MULTIPLE COMPARISONS WITH A CONTROL UNDER A GROUP TESTING SCENARIO YANINA GRANT. Bachelor of Science. Technical Institute of Advanced

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1 MULTIPLE COMPARISONS WITH A CONTROL UNDER A GROUP TESTING SCENARIO By YANINA GRANT Bachelor of Scece Techcal Ittute of Adaced Stude of Moterrey Moterrey Nueo Leó Meco Submtted to the Faculty of the Graduate College of the Olahoma State Uerty artal fulfllmet of the requremet for the Degree of MASTER OF SCIENCE May 6

2 MULTIPLE COMPARISONS WITH A CONTROL UNDER A GROUP TESTING SCENARIO The Aroed: The Ador Dr Melda H McCa Dr Stehae A Mo Dr Wllam D Warde Dea of Graduate College Dr A Gordo Emle

3 ACKNOWLEDGMENTS I would le to ere my grattude to my academc ador Dr Melda McCa for her commtmet to helg ee th roect trough to t fal comleto ad her we gudace durg t deelomet She alway roded tmely ad tructe commet ad ealuato at eery tage of the the roce allowg me to comlete th roect o chedule My cere arecato eted to my other commttee member Dr Wllam Warde ad Dr Stheae Mo for ther d atace ad uort I would alo le to ge my ecal arecato to my famly ecally my father ad mother who tlled me from a early age the dere ad ll to comlete th Mater Degree Further tha go to my ecal boyfred Ahw Bhuaedra for h uort ad ecouragemet at tme of dffculty Fally I would le to tha the Deartmet of Stattc for uortg durg thee three year of my Mater Program

4 TABLE OF CONTENTS Chater Page INTRODUCTION THEORETICAL BACKGROUND 6 Tuey-Kramer cofdece teral 7 Jeffrey-Per cofdece teral 8 3 MULTIPLE COMPARISONS WITH A CONTROL 4 4 SIMULATIONS 5 APPLICATION 4 6 SUMMARY AND CONCLUSIONS 8 BIBLIOGRAPHY 3 APPENDIXES APPENDIX A: The mamum lelhood etmator of 38 APPENDIX B: The dtrbuto of 39 APPENDIX C: The etmated aymtotc arace of b 4

5 LIST OF TABLES Table Page Etmated multaeou coerage robablte for the Duett ad Jeffrey-Per rocedure wth a gfcace leel of 5 3 Percetage of tme the adutmet for wa made durg the mulato for Duett ad Jeffrey-Per 33 3 Percetage of tme the adutmet for wa made durg the mulato for Duett ad Jeffrey-Per 34 4 Bawde ad Kaa 946 Obered roorto of fected lat wth the otato ru Y for dfferet arete 35 5 Multle-ector trafer deg tudy of Bawde ad Kaa 946 Duett ad Jeffrey-Per 95% multaeou cofdece teral for - whe two ahd are ued to coloze the tet lat 36 6 Multle-ector trafer deg tudy of Bawde ad Kaa 946 Duett ad Jeffrey-Per 95% multaeou cofdece teral for - whe four ahd are ued to coloze the tet lat 37

6 CHAPTER INTRODUCTION Whe a reearcher erformg a tudy wth more tha two leel of a factor arable ofte the uroe to quatfy the dfferece amog the factor leel mea We refer to the roblem of determg multaeouly whch leel of a factor dffer gfcatly from a cotrol a multle comaro wth a cotrol MCC Duett 955 deeloed coerate multaeou -I% cofdece teral to etmate thee dfferece multaeouly uder ormalty wth deedet amle Aumg that µ the mea of the cotrol ad µ µ are the mea of the o-cotrol leel of the factor arable the arameter of teret are µ for Sometme howeer the reoe of teret bary For eamle µ whe determg huma mmuodefcecy ru HIV erootty ddual are clafed a ether ote or egate ad whe detectg defecte tem large oulato tem are clafed a ether defecte or o-defecte I cae le thee rather tha etmatg the dfferece betwee factor leel mea the goal of the reearcher mght be to etmate dfferece betwee roorto of deedet bomal arable wth for a the arameter of teret where the roorto of the leel codered the cotrol ad are the roorto of the o-cotrol leel of the factor of teret A ueful alcato of the later deg occur the aaly of realece of a ge deae huma where the oulato of teret dded to ubgrou that hare the ame charactertc uch a age race or ocal tatu Smlarly lat/amal deae aemet t of teret to ow the roorto of ector orgam uch a ect

7 caable of tramttg a ru th cae the oulato ca be ubdded by ector ece lat arety or erometal codto Coder ecfcally a eermet olg determato of HIV erootty The HIV ru ca be tramtted dfferet way through eual cotact wth other that are fected from a fected mother to her ubor baby ad through blood; the latter occur whe eole hare eedle or from blood trafuo hotal Curretly effecte method are aalable for reeto of HIV tramo through blood a a reult the r ha bee reduced to low leel the deeloed coutre; howeer creeg tet ca be relately eee for deelog coutre where the goermet face erou facal retrcto Oe oble way to reduce the cot of HIV tetg ole oolg the era from eeral ddual ad the tetg the ool for HIV fecto; we refer to th methodology a grou-tetg Grou-tetg ole oolg ddual to grou tetg the grou ad clafyg each grou a ether ote or egate A egate reult for the grou occur whe oe of the ddual oee the charactertc of teret; a ote reult for the grou occur whe at leat oe of the ddual oee the charactertc Th tye of amlg rode ubtatal beeft amog them the reducto of cot of clafyg all ddual of a oulato accordg to whether or ot they oe a certa trat whe the cdece rate ery low Beeft from grou tetg deed o the ze of the ool Ufortuately otmum electo of grou ze dffcult becaue t requre owledge of the uow realece Hughe-Oler & Roeberger The org of grou-tetg credted to Robert Dorfma 943; he rooed the ue of a mle bomal model to reduce the umber of medcal tet whle detectg all member of a oulato that hae the yhl atge Dorfma uggeted oolg eeral blood amle ad tetg the ool for the yhl atge; ad the cae where a grou teted ote all the ubect from that artcular grou would be re-teted o a ddual ba Dorfma goal wa to clafy each of the ddual the oulato a fected or ot whle reducg the eected

