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Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2008-02-29.

Permanent Link: http://ufdc.ufl.edu/UFE0019743/00001

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Title: Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2008-02-29.
Physical Description: Book
Language: english
Creator: Ryu, Euijung
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Statement of Responsibility: by Euijung Ryu.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Agresti, Alan G.
Electronic Access: INACCESSIBLE UNTIL 2008-02-29

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0019743:00001

Permanent Link: http://ufdc.ufl.edu/UFE0019743/00001

Material Information

Title: Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2008-02-29.
Physical Description: Book
Language: english
Creator: Ryu, Euijung
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Statement of Responsibility: by Euijung Ryu.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Agresti, Alan G.
Electronic Access: INACCESSIBLE UNTIL 2008-02-29

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0019743:00001


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FirstofallIexpressmydeepestgratitudetoDr.AlanAgrestiforservingasmydissertationadvisorandoeringendlesssupportandcontinuousencouragement.Also,Ithankdrs.RonaldRandles,MichaelDaniels,BabetteBrumback,andJamesAlginaforservingonmycommittee.Iamverygratefultomyparentsforconstantloveandcondenceinmysuccess.Withouttheirsupport,IwouldnotbethepersonIamnow.Ialsothankmysistersandbrotherforendlessemotionalsupport. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 11 ABSTRACT ........................................ 12 CHAPTER 1INTRODUCTION .................................. 13 1.1AnOrdinalEectSizeMeasure ........................ 14 1.1.1PropertiesoftheMeasure ....................... 15 1.1.2AreaUnderReceiverOperatingCharacteristicCurve ........ 16 1.2Mann-WhitneyStatisticanditsVarianceExpression ............ 18 1.2.1ContinuousCase ............................. 18 1.2.2CategoricalCase ............................. 19 1.3ExistingMethodstoFindCondenceIntervals ................ 21 1.3.1Halperin,Hamdy,andThall(HHT)CondenceInterval ...... 22 1.3.2Newcombe'sScoreCondenceInterval ................. 25 1.4OutlineofDissertation ............................. 27 2CONFIDENCEINTERVALSUNDERANUNRESTRICTEDMODEL ..... 31 2.1BasicIntroductionofFourCondenceIntervals ............... 31 2.1.1WaldCondenceInterval ........................ 31 2.1.2LikelihoodRatioTest(LRT)-basedCondenceInterval ....... 32 2.1.3ScoreCondenceInterval ........................ 32 2.1.4PseudoScore-typeCondenceInterval ................ 32 2.2WaldCondenceIntervalfor 33 2.2.1WaldCondenceIntervalbasedontheLogitTransformation .... 35 2.2.2ComparisonwithNewcombe'sWaldCondenceInterval ....... 35 2.3RestrictedMLEstimates ............................ 37 2.3.1Multinomial-PoissonHomogeneousModels .............. 38 2.3.2AlgorithmstoFindtheRestrictedMLEstimates ........... 43 2.4LikelihoodRatioTest-basedCondenceInterval ............... 46 2.5ScoreCondenceInterval ............................ 46 2.6PseudoScore-typeCondenceInterval .................... 47 3CONFIDENCEINTERVALSUNDERAPARAMETRICMODEL ....... 49 3.1WaldCondenceIntervals ........................... 50 3.2RestrictedMLEstimates ............................ 53 3.3CondenceIntervalsbasedonRestrictedMLEstimates ........... 55 5

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..................................... 56 4SIMULATIONSTUDIES .............................. 58 4.1FactorialDesignofConditions ........................ 58 4.2EvaluationCriteriaforCondenceIntervals ................. 61 4.2.1CoverageProbability .......................... 61 4.2.2ExpectedLength ............................ 61 4.2.3OverallSummaries ........................... 62 4.3ComparisonofMethods ............................ 62 5CONFIDENCEINTERVALSFORMATCHED-PAIRSDATA .......... 69 5.1TwoOrdinalEectSizeMeasures ....................... 71 5.2WaldCondenceIntervals ........................... 73 5.3RestrictedMLEstimation ........................... 74 5.4LRT-basedCondenceInterval ........................ 75 5.5ScoreCondenceInterval ............................ 75 5.6PseudoScore-typeCondenceInterval .................... 76 5.7Example:DataAnalysis ............................ 76 5.8SimulationStudy ................................ 77 6CONFIDENCEINTERVALSFORFULLY-RANKEDDATA .......... 81 6.1PerformanceoftheMethods .......................... 83 6.2ConnectionswithanEectSizeMeasureforNormalDistributions ..... 84 7MODELINGTHEORDINALEFFECTSIZEMEASUREWITHEXPLANATORYVARIABLES ..................................... 87 7.1Fixed-EectsModellingwithCategoricalCovariates ............. 89 7.1.1MaximumLikelihoodEstimation ................... 91 7.1.2ScoreCondenceInterval ........................ 95 7.1.3Goodness-of-tTests .......................... 95 7.1.4Example:DataAnalysis ........................ 96 7.1.5Simpson'sParadox ........................... 96 7.2Fixed-EectsModellingwithContinuousCovariates ............. 97 7.2.1CumulativeLogitModelswithContinuousCovariates ........ 97 7.2.2Example:DataAnalysis ........................ 99 7.3Random-EectsModelling ........................... 101 7.3.1CumulativeLogitsModelswithRandomEects ........... 103 7.3.2EstimationandPrediction ....................... 105 7.3.3Example:DataAnalysis ........................ 111 8SUMMARYANDFUTURERESEARCH ..................... 114 8.1Summary .................................... 114 8.2FutureResearch ................................. 116 6

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APROOFS ....................................... 117 BRCODES ....................................... 130 B.1RCodestoCalculateScoreCondenceIntervalfora2cTable ...... 130 B.2RCodestoCalculateLogitWaldCondenceIntervalforMatched-PairsData ....................................... 136 COPINIONABOUTSURROGATEMOTHERHOODANDTHELIKELIHOODOFSELLINGAKIDNEY .............................. 139 REFERENCES ....................................... 143 BIOGRAPHICALSKETCH ................................ 147 7

