D AO ~ FLORIDA STATE UNIV TALLAHASSEE DEPT OF STATISTICS ~ F/B 12/1

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1 I r - D AO ~ FLORIDA TATE UNIV TALLAHAEE DEPT OF TATITIC ~ F/B 12/1! OlE OPTIMAL DEIGN REULT IN PAIRED COMPARION.(IJ) MAY fl R A BRADLEY, A t EL IIELBAWY N C 0605 UNCLAIFIED FU TATITIC M 4114 ONR TR j14 Pt or kflbo4 ~~~ t2 END 8-77 I

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4 .-F ~~~~ --i: ~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \ OME th TIMAL REIGN JEULT i ~ PAIRED COMPARION ~ by / ~~ Ralph A./Bradley ZI bda11a ~~~~ 1-Re1bawy,i ~~ PU Technical Report No. 414 ONR Technical Repcrt No. 114 / 7 1 ~~~~~~~~~~ ft / -. ~~~~~~~~~ / -._5 j_ / ;1/ ~~~~~~ / ~~~~~~~~~~~~~~~~~~~~~~~~ T 7 7 I - / 1 1 ~~~~~~. ~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The Florida tate Uni ~~ reity Department of tatistics Tal1aji ~ ssee, Florida D D C ~ r [~~~~~~ nn ~~~fl MW f., 1 Prepared for presentation ati the 41st ession of the International tatistical Institute, New Delhi, India? December, Research supported at the 1~ pr(da tied University by the Ansy, Navy and Air Force th rough ONR Contract N C 5 06_8 Reproduction in whole or in part is permitted for any purpose of t hi U ~~ ttd ates Government. ~~~~~

5 ~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ - (i) Preface: This technical report is a short manuscript prepared for the proceedings of the 41st ession of the International tatistical Institute meeting in New Delhi, December, 1977 and to be presented at that meeting. Results summarized have h ~~ e been developed in detail in ONR Technical Reports, No. s 99, 100 and 102 submitted earlier. Nfl UIwm ~~ ) JUsTmc ~ : _ / R.A.B. ~~~~~~~~~ co ~ D ~ t. r ~~~~~ I U. -. ~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~ ~

6 - n ~~~~~~~~ ~~~~~~~~~~ ~~~~ ~ = ~~~~~~~~~ ~~~~~~~~ OME OPTIMAL DEIGN REULT IN PAIRED COMPARION Ralph A. Bradley and Abdalla T. El Helbavy Florida tate University, Tallahassee, U..A. and University of Cairo, Cairo, A.R. E. - ~ INTRODUCTION The authors, B ~,adley and El } ~ e1bawy (1976), E1 Helbawy T L c i..,,~ - ~~~~~~, - 5. ~. ~ - s and Bradley (1977a,b), have developed the methodology for consideration of specified treatment contrasts in paired comparisons. The procedures developed give much new flexibility to the use of paired comparisons and, in particular, to the use of factorial treatment combinations in such experiments. ~. The probability model developed by Bradley and Terry (1952), originally proposed by Zermelo (1929), is used. Many additional references are given by Davidson and Farquhar (1976) in their bibliography and Bradley (1976) reviews various approaches to the model and its extensions. In this short presentation, we summarize important results on treatment contrasts and indicate how they may be used to consider optimal design questions. ome simple optimal design results are given. UMMARY OF METHODOLOGY uppose that the paired comparisons experiment has t treatments T,...,T, with n comparisons of T 1 and T, 1 t ij Li