8 umber of tet th ow a the clafcato roblem Oe ca alo ue grou tetg to etmate the roorto fected a oulato wthout ecearly detfyg the fected ddual whch ow a the etmato roblem I fact Dorfma dea of oolg blood amle for yhl creeg wa ot ued; but ome year later the cocet wa aalyzed aga ad today ueful alcato ca be foud the lterature of grou-tetg Amog thoe alcato are determato of HIV erootty Gatwrth & Hammc 988 Gatwrth & Joho 994 Kle Brother Broomeyer Zeger & Qu 989 We & Zeo 996 etmato of ru fecto rate lat Bhattacharyya Karado & DeFollart 979 Heworth 996 Swallow 985 Tebb & Blder 4 Thomo 96 lea tetg of ealed radoacte ource Thoma Paterac Vacrca & Thomo 973 ad geetc Chc 996; whle grou tetg ha bee how to hae ubtatal beeft multlcty adutmet for eermet coducted uder th rotocol ha bee lmted McCa ad Tebb 6 dere two multaeou cofdece teral aroache for all arwe roorto dfferece whe the data obtaed uder a grou-tetg cearo Both of ther rocedure Tuey-Kramer ad Jeffrey-Per are baed o aymtotc reult; the author ealuate both aroache term of multaeou coerage robablty ad mea teral legth for mall amle cae McCa ad Tebb coclude that both rocedure hae good erformace for mall amle recommedg the Tuey-Kramer aroach for large grou of about ddual or more; ad the Jeffrey-Per rocedure for mall grou of le tha ddual McCa ad Tebb alo llutrate the alcato of both rocedure ug data from a HIV tudy olg drug uer Houto The oulato of teret clafed baed o two dfferet factor race ad tye of drug ected wth three leel for the race factor Hac whte ad blac ad four leel for the tye of drug ued by the ddual hero cocae/heroe cocae ad cocae/amhetame Parwe dfferece are etmated betwee the roorto of ote at each leel wth each factor Th methodology allow the 3

9 reearcher to ra the dfferet leel of a factor baed o HIV realece whle alo retag the ablty to ae ractcal gfcace a the teral etmate I ome cae howeer dfferece wth a cotrol are of teret The goal of the curret aer to rode wth a grou-tetg cotet multaeou cofdece teral for roorto dfferece betwee the cotrol ad all other leel Both multaeou feretal rocedure reeted by McCa ad Tebb 6 wll be modfed for th cearo ad ubequetly the Duett ad Jeffrey-Per multaeou cofdece teral wll be geerated ad llutrated ug data from a multle-ector trafer deg tudy For referece throughout the aer we ow coder a ecfc eamle where MCC clearly wated ad the data are correlated a a grou tetg rotocol It well ow that lat rue are the caue of bg loe cro roducto ad qualty all oer the world therefore deelog treatmet to cotrol the read of rue of rme mortace Normally orgam uch a ect tramt rue A ect that caable of tramttg the cauate aget of deae ow a ector I athology ofte t of teret to etmate the robablty of ru tramo by a gle ector; oe way to do t cot of collectg amle of ect ad cagg ect ddually wth a healthy lat If after a ru tet the lat tet ote the the ect clafed a a ector; ad the roorto of fected lat ca be ued a a etmate of the roorto of ector the oulato Uually the oulato of ect are ery large ad the reou rocedure may be cot rohbte due to the requred umber of lat cage ad ace; thu t coeet to tet eeral ect o each healthy lat; th ow the lterature a multle-ector trafer deg Thomo 96 Coder for tace a tudy by Bawde ad Kaa 946 to comare the ucetblty of fe dfferet arete of otato to the tramo of otato ru Y I the ame reearch the author erformed a eermet whch howed that tobacco more ealy fected tha otato by the ru Y ad thu t ca be ued a a referece whe meaurg 4

10 fecto rate For th reao whe llutratg the Duett ad Jeffrey-Per aroache uder the MCC modalty tobacco ued a the cotrol leel ad fe dfferet arete of otato are ued a the o-cotrol leel Bawde ad Kaa frt ued two ect to tramt the ru from fected tobacco lat to 36 healthy lat of each of the arete uder aaly ad the roorto of healthy lat that howed ymtom of the deae wa recorded a a meaure of the ucetblty of a lat to the ru Y fecto I a ecod relcate of the eermet four ect were ued to tramt the ru to aother 36 healthy lat of each arety ad aga the roorto of tet lat that became fected wa recorded The remader of the aer orgazed a decrbed et I Chater we reew the methodology of the Tuey-Kramer ad Jeffrey-Per rocedure for all arwe roorto dfferece whe the data collected uder a grou-tetg cotet I Chater 3 we reet the theoretcal bacgroud that mae oble the trato from all arwe comaro to multle comaro wth a cotrol I Chater 4 we ealuate the erformace of both the Duett ad Jeffrey-Per aroache baed o multaeou coerage robablty for arou cearo I Chater 5 we llutrate the aboe rocedure ug data from a multle-ector trafer deg tudy I Chater 6 we dcu the geeral cocluo of th wor ad coder ome future eteo 5

11 CHAPTER THEORETICAL BACKGROUND For the uroe of th aaly we aume that the oulato of teret dded to ubgrou that hare a ecfc charactertc; heceforth the ubgrou wll be referred a trata Suoe there are a total of trata; ddual are the ooled to grou wth each tratum wth the th tratum cotag grou all of amle ze I grou-tetg each ut aumed to rereet a deedet Beroull arable where the robablty that a radomly elected ubect oee the charactertc of teret ad the robablty of radomly electg a ubect ot oeg the charactertc Thu the robablty of radomly electg a grou of ubect all of whom do ot oe the charactertc of teret ad coequetly the robablty of obtag a grou of ubect where at leat oe ubect oee the trat of teret Let Y l f the lth grou the th tratum oee the charactertc of teret ad Y otherwe l l It aumed that the Y l are alo deedet ad detcally dtrbuted Beroull radom arable wth mea Ug the arace roerty of mamum lelhood etmator ad the uual bomal MLE for the MLE of ca be wrtte a / for all ; where l y l / correod to the obered roorto of grou oeg the charactertc of teret tratum ee Aed A for ecfc detal 6