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Table page 1-1Shouldertippainscoresafterlaparoscopicsurgery ................. 13 1-2Twosetsofcellprobabilitieswith=0:5 ..................... 16 1-3Frequenciesoftwocategoricalvariableswithccategories ............. 19 2-1RestrictedMLestimatesunderH0:=0:5 .................... 43 2-2Non-zerocountssummarizedina23table .................... 44 2-3RestrictedMLestimatesofcellprobabilitiesinTable2-2 ............. 45 3-1CondenceintervalsforinTable1-1withandwithoutassumingacumulativelogitmodel ...................................... 56 4-1Cellprobabilitiesfordierentconditions ...................... 60 4-2Withsamplesizes100each,coverageprobabilities(CP)andoverallsummariesfromsimulationstudyforcasesinwhichcumulativelogitmodelholds(CL)ordoesnothold(NotCL) ............................... 65 4-3Withsamplesizes50each,coverageprobabilities(CP)andoverallsummariesfromsimulationstudyforcasesinwhichcumulativelogitmodelholds(CL)ordoesnothold(NotCL) ............................... 65 4-4Withsamplesizes(50,100),coverageprobabilities(CP)andoverallsummariesfromsimulationstudyforcasesinwhichcumulativelogitmodelholds(CL)ordoesnothold(NotCL) ............................... 66 4-5Withsamplesizes(10,100),coverageprobabilities(CP)andoverallsummariesfromsimulationstudyforcasesinwhichcumulativelogitmodelholds(CL)ordoesnothold(NotCL) ............................... 66 4-6Withsamplesizes(10,50),coverageprobabilities(CP)andoverallsummariesfromsimulationstudyforcasesinwhichcumulativelogitmodelholds(CL)ordoesnothold(NotCL) ............................... 67 4-7Withsamplesizes(10,10),coverageprobabilities(CP)andoverallsummariesfromsimulationstudyforcasesinwhichcumulativelogitmodelholds(CL)ordoesnothold(NotCL) ............................... 67 4-8Meancoverageprobabilities(CP)fortwelve,averagedoversamplesizes(100,100)or(50,50),c=3and6,=0:5and0:8 ................... 68 4-9Overallperformancesummariesofcoverageprobability(CP)fromsimulationstudyfortwelvemethods,averagedoverseveralsamplesizes,c=3and6,=0:5and0:8,andwhetherornotacumulativelogitmodelholds ......... 68 8

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........................ 69 5-2Jointcellprobabilitiesofthematched-pairsdata .................. 70 5-32ctablewithmarginalrowtotalsandcolumntotalsfromTable5-1 ..... 71 5-4Opinionaboutpremaritalandextramaritalsex .................. 76 5-595%CondenceIntervalsforMP1 77 5-6Jointcellprobabilitieswithc=6,=0:5,and=0:8 .............. 77 5-7Overallperformancesformatched-pairsdataoverallconditionsconsidered ... 78 5-8Coverageprobabilitiesformatched-pairsdatawithc=6andsamplesizes=25 79 5-9Coverageprobabilitiesformatched-pairsdatawithc=3andsamplesizes=25 79 5-10Coverageprobabilitiesformatched-pairsdatawithc=6andsamplesizes=50 79 5-11Coverageprobabilitiesformatched-pairsdatawithc=3andsamplesizes=50 80 5-12Coverageprobabilitiesformatched-pairsdatawithc=6andsamplesizes=75 80 5-13Coverageprobabilitiesformatched-pairsdatawithc=3andsamplesizes=75 80 6-1Fully-rankeddatawithc=5 ............................. 81 6-2Coverageprobabilities(CP)andoverallsummariesforfully-rankeddatawithsamplesizes(10,10)and(20,30) .......................... 84 6-3Relationshipbetweenand 84 7-1Shouldertippainscoresstratiedbygenderandage ............... 87 7-2MLestimatesofkandjparameters,withtheir95%scorecondenceintervals 96 7-3Opinionaboutsurrogatemotherhoodandthelikelihoodofsellingakidneyatage87 ......................................... 100 7-4Pointestimatesandcondenceintervalsof(x)asage(x)varies,undermain-eectsmodel ......................................... 101 7-5Clinicaltrialrelatingtreatmenttoresponseforeightcenters ........... 102 7-6MLestimatesofparametersandtheirstandarderrors .............. 111 7-7EBestimatesof'sandtheirstandarderrorsofprediction ............ 113 C-1Opinionaboutsurrogatemotherhoodandthelikelihoodofsellingakidneywithagebetween18and35 ................................ 139 9

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................................ 140 C-3Opinionaboutsurrogatemotherhoodandthelikelihoodofsellingakidneywithagebetween56and75 ................................ 141 C-4Opinionaboutsurrogatemotherhoodandthelikelihoodofsellingakidneywithagebetween76and89 ................................ 142 10

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Figure page 1-11-specicity=P(Y1>k)andsensitivity=P(Y2>k),k=1;2;3,and4. ...... 17 6-1PlotofasymptoticeciencyofMann-Whitneyestimaterelativetoparametricestimate ........................................ 86 11

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Anordinalmeasureofeectsizeisasimpleandusefulwaytodescribethedierencebetweentwoorderedcategoricaldistributions.Thismeasuresummarizestheprobabilitythatanoutcomefromonedistributionfallsaboveanoutcomefromtheother,adjustedforties.Theordinaleectsizemeasureissimpletointerpretandhasconnectionswithacommonlyusedeectsizemeasurefornormaldistributions.Wedevelopandcomparecondenceintervalmethodsforthemeasure.Simulationstudiesshowthatwithindependentmultinomialsamples,condenceintervalsbasedoninvertingthescoretestandapseudo-scoretestperformwell.Thisscoremethodalsoseemstoworkwellwithfully-rankeddata,butfordependentsamplesasimpleWaldintervalonthelogitscalecanbebetterwithsmallsamples.Wealsoexplorehowtheordinaleectsizemeasurerelatestoaneectmeasurecommonlyusedfornormaldistributions,andweconsideralogitmodelfordescribinghowitdependsoncategoricalexplanatoryvariables.Themethodsareillustratedforseveralstudiesthatcomparetwogroups,includingastudycomparingtwotreatmentsforshouldertippain. 12

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Whenweareinterestedincomparingtwogroups,itisusefultoknownotonlywhetherornottheyhaveastatisticallysignicantdierence,butalsotheeectsize,whichisameasuretodescribethedierencebetweenthetwogroups.Forinstance,considerthesituationinwhichascoreissampledfromonedistributionandascoreissampledfromanotherdistribution.Ifweconsideracontinuousoutcomehavingadistributionsuchasthenormal,thentherearetwokindsofmeasurestoexplaintheeectsize:anabsolutemeasureandarelativemeasure.Anexampleofanabsolutemeasureisthedierenceinthetwogroupmeans,andanexampleofarelativemeasureisthestandardizeddierence,obtainedbydividingthedierenceoftwomeansbythepooledstandarddeviationforthosemeans,whichisreferredasCohen'sd(Cohen1992).Iftheoutcomesarebinary,thedierenceofproportionsisusedasanabsolutemeasureandtherelativeriskandtheoddsratioareusedasrelativemeasures. Acommoninterestinmanyresearchareasistocomparetwogroupswhenameasurementisonanorderedcategoricalscale.Forinstance,weuseastudy(Lumley1996)thatcomparesanactivetreatmentwithacontroltreatmentforpatientshavingshouldertippainafterlaparoscopicsurgery(Table1-1). Table1-1. Shouldertippainscoresafterlaparoscopicsurgery Treatments12345 Active192100Control73432 Thetwotreatmentswererandomlyassignedto41patients.Thepatientsratedtheirpainlevelonascalefrom1(low)to5(high)onthefthdayafterthesurgery.Theresponsesareorderedcategoricalandcanbesummarizedbyfrequenciesina25contingencytable.Theabsolutedistancesbetweenthecategoriesareunknown.Whenapatient'spainlevelis\high,"itisclearthatthepatienthasmorepainthanapatientwhorateshis/herpainlevellowerthanthat.Butitisunclearhowtoassignanumerical 13

