Huyett selection procedure. Keywords: MATCHED SAMPLES DESIGN; INDEPENDENT SAMPLES DESIGN; BERNOULLI TREATMENTS; RANKING AND SELECTION; INDIFFERENCE-ZONE APPROACH 1. INTRODUCTION IN this paper we consider the problem of selecting the "better" of the two Bernoulli populations (i.e. the one having the larger success probability) when a matched samples design is used. The corresponding problem when the independent samples design is used has been considered by Sobel and Huyett (1957). It should be noted that although considerable literaturexists on the problem of comparing matched proportions (see McNemar, 1947; Cochran, 1950; Bennet, 1967, 1968; Bhapkar, 1973; Bhapkar and Somes, 1977), mostly it deals with tests of homogeneity. However, in many practical situations the experimenter's goal is to select the "best" treatment; a test of homogeneity does not provide the information which the experimenter truly seeks in such situations. This paper provides an appropriate formulation of the selection problem and gives a procedure for attaining this goal. 2. ASSUMPTIONS, NOTATION AND PROBLEM FORMULATION Consider two treatments T, and T2 and let 7Tij denote the probability that a matched observation on T, and T2 results in outcome i with T, and outcome j with T2 (i,j = 0,1) where 1 denotes success and 0 denotes failure. We have I E 7Tij = 1. We assume that the 7Tij remain constanthroughouthe trial. Thus each matched observation can be thought of as a realization from a fixed multinomial distribution with four cells: (1, 1), (1,0), (0, 1) and (0,0); the corresponding probabilities are 7T11, 7T10, 7TO, and 7TOO, respectively. Let Pt = T10 and P2 = To be the success probabilities of T, and T2 respectively and let P[t]!P[2j denote the ordered values of the pi. We assume that the 7Tij are unknown, but the experimenter is able to specify an upper limit 7T*(0 < 7T* < 1) on 7T10 + 7TOl = 7T (say). (In general, if matching is properly done and T, and T2 are comparable to each other then 7i, the probability of different outcomes on T, and T2 with the same matched observation, will be small; see Section 5 for further discussion.) The experimenter's goal is to select the treatment associated with p' p such a selection is referred to as a correct selection (CS) and the corresponding probability is denoted by PCS. The experimenter restricts consideration to procedures which guarantee the probability requirement: PCSkP* wheneverp[21-p[1,= 8>8*, and 7Tto+roj=7T-7r, (2.1) where {7T8 **,P* are constantspecified before experimentation starts; 0 < 7T* < 1, 0 < 8* < 7* and -<P*< 1.

3 1980] TAMHANE - The Better Bernoulli Treatment THE PROPOSED PROCEDURE AND ITS PCS We propose a natural selection procedure which takes n matched observations on T1 and T2. Let the outcomes be represented in a 2 x 2 table: T2 Success Failure Success x11 x10 T Failure xoi x00 The decision rule is: select T1 if x10 >x01; select T2 if x10< x01; and select T1 or T2 at random assigning equal probability to each one if x10 = x01. It might be noted that x1l and x00 play no role in the selection procedure. Our main problem is to find the minimum value of n which guarantees (2.1). Without loss of generality assume that T1 is the better treatment, i.e. P1 >P2 or equivalently V10 > Tro1. Then PCS = P{X10> X01} + vp{x10 = X01} = E[P{X10> xolxo = x} + P{X10 = Xoi X1o + Xo0=x}] (= ) x(l -( 7r)nx. (3.1) Note that the quantity inside the squawre brackets in (3.1) is just P( Y> 2x) + P( Y = lx) if x is even, and P{ Y) -(x + l)} if x is odd, where Y has a binomial distribution with parameters x and A = vlolv. Denoting it by g(x, A) we have where g(x, A) = IA{2(x+ 1), '(x+ 1)} for odd x> 1, (3.2a) = la(1x, 2X) for even x > 2, (3.2b) 2 1 for x = 0, (3.2c) I(a, b) = r(a+b) Pua-l(l-u)bdu F (a)rf(b) Jo denotes the usual incomplete beta function. To guarantee (2.1) it is necessary to find the infimum of the PCS over the region PI P2 = Tlh-rol > 8, *10 + v &ol'<*; the minimum value of n which makes this infimum > P* will be the desired sample size. To find the infimum, represent PCS = Eff{g(X, A)} where X is a binomial random variable with parameters n and Xr and E., denotes the expectation evaluated at parameter value v. Intuitively the infimum of the PCS over the specified region will occur when the pi are as close as possible and the matching is as ineffective as possible, i.e. when p1-p2 = = 8* and 7r10+ vol = 7r* which is referred to as the least favourable configuration (LFC). However, note that it is not completely obvious that the PCS decreases with increasing 7T since this also corresponds to an increase in the number of "effective" observations xl0 and x01. A formal proof of the LFC is thus needed and is given in the Appendix. Note that at the LFC, we have PCSLFC = Elr*{g(X, 1 + 8*/2X*)} nn 1 - zg(x,, 2+ 8*/27 ( )(7,*)X(1-7T*)n-x.(3.3) x=o It is fairly straightforward to verify that the right-hand side of (3.3) is increasing in n and tends to 1 as n tends to of. Thus any desired value of P* can be attained by choosing n large enough. It should be noted that, when S*= 7r*, (3.3) simplifies to 1 -'(1 *)n and when T= 1, (3.3) simplifies to g(n, +.*).

