IJPSS Volume 2, Issue 7 ISSN:

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1 IJPSS Volue Issue 7 ISSN: A BAYESIAN LOOK AT THE BRADLEY-TERRY PAIRED COMPARISON MODEL Noshee Altaf* Ghausa Masood Gla** Muhaad Asla*** _ ABSTRACT I the preset study we have preseted a statstcal aalyss for pared coparso data through Bayesa approach. I the ethod of pared coparsos obects are preseted pars to the udges. The Bradley-Terry odel for pared coparsos s cosdered for Bayesa aalyss. The pror dstrbuto for the paraeters of the odel s supposed to be beloged to the eber of the Drchlet faly ad the ethod for elctato of hyperparaeters s based o the pror predctve dstrbuto. For Bayesa aalyss the posteror dstrbuto of the paraeters s derved the preferece probabltes usg the posteror eas ad the predctve probabltes for parwse coparsos a sgle future coparso are obtaed. The posteror probabltes for the hypotheses of coparg two paraeters are also coputed. The goodess of ft test for the approprateess of the odel s preseted too. KEYWORDS: Pared coparso ethod; Bradley-Terry odel; Drchlet dstrbuto; Hyperparaeters; Elctato; Posteror dstrbuto; Bayesa hypothess testg; Predctve probabltes; Preferece probabltes. * Rphah Iteratoal Uversty Islaabad-000 Paksta. ** College of Statstcal ad Actuaral Sceces Uversty of the Puab Lahore Paksta. *** Departet of Statstcs Quad--Aza Uversty Islabad-000 Paksta. A Mothly Double-Bld Peer Revewed Refereed Ope Access Iteratoal e-joural - Icluded the Iteratoal Seral Drectores Idexed & Lsted at: Ulrch's Perodcals Drectory U.S.A. Ope J-Gage Ida as well as Cabell s Drectores of Publshg Opportutes U.S.A. Iteratoal Joural of Physcal ad Socal Sceces

2 IJPSS Volue Issue 7 ISSN: INTRODUCTION I the ethod of pared coparso obects are preseted pars to oe or ore udges for the purpose of coparso. The obect ay be a perso a treatet stul ad the lke. The basc experetal ut s the coparso of two obects. K ad K (00 preset a Bayesa approach to pared coparso of several products of Posso Rates. Adas (005 descrbes Bayesa versos of the ethod of pared coparso ad ther advatages of behavoral studes. Ja (007 has developed statstcal odel ad procedure for slarty tests usg pared coparsos ethod. Mrada et al (009 propose the use of a odfed pared coparso ethod whch a reduced uber of coparsos s selected accordg to a coplete cyclc desg. Causeu ad Husso (005 propose a -desoal exteso of the Bradley Terry odel. The detal about the pared coparsos ethod ca be see Davd (988. The Bradley-Terry (95 odel s defed Secto wth the otatos. Dscusso about the foratve pror ad elctato of hyperparaeters are preseted Secto. Bayesa aalyss of the Bradley-Terry odel for the uber of treatets = s gve secto ths secto cludes forato of the posteror dstrbuto the argal posteror destes the posteror eas ad the odes of the paraeters the posteror probabltes of hypotheses for coparg two paraeters the predctve probabltes that oe treatet would be preferred to aother treatet a future sgle coparso ad the preferece probabltes. Approprateess of the odel s also tested Secto 5..THE BRADLEY-TERRY MODEL FOR PAIRED COMPARISON The Bradley-Terry odel troduced by Zerelo (99 ad developed by Bradley ad Terry (95. Ths odel ples that the dfferece betwee two latet varables ( X X has a logstc desty wth paraeter ( l l. If deotes the probablty P( X X that the treatet copared the T s preferred to the treatet T ( whe the treatets T ad T are A Mothly Double-Bld Peer Revewed Refereed Ope Access Iteratoal e-joural - Icluded the Iteratoal Seral Drectores Idexed & Lsted at: Ulrch's Perodcals Drectory U.S.A. Ope J-Gage Ida as well as Cabell s Drectores of Publshg Opportutes U.S.A. Iteratoal Joural of Physcal ad Socal Sceces