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Whenresponsesareordinal,acommonmethodistoassignscorestothecategoriesandndthemeans.Thescoresassigndistancesbetweencategoriesandtreatthemeasurementscaleasinterval.Ifamodelingapproachsuchasthecumulativelogitmodelwithproportionaloddsstructureisused,thenthegroupeectisexplainedbyanoddsratiosummaryusingcumulativeprobabilities(Agresti2002).Inourdissertationweuseanalternativemeasurethattreatstheresponseasordinalbutissimplertointerpretforanaudiencenotfamiliarwithoddsratiosandthathasconnectionswithaneectsizefornormaldistributions. 2P(Y1=Y2); 14

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2(Y1=Y2)=cXa=1P(Y2>a)P(Y1=a)+1 2cXa=1P(Y2=a)P(Y1=a)=c1Xa=1cXb>aab+1 2cXa=1aa=TA;(1{1) whereA=0BBBBBBBBBBBBBB@0:50010:500......1110:501110:51CCCCCCCCCCCCCCA: 15

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FromProperty1.3,twoidenticalvariablesgive=0:5.However,manydistributionsthatarenotidenticalcangive=0:5iftheygivenopreferenceintermsoflikelyhavinglargervalues(Troendle2002),whichincludessymmetricdistributions.Forinstance,cellprobabilitiesinTable1.2give=0:5. Table1-2. Twosetsofcellprobabilitieswith=0:5 Identicaldistribution 12345 12345 16

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1-specicity=P(Y1>k)andsensitivity=P(Y2>k),k=1;2;3,and4. itsverticalcoordinate\sensitivity"foreachpointk,includingthepoint(1;1).Inotherwords,theROCcurvehaspointsf(P(Y1>c);P(Y2>c));(P(Y1>c1);P(Y2>c1));;(P(Y1>1);P(Y2>1));(1;1)g; 2P(Y1=Y2)=:

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whereQ1=P[Y1i
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Table1-3. Frequenciesoftwocategoricalvariableswithccategories 12c1c Y1n11n12n1(c1)n1cY2n21n22n2(c1)n2c 2ifx=y0ifx>y: 2cXa=1n1an2a: 4cXi=1ii#;(1{3) 19

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LetY1;Y1;Y2,andY2beindependentlydistributed.FromNoether(1967),thevarianceofUisVar(U)=n1n2

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4#: 4#: Whenc=2,whichrelatestothedierenceofproportion,therearemanypossibilitiesforcondenceintervals,includingelevenmethodsevaluatedbyNewcombe(1998).For 21

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Halperin,Hamdy,andThall(1989)providedadistribution-freecondenceintervalfor,whichisbasedonthepivotalquantityZ2HHT=(^)2=^VHHT;where^VHHTusesestimatesofsomeparametersbutisanexplicitfunctionof.Basedonasimulationstudy,Halperin,Hamdy,andThall(1989)mentionedthattheirapproachisasgoodasorbetterthanthatofHochberg's(1981)U-statistic-basedmethod,andisespeciallybetterforextremevaluesof,i.e.,relativelyfarawayfrom0.5,intermsofdeviationfromnominalcoverageprobability.Below,wewilldiscusstheHalperin,Hamdy,andThall(1989)method. ThemainpartoftheHalperin,Hamdy,andThall(1989)methodistond^VHHT.TheideacamefromHalperin,Gilbert,andLachin(1987),inwhichtheyusedapivotalquantitytoobtainadistribution-freecondenceintervalforP(Y1
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Fromequation(1-3),thevarianceof^isVd=(n1n2)2=1 4cXi=1ii#: 4cXi=1ii;(1{4) sothatVd=(n1n2)=V1+V2andclearlyV1isafunctionofalone.NotethatV2isboundedaboveandbelowbyexplicitfunctionsofsince2C;D.Thatis,LV2U,whereL=(n1+n22)21 4andU=(n1+n22).Thus,thereexistsawhichisbetween0and1suchthat Assumingminfn1;n2g!1,then1 4Pci=1ii=o(minfn1;n2g)inequation(1-4)and 23

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4(n21)cXi=1^i^i(1^i)~D=^D1 4(n11)cXj=1^j^j(1^j)^(1)=(n1n2n1n2+2)^n1n2^2 n1+~D n2: Thismethodisrobusttodistributionalassumptions.However,ithasthedrawbackofignoringanyinformationthatmightbeavailableontheunderlyingdistribution.Anotherproblemofthismethodisthattheyignoredsomelower-ordertermsinV2tondtheestimatedvariance^VHHT.Althoughitgivesasimpleformfor^VHHT,itispossiblethat 24

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Recallthatanonparametricestimatorofis^=U=n1n2andassumingbothY1andY2havecontinuousdistributions,equation(1-2)givesastandarderrorof^,s:e:(^)=s Wenowdiscusstheirsecondmethod.Firstnotethats:e:(^)involvesthreeparameters,,Q1,andQ2.AlthoughQ1andQ2dependontheunderlyingdistributionweassume, 25

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Thereisnoclosedformsolutionofthisequation.But,squaringbothsides,itisaquarticfunctionfor.SoitcanbesolvedbyiterativemethodssuchastheNewton-Raphsonalgorithm. BlythandStill(1983)mentionedthatanequivariancepropertyisdesiredforabinomialcondenceinterval.Themeaningofequivarianceintheirpaperisasfollows.SupposethatYwithBinomial(n;p)distributionhasacondenceinterval(L;U)forpandconsidertransformationsY!nYandp!1p.Thecondenceintervalissaidtohavetheequivariancepropertyifthecondenceintervalforpisequalto(1U;1L).Forordinalcategoricaldata,theconceptoftheequivarianceisrelatedtoreversingthecategoriesfrom(1;2;;c)to(c;;2;1).FromProperty1.2inSection1.1,thecorrespondingisequalto1.Becauseoftheasymmetryoftheexponentialdistributions,thecondenceintervalfromequation(1-6)doesnotpossesstheequivarianceproperty.Newcombe(2006b)modiedthiscondenceintervaltosatisfyanequivarianceproperty,suggestingtousen1+n2 26

