Memorandum COSOR 97-??, 1997, Eindhoven University of Technology

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1 Meoandu COSOR 97-??, 1997, Endhoven Unvey of Technology The pobably geneang funcon of he Feund-Ana-Badley ac M.A. van de Wel 1 Depaen of Maheac and Copung Scence, Endhoven Unvey of Technology, Endhoven, The Neheland Abac We deve a expeon fo he pobably geneang funcon of he dbuon-fee Feund- Ana-Badley cale ac. Fo h geneang funcon we how how o yeacally copue he exac null dbuon and he oen of he ac whn he copue algeba ye Maheaca. Fnally, we gve a able wh ccal value whch exend he exng able. Keywod Feund-Ana-Badley e, geneang funcon, exac dbuon, oen, copue algeba, nonpaaec ac. 1 Inoducon In h acle we deve a expeon fo he pobably geneang funcon of he dbuon-fee Feund-Ana-Badley ac, whch abbevaed a he FAB ac. Sacally euvalen veon of h ac wee noduced by Feund and Ana (1957) and Ana and Badley (1960). We pleened h geneang funcon n he copue algeba ye Maheaca fo copung he dbuon of he e ac vey uckly. We alo how how o ue he geneang funcon fo copung (hghe) oen of he e ac. In he appendx we peen he Maheaca code fo expandng he geneang funcon, n he econd appendx we gve a able wh ccal value fo he balanced cae whch exend he exng able fo N 0 o N 80. Fo geneal deal abou he FAB ac we efe o Gbbon and Chakabo (199). Seveal ehod have been developed fo copung he null dbuon of he FAB ac. Ana and Badley (1960) and Kanneann (1983) deve ecuence elaon baed on a wodenonal geneang funcon of Eule (1748, anl. 1988). Ohe ehod ae geneal ehod fo copung null dbuon of lnea wo-aple ank e and hence do no ue he pecc chaacec of he FAB ac. Exaple of hee ehod nclude he Pagano and Tchle (1983) appoach baed on chaacec funcon and fa Foue anfo and he newok algoh developed by Meha e al. (1987). Fo a ho decpon of hee ehod we efe o Good (1994). We gve an alenave ehod baed on he wo-denonal geneang funcon n Van de Wel (1996). Th la ehod fa and, alhough no val, vey nuve. We ue exac expeon and do no deal wh oundng eo a oe ecuve ehod do. The Feund-Ana-Badley e Le (X 1 ::: X )and(y 1 ::: Y n ) be ndependen aple fo connuou dbuon funcon. Thu we ay and wll aue ha e do no occu. We conde he cobned aple (X 1 ::: X Y 1 ::: Y n ), N +n. TheFeund-Ana-Badley e a wo-aple cale e. The coepondng e ac dened by: A N NX `1 ` ; N +1 Z` (1) 1 Mak van de Wel, Depaen of Maheac and Copung Scence, Endhoven Unvey oftechnology P. O.Box MB Endhoven, The Neheland akvdw@wn.ue.nl 6