4 28 TAMHANE - The Better Bernoulli Treatment [No. 1, 4. TABLES OF SAMPLE SIZES The values of n which guarantee (2.1) were found using (3.3) for n ( 35. For n> 35, the following normal approximation was used. Note that since X1o, X0l are multinomial frequencies, for large n, (X1O-X0l) can be regarded as a normal random variable with mean = n(71o - 7T) and variance = n{i71o + i7t1 - (7-70-ol)2}. Therefore the PCS under the LFC can be written as where (D(.) denotes the standard normal distribution function. From this we obtain n L (v* 8*2){(D l(p*)}2 (4.1) This approximation is useful when P* is large and/or 8* is small and/or 7&* is large. The values of n obtained from (4.1) were rounded upwards. The calculations for P* = 0 90 and 0 95 appear in Table 1; the values of n for P* = 0 99 are almost exactly double those for P*= 0 95 and hence are not given here. TABLE 1 Values of n P* = * 7T* \ \8* P*= o To check the accuracy of the normal approximation we computed exact and approximate n-values for 20 < n < 35 and found that the approximate n is always within + 1 of the exact n; the accuracy of the approximation improves with increasing n. Thus the normal approximation should be very good for n > 35.

5 1980] TAMHANE - The Better Bernoulli Treatment COMPARISON OF MATCHED SAMPLES DESIGN WITH INDEPENDENT SAMPLES DESIGN In the case of independent samples design the goal of the experimenter is the same as before, namely to select the treatment having the larger success probability. However, since no r are present here, the probability requirement simply reads PCS >P* whenever P[2]-P[1] = 8> 8 *9 (5.1) where {8*,P*} are constants specified beforexperimentation starts; 0 < 8 < 1 and i < P* < 1. Sobel and Huyett (1957) proposed a single-stage procedure which guarantees (5.1) and showed that the optimal sample size per population, in large samples, is n (I - 2*2){ D-l(p*)}2 (5.2) The relativefficiency (RE) of the matched samples design in terms of the ratio of sample sizes obtained from (4.1) and (5.2) is RE *2 ~~~~~~~~~~~(5 2(ta h s 1r- *2(3 Note that we assume the values of 83* and P* specified by (2.1) and (5.1) are the same and that RE does not depend on P*. Furthermore, RE> 1 if it* < 2(1 + 8 *2). Also if &* = I (i.e. 1T=-10 +o701 is not constrained by any prior knowledge about its value) then RE = 0 5. Thus for r* > 1(1 + 8*2), the matched samples design is less efficient in the large sample case than the independent samples design. If the experimenter does not assume any prior knowledge concerning the value of 7T, then in the large sample case the matched samples design requires twice as many observations as the independent samples design to guarantee the same probability requirement. These results give a quantitative measure of how effective the matching must be (in our notation how small 1T must be) so that the matched samples design is more efficient than the independent samples design. The main conclusion to be drawn from this discussion that, in the design of a matched samples experiment a high level of matching should be ensured. This can be achieved by choosing the matching variableso that they are highly correlated with the outcome variables. If the matching is ineffective then there can be considerable loss in efficiency relative to the independent samples design. The results obtained here are in broad agreement with the similar work done for the testing problem in 2 x 2 tables by several authors, see, for example, Youkeles (1963), Worcester (1964) and Miettinen (1968). These authors also reach the conclusion that for testing the homogeneity of two proportions, matched samples design can be disadvantageous if the matching is not effective and the advantage is not substantial unless the matching is highly effective. For additional references and also for some practical aspects of matching see McKinlay (1977). ACKNOWLEDGEMENTS The author is thankful to three referees for making many useful suggestions for improvement of the presentation. This research was supported by NSF Grant No. ENG REFERENCES BECHHOFER, R. E. (1954). A single-sample multiple-decision procedure for ranking means of normal populations with known variances. Ann. Math. Statist., 25, BENNET, B. M. (1967). Tests of hypotheses concerning matched samples. J. R. Statist. Soc. B, 29, (1968). Note on x2 tests for matched samples. J. R. Statist. Soc. B, 30, BHAPKAR, V. P. (1973). On the comparison of proportions in matched samples. Sankhyd A, 35, BHAPKAR, V. P. and Somes, G. W. (1977). Distribution of Q when testing equality of matched proportions. J. Amer. Statist. Ass., 72,