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Tohavetheequivarianceproperty,therighthandsidesinequation(1.6)and(1.7)shouldbethesame.Thisissatisedifbothn1andn2areequal,buttheyarenotequalingeneral.Eventhoughthesearenotequalfromthedata,thepropertyholdsifthesamevalueforn1andn2isusedinthenumerator.Forinstance,ifthemidpoint,(n1+n2)=2,ofn1andn2issubstitutedforn1andn2inthenumerator,thepropertyholds.However,itshouldbenotedthatthiscondenceintervalisvalidonlyifn1andn2growatthesamerate.Newcombe(2006b)mentionedthatitmayperformadequatelyunderamisspecicationofthetruedistributions,butitsvalidityisquestionablesinceitisbasedonastrongassumptionaboutthedistributionswhichmaynothold. InmyPhDresearch,Iinvestigateothermethodstondcondenceintervalsfor.Usingalikelihoodfunction,whichdependsonthemodelstructureweuse,weconsiderseveralasymptoticcondenceintervals:Waldcondenceintervals,thelikelihoodratiotest(LRT)-basedcondenceinterval,thescorecondenceinterval,andapseudoscore-typecondenceinterval.ThesecondenceintervalsareobtainedbyinvertingthecorrespondingteststatisticsforH0:=0.Sincecanbeexpressedintermsoftheparametersinthemodel,thenullhypothesisH0:=0givesaconstraintfunctionoftheparameters. 27

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Whenwehavea2ccontingencytable,itisnaturaltoassumethatthecountsineachrowhaveamultinomialdistributionandthetwodistributionsareindependent.InChapter2,wedevelopthecondenceintervalsunderanunrestrictedmodelinwhichwedonotassumeanyrelationshipbetweentheprobabilitiesintherstandsecondrows.Underthisassumption,wecanusealikelihoodfunctionthatisafunctionofthe2(c1)nonredundantcellprobabilities.Then,isexpressedintermsofthosecellprobabilities,andsotheconstraintfunctioncorrespondingtothenullhypothesisisalsoafunctionoftheseprobabilities. BasedontheAitchisonandSilvey(1958)methods,Lang(2004)showedauniedtheoryofMLinferenceincontingencytableswithaconstraintthatissucientlysmoothandhomogeneous.AdenitionofbeinghomogenousisdescribedinLang(2004).Althoughheexpressedthelikelihoodandtheconstraintintermsofexpectedcellcountsratherthanactualcellprobabilities,wecanusehisresulttondtherestrictedMLestimatesofcellprobabilitiesaftershowingthatoursituationisaspecialcaseofLang's.WendtheLRT-basedcondenceinterval,thescorecondenceinterval,andapseudoscore-typecondenceintervalusingtheserestrictedMLestimates.ForWaldcondenceintervals,wewillusetheactualvarianceinsteadofanasymptoticformofthevariance. 28

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Waldcondenceintervalscanbeobtainedbyusingparticularalgorithmsincludedinstandardsoftwares,becausetheygiveMLestimatesfortheparametersandsowhatweneedtodotondthecondenceintervalistosubstitutetheestimatesintheformofanditsvariance.FortheLRT-basedcondenceintervalandthescorecondenceinterval,andapseudoscore-typecondenceinterval,weagainusetheAitchisonandSilvey(1958,1960)methodstondrestrictedMLestimatesofparametersinthecumulativelogitmodelunderthenullhypothesis=0.WecanuseeitheraNewton-RaphsonalgorithmorLang'salgorithm. Chapter4willdiscussaplannedsimulationstudytocomparetheproposedcondenceintervalsforina2ctabletoexistingmethods.Forthispurpose,wegeneratedataunderseveralconditionsbychangingthesamplesize,thenumberofcolumns,thetruevalue,andwhethertheparametricmodelholds.Forevaluationcriteria,wewillusethecoverageprobabilityandthreeoverallsummaries. InChapter5and6,weconsidertwodierentdatastructuresthatarenotindependent.Matched-pairsdataandfully-rankeddatacasesareconsideredinChapter5and6,respectively.Formatched-pairsdatainChapter5,wewilldealwithdatathataresummarizedinaformofacccontingencytableinsteadof2ctable,whichproducesadependencyinthetwosamples.Tocomparethetwosamples,wecanusetheparameter,whichissameastheoneusedintheprevioussections,exceptmarginalrowtotalsandcolumntotalsareused.InChapter6forfully-rankeddata,a2ctableisconsidered,buteachcolumnreferstoasingleobservation.Forbothchapters,wewilldiscusshowtouse 29

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InChapters2and3,wehavefocusedonndingcondenceintervalsforinasingle2ctablewhentherearenoexplanatoryvariables.InChapter7,weproposemodellingwhenexplanatoryvariablesexist.Iftheyexist,weshouldconsideratleasttworelatedcontingencytablessimulatively.Weconsideronlythecaseinwhichthereareexplanatoryvariablesforanunrestrictedmodel.Theparameterineach2ctableisexpressedintermsofunrestrictedcellprobabilities,andthevectorof's(orthelogitscalesof's)fromthetablesismodelledbyalinearformoftheeectparametersofexplanatoryvariables.Inthiscase,thelikelihoodfunctioncanbeexpressedasafunctionofthecellprobabilities,andthecorrespondingconstraintfunctioncanbeexpressedintermsofthecellprobabilitiesaswellastheeectparameters.Lang(2005)developedatheoryforaso-calledhomogeneouslinearpredictor(HLP)model.WewillusehisresultaftershowingthatourmodelisaspecialcaseoftheHLPmodel.Toestimate'sandtheeectparameters,wewillusethescorecondenceintervalssincethemethodperformswell,basedonsimulationstudiesdiscussedinChapter4. 30

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AssumethatthatY1andY2areindependentrandomsamplesfrommultinomialdistributionsofsizesn1andn2withcellprobabilities=(1;2;;c)Tand=(1;2;;c)Twithfrequenciesfnijg.SinceY1andY2areindependent,thelog-likelihoodisl(;)=yT1log()+yT2log();

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32

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Dene=fvec(;):1T=1;1T=1gwhichisasubsetof[0;1]2c.Thenwehavethefollowinglemmaswhoseproofsareintheappendix. Theselemmasimplythatthereexistsauniqueglobalmaximumofl(;)onthedomain.Bytakingapartialderivativeforeachparameter,wegetthefollowingmaximumlikelihoodestimators(MLE)forand:^a=n1a n1n2:

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4Xi^i^i#: 34

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AmorepromisingWaldapproachconstructstheintervalforatransformationof,suchaslogit(),andtheninvertsittothescale.Fromthedeltamethod,theWaldcondenceintervalforlogit()islogit(^)z=2q 35

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4Xi^i^i#: Toseetherelationshipbetween^V^and^VN,let'sndtherelationshipbetween^Cand^Q1rst.Recallthat^C=Pc1a=1^a(Pcb=a+1^b+^a 2^a+Pb>a^b)2,lettingPb>c^b=0.Then^C=cXa=1^a(1 2^a+Xb>a^b)2=cXa=1n1a 2n2a 4n1n2Pi^i^i.Thismeansthatthe 36

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LetH0=f(;):TA=0;1T=1;1T=1g: Thislemmaimpliesthatthereexistsvec(~;~)2H0suchthatsupvec(;)2H0l(;)=yT1log(~)+yT2log(~): [Diag()Diag()]10B@y1y21CA+T=0TA0=01T1=01T1=0;(2{1) 37

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AitchisonandSilvey(1958,1960)showedthat,undercertainfairlygeneralconditions,ifvec(y1;y2)belongstoasetwhoseprobabilitymeasuretendsto1asngoesto1,theequationshaveasolutionvec(~;~)thatmaximizesl(;)underthenullhypothesis.WecanndasolutionusingtheNewton-Raphsonalgorithm.Fromequation(2-1),therearenorestrictionstotherangeofthecellprobabilitiesand,whichmustbebetween0and1.Becauseofthis,sometimestheequationshaveasingularHessianmatrix,andsowecannotndsolutionsusingtheNewton-Raphsonalgorithm.Also,sometimestheequation(2-1)givessolutionsthatareoutsidethepermissiblerange,althoughtheNewton-Raphsonalgorithmconverges. Thentheparameterspace0isequivalenttothesetfvec(m1;m2):m1>0;m2>0;mT11=n1;mT21=n2;h(m1;m2)=0g: 38

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AMPHmodelischaracterizedbyanindependentsamplingplan(Z;ZF;n)andasystemofhomogeneousconstraintsh(m)=0.ThematrixZ,whichisregardedasapopulationmatrix,determinesthestratafromwhichthesamplesaretakenandZF,whichisregardedasasamplingconstraintmatrix,determineswhichstratasamplesizesarexedapriori.nisavectorofapriorixedsamplesizes.Bothmatricesaredenedasfollows. if(i)Zik2f0;1g,(ii)Zi+=1and(iii)Z+k1. Conditions(ii)and(iii)areequivalenttoassumethateachobservationoccursinoneandonlyonestratum,andeachstratumhasatleastoneobservations. withrespecttopopulationmatrixZifZF=ZQF,whereQF2=fQ:Qij2f0;1g;Q+j=1;Qi+1g[f0g. 39

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FromLang(2004),amodelisaMPHmodelifthereexistasamplingplanandtheconstraintfunctionhisZ-homogeneous,whichisdenedinDenition2.3,andsatisesthefollowingthreeconditions: (1)fm:m>0;h(m)=0g6=?, (2)hhascontinuoussecond-orderderivatives, (3)H(m)hasfullcolumnranku.

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Ontheotherhand,thefunctionh(x)=(x1x3;x22x2x4)TisnotZ-homogeneous,sinceh(D(Z)x)=0B@1x12x321x2212x2x41CA Lemma2.4saysthattheconstraintfunctiondiscussedinthissectionisZ-homogenousoforder0.Theproofisintheappendix. 41

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(1Tm1)(1Tm2)0B@ATm2mT2Am1 Theestimator(~;~)canbecalculatedbyusingtheMPHmodel-ttingprogram,whichisavailablebyrequestfromDr.JosephB.Lang(\joseph-lang@uiowa.edu").Unlike 42

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ThebasicassumptionbehindndingtherestrictedMLestimatesforbothmethodsisthatallcellprobabilitiesshouldbestrictlypositive.Whenzerocountsexistinthetable,Lang(2004)introducedanextendedMLestimatethatallowszeroastheMLestimateandhisalgorithmusingtheextendedMLestimatesoftenconvergestoasolution.Alsoheusedtuningparameterstohelpthealgorithmtomoveawayfromtheboundaryoftheparameterspace,andsothealgorithmoftenconvergestoasolution.Forinstance,headdedasmallvalue,say0.1,totheoriginalcountsfortherstveiterations,andthenusedtheoriginalcountsafterthefthiteration. Table2-1. RestrictedMLestimatesunderH0:=0:5 FirstRowCounts10020RestrictedMLEstimates0.4070.0000.593 SecondRowCounts203040RestrictedMLEstimates0.2120.3290.459 43

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Example2.3showsthatLang'salgorithmcanperformbetterwhenzerocountsexist.Fordatathatdonothavezerocounts,boththestandardNewton-RaphsonalgorithmandLang'salgorithmcanbeused.Theperformancesofthetwoalgorithmsdependontheconstraint=0.If0isclosetotheoverallMLestimateof,bothalgorithmsconvergeandgivethesamesolution.Whenthedistancebetween0andtheoverallMLestimateislargeand0isclosetotheboundaries,either0or1,thetwoalgorithmsbehavedierently,andneitherisuniformlybetteratproducingtheMLsolution. Table2-2. Non-zerocountssummarizedina23table 123 tableis0:319.Toseehowthetwoalgorithmsworkfordierent0values,weconsiderfourcases:0=0.15,0.5,0.9,and0.95.The~and~inTable2-3arerestrictedMLestimatesofandforthefourcases,and~0isusedtodenote~TA~.Underproperconvergence,~0=0.Thelastrowisthenumberofiterationsuntilthealgorithmsconverge.Althoughwecouldincreasethemaximumnumberofiterations,weused100,whichisadefault,becauselargeriterationnumbersdidnothelpthealgorithmconvergeforourdata.A\-"isusedfortherestrictedMLestimatesifthestandardNewton-Raphsonalgorithmdoesnotconverge.IfLang'salgorithmdoesnotconverge,~0isnotequalto0.\N-R"representsthestandardNewton-Raphsonalgorithmand\Lang"representsLang'salgorithm. 44