12 Recall that our goal to obta multaeou cofdece teral for all where rereet the realece for the ubgrou deemed the cotrol th ettg McCa ad Tebb 6 coder a mlar roblem ad deelo two multaeou cofdece teral aroache for all arwe roorto dfferece whe the data obtaed uder a grou-tetg cearo Both of ther rocedure the Tuey-Kramer ad Jeffrey-Per are baed o aymtotc reult ad both hae good erformace for mall amle cae term of multaeou coerage robablty ad mea teral legth McCa ad Tebb aly both rocedure to cae where the reearcher tereted all arwe dfferece; for eamle f the uroe to ra dfferet race baed o the roorto of HIV ote whle retag the ablty to etmate the dfferece thee roorto all arwe dfferece ca be etmated a multaeou cofdece teral ad the rag determed a a reult of the arwe comaro Thee rocedure ca be modfed arorately for our all comaro wth a cotrol cearo The Tuey-Kramer ad Jeffrey-Per aroache dered by McCa ad Tebb are reeted et Tuey-Kramer cofdece teral The Tuey-Kramer method a arorate multle comaro rocedure whe all arwe dfferece are of teret Hochberg ad Tamhae 987 rode the detal of th multle comaro rocedure for the ddual tetg cae Th aroach ca be alo aled to the grou-tetg cearo ce we hae deedet grou bary reoe for all trata Hochberg ad Tamhae recommed the ue of the Tuey-Kramer teral otg that they are coerate for large amle ze 3 ad hae maller wdth tha other tycal rocedure; howeer for mall amle ze the rocedure ehbt oor coerage charactertc ad alterate rocedure hould be codered 7

13 Recall that the uual bomal MLE ha a aymtotc ormal dtrbuto wth mea ad arace / / A g the delta method yeld that the aymtotc dtrbuto of alo ormal wth mea ad arace [ ] / ee Aed B for ecfc detal By the roerte of the MLE a cotet etmator of ad ce a fucto of we ca alo ay that a cotet etmator of Hece t arorate to ue a a etmate of the uow quatty Thu for the grou-tetg cae the Tuey-Kramer multaeou ercet aymtotc cofdece teral for all arwe dfferece < are ge by ± q * where ad are the aymtotc etmated arace of ad * reectely ad q a decrbed ecto 5 of Hu 996 McCa ad Tebb 6 Sce th tattcal rocedure ole roorto ome dffculte occur whe oe of the grou tratum ad tratum hae the charactertc of teret ; th cae a degeerate teral roduced There alo a roblem whe all the grou oe the charactertc of teret tratum or tratum or ; uder th cearo a oformate teral ge [-] The ecod rocedure dered by McCa ad Tebb 6 the Jeffrey-Per cofdece teral reeted et Jeffrey-Per cofdece teral Beal 987 aalye the ue of aymtotcally-baed cofdece teral for the dfferece betwee the robablty of ucce for two bomal oulato She ealuate the 8

14 tattcal behaor of fe uch teral Wald Mee Mette & Nurme Haldae ad Jeffrey-Per Ecet for the Jeffrey-Per teral thee rocedure wll ot be llutrated th aer but they are decrbed Beal 987 Beal recommed the Jeffrey-Per teral f oe wat to comute a mle teral The Jeffrey-Per teral how a coderable mroemet oer other oto whe the alue of both ad are ot ery mall or ery large Pegorch 99 coder Beal formulato of a Jeffrey-Per teral but rather tha etmatg the dfferece betwee the roorto of two bomal oulato Pegorh coder multaeou teral for all ad for all comaro wth a cotrol He how that the Jeffrey-Per rocedure ehbt geerally omal emrcal coerage charactertc ad recommed t for ue wth mall to moderate amle ze Both Beal 987 ad Pegorch 99 oly coder the cae whe Howeer McCa ad Tebb 6 geerale the Jeffrey-Per rocedure for the cae whe > ad > whch correod to the grou-tetg cearo They utle Beal rearameterato a ad b Howeer Beal coder a ror dtrbuto f P P that roortoal to / { } for < ad < ; < < but McCa ad Tebb lace a o-formate ror dtrbuto o the roorto of grou oeg the charactertc of teret ce grou reoe are obered uder the grou-tetg cearo Thu the ror f / roortoal to { } / for < ad < Coequetly f we coder ad < < ole for / ad reectely ad erform 9

15 a barate traformato wth J the we ca wrte the ror dtrbuto o a ddual cale a J f f P P or / / / / } { } { P P f for alue of < < ad < < Now let X be the umber of grou tratum that oe the trat of teret thu l X Y l The X X X are deedet ad detcally dtrbuted Bomal radom arable hece the lelhood fucto P P X X f for ad roortoal to } { } { Coequetly the oteror dtrbuto of ad ge by P P P P X X P P P P X X X X P P d d f f f f f Frt coder the deomator where the double tegral ca be ereed a follow: / / / / } { * } { d d Sce ad are deedet ad detcally dtrbuted the wthout lo of geeralty we ca wor wth ddually The tegral olg ca be rewrtte a / } { d Now utlze the chage of arable t wth d dt Thu the tegral equalet to

16 t t dt t t dt 3 Let a / ad b / After addg the arorate cotat equato 3 ca be ereed a b a a b b a t t dt a b 4 b a where ymbolze the regular gamma fucto Now the oluto of the tegral equato 4 oe ad thu the oteror dtrbuto of ca be wrtte a { { } / / } b a a b Coequetly the oteror dtrbuto of ad ge by c { } { } for < ad < where the cotat < < c / / / / Now let ad b be the mamum lelhood etmate of ad b reectely The the Jeffrey-Per multaeou ercet cofdece teral for all arwe dfferece < are cotructed by olg * b b q ; ~ V b a a 5 * for b where q decrbed equato ad V b; a a ~ the aymtotc arace of b wth a ~ the oteror mea of a erted a a etmate of a