2 whee Z` 1fhe`h ode ac n he cobned aple an X-obevaon and Z` 0 ohewe. Thu, fo h ac he ank coe ae ` 1 ::: N. a(`) ` ; N +1 () Duan (1976) copae he FAB ac wh ohe cale ac lke he Mood cale ac, he Segel-Tukey ac and he Kloz ac (fo deal abou hee e ac: ee Gbbon and Chakabo (199)). I un ou ha he Segel-Tukey ac and FAB ac ae aypocally euvalen n e of (Pan) aypoc elave ecency. The Mood ac and Kloz ac ae oe ecen when he alenave noal (o lgh-aled), whle he Segel-Tukey ac and FAB ac ae oe ecen fo heavy-aled dbuon. The FAB ac aan uch le value han he Mood and Kloz ac. Th boh a copuaonal advanage and a poenal dadvanage egadng he nube of gncance level (fo all aple ze). 3 The pobably geneang funcon Fo () we ee ha fo N even he ank coe ae of he fo ; 1 1 :::N.Wewll nd o be convenen ha he coe ae nege and heefoe we noduce adjued FAB coe: a 0 (`) 8 >< >: ` ; N +1 ` ; N f N even (3) f N odd, ` 1 ::: N: We dene he adjued FAB e ac A 0 N a A N wh he FAB coe eplaced by he adjued FAB coe. The ac A 0 N, of coue, acally euvalen oa N.Fo N even, we ay pl he e of adjued coe fa 0 (1) ::: a 0 (N)g no wo e of Wlcoxon coe f1 ::: 1 Ng, whch he an dea behnd ou appoach. Wh he ad of h obevaon we found he followng heoe fo he cae N even: Theoe 3.1 Unde H 0, N + n even and n he pobably geneang funcon of he Feund-Ana-Badley ac whee 0 1 k0 P(A N k) x k 1 ; N X 0 c() c( ; ) N Q `1 (1;`) Q `1 (1;`)Q ; `1 (1;`) fo >0 and c() (+1) : N (4) ; Poof: Aue, whou lo of genealaon, ha n. We dene R a he ank coe n he pooled aple coepondng o X 1 :::. Slaly, S j he ank coe coepondng o Y j j 1 ::: n. We know ha unde H 0 evey conguaon of R 1 ::: R S 1 ::: S n eupobable. Le T (a) be he half of a conguaon a. We can paon hee conguaon no clae of conguaon wh R' n T (a),. The cla of conguaon wh R' n T (a) called C. We denoe he eleen of T (a) by R u1 :::R u S v1 ::: S vn;, whee u 1 ::: u and v 1 ::: v N; ae ubeuence of 1 ::: and 1 ::: n, epecvely. We dene he followng Wlcoxon ac: W N; X j1 R uj and W ; N;+ ; X j1 R u+j : 7

3 We need o pove he condonal ndependence of W N; and W ; N;+, gven C. Le K 1 (a) be he cla of ' fo whch he half eual he half of a conguaon a. Slaly, le K (a) be he cla of ' fo whch he econd half eual he econd ; ; half of a. We noe ha N he even a euvalen ok 1 (a) \ K (a). Thee ae N ; conguaon n C. They ae eupobable unde H 0,o Unde H 0 wehave, P( K 1 (a)j C ) P( K (a)j C ) P( aj C ) 8 < : ( 1 f a C ( N )( ;) N 0 f a 6 C. #(:K 1(a)\C ) #(:C ) ( ;) N ( N )( ;) 1 f a C N ( N ) 0 f a 6 C, ( 1 ( N f a C ;) 0 f a 6 C. (5) (6) Cobnng (5) and (6) we conclude ha P( K 1 (a) \ K (a)j C )P( aj C ) P( K 1 (a)j C )P( K (a)j C ): (7) Le 1 be a conguaon of R' and N ; S' and le be a conguaon of ; R' and N ; + S'. Fo j 1 : P( K j (a)j C )f P( j a j) (8) whee a j he jh (j 1 ) half of he conguaon a. The ybol P denoe he pobably eaue on he pace wh