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RestrictedMLestimatesofcellprobabilitiesinTable2-2 Sample Proportions N-RLang N-RLang N-RLang N-RLang ~1 0.0660.055 0.3110.311 -0.797 0.1140.876~2 0.2420.278 0.3420.342 -0.137 1.2470.086~3 0.6920.695 0.3470.347 -0.067 -0.3620.041 ~1 0.4680.447 0.2350.235 -0.052 -0.2440.025~2 0.4580.450 0.4890.489 -0.364 1.1400.336~3 0.0740.126 0.2760.276 -0.584 0.1030.646 ~0 0.50.5 -0.9 0.950.954 No.iter 6100 415 -93 8100 When0=0.5,thetwoalgorithmsgivethesamerestrictedMLestimates.If0=0:15,however,thestandardNewton-Raphsonalgorithmconverges,whileLang'salgorithmdoesnotconverge,whichgives~0=0.185,whichissupposedtobe0.15.When0=0:9,thestandardNewton-Raphsonalgorithmdoesnotconverge,whileLang'smethodsucceedsinndingtherestrictedMLestimates.ThelasttwocolumnsshowthatLang'smethoddoesnotconvergeforthesamereasonwhyitdoesnotconvergewhen0=0:15,whilethestandardNewton-Raphsonalgorithmdoesconverge,butthesolutionsareoutofbounds.Therefore,inthiscase,bothalgorithmsfailtondtherestrictedMLestimates.ItshouldbenotedthatthestandardNewton-RaphsonalgorithmrequiresfeweriterationstoconvergethanLang'salgorithmwhentheybothconverge. Hence,asensiblewaytondtherestrictedMLestimatesofcellprobabilitiesistousethestandardNewton-Raphsonalgorithmrst,becausebothalgorithmsconvergeif0iscloseenoughtotheoverallMLestimatesofandthestandardNewton-Raphsonalgorithmisfaster.IfthestandardNewton-Raphsonalgorithmfailstoconvergeorgivessolutionsthatareoutofbounds,wecantryLang'salgorithmbecausesometimesitmaygivesolutions.However,itshouldbenotedthatthereareseveralcasesthatneithertheNewton-RaphsonalgorithmnorLang'salgorithmwork,especiallywhen0isclosetotheboundaries.Forfutureresearch,weplantotrytomodifythealgorithmstoovercometheseproblematicsituations. 45

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46

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(n1n2)2Vd=1 4Xiii#: Lang(2004)provedthattherestrictedMLestimators~and~arestronglyconsistentintheMPHmodel.Bythecontinuousmappingtheorem,wecanseethatM~V^(0)p!1(C2)+(1)1(D2): Generally,thismethoddiersfromthescorecondenceintervalmethodinSection2.5.Forexample,whenc=2with0<0<1,thescoreteststatisticisS2(0)=n1(^1~1)2

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48

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Intheprevioussection,wedidnotassumeanyrelationshipbetweenthecellprobabilitiesintherstrowandthoseinthesecondrowofthe2ctable.So,therewere2(c1)unknownparameterstobeestimated.Analternativemodelthatisfrequentlyusedforordinalresponsesisthecumulativelogitmodel.Itusesthelogitsofcumulativeprobabilitiesinsteadofcellprobabilities.Forthis2ctable,themodelhasthefollowingform:logit(P[Ykj])=j(k1);j=1;;c1;k=1;2: Onewaythismodelarisesisbyassumingthattherstrowhasalogisticdistributionandthesecondhasthesamedistributionexceptforalocationshift.A2ccontingencytableisconstructedbychoppingthemintocintervalsusingc1cutopoints.Inotherwords,Y1andY2canbeconstructedbyY1j,Y1jandY2j,Y2j;j=1;c; Tondanexpressionofunderthemodel,denote11;;1(c1)and21;;2(c1)by1j=P(Y1j)=exp(j) 1+exp(j)and2j=P(Y2j)=exp(j) 1+exp(j);j=1;;c1:

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1+exp(j(k1))exp(j1(k1)) 1+exp(j1(k1));(3{1) where=(1;;c1)T. @=cXj=1n2j(j)(j1) (j)(j1)=0;Foreachj=1;;c1;@l @j=2Xk=1(j(k1))nkj

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@1;@l @c1;@l @T: @2=cXj=1n2[(j)(j1)][(j)+(j1)]E@2l @j@=n2[(j+1)(j1)](j);j=1;;c1E@2l @21=2Xk=1nk(2(k1))(1(k1))+2Xk=1nk(1(k1))(2(k1)) (2(k1))(1(k1))E@2l @j@j+1=2Xk=1nk(j(k1))(j+1(k1)) (j+1(k1))(j(k1));j=1;;c2E@2l @2j=2Xk=1nk[(j+1)(j1)](j(k1))+2Xk=1nk(j1(k1))(j(k1)) [(j(k1))(j1(k1))+2Xk=1nk(j(k1))(j+1(k1)) (j+1(k1))(j(k1));j=2;;c2 51

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@2c1=2Xk=1nk[1(c2(k1))](c1(k1))+2Xk=1nk(c1(k1))(c2(k1)) (c1(k1))(c2(k1)): @1;;@ @c1;@ @; @1=0:5[(2)(1)+(1)(2)]@ @j=0:5(j)[(j+1)(j1)]+0:5(j)[(j1)(j+1)];j=2;;c2@ @c1=0:5[(c1)(c2)+(c1)(c2)+(c1)(c1)@ @=()TD()+0:5(c1);

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1+exp(LB);exp(UB) 1+exp(UB))asthe100(1)%logitWaldcondenceintervalfor. satisfytheregularityconditionsofAitchisonandSilvey(1958,1960).Thus,bytheirresults,vec(~;~)and~haveasymptoticmultivariatenormaldistributions,withmeansequaltovec(;)and,respectively.Furthermore,vec(~;~)and~areasymptotically 53

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@21=0:5[(2)(1)(12(1))+(1)(12(1))(2)]@2g @2j=0:5(j)[12(j)][(j+1)(j1)]+0:5(j)[12(j)][(j1)(j+1)];j=2;;c2@2g @2c1=0:5[(c1)(12(c1))(c2)+(c1)(12(c1))(c2)]+0:5[(c1)(12(c1))(c1)(12(c1))]@2g @2=()TD[()(12())]0:5(c1)[12(c1)]@2g @j@j+1=0:5(j)(j+1)0:5(j)(j+1);j=1;;c2@2g @@1=0:5[(2)(1)(12(1))(1)(2)]@2g @@j=0:5[((j1(j+1))(j)(12(j))+(j)(j+1)]@2g @@c1=0:5[(c1)(c2)+(c1)(12(c1))(1(c2))]:

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Ifl0(0)isthevalueoflevaluatedat~1;;~c1and~when=0andifl1isthevalueoflevaluatedattheoverallMLestimators,thenthe100(1)%LRT-basedcondenceintervalisthesetf0:2(l0(0)l1)2(1);1g:

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Table3-1. CondenceintervalsforinTable1-1withandwithoutassumingacumulativelogitmodel AssumemodelWaldLogitWaldLRTPseudoScoreScore NoLowerEndpoint0.6440.6210.6350.6280.633UpperEndpoint0.9000.8740.8820.8740.875 YesLowerEndpoint0.6450.6210.6320.6270.629UpperEndpoint0.9010.8760.8850.8760.876 56