3 IJPSS Volue Issue 7 ISSN: = (l l = y sec h ( y / dy =. y e ( dy (l l e =. (. The Bradley-Terry odel s defed by (. The followg otatos are used the aalyss: k = or 0 accordg as treatet T s preferred to treatet (k= r of the coparso. r = the uber of tes treatet T s copared wth treatett. T or ot the k th repetto = k k = the uber of tes T s preferred to treatet T. = = the total uber of tes T s preferred to ay other treatet. Here ad r k k Hece the lkelhood fucto of the observed outcoe x whch represets the data ( r of the tral s: l( x ;... Pk k r ( r. = r r. (. where (= are the treatet paraeters. We clude a costrat o the paraeters of the odel that they are postve ad su to uty.e. paraeters are well defed ad detfable. ths codto esures that A Mothly Double-Bld Peer Revewed Refereed Ope Access Iteratoal e-joural - Icluded the Iteratoal Seral Drectores Idexed & Lsted at: Ulrch's Perodcals Drectory U.S.A. Ope J-Gage Ida as well as Cabell s Drectores of Publshg Opportutes U.S.A. Iteratoal Joural of Physcal ad Socal Sceces 5

4 IJPSS Volue Issue 7 ISSN: CHOICE OF AN INFORMATIVE PRIOR Geerally the pror dstrbuto s chose wth accord to the rage of the paraeter. Davdso ad Soloo (97 assue the atural cougate faly of pror for the paraeters of the Bradley-Terry odel. Leoard (977 assues the ultvarate oral dstrbuto as a pror dstrbuto for the logarth of the paraeters. But ether dscusses the ethod of elctato of the hyperparaeters. I preset study pror dstrbuto of the paraeters: beloged to the eber of the Drchlet faly..e.... s supposed to be p( ( a... a θ a ( a... ( a = a 0 ;... (. where θ (... are the treatet paraeters ad a (... are the hyperparaeters.. Elctato of Hyperparaeters Asla (00 descrbes three ethods of elctg hyperparaeters va pror predctve dstrbuto (PPD. The other authors who adopted the PPD for elctato of pror dstrbutos are Kadae et. al. (980 Wkler (980 ad Chaloer ad Duca (98. Oe of the ethods to elct the hyperparaeters based o cofdece levels of the PPD. The followg fucto s used to elct the hyperparaeters: ( a... a ( CCL ( ECL (. a... a k l k where k s the uber of tervals cosdered the elctato CCL s the cofdece level characterzed by the hyperparaeters ad ECL s the elcted cofdece level. The set of hyperparaeters wth u value of ( a... a (. s cosdered as the elcted values of hyperparaeters. For the uber of tes treatet T s preferred to treatet T whe a par of treatets (T T s beg copared wth the uber of tes r the pror predctve dstrbuto { p ( } of the Bradley-Terry odel for a par (T T {usg (. ad (.} s: A Mothly Double-Bld Peer Revewed Refereed Ope Access Iteratoal e-joural - Icluded the Iteratoal Seral Drectores Idexed & Lsted at: Ulrch's Perodcals Drectory U.S.A. Ope J-Gage Ida as well as Cabell s Drectores of Publshg Opportutes U.S.A. Iteratoal Joural of Physcal ad Socal Sceces k 6

5 IJPSS Volue Issue 7 ISSN: p ( = a a ck ( r 0 0 ( K d d 0... r (. where K= ( r ( a a c k ( (. ( a ( a ( c k r ad k c a. l p ( = ( r ( B a a ( (. B( a a (. Here B stads for Beta fucto. [For ore detal see Asla (00]. For elctato of hyperparaeters we assue a balaced desg that the uber of coparso for each par s equal ( r 0 ; (<= ad the expert cofdece levels are gve rght sde of the followg equatos. These equatos are derved usg the PPD (.: 0 r! B( a a =0.5 (.5!( r! B( a a 7 r! B( a a!(! ( 0 r B a a 0 r! B( a a!( r! B( a a 7 r! B( a a!( r! B( a a 0 r B a a! (!( r! B( a a 0 0 r! B( a a!( r! B( a a (.6 (.7 (.8 (.9 (.0 A Mothly Double-Bld Peer Revewed Refereed Ope Access Iteratoal e-joural - Icluded the Iteratoal Seral Drectores Idexed & Lsted at: Ulrch's Perodcals Drectory U.S.A. Ope J-Gage Ida as well as Cabell s Drectores of Publshg Opportutes U.S.A. Iteratoal Joural of Physcal ad Socal Sceces 7