conguaon of lengh N, wheea P f denoe he pobably eaue on he pace wh conguaon of lengh N: The ac W N; and W ; N;+ ae funcon of 1 and,epecvely. Becaue of eualy (8)weayegad W N; and W ; N;+ a funcon of all ' fo whch C. Snce euaon (7) ell u ha, gven C, he even f K 1 (a)g and f K (a)g ae ndependen, we conclude ha W N; and W ; N;+ ae alo ndependen, gven C. The e of adjued FAB coe con of wo dencal e of Wlcoxon coe. When coe of he e ae agned o he X', we knowha ; coe of he econd e ae agned o he X', 0. The u of he coe agned o X' eual A 0 N and alo eual he u of W N; and W ; N;+. Theefoe, #(A 0 N k) P 0 #(W N; + W ; N;+ k). Le H Z be a geneang funcon fo he nube of way a ac Z can each a cean value. Then, H A 0 N X k0 X 0 k0 0 #(A 0 N k) x k X k0 0 #(W N ; + W ; N ;+ k) xk H W N ; H W ; N ;+ #(W N ; + W ; N ;+ k) xk X 0 H W N ; +W ; N ;+ whee n he la ep we ued ha W N ; and W ; N ;+ ae ndependen. The geneang funcon H Wa b can ealy be deved fo he geneang funcon of he euvalen Mann-Whney (9) 8

4 ac M a b.weknowhaw a b M a b + 1 a(a+1) and fo Andew (1976, Ch. 3) and Davd and Baon (196, pp ) we knowha Theefoe, H Ma b H Wa b c(a) a + b : a a + b : (10) a Subung (10) no (9) wh (a b) ( N ; ) and(a b) ( ; N ; + ) gve u H A 0. N We coplee ou poof by eakng ha fo N even, A N A 0 N ;. Theoe 3. Unde H 0, N + n odd and n he pobably geneang funcon of he Feund-Ana-Badley ac whee k0 P(A N k) x k ; 1 X N 1 j0 and c() a n Theoe 3.1. ;j X 0 N;1 c() c( ; j ; ) N;1 ; j ; Poof: The poof la o ha of Theoe 3.1, bu now we deal wh wo dencal e of Wlcoxon coe and one coe ha eual o zeo. Th poble olved by ung ove a vaable j ha eual 1 f zeo agned o an X-obevaon and ha eual 0 ohewe. So fo j 0we oban all poble value of A 0 N wh coe n he wo dencal e and fo j 1 we oban all poble value of A 0 N wh ; 1 coe n he wo dencal e. 4 Moen of he FAB ac Foula (4) and (11) ell u ha fo copung oen of he FAB ac uce o copue devave of he expeon V () d() whee d() a ne u of e of he fo c z whee c and z ae aonal. If we denoe he kh devave off() by f (k) () hen we ee ha V (k) () Xk; kx j0 0 k k ; ; j j d (k;;j) () () (j) Snce d() a ne u of e of he fo c z whee c and z ae aonal, aghfowad o copue d (`) (1) fo ` abay lage. The wo ohe e n he gh hand de of (1) ae polynoal, o we ayake he! 1 and we nd he followng expeon fo he kh devave ofv () a 1: V (k) (1) Xk; kx j0 0 k k ; ; j j d (k;;j) (1)!1 ()!1 (j) D Bucchanco (1996) povde a ehod fo copung expeon of he fo ()!1 wh he ad of he copue algeba package Maheaca. We ued h ehod fo copung oen of he FAB ac. (11) (1) (13) 9