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Whendataaresummarizedinaformofa2ctable,weestimatedtheordinaleectsizemeasureforthedata,usingseveraltypesofcondenceintervalsinthepreviouschapters.InChapters2and3,wedevelopedveasymptoticcondenceintervalmethodsunderanunrestrictedmodelandacumulativelogitmodel,respectively.Ascompetitors,inChapter1wereviewedtwocondenceintervalmethodsthatalreadyexist:themethoddevelopedbyHalperinetal.andNewcombe'sscore-typemethod.Inthischapter,wedoasimulationstudytoevaluateandcomparethecondenceintervalmethodspresented. 58

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TheprimarydierencebetweenthemethodsinChapter2andthoseinChapter3isthatthemethodsinChapter3areconstructedunderaparametricmodel.Hence,ifthedatasatisfythestructureintheparametricmodel,weexpectcondenceintervalsfromChapter3toperformbetter.Ifnot,thoseinChapter2maybebetter.Sowewillcomparetheperformanceofeachmethodundereithercase. Anotherthingtobeconsideredinasimulationstudyisthenumberofcategories,c.Thelargercis,themoreparametersmustbeestimatedinordertondcondenceintervals,althoughtheparameterofinterestis.Sincemethodsusingfewerparametersareexpectedtoperformbetter,wewillcomparetheperformancesofmethodsforc=3tothoseforc=6.Ifthenumberofcategoriesislargerthan2,theparametricmodelhasfewerparameterstobeestimatedthananunrestrictedmodel.Wewillcomparetheperformancesofthemethodsundertheparametricmodeltothoseunderanunrestrictedmodelforbothc=3andc=6. Thelastissueisthetruevalueof.Wewillseeiftherearesomedierencesofcoverageprobabilitiesbychangingthetruevalue.Wewilltakethetruevaluetobeeither0:5(noeect)or0:8(strongeect).Insummary,wegeneratedataunder16(=2222)conditionsforthemethodsundertheparametricmodeland48(=6222)conditionsfortheothersixmethodsbychangingthesamplesize,thenumberofcolumns,thetruevalue,andwhethertheparametricmodelholds, Togeneratedatafrommultinomialdistributions,werstneedtosetthecellprobabilitiesforeachofthe8(=222)conditions,bychangingnumberofcolumnsc, 59

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Weobtainthecellprobabilitiesbycategorizingtwologisticdistributions.Fortherstrow,weuseequalcellprobabilities.Thesecondrowprobabilitiesareobtainedbypossiblychangingthelocationandscaleparameters.Usingthestandardlogisticdistributionwithlocation=0andscale=1fortherstrow,wedeterminec1cutopointsofthe2ctable.Forthesecondrowwesetthescaleparametertoequaleither1(cumulativelogitmodelholds)or2(modeldoesnothold).Thelocationparameterforthesecondrowisdeterminedsothateither=0.5(noeect)or0.8(strongeect).Table4-1showsthecellprobabilitiesthatareusedforthesimulationstudy.\CL"and\NotCL"inthetablerepresentthecasethatacumulativelogitmodelholdsandthecasethatthemodeldoesnothold,respectively. Table4-1. Cellprobabilitiesfordierentconditions 3;1 3;1 3)Secondrow(1 3;1 3;1 3)NotCLFirstrow(1 3;1 3;1 3)Secondrow(0:414;0:172;0:414) 3;1 3;1 3)Secondrow(0:021;0:058;0:921)NotCLFirstrow(1 3;1 3;1 3)Secondrow(0:034;0:032;0:934) 6;1 6;1 6;1 6;1 6;1 6)Secondrow(1 6;1 6;1 6;1 6;1 6;1 6)NotCLFirstrow(1 6;1 6;1 6;1 6;1 6;1 6)Secondrow(0:309;0:105;0:086;0:806;0:105;0:309) 6;1 6;1 6;1 6;1 6;1 6)Secondrow(0:021;0:030;0:046;0:080;0:173;0:649)NotCLFirstrow(1 6;1 6;1 6;1 6;1 6;1 6)Secondrow(0:061;0:032;0:034;0:044;0:075;0:754) 60

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convergestoCP()inprobability.Here,Iisanindicatorfunctionandnisthenumberofsimulations,say10,000.Itisdesirablethatthiscoverageprobabilitybeclosetothenominallevel.Ifthecoverageprobabilityis0.95,thesampleaveragecoverageproportionhasstandarderrorq convergestoEL()inprobability.Itisdesirablethatthisbeshort.Usingthesetwocriteria,wecandeterminewhichcondenceintervalmethodisbestundereachcondition. 61

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Thesampleoverallcoverageproportionthatisanestimateoftheoverallcoverageprobabilityistheaverageofthesamplecoverageproportionsoverallconditions.Forthevemethodsunderaparametricmethod,itistheaverageoverall12conditions.Ontheotherhand,itistheaverageoverall48conditionsfortheothersevenmethods.WewillusetheBonferronimultiplecomparisonmethodwith=:05=66toseeifeachpairofoverallcoverageprobabilitiesamong122(=66)aredierentusingthestandarderrorq Thesecondcriterionistheproportionofcoverageprobabilitiesthatareeithertoosmallortoolarge,denedastheproportionofthecoverageprobabilitieswhoseabsolutedistancesfromthenominal0.95arelargerthan0.02.Thatis,itistheproportionofcaseswhosecoverageprobabilitiesareeitherlessthanorequalto0.93,orgreaterthanorequalto.97amongallconditions.Thelastcriterionistheabsolutedierencebetweencoverageprobabilityandthenominalvalue0.95. 62

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Toconsidertheeectofsmallsamplesizesonthecondenceintervalmethods,wecomparethesamplesizes:(n1;n2)=(100,100)and(50,50).BasedonTables4-2and4-3,exceptfortheoriginalWaldcondenceintervalunderanunrestrictedmodel,performancesofalltheothermethodsarenotaectedbyreducingthesamplesizesto(50;50).Thatis,fortheelevenmethodsundertheeightconditions,theabsolutedierencesbetweenaveragesofsamplecoverageproportionswith(100,100)andthosewith(50,50)arelessthanq 810000z1=2=0:004. Itwasexpectedthatmethodsdevelopedunderaparametricmodelwouldperformbetterifthedatasatisfythestructureinthemodel.However,ifthemodelholds,Tables4-2and4-3showthatthecoverageprobabilitiesofmethodsunderanunrestrictedmodelaresimilartothoseunderaparametricmodel.Ontheotherhand,methodsundertheunrestrictedmodelperformbetteriftheparametricmodeldoesnothold.Withstandarderror=0.0001andforaveragingovereightconditions,basedonTable4-8,weseethatallmethodsundertheparametricmodelhavelargercoverageprobabilitiesifthemodelholds,whilethoseunderanunrestrictedmodelarenotaectedbytheparametricmodelassumption.Alsowhentheparametricmodeldoesnothold,theparametricmethodshavemuchpoorercoverageprobabilitieswhenc=6thanwhenc=3.Hence,wecansaythatthemethodsunderanunrestrictedmodelaresafertouseinpracticebecausewedonotknowwhethertheparametricmodelholdsornot.Becauseofthisreason,wedonotconsiderthesemethodsforfurtherconsideration. Tondthebestcondenceinterval,Table4-9summarizesthesimulationresultsforthetwoexistingmethodsandtheveunrestrictedmethodsproposedinthisdissertation,averagedoverallthesamplesizecases,overc=3and6,over=0.5and0.8,andoverwhetherthemodelholds.Thetablereportsthreeoverallsummariesofperformance:themeancoverageprobability,themeanoftheabsolutedierencesbetweencoverage 63