7 F 2 n 1 ~ 0, n ~ fl 1, j l,...,t. A parameter jj~ is associated with ~~ w > 0, such that the 1 probability of selection of T when 1 compared with T ~ is pr(t ~ > T ~ ) w ~ /(w 1 + ir ~~ ) ~ j j. (1) The convenient scale determining constraint is 1 1 ~ 0, log ir k, I 1,...,t, (2) different from that used by Bradley and Terry. On the assumption of independence of selection judgments, the likelihood function is is a n L(n) 11w ~ / 11 (ir ~ + w,) ~~~~~~ (3) i i J i<j where a ~ is the total number of selections of T 1, ~ a 1 N. ~ I i<j w is the column vector with typical element w and other vectors ~ below are defined similarly. Treatment contrasts are specified as linear, orthonormal contra8ts on the y ~. The typical estimation problem is to maximize L subject to (2) and.! 1~ X!) 2 ~ ~ where 0 is a column vector of zeros and B consists of in, zero - ~~~ sum, orthonorizal rows. The resulting likelihood equations are Zaj I fl jk P j /(P j + P k )JD 0, i l,...,t, ij I Y (!) 0, and ~! x(2) o, ~ where D 1 ~ is the (i,j) element of ~~~~~~~~~~~ ~~~~~, the t square L P identity matrix, ~ Lathe estimator of ~~~, and x ~ ) of x(j!) El Helbawy and Bradley (197Th) show that 1I ~ [ ~ (p) i (i ~ )1 has the singular, t-variate normal limiting distribution function in (t m-l) dimensions with zero mean vector and dispersion matrix given in the references. El Helbawy and Bradley (l977a) examine the solution of (5) and convergence properties of a suggested iterative scheme.

8 -- -- ~~~~~~~~~~~ ~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - 3 The typical testing situation assumes (4) and specifies H 0 : ~~~ (w) against the alternative, H 8 : ~~ y(ir ) 0 and uses the likelihood ratio statistic, A N (H o, H ). a It is shown in the (1977b) paper that 2log A N (H O) H ) a has the chi square limiting distribution with n degrees of freedom, central under H 0 and non-central under H with non-centrality parameter, (6) where E is a dispersion matrix dependent on H given in the ref erences and ha ~~~~~, ~~1(1 N ) N ~~~ N and is a N4CD equence of local alternatives to 11 satisfying (2) and (4) o. Bradley and El Helbawy (1976, 1977b) show how the contrasts described by ~~ and ~~ may be related to factorial effects when the treatments are factorial treatment combinations and, indeed, give a reparamet ~~ tion of the problem for factorials. OME OPTI? IAL DEIGN REULT The results summarized above for the first time provide means of considering asymptotically - optimal design of paired comparisons experiments. We limit consideration to two examples I with t 8 and a 2 3 factorial. T is associated with T 1 a ~ (a 1, a 2, a ), cz 3 ~ 0,1, a 1, 2, 3, a designating the level -~~~ of Factor a in the treatment combination. Consider a test of no interaction between Factors 1 and 2; B in (4) does not exist and B describes the usual analysis of variance contrast for the specified treatment contrast, now in terms of the y. The objective is to maximize asymptotic 1 power, that is, to maximize in (6) for the desired test. in (6) depends on and A ~~ him n 1 ~ /N ~ I J. The ~ maximization is with respect to the )t jj and is taken to!o be ~ 8 a column vector of unities, consistent with H and the 0 concept that any other effects present are of the same order of magnitude relative to N as the contrast under test. The experiment is assumed to be as balanced as possible but to A

9 ~~~~~~~~~~~~~ I ~ 7 ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ P permit optimality consideration ; we take A 1 ~ a or b respectively as T and T represent factorial treatment combinations with i j factor levels a and 0 such that ( 1) does or does not 1 2 have the same sign for the two treatments, 12a + 16b 1. Maximization of A 2 with respect to a and b, 12a + 16b 1, yields a 0, b 1/16; no observations are taken on comparisons that yield no Information on the two factor interaction under test. The same result occurs, for example, for the same tea t with chosen to assume that the three factor and other two factor interactions are null. uppose that all factorial effects are assumed null except the three interactions involving Factor 1. Then B has four rows. We take w a central value satisfying (2) and (4) and make the simplif ying assumption of as much balance In the experiment as possible but permitting optimahity considerations. We are concerned with the dispersion matrix and show that in one should take all ~~ 0 except for those treatment comparisons yielding information on all of the F 1 F 2, F 1 F 3 and F 1 F 2 F 3 interactions for A, D and E optimahity minimizing respectively t r ~~~~~, I ~~ I and the largest variance of ome other examples are given by El Helbawy and Bradley (1977b). While the results noted are consistent with intuition, formal demonstration is given for the first time and the way is open for more general consideration of optimal design in paired comparisons. UMMARY The authors have shown (Biometrlk a, 1976) how to consider specif ied treatment contras ts in paired comparisons and given applications to factorials. In a subsequent paper, pending publication, they consider asymptotic theory and applications to optiasi design when the treatments are factorial treatment combinations. This paper is a summary of some of the main results. - 5 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -

10 F~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ OMMAIRE 5 Lea auteurs ont démontrê (Biometrika, 1976) comment ~ considérer des contrastes specifi ~ s entre traitements en comparaisons par paires et donné des applications pour traitenients factoriehs. Dane des subséquentes recherches, ne pea encore pubhiées, us ont considéré la théorie asymptotique et lea applications ~ dessein optimal quand lea traitements sont des combinaisons factoriels. Ce papier eat un soixiaire des résultats principaux. ACKNOWLEDQIENT A This research was supported at the Florida tate University by the Army, Navy and Air Force through ONR Contract N C BIBLIOGRAPHY Bradley, R.A. (1976). cience, statistics and paired comparisons, Biometrics 32, Bradley, R.A. & El Helbawy, A.T. (1976). Treatment contrasts In paired comparisons : Basic procedures with application to factorials. Biometrika 63, Bradley, R.A. & Terry, M.E. (1952). Rank analysis of incomplete block designs. I. The method of paired comparisons. Biometrika 39, Davidson, R.R. & Farquhar, P.14. (1976). A bibliography on the method of paired comparisons. Biometrics ~~~~~~ El Relbawy, A.T. 6 Bradley, R.A. (1977a). Treatment contrasts in paired comparisons: Convergence of a basic iterative scheme for estimation. Connun. tatist. Theor. Meth. ~~~~~~, El-flelbawy, A.T. & Bradley, R.A. (197Th). Treatment contrasts paired comparisons: Large sa ~~ le results, applications and some optimal designs. Tallahassee, Florida tate University tatistics Technical Report M368. Zermelo, E. (1929). Die Berechning der Turnier Ergebntsse ala em Maximum problem der Wahrecheinhichkeit ~ êthnung, Math. Zeit. 29, (Key words: Optimahity, paired comparisons, factorials, contrasts). I.

11 - - -~ - - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ - --s UNCLAIFIED ECURITY CLAIFICATION OF THI PAGE (When Data Entered) REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM 1. REPORT NUMBER 2. GOVT. ACCEION NO. 3. RECIPIENT CATALOG NUMBER ONR Report No TITLE (and subtitle) 5. TYPE OF REPORT & PERIOD COVERED ome Optimal Design Results in Paired Comparisons Technical Report 6. PERFORMING ORG. REPORT NUMBER FU tatistics Report M4l4 1 7 AUTHOR (a) 8. CONTRACT OR GRANT NUMBER(s) Ralph A. Bradley and Abdahla T. El Helbawy N C PERFORMING ORGANIZATION NAME AND ADDRE 10. PROGRAM ELEMENT, PROJECT, TAK Florida tate University Department of tatistics Tallahassee, Florida AREA & WORK UNIT NUMBER 11. CONTROLLING OFFICE NAME AND ADDRE 12. REPORT DATE Office of Naval Research May, 1977 tatistics & Probability Program 13. NUMBER OF PAGE Arlington, Virginia MONITORING AGENCY NAME & ADDRE (1f 15. ECURITY CLA (of this reportj different from Con trollir ~ g Oi_ r ~~ ~~ ) Unclassified T5a. DECLAIFICATION/r&WNGRADING CHEDULE 16. DITRIBUTION TATEMENT (of this Report) Approved for public release ; distribution unlimited. 17. DITRIBUTION TATEMENT (of the abstract entered in Block 20, if different from re,rt) 18. UPPLEMENTARY NOTE 19. KEY WORD Optmmality, paired comparisons, fac tor ials, contrasts ~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~ 5 ~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~

12 - ~ -.~~ ABTRACT The authors have shown (Biometrika, 1976) how to consider specified treatment contrasts in paired comparisons and given applications to factorials. In a subsequent paper, pending publication, they consider asymptotic theory and applications to optimal design when the treatments are factorial treatment combinations. This paper is a summary of some of the main results.

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