17 The oteror mea of a calculated by tag the eected alue wth reect to the oteror dtrbuto X X P P f a follow ~ X X P P X X P P d d f X X E a Aga becaue of deedece ad detcal dtrbuto we ca wor wth ad earately The oteror mea of ca be etmated by olg the followg equato / / / / d If we ue the ame chage of arable a before t wth d dt the we hae / / / / / dt t t t / / / / / dt t t 6 Now let / / ' a ad ' / b the equato 6 become ' ' ' ' ' ' ' ' / ' ] [ a b dt t t b a b a b a a X E The tegral th ereo equal oe ad coequetly the oteror mea of ge by / / / / Thu the oteror mea of a ca be wrtte a / / / / / / / / ~ a

18 Whe a ~ ued a a etmate of a the etmated aymtotc arace of b a follow a ~ b g a ~ b g a ~ b g a ~ } { } b { g V b a a ~ ; where g a ~ b a~ / ad g a ~ b a~ / ee Aed C for ecfc b b detal A wth the Tuey-Kramer teral ome dffculte ca be foud wth calculato of thee teral whe all the grou wth trata ad oe the charactertc of teret or whe oe of the grou oe uch a trat Whe ad equato 5 may ot hae ay oluto th cae a o-formate teral [-] et for Whe ofte oly oe oluto reet I th cae a oe-ded teral wll be roded Both rocedure Tuey-Kramer ad Jeffrey-Per a reeted by McCa ad Tebb * q clude a crtcal alue whch the uer I crtcal ot decrbed Secto 5 of Hu 996; th crtcal alue arorate whe all arwe comaro are of teret; howeer multle comaro wth a cotrol are of cocer the curret aer ad therefore a * d Duett crtcal alue requred Net we decrbe how th Duett crtcal ot etmated whe the arameter of teret where rereet the cotrol leel ad we alo reet the Duett ad Jeffrey-Per cofdece teral utlzg a Duett crtcal ot whch arorate for multaeou ferece wth a cotrol for the grou tetg cearo 3

19 CHAPTER 3 MULTIPLE COMPARISONS WITH A CONTROL Whe a reearcher tereted comarg all treatmet grou wth a cotrol grou the multle comaro rocedure due to Duett ofte arorate Duett 955 rode the detal o calculatg the arorate crtcal alue for multaeou cofdece teral for the dfferece betwee all o-cotrol leel factor mea wth the mea for the cotrol leel The arameter of teret th cae are µ µ where µ rereet the mea of the cotrol factor ad µ the mea of the th o-cotrol factor for - The ot etmate of µ µ the dfferece betwee the amle mea u u where u the amle mea of the cotrol factor ad u the amle mea of the th o-cotrol factor - Uder the aumto that u ad u hae deedet ormal dtrbuto wth cotat arace u u ormally dtrbuted wth mea µ µ ad arace where ad are the amle ze of the cotrol factor ad the th o-cotrol factor - reectely If we let z u u µ µ the z ha a ormal dtrbuto wth mea zero ad arace oe A we are cocered wth multaeou teral we eed to eame the ot dtrbuto of u * u u - The ot dtrbuto of the u * - a multarate ormal dtrbuto wth mea ector 4

20 5 µ ' µ µ µ µ µ µ ad arace-coarace matr ge by Thu the correlato coeffcet betwee * u ad * u obtaed a follow! " " # $! " " # $ / / where /! " " # $ for - Duett further aume that a etmate of aalable whch deedet of u u ; for eamle he rooe ug / where ; hece ha a ch-quare dtrbuto wth degree of freedom ad / z z t follow a tudet t-dtrbuto wth degree of freedom The requred crtcal alue d eeded to cotruct multaeou cofdece teral for µ µ - are the choe to atfy P % & d t P P d z % & where P the

21 ot cofdece coeffcet The oluto are foud by olg the followg equato for d - $ F" d d d! d # where F z z z the multarate ormal cdf of the z ad the robablty dety fucto of Stadard ractce to requre d d d d The Duett rocedure et the aboe tegral equal to P ad ole for d Howeer for the uroe of the curret aer the arameter of teret are ot µ µ - but rather - a defed Chater Recall that Chater the MLE of / ad t aymtotc dtrbuto were dered A the trata are deedet the ot aymtotc dtrbuto of the ector ' multarate ormal wth mea ' ad arace-coarace matr ge by ' where { } / By argumet mlar to thoe Aed B Sce our goal to etmate multaeou cofdece teral for each of the dfferece we eed to determe the ot dtrbuto of g g g ' where aga deote our cotrol leel By the multarate delta method g g g ' ha a aymtotc ormal dtrbuto wth mea g ad arace V C' C' where C a - by matr of artal derate wth elemet ge by 6

22 7 dg dg dg dg C Thu V a - by - matr ge by V where / } { The correlato betwee g ad g ca be calculated a the followg equato:! " " # $! " " # $ / / 7 where / } { } {! " " # $ Now ule Duett 955 we do ot hae a deedet etmate for the arace comoet ad coequetly we caot utlze a tudet t-dtrbuto Itead we ca cotruct tet tattc wth a aymtotc multarate ormal dtrbuto wth a mea ector of zero ad a coarace matr wth oe o the dagoal a tadardzato; e d z ~ N - 8 Coequetly fty degree of freedom are ued the reou Duett formulato a a t Y dtrbuto correod to a z dtrbuto