6 IJPSS Volue Issue 7 ISSN: A progra s desged SAS package to solve above equatos to elct the values of hyperparaeters a a a ad a whch are foud to be ad respectvely.. BAYESIAN ANALYSIS FOR THE BRADLEY-TERRY MODEL (= Let the uber of treatets be here the treatet paraeters are ad usg the costrat: ow the lkelhood s: l(; x ( ( ( ( ( ( ( r r r r r r where ad r.. so o. (. The pror dstrbuto for = usg the costrat: s: ( a a a a p θ (. ( a a a ( a ( a ( a ( a ( a where θ = ( Now the posteror dstrbuto of the paraeters ad usg (. ad the lkelhood (. s: p( x = ( a a a a r r r r r r K( ( ( ( ( ( where K s oralzg costat ad <. (. The argal posteror destes for the paraeters are obtaed by usg (.. The graphs of the argal posteror destes of ad are preseted Fg.. A Mothly Double-Bld Peer Revewed Refereed Ope Access Iteratoal e-joural - Icluded the Iteratoal Seral Drectores Idexed & Lsted at: Ulrch's Perodcals Drectory U.S.A. Ope J-Gage Ida as well as Cabell s Drectores of Publshg Opportutes U.S.A. Iteratoal Joural of Physcal ad Socal Sceces 8

7 IJPSS Volue Issue 7 ISSN: We sulate data (gve Table. for the pared coparso of four treatets assug the Bradley-Terry odel wth the paraeters values as: =0.6 =0. =0. =0.0 ad the uber of coparsos ( r 0 : Table.:-Sulated Data for = Pars ( ( ( ( ( ( r Posteror (Margal Destes for the Paraeters of the Bradley-Terry Model p ( x p ( x A Mothly Double-Bld Peer Revewed Refereed Ope Access Iteratoal e-joural - Icluded the Iteratoal Seral Drectores Idexed & Lsted at: Ulrch's Perodcals Drectory U.S.A. Ope J-Gage Ida as well as Cabell s Drectores of Publshg Opportutes U.S.A. Iteratoal Joural of Physcal ad Socal Sceces 9

8 IJPSS Volue Issue 7 ISSN: p ( x p ( x Fgure... Posteror Estates Usg the posteror dstrbuto (. two progras are desged SAS package to fd the posteror estates: eas ad odes. Havg ru the progras the obtaed results are preseted Table. for the data set gve Table. Table.:-Posteror Estates Paraeters Posteror Meas Posteror Modes It s observed that the ot posteror ode s slar to the posteror eas ad dcate the slar rakg of the treatets.e. T T T T. A Mothly Double-Bld Peer Revewed Refereed Ope Access Iteratoal e-joural - Icluded the Iteratoal Seral Drectores Idexed & Lsted at: Ulrch's Perodcals Drectory U.S.A. Ope J-Gage Ida as well as Cabell s Drectores of Publshg Opportutes U.S.A. Iteratoal Joural of Physcal ad Socal Sceces 0

9 IJPSS Volue Issue 7 ISSN: Preferece Probabltes As we kow that preferece probabltes are the probabltes of preferrg treatet T to treatet T. For deterg ad we use the values of the posteror eas: =0.60 =0.9 =0.97 ad =0.90. The preferece probabltes are show Table.: Table.:- Preferece Probabltes Preferece Prob Values Preferece Prob Values Posteror Probabltes of Hypotheses The followg hypotheses : H ad H ad H :. H (<= are copared: The posteror probablty p for H s p p ( ad q p s the posteror probablty for H. The posteror probablty { p } for H s obtaed as: p p( p( 0 p ( 0 = a a a a ( ( 0 0 KD d d d (. where ad K s oralzg costat ad D s defed as: r r r r r r D= ( (( ( ( ( ( ( By SAS package we get posteror probabltes show Table.:. A Mothly Double-Bld Peer Revewed Refereed Ope Access Iteratoal e-joural - Icluded the Iteratoal Seral Drectores Idexed & Lsted at: Ulrch's Perodcals Drectory U.S.A. Ope J-Gage Ida as well as Cabell s Drectores of Publshg Opportutes U.S.A. Iteratoal Joural of Physcal ad Socal Sceces