5 4.1 Exaple A an lluaon we copue he ean fo he cae N even. In h cae k 1, o afe expandng (13) we ge 0 V 0 (1) d(1)!1 ; N ;1 1 In h cae d() d 0 (1)!1!1!1 + d(1)!1 (+1)+ 1 (;)(;+1);,od(1) ; N!1 0 + ;1 and d 0 (1) ; N (14) ;1( 1 ( + 1) + 1 ( ; )( ; +1); ) 1 ; N ;1( + ; ). Fo he exaple n D Bucchanco (1996,p. 9) we exac ha!1 and 0 1!1 ( ; ): The ae euaon hold fo eplaced by. In euaon (4) we eeha N and ;. Subung n (14) leave u afe oe labou ; V 0 (1) N;N ; N ; N (15) 4 The la hng we have odoungove and we ge he ean AN AN X 0 N ; N ; N 4 ; N fo N even: ; N 4 (16) P ; N ; N ; whee we ue he fac ha 0 ; N whch a pecal cae of he Chu-VandeMonde foula. Fo a coplee poof: ee (Chu 1303, anl. 1959) o Rodan (1979).. 4. Exaple of a hghe oen We gve EA 5 N, he fh oen ofhefab ac fo N even. I ook 80 econd o copue h oen on a SunSpac 10. EA 5 N P( n) 307 ( + n ; 3) ( + n ; 1) wh P ( n) (;1+1n)+ 9 ; ;111 ; 6 n +63n + 8 ; 480 ; 705 n ; 10 n +105n ; n ; 1855 n ; 90 n n ; ; n +340n ; 585 n 3 +0n 4 +63n ; ;4560 ; 416 n n n 3 ; 005 n 4 +78n 5 +1n ; ; 3196 n ; n +1975n 3 ; 48 n 4 ; 811 n 5 +48n 6 +3n ; ; n ; 6896 n ; 1364 n n 4 ; 8 n 5 ; 15 n 6 +10n 7 + ; ; n +9075n ; 5051 n n n 5 ; 60 n 6 +5n 7 + ; 5596 ; n n +0768n 3 ; n n 5 ; 36 n 6 ; n n ; n n 3 ; 364 n 4 ; 3 n 5 +16n 6 Skech of he poof: dene a geneang funcon wh coecen eual o he lef-hand de of he eualy and wh ung vaable. Then ue he convoluon heoe o pl he u no a poduc of wo u. Fnally, ue he bnoal heoe o oban he deed eul. 10

6 5 Copue algeba Th econ conan he ex of he Maheaca package we ued fo copung he dbuon of he FAB ac by ung Theoe 3.1 and 3.. We alo gve a all exaple. FABevengf[N_,_]: Expand[Splfy[1/Bnoal[N,]*^(-(/))* (Su[c[]*c[-]*Poduc[1-^l,{l,N/}]/(Poduc[1-^l,{l,}]* Poduc[1-^l,{l,N/-}])*Poduc[1-^l,{l,N/}]/ (Poduc[1-^l,{l,-}]*Poduc[1-^l,{l,N/-(-)}]),{,-1}] + (*c[])*poduc[1-^l,{l,n/}]/ (Poduc[1-^l,{l,}]*Poduc[1-^l,{l,N/-}]))]] FABoddgf[N_,_]: Expand[Splfy[1/Bnoal[N,]* Su[Su[c[]*c[-j-]*Poduc[1-^l,{l,(N-1)/}]/ (Poduc[1-^l,{l,}]*Poduc[1-^l,{l,(N-1)/-}])* Poduc[1-^l,{l,(N-1)/}]/(Poduc[1-^l,{l,-j-}]* Poduc[1-^l,{l,(N-1)/-(-j-)}]),{,-j-1}] + (*c[-j])*poduc[1-^l,{l,(n-1)/}]/(poduc[1-^l,{l,-j}]* Poduc[1-^l,{l,(N-1)/-(-j)}]),{j,0,1}]]] c[_]: ^((1/)**(+1)) Noe ha he cae 0 and ;j ae pl o, becaue we have o ell Maheaca explcly ha 0 1. The conbuon of hee cae o he u eual. The Maheaca funcon Expand and Splfy ae ued o copue he full polynoal whch epeen he dbuon of he FAB ac. The followng exaple gve he dbuon of he FAB ac fo n 4. FABevengf[8,4] Table of ccal value Wh he ad of Theoe 3.1 and 3. we wee able o exend he exng able of ccal value. Ana and Badley (1960) gve ccal value fo N 0. We gve able fo n N 80. Fo paccal eaon we dd no pn he unbalanced cae. Anyone neeed n ccal value fo an unbalanced cae ay conac he auho. Acknowledgeen I would lke o hank Aleando D Bucchanco fo he dcuon and coecon and fo povdng he Maheaca package fo copung oen of he Mann-Whney ac. 11