23 Sce the are uow quatte we caot obta the eact crtcal alue d Ee f we ew the correodg d would tll yeld oly aymtotcally correct teral a oly aymtotcally ormal Howeer a a cotet etmator of by the roerte of the MLE ; we ca ue equato 7 ad 8 for ad obta a cotet etmator of the aymtotcally correct crtcal alue d Coequetly are ued a etmate of I order to etmate the arorate Duett crtcal alue requred to cotruct multaeou cofdece teral for where the cotrol leel a Fortra rogram roded by Hu 996 wa adated for our ecfc grou tetg cearo The ut for th comuter rogram are the gfcace leel the degree of freedom ad After calculatg the arorate Duett crtcal ot the multaeou cofdece teral are etmated Thee teral ole adutg the multaeou ferece rocedure dered by McCa ad Tebb 6 for the MCC cearo ad are ued to multaeouly etmate roorto dfferece betwee a cotrol grou ad eeral o-cotrol grou Thu for the grou-tetg cae the Duett multaeou -I ercet cofdece teral for the dfferece where rereet the grou that act a cotrol are ge by ± d 9 * where ad are the aymtotc arace of ad * ad d the Duett crtcal alue calculated ug reectely Addtoally the Jeffrey-Per multaeou -I ercet cofdece teral for the dfferece where aga the grou that act a cotrol are cotructed by olg 8

24 * b b d ; ~ V b a a * for b recall b where d a equato 9 ad V b; a a ~ the aymtotc arace of b wth a ~ the oteror mea of a erted a a etmate of a See Chater for a detaled derato of the Jeffrey-Per aroach 9

25 CHAPTER 4 SIMULATIONS The Duett ad Jeffrey-Per aroache dered Chater 3 for multle comaro wth a cotrol uder a grou tetg cearo are ow ealuated ad comared term of multaeou coerage robablty Uually whe worg wth roorto ome comlcato are whe ether all reoe are ote or whe all are egate For eamle our ettg uoe that the umber of grou oeg the trat of teret tratum zero; that ugget that the obered roorto of grou wth the charactertc of teret tratum zero ad coequetly / where ad are a defed reou Chater Whe the the etmated arace { } / ad a a reult the correlato betwee g g ad o-etmable; recall that th correlato etmated by / / $! $! " " # # See equato 7 Chater 3 for more detal If caot be etmated the we alo caot etmate the Duett crtcal ot ad therefore the cofdece teral are ot aalable A mlar roblem occur whe Hag all reoe ote ot oly a roblem for the cotrol leel but alo for the o-cotrol leel I order to calculate the Duett crtcal alue we eed to ole a equato that ole the factor c ; howeer recall that mle that / / ad

26 ad coequetly the factor c o-etmable ad oce aga the cofdece teral would ot be aalable To aod thee roblem we mae ome adutmet eery tme the obered roorto of ote ether zero or oe Whe we et 5 ad whe we et 5 Whle thee etmate are ot techcally the MLE otce that a the robablty of mag uch adutmet roded < wll coerge to zero Thu the adutmet hould ot gfcatly chage the oerall behaor of our etmator for large amle The ercetage of tme uch adutmet were made cluded the mulato reult To ru the mulato radom draw were tae from a bomal dtrbuto wth arameter ad ; recall from Chater that a bomal radom arable A gfcace leel of 95 wa ued throughout Now let be the umber of trata the umber of grou wth each tratum the ector of roorto ad the reecte grou ze I our ettg the frt tratum codered the cotrol leel Now ug dfferet alue of ad ad eeg for mlcty ad cotat all trata; we ra mulato for each et of data ad etmated the multaeou coerage for the < cofdece teral dered equato 9 ad Chater 3 I order to etmate th multaeou coerage robablty we mly couted the umber of tme the teral multaeouly cotaed the true dfferece - ad dded that umber by We alo et trac of the umber of tme we made the adutmet reouly dcued to aod or The ercetage of tme the adutmet were made to aod for each et of data roded Table The

27 ercetage of tme the adutmet were made to aod for each et of data roded Table 3 ad the etmated multaeou coerage robablty reult are how Table Sce the omal coerage 95 we eect that whe the umber of grou wth each tratum creae the the etmated coerage robablty coerge to the 95 omal leel I Table the etmated coerage robablte bold are thoe wth two etmated tadard deato ug 95 a the true roorto alue From Table we otce that the cae of ddual tetg both the Duett ad Jeffrey-Per rocedure are ery coerate a the coerage robablte are more tha two tadard deato aboe 95 for mot of the cae Th robably due to the large ercetage of tme we mae the adutmet thee cae We alo otce that whe > the Jeffrey- Per rocedure ofte lghtly more coerate tha the Duett rocedure ce more of the tme the coerage robablty larger for the Jeffrey-Per cae From Table we ca alo obere that whe 5 both rocedure erform farly well a the etmated coerage robablty ery cloe to the omal coerage of 95 Whe 5 or the Jeffrey-Per rocedure ted to erform better tha the Duett rocedure a t ofte cloer to the omal coerage leel I the tuato where oe of the aroache cloe to the omal leel Jeffrey-Per rode coerate cofdece teral whle the Duett rocedure atcoerate th alo ugget that t better to ue the Jeffrey-Per aroach I cocluo both the Duett ad Jeffrey-Per aroache erform well for large grou ze [ 5 For maller grou ze 5 < < the Jeffrey-Per methodology recommeded I Table ad 3 we reet the umber of tme a adutmet wa made due to ad reectely A eected the frt tye of adutmet whe occurred much more ofte tha the ecod tye of adutmet whe Th becaue the true alue of the our mulato are tycally ery mall -3

28 coequetly the are alo mall ad t reaoable to obta a obered umber of grou wth the charactertc of teret of zero 3