10 IJPSS Volue Issue 7 ISSN: Table.:- Posteror Probabltes Pars( ( ( ( ( ( ( p q The hypothess wth greater probablty wll be accepted. Let s= ( p q f s>0. the decso s coclusve. Here H ad H are accepted wth hgh probablty ad all other hypotheses see coclusve... Predctve Probabltes Let the predctve probablty P ( that treatet T would be preferred to treatet T a future sgle coparso of two treatets s{usg (. ad (.9} s: P ( = P( T T p( x d d d ( Usg SAS package the predctve probabltes are gve as follows Table.5: Table.5:- Predctve Probabltes Pars( ( ( ( ( ( ( P ( P ( We would always expect that the predctve probabltes are ear to Approprateess of the Model For the data gve Table. we cosder ch-square goodess of ft test for the odel fttg. The expected ubers of prefereces are calculated by ultplyg preferece A Mothly Double-Bld Peer Revewed Refereed Ope Access Iteratoal e-joural - Icluded the Iteratoal Seral Drectores Idexed & Lsted at: Ulrch's Perodcals Drectory U.S.A. Ope J-Gage Ida as well as Cabell s Drectores of Publshg Opportutes U.S.A. Iteratoal Joural of Physcal ad Socal Sceces

11 IJPSS Volue Issue 7 ISSN: probabltes (gve Table. wth uber of coparsos ( r =0. The followg chsquared test s used to test the approprateess of the odel: {( ˆ wth ( ( degrees of freedo. Table 5.:- Observed ad Expected Nuber of Preferece Pars ( ( ( ( ( (. Observed Expected We fd =.699 wth p-value dcates that there s o evdece that the odel does ot ft the data. Refereces Ada E. S. (005. Bayesa Aalyss of Lear Doace Herarches. Aal Behavor. Vol 69. ssue Asla M. (00. A applcato of pror predctve dstrbuto to elct the Pror desty. Joural of Statstcal Theory ad Applcatos Vol Bradley R. A. ad Terry. M.E. (95. Rak Aalyss of Icoplete Block Desg:І. The Method of Pared Coparsos. Boetrka 9-5. Causeur D. & Husso F. (005. A desoal exteso of the Bradley-Terry odel for pared coparsos. Joural of Statstcal Plag ad Iferece. 5. pp Chaloer K. M. ad Duca G. T. (98. Assesset of a beta pror dstrbuto: PM elctato. The Statstca ½7-80. Davd H. A. (988. The ethod of pared coparsos d ed. Charles Grff ad Copay-Lodo. A Mothly Double-Bld Peer Revewed Refereed Ope Access Iteratoal e-joural - Icluded the Iteratoal Seral Drectores Idexed & Lsted at: Ulrch's Perodcals Drectory U.S.A. Ope J-Gage Ida as well as Cabell s Drectores of Publshg Opportutes U.S.A. Iteratoal Joural of Physcal ad Socal Sceces

12 IJPSS Volue Issue 7 ISSN: Davdso R. R. ad Soloo D. L. (97. A Bayesa Approach to Pared Coparso Experetato. Boetrka Ja B. (007. Slarty testg usg pared coparso ethod Food Qualty ad Preferece Volue 8 Issue Kadae J. B. (966. Soe equvalece classes pared coparsos. The Aals of Matheatcal statstcs vol.7 No Kadae J. B. Dckey J. M. Wkler R. L. Sth W. ad Peter S.C. (980. Iteractve elctato of opo for a oral lear odel. JASA Leoard T. (977. A alteratve Bayesa approach to the Bradley-Terry odel for pared coparsos. Boetrcs -. K D-H ad K H-J. (00. A Bayesa Approach to Pared Coparso of Several Products of Posso Rates. Proceedgs of the Autu Coferece Korea Statstcal Socety. Mrada. E Bourque.P Abra. A(009 Iforato ad Software Techology Volue 5 Issue 9 Septeber Wkler R. L. (980 Pror Iforato Predctve Dstrbutos ad Bayesa Model Buldg. Bayesa aalyss ecooetrcs ad statstcs Zeller A (Ed. North- Hollad Asterda Zerelo E. (99 De Berechug der Turer-Ergebsse als e Maxuproble der Warschelchketsrechug. Math. Zet A Mothly Double-Bld Peer Revewed Refereed Ope Access Iteratoal e-joural - Icluded the Iteratoal Seral Drectores Idexed & Lsted at: Ulrch's Perodcals Drectory U.S.A. Ope J-Gage Ida as well as Cabell s Drectores of Publshg Opportutes U.S.A. Iteratoal Joural of Physcal ad Socal Sceces