7 n 0:005 0:01 0:05 0:05 0:1 0:1 0:05 0:05 0:01 0: :5 7:5 7:5 8:5 16:5 17:5 17:5 18: :5 14:5 15:5 17:5 18:5 30:5 31:5 33:5 34:5 35: :5 6:5 8:5 30:5 3:5 48:5 50:5 5:5 54:5 55: :5 4:5 44:5 46:5 49:5 71:5 74:5 76:5 78:5 80: :5 61:5 64:5 67:5 70:5 98:5 101:5 104:5 107:5 109: :5 83:5 88:5 91:5 95:5 19:5 133:5 136:5 141:5 143: :5 110:5 115:5 119:5 14:5 164:5 169:5 173:5 178:5 18: :5 140:5 146:5 151:5 157:5 03:5 09:5 14:5 0:5 4: :5 173:5 180:5 186:5 193:5 47:5 54:5 60:5 67:5 71: :5 11:5 19:5 6:5 34:5 94:5 30:5 309:5 317:5 33: :5 5:5 61:5 69:5 78:5 346:5 355:5 363:5 37:5 379: :5 96:5 307:5 315:5 36:5 40:5 413:5 41:5 43:5 439: :5 345:5 356:5 366:5 378:5 46:5 474:5 484:5 495:5 503: :5 397:5 410:5 41:5 433:5 57:5 539:5 550:5 563:5 57: :5 453:5 467:5 479:5 493:5 595:5 609:5 61:5 635:5 645: :5 513:5 58:5 541:5 556:5 668:5 683:5 696:5 711:5 7: :5 576:5 593:5 607:5 64:5 744:5 761:5 775:5 79:5 803: :5 643:5 661:5 677:5 695:5 85:5 843:5 859:5 877:5 889: Table 1: Lef and gh ccal value fo he Feund-Ana-Badley e, n. 1

8 Refeence Andew, G.E. (1981), The heoy of paon, Encyclopeda of aheac and applcaon (Addon-Weley, Readng, nd ed.). Ana, A.R. and R.A. Badley (1960), Rank u e fo dpeon, Ann. Mah. Sa. 31, Chu Ch-Ke (1303), Manucp, Chna, n: J. Needha, Scence and Cvlzaon n Chna 3 (Cabdge Unvey Pe, Cabdge, 196). Davd, F.N. and D.E. Baon (196), Cobnaoal chance (Chale Gn & Co., London) D Bucchanco, A. (1996), Cobnaoc, copue algeba and he Wlcoxon-Mann-Whney e, Tech. Rep. COSOR 96-4 (Endhoven Unvey of Technology, Endhoven). Duan, B.S. (1976), A uvey of nonpaaec e fo cale, Co. Sa.{Theoy Mehod A5, Eule, L. (1748), Inoduco n analyn nnou, n: Inoducon o analy of he nne (Spnge, New Yok, 1988). Feund, J.E. and A.R. Ana (1957), Two-way ank u e fo vaance, Vgna Polyechnc Inue Techncal Repo o Oce of Odnance Reeach and Naonal Scence Foundaon 34. Gbbon, J.D. and S. Chakabo (199), Nonpaaec acal nfeence (Macel Dekke, New Yok, 3d ed.). Good, P.I. (1994), Peuaon e, a paccal gude o eaplng ehod fo eng hypohee (Spnge-Velag, New Yok). Kanneann, K. (1983), A unfyng, analyc odel fo cean lnea ank e II. Exac dbuon algoh, Boecal J. 5, Meha, C.R., N.R. Pael and L.J. We (1987), Conucng exac gncance e wh eced andozaon ule, Boeka 75(), Pagano, M. and D. Tchle (1983), On obanng peuaon dbuon n polynoal e, J. Ae. Sa. Aoc. 78, Rodan, J. (1979), Cobnaoal dene (Wley, 1979, nd ed.). van de Wel, M.A. (1996), Exac dbuon of nonpaaec ac ung copue algeba, Mae' he, (Endhoven Unvey of Technology, Endhoven). 13