29 CHAPTER 5 APPLICATION I th Chater a ractcal alcato of the Duett ad Jeffrey-Per rocedure wll be reeted ug data from a multle-ector trafer deg tudy I order to cotrol the read of lat deae Pathology ofte t of teret to etmate the roorto of a oulato that fected wth a lat ru A multle-ector trafer deg wdely ued for th uroe Th rocedure cot of collectg a amle of ahd or ect feedg them o a fected lat ad the earatg thee ect to grou ad cagg each grou wth a healthy or tet lat Subequetly the reearcher obere f the tet lat deelo the ymtom of the deae or ot ad th how the roorto of fected lat etmated To llutrate the Duett ad Jeffrey-Per cofdece teral data obtaed from a multle-ector trafer deg wa ued The obecte of th tudy wa to comare the ucetblty of dfferet arete of otato to the tramo of otato ru Y Bawde ad Kaa 946 The dfferet arete of otato aalyzed were arra baer maetc arra coul arra lot ad may quee Bawde ad Kaa wated to detfy the retace of arou otato arete to the otato ru Y For that uroe they erformed feld ad glahoue eermet comarg eeral otato derte The former tye of aaly oled lot of dfferet arete whch were eoed equally to chace of fecto The lot coted of fe row of fe lat each where the lat located at the ceter wa fected wth ru Y The reult of th eermet are ot roded the curret aer howeer they ca be foud o Bawde ad Kaa 946 The glahoue eermet are decrbed et 4

30 Bawde ad Kaa aalyzed the fecto rate of dfferet otato arete ug frt a oculato rocedure ad the ug ect to tramt the ru Y The oculato method cot of tag a from fected lat ad rubbg healthy lat wth the ubtace Bawde ad Kaa obtaed a from lat fected wth the ru Y ad ued t a oculum after dluto wth water Te lat of each arety of otato ad tobacco were oculated Two leaflet of aromately the ame ze were rubbed oer ther urface wth the oculum The reult howed that % of the tobacco lat were fected but the obered roorto that wa fected wa much lower wth the otato arete From th tudy Bawde ad Kaa cocluded that tobacco more ealy fected tha otato Therefore uder our multle comaro wth a cotrol ettg tobacco ued a the cotrol leel Whe Bawde ad Kaa aalyzed the fecto rate of the arete of otato uder a multle-ector trafer deg rotocol they ued the ahd Myzu ercae Sulz to tramt the ru The ahd were raed o tur or radhe ad tared for 4 hour The they were fed for 3 or 4 mute o the uderurface of a fected tobacco leaf ad mmedately traferred to the healthy otato ad tobacco lat Thrty- lat of each arety were colozed wth two ahd ad the umber of fected lat wa couted To ae the dfferece amog the arete Bawde ad Kaa mly obered the roorto of lat er arety that became fected after eoure to the ru Y; thee obered roorto are ge Table 4 The reearcher dd ot adut ther aalye for ether the grou-tetg ettg or multlcty robably becaue thee rocedure were formally dered year later Duett rocedure for mea aeared 955 ad formal aalye of grou tetg eermet had oly ut bega to aear by 946 Regardle mly reortg the ercetage of fected lat doe ot ge accurate etmate of the ercetage of ahd that tramt the deae or doe t allow for accurate determato of the otato arete whoe tramo rate dffer gfcatly from tobacco the cotrol 5

31 I our curret aer the data from the Bawde ad Kaa tude were eteded for our MCC cae uder a grou-tetg rotocol The dfferece betwee the roorto of fected lat of each of the arete of otato ad the roorto of fected tobacco lat were multaeouly etmated ug the Duett ad Jeffrey-Per rocedure wth 95% cofdece The etmated cofdece teral are how Table 5 From Table 5 we otce that the Duett ad the Jeffrey-Per aroache geerated ery mlar reult We ca coclude that the roorto of fected tobacco lat gfcatly greater tha the roorto of fected lat for the followg otato arete: arra baer maetc arra coul ad arra lot We dd ot fd a gfcat dfferece betwee tobacco ad the may quee otato arety Bawde ad Kaa aga erformed the ame geeral eermet ug ect but th tme they colozed the healthy lat wth four ahd rather tha two; the other codto rema the ame I Table 4 we reet the obered roorto of tet lat that were fected for the dfferet arete whe 4 A we eected creag the umber of ahd er tet lat grou ze reulted a creaed obered roorto of fected lat Sce tobacco ery ete to the Y ru; wth four ahd er lat all of the tobacco tet lat acqured the deae Now the dfferece betwee tobacco ad each of the otato arete a far a fected lat cocered larger; whch agree wth the ma cocluo roded by Bawde ad Kaa Table 6 how the Duett ad Jeffrey-Per 95% cofdece teral whe 4 Sce there a larger obered dfferece betwee tobacco ad each of the otato arete the cofdece teral are wder tha the reou cae whe Howeer from th tudy we ca coclude that tobacco gfcatly more ucetble to fecto tha all of the otato arete codered th tudy We alo obered that the Jeffrey-Per rocedure foud oly oe root for all teral wth the [-] rage ad therefore the uer lmt wa et to ; coequetly the Jeffrey-Per aroach geerated wder teral tha Duett Th ot 6

32 urrg a the Jeffrey-Per rocedure ca yeld oly oe root whe the obered roorto for oe grou at or ear zero or oe Recall that here for tobacco oe I th ecto we reeted the reult of two geeral eermet erformed by Bawde ad Kaa 946 Ther goal wa to aalyze the ucetblty of otato arete to the ru Y The eermet were coducted uder a grou-tetg ettg ad ued tobacco a a cotrol leel to comare eeral otato arete wth reect to the tramo rate The author dd ot ue ay tattcal rocedure that aduted for the grou tetg or multlcty oled th eermet I cotrat our aaly of the wor of Bawde ad Kaa ued the MCC ettg uder a grou-tetg rotocol ad cocluo that are more accurate are roded 7