6 30 TAMHANE - The Better Bernoulli Treatment [No. 1, COCHRAN, W. G. (1950). Comparison of percentages in matched samples. Biometrika, 37, GUPTA, S. S. and PANCHAPAKESAN, S. (1972). On a class of subset selection procedures. Ann. Math. Statist., 43, McKINLAY, S. M. (1977). Pair-matching-a reappraisal of a popular technique. Biometrics, 33, McNEMAR, Q. (1947). Note on the sampling error of the difference between correlated proportions or percentages. Psychometrika, 12, MIETTINEN, 0. S. (1968). The matched pairs design in the case of all-or-none response. Biometrics, 24, SOBEL, M. and HuYETT, M. J. (1957). Selecting the best one of several binomial populations. Bell System Tech. J., 36, WORCESTER, J. (1964). Matched samples in epidemiologic studies. Biometrics, 20, YOUKELES, L. H. (1963). Loss of power through ineffective pairing of observations on small two-treatment all-or-nonexperiments. Biometrics, 19, APPENDIX To prove the assertion regarding the LFC, we first keep Xf" fixed and regard the PCS as a function of A = 7r1Tol. To show that, subject to irl+vil = ir (fixed), the PCS is minimized at ilo-vrl = 8*, it suffices to show that g is a non-decreasing function of A for each x. But this follows immediately from (3.2); in fact- g is a strictly increasing function of A> l for each x > 1. Our next task is to find the infimum over X < w* of where A* = i(f?+ inf PCS = EJ,{g(X, A*)}, ffo10to14>8* (A.1) 8*)/lr. To show that this infimum occurs at X = 1T* we prove the following. Theorem. E,,(g(X, A*)} is a decreasing function of v. Proof. Denote the binomial probability function (f)px(i p)n-x by b(x; n,p) and the corresponding distribution function by B(x; n,p). A "discrete analog" of Theorem 2.1 of Gupta and Panchapakesan (1972) shows that the condition to be verified for the monotonicity of Ef,(g(X, A*)} relative to 7r is {(O/ a)g(x, A*)}b(x; n, m) )-{g(x, A*)-g(x- 1, A *)}(a/lav) B(x- 1; n, f) < 0 for 1 (x < n; for x = 0 the left-hand side of (A.2) is 0. Substitute in (A.3), (al/a) B(x- 1; n, r) = -(x/it) b(x; n, it) and find that the condition to be verified becomes (a/av)g(x, A*)+ (x/t){g(x, A*) -g (x- 1, A*)} < 0. With some algebraic manipulations it can be shown that, for x odd, g(x, A*)-g(x- 1, A*) = -(i/x)(a/la)g(x, A*). (A.2) (A.3) (A.3) Thus, we find that the left-hand side of (A.3) is 0 for x odd. For x even > 2 it follows from (3.2a) and (3.2b) that g(x, A*)-g(x-1, A*) = 0. Furthermore, it can be easily verified that (a1/a,)g(x, A*) <0. Thus (A.3) is verified in all the cases. Because of the strict inequality for x even > 2, it follows that Eff{g(X, A*)} is strictly decreasing ff.

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