33 CHAPTER 6 SUMMARY AND CONCLUSIONS McCa ad Tebb 6 reeted two aroache the Tuey-Kramer ad Jeffrey- Per for obtag arwe multaeou cofdece teral for all comaro of roorto uder a grou-tetg rotocol I ome cae howeer dfferece wth a cotrol rather tha all arwe comaro are of teret The uroe of the curret aer to rode wth a grou-tetg cearo multaeou cofdece teral for roorto dfferece betwee the cotrol ad all other leel Coequetly the McCa ad Tebb aroache were aduted for the MCC cae; wth the ma dfferece here beg that a Duett crtcal alue requred to cotruct the cofdece teral Th crtcal ot deed o the correlato betwee g ad g where the ub de rereet the cotrol leel The two rocedure dcued th aer for the MCC cae Duett ad Jeffrey- Per were dered Chater 3 ad ealuated ad comared term of coerage robablty Chater 4 Te thouad mulato were ra for each ecfc cearo whch coted of ecfed alue for ad Three four ad fe trata were alo codered wth a cofdece leel of 95 The coerage robablty wa the etmated for both rocedure thee cearo The reult how that for large grou ze [ 5 the Duett ad the Jeffrey-Per aroache both erform ery well A the Duett rocedure much le tee comutatoally we recommed Duett for thee tuato For maller grou ze the Jeffrey-Per methodology recommeded ce t cloer to the omal leel of -I ad a coerate rocedure 8

34 Both rocedure Duett ad Jeffrey-Per were alo ued to ealuate real data from a multle-ector trafer deg tudy Comarg the ucetblty of tobacco ad fe dfferet otato arete to the ru otato Y Whe a grou ze of two wa ued tobacco roed more ete tha four of the otato arete Whe a grou ze of four wa ued tobacco wa more ucetble to fecto tha all of the otato arete codered th tudy The dea reeted th aer ca alo be eteded to other cae For eamle f the etgator wat to elect treatmet or trata from a et uch that the bet treatmet or oulato cluded a wg ubet; the multle comaro wth the bet treatmet or bet trata leel could be of teret A Duett crtcal ot alo utlzed th cearo o the reult of our ettg could ealy be adated I addto although the calculato are much more tee t alo oble to eted thee reult to other et of defed cotrat whch may be requred to awer reearch queto ecfc to a artcular eermet I th cae we would ue a crtcal alue from a coerate method uch a Boferro Sda McCa- Edward or Huter-Worley aga wth fty degree of freedom a we are aalyzg roortoal data 9

35 BIBLIOGRAPHY Bhattacharyya M Karado M & DeFolart G 979 Pot etmate ad cofdece teral for fecto rate ug ooled orgam edemologc tude Amerca Joural of Edemology Bawde F & Kaa B 946 Varetal ucetblty to otato ru Y Aal of Aled Bology Beal S 987 Aymtotc cofdece teral for the dfferece betwee two bomal arameter for ue wth mall amle Bometrc Chc S 996 Bayea model for lmtg dluto aay ad grou tet data Bometrc Dorfma R 943 The detecto of defecte member of large oulato Aal of Mathematcal Stattc Duett C 955 A multle comaro rocedure for comarg eeral treatmet wth a cotrol Joural of the Amerca Stattcal Aocato Gatwrth J & Hammc A 988 Etmato of the realece of a rare deae reerg the aoymty of the ubect by grou tetg: alcato to etmatg the realece of AIDS atbode blood door Joural of Stattcal Plag ad Iferece 5-7 Gatwrth J & Joho W 994 Screeg wth cot-effecte qualty cotrol: otetal alcato to HIV ad drug tetg Joural of the Amerca Stattcal Aocato Heworth G 996 Eact cofdece teral for roorto etmated by grou tetg Bometrc Hochberg Y & Tamhae A 987 Multle Comaro Procedure New Yor: Wley Hu J 996 Multle Comaro: Theory ad Method Boca Rato: Chama & Hall/CRC Hughe-Oler J & Roeberger W Effcet etmato of the realece of multle rare trat Bometra Kle R Brother T Broomeyer R Zeger S & Qu T 989 Ealuato of huma mmuodefcecy ru erorealece oulato urey ug ooled era Joural of Clcal Mcrobology

36 Lehma E & Caella G 998 Theory of ot etmato Srger Tet Stattc Secod Edto McCa M & Tebb J 6 Parwe comaro for roorto etmated by ool tetg Joural of Stattcal Plag ad Iferece re Pegorch W 99 Multle comaro for aalyzg dchotomou reoe Bometrc Swallow W 985 Grou tetg for etmatg fecto rate ad robablte of deae tramo Phytoathology Tebb J & Blder C 4 Cofdece teral rocedure for the robablty of deae tramo multle-ector-trafer deg Joural of Agrcultural Bologcal ad Erometal Stattc 9-6 Thoma J Paterac B Vacrca S & Thomo D 973 Alcato of grou tetg rocedure radologcal health Health Phyc Pergamo Pre Thomo K 96 Etmato of the roorto of ector a atural oulato of ect Bometrc We L & Zeo S 996 Pooled tetg for HIV creeg: caturg the dluto effect Oerato Reearch

37 Table : Etmated multaeou coerage robablte for the Duett D ad Jeffrey-Per JP rocedure wth I5 where the umber of tet ad the grou ze are cotat for all trata ad the frt tratum codered the cotrol leel *The etmated coerage robablte bold are thoe wth two etmated tadard deato ug 95 a the true roorto D JP D JP D JP D JP D JP D JP D JP D JP D JP D JP D JP D JP 3

38 Table : Percetage of tme the adutmet for Duett D ad Jeffrey-Per JP wa made durg the mulato for D JP D JP D JP D JP D JP D JP D JP D JP D JP D JP D JP D JP 33

39 Table 3: Percetage of tme the adutmet for Duett D ad Jeffrey-Per JP wa made durg the mulato for D JP 5 D JP D JP 5 D JP D JP 5 D JP D JP 5 33 D 37 JP D JP 5 D JP 9 D JP 5 44 D 4 9 JP 34

40 Table 4: Bawde ad Kaa 946 Obered roorto of fected lat wth the otato ru Y for dfferet arete *Ug two ahd to coloze the healthy lat Tet lat Proorto of fected lat Arra Baer 3/3683 Maetc 5/3639 Arra Coul /3678 Arra Plot /36333 May Quee 5/3647 Tobacco 5/36694 *Ug four ahd to coloze the healthy lat Tet lat Proorto of fected lat Arra Baer /3678 Maetc 7/3647 Arra Coul 9/3658 Arra Plot 5/36694 May Quee /36583 Tobacco 36/36 35

41 Table 5: Aaly of the multle-ector trafer deg tudy of Bawde & Kaa 946 Duett ad Jeffrey-Per 95% multaeou cofdece teral for - where roorto of tobacco fected lat ad roorto of fected lat for the dere arete of otato *Two ahd were ued to coloze each tet lat Dfferece betwee Pot Etmate Cofdece Iteral arete Duett Jeffrey-Per Tobacco - Arra Baer Tobacco - Maetc Tobacco - Arra Coul Tobacco - Arra Plot Tobacco - May Quee

42 Table 6: Aaly of the multle-ector trafer deg tudy of Bawde & Kaa 946 Duett ad Jeffrey-Per 95% multaeou cofdece teral for - where roorto of tobacco fected lat ad roorto of fected lat for the dere arete of otato *Four ahd were ued to coloze each tet lat Dfferece betwee Pot Etmate Cofdece Iteral arete Duett Jeffrey-Per Tobacco - Arra Baer Tobacco - Maetc Tobacco - Arra Coul Tobacco - Arra Plot Tobacco - May Quee

43 APPENDIX A: The mamum lelhood etmator of Recall that the robablty that a grou of ze ubect ha at leat oe ubect oeg the trat of teret The uual bomal MLE of etmated wth the followg rocedure The lelhood ge by: where ge by X Y l l wth Yl! L!! ; defed Chater Thu the atural logarthm of the lelhood $!! l L l " l!! # l ad the frt derate of the l L ge by d l L d To fd the MLE we et the frt derate of l L to zero ad ole for To eure that we hae a mamum we eed to fd the ecod derate of l L ealuate th for ad erfy that the reult egate Here d l L d d l L ad < d ce ad ; hece a mamum Thu by the arace roerte of mamum lelhood etmator the MLE of / / where 38

44 39 APPENDIX B: The dtrbuto of A the bomal lelhood a member of the regular eoetal cla REC the regularty codto regardg the lmtg dtrbuto of the MLE are atfed ad the dtrbuto of aymtotcally ormal wth aymtotc mea ad aymtotc arace ge by l L d d E To calculate the aymtotc arace we frt coder the deomator where the lelhood ge by L!!! ; ad the atural logarthm of the lelhood l l!!! l l L! " " # $ The frt ad ecod derate of l L are ge by the followg equato: l d L d ad l d L d The eected alue of the ecod derate of l L ge by ] [ ] [ l E E d L d E ce E ] [ after multlyg by - the deomator of the aymtotc arace become l d L d E Coequetly the aymtotc arace of L d d E l

45 Now the delta-method tate: If T N et ad t ot zero Recall that the { h T h } N { h' } g / roded h' ad ce aboe we dered the aymtotc ormal dtrbuto of the alyg the delta-method we ca dere the dtrbuto of follow: h' / a / [ h' ] / [ h' ] / / ce Thu alo follow a aymtotc ormal dtrbuto wth mea ad arace Lehma E & Caella G 998 4

46 4 APPENDIX C: The etmated aymtotc arace of b I aed B we howed that ha a aymtotc ormal dtrbuto wth mea ad arace / ] [ Sce b a lear combato of arable wth deedet aymtotc ormal dtrbuto; the t dtrbuto alo aymtotc ormal wth mea ad arace Recall that a ad b ad ote that b a ad alo that b a Now ug a ~ the oteror mea of a dered Chater a a etmate of a ad lettg ~ ~ b a b a g ad ~ ~ b a b a g we ca rewrte the aymtotc arace of b a: b a g b a g b a g b a g a a b V ~ } ~ { ~ } ~ { ~ ;

47 VITA Yaa Grat Caddate for the Degree of Mater of Scece The: MULTIPLE COMPARISONS WITH A CONTROL UNDER A GROUP TESTING SCENARIO Maor Feld: Stattc Bograhcal: Peroal Data: Bor Torreó Coahula Meco Educato: Receed Bachelor of Scece degree Ecoomc wth a mor Face from the Techcal Ittute of Adaced Stude of Moterrey Moterrey Nueo Leó Meco May Comleted the requremet for the Mater of Scece degree wth a maor Stattc at Olahoma State Uerty Deartmet of Stattc May 6 Eerece: Emloyed a Maretg Aalyt by the Deartmet of Maretg Vllacero Grou Moterrey Nueo Leó Meco from Jauary to Arl Itered at the Neuroharmacology Deartmet Scr Reearch Ittute from Jue 5 to Augut 5 Profeoal Memberh: Amerca Stattcal Aocato Ittute of Mathematcal Stattc 4

48 Name: Yaa Grat Date of Degree: May 6 Ittuto: Olahoma State Uerty Locato: Stllwater Olahoma Ttle of Study: MULTIPLE COMPARISONS WITH A CONTROL UNDER A GROUP TESTING SCENARIO Page Study: 4 Caddate for the Degree of Mater of Scece Maor Feld: Stattc Scoe ad Method of Study: The obecte of th roect to dere two multaeou ferece aroache Duett ad Jeffrey-Per to quatfy roorto dfferece betwee a cotrol ad eeral o-cotrol leel uder a grou tetg cearo Both rocedure were ealuated term of etmated coerage robablty a mulato Te thouad mulato were ru for dere data cearo ad the coerage robablty wa etmated by calculatg the roorto of tme the cofdece teral multaeouly cotaed the real roorto dfferece Fdg ad Cocluo: It wa foud that whe ug large grou ze of about 5 or more oberato both the Duett ad Jeffrey-Per aroache erform farly well For mall amle ze betwee 5 ad oberato the Jeffrey-Per teral are recommeded a ther coerage cloer to the omal -I leel ad t alo a coerate aroach whle the Duett rocedure atcoerate We alo foud that grou tetg rode more accurate reult tha ddual tetg ce more coerage robablte are cloer to the omal whe ug grou ze greater tha oe ADVISOR S APPROVAL: Dr Melda McCa 43