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Withsamplesizes100each,coverageprobabilities(CP)andoverallsummariesfromsimulationstudyforcasesinwhichcumulativelogitmodelholds(CL)ordoesnothold(NotCL) ExistingHalperinetal.0.9500.9520.9520.9530.9490.9500.9520.9520.9510.002NewcombeScore0.9630.9630.9770.9770.9520.9490.9580.9520.9610.012 UnrestrictedWald0.9470.9480.9430.9460.9460.9470.9430.9450.9460.004LogitWald0.9500.9520.9510.9530.9500.9510.9530.9520.9520.002LRT0.9490.9500.9490.9510.9480.9490.9500.9500.9500.001Score0.9490.9510.9500.9530.9490.9500.9530.9520.9510.001PseudoScore0.9500.9510.9500.9520.9490.9510.9530.9520.9510.001 ParametricWald0.9470.9430.9430.9430.9440.9330.9450.9250.9400.010LogitWald0.9490.9470.9510.9500.9480.9390.9510.9370.9470.004LRT0.9490.9470.9480.9470.9470.9400.9500.9330.9450.005Score0.9490.9470.9500.9490.9480.9380.9510.9380.9460.004PseudoScore0.9490.9460.9500.9490.9470.9340.9500.9360.9450.005 Withsamplesizes50each,coverageprobabilities(CP)andoverallsummariesfromsimulationstudyforcasesinwhichcumulativelogitmodelholds(CL)ordoesnothold(NotCL) ExistingHalperinetal.0.9510.9500.9550.9520.9500.9490.9480.9510.9510.002NewcombeScore0.9640.9610.9790.9760.9540.9480.9520.9510.9610.011 UnrestrictedWald0.9420.9420.9390.9380.9450.9410.9340.9430.9410.010LogitWald0.9510.9500.9530.9500.9520.9500.9510.9540.9510.001LRT0.9470.9460.9490.9480.9490.9470.9460.9500.9480.002Score0.9490.9480.9500.9470.9500.9490.9490.9510.9490.001PseudoScore0.9500.9490.9520.9470.9520.9490.9500.9530.9500.001 ParametricWald0.9400.9350.9370.9360.9420.9320.9370.9200.9390.015LogitWald0.9460.9430.9510.9510.9490.9400.9510.9400.9460.004LRT0.9450.9410.9490.9490.9480.9400.9490.9330.9440.006Score0.9460.9410.9460.9470.9480.9370.9490.9360.9430.006PseudoScore0.9460.9410.9470.9470.9490.9390.9510.9380.9450.006

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Withsamplesizes(50,100),coverageprobabilities(CP)andoverallsummariesfromsimulationstudyforcasesinwhichcumulativelogitmodelholds(CL)ordoesnothold(NotCL) ExistingHalperinetal.0.9490.9510.9500.9500.9470.9480.9480.9500.9490.001NewcombeScore0.9620.9700.9630.9650.9520.9640.9530.9580.9610.011 UnrestrictedWald0.9440.9460.9410.9410.9400.9430.9390.9420.9420.008LogitWald0.9490.9510.9500.9500.9470.9490.9500.9530.9500.001LRT0.9480.9490.9490.9470.9460.9470.9470.9470.9480.003Score0.9490.9510.9500.9480.9470.9480.9510.9500.9490.001PseudoScore0.9490.9510.9500.9490.9470.9490.9520.9510.9500.001 Withsamplesizes(10,100),coverageprobabilities(CP)andoverallsummariesfromsimulationstudyforcasesinwhichcumulativelogitmodelholds(CL)ordoesnothold(NotCL) ExistingHalperinetal.0.9320.9330.9420.9430.9350.9380.8780.8800.9230.027NewcombeScore0.9640.9810.9510.9530.9560.9890.9520.9720.9650.015 UnrestrictedWald0.9080.9140.8780.8800.9120.9200.8690.8710.8940.056LogitWald0.9270.9280.9360.9360.9330.9340.8790.8790.9190.031LRT0.9370.9390.9550.9530.9350.9330.9170.9190.9360.016Score0.9500.9490.9550.9540.9540.9480.9630.9610.9540.005PseudoScore0.9500.9500.9550.9550.9550.9520.9630.9610.9550.005

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Withsamplesizes(10,50),coverageprobabilities(CP)andoverallsummariesfromsimulationstudyforcasesinwhichcumulativelogitmodelholds(CL)ordoesnothold(NotCL) ExistingHalperinetal.0.9380.9400.9430.9440.9400.9430.9010.9110.9330.018NewcombeScore0.9650.9780.9540.9570.9570.9830.9540.9710.9650.015 UnrestrictedWald0.9160.9210.8860.8900.9190.9280.8810.8940.9040.046LogitWald0.9340.9370.9400.9410.9410.9430.9060.9150.9320.018LRT0.9390.9410.9470.9440.9400.9410.9240.9280.9380.012Score0.9460.9470.9550.9550.9450.9430.9650.9620.9550.007PseudoScore0.9480.9490.9550.9540.9490.9490.9650.9630.9540.005 Withsamplesizes(10,10),coverageprobabilities(CP)andoverallsummariesfromsimulationstudyforcasesinwhichcumulativelogitmodelholds(CL)ordoesnothold(NotCL) ExistingHalperinetal.0.9580.9570.9640.9660.9470.9500.9310.9450.9520.009NewcombeScore0.9670.9640.9740.9740.9560.9530.9650.9550.9640.014 UnrestrictedWald0.9180.9140.8990.9030.9160.9180.8820.8790.9040.046LogitWald0.9580.9570.9520.9560.9590.9610.9480.9560.9560.006LRT0.9400.9400.9400.9380.9430.9440.9210.9110.9350.015Score0.9480.9460.9560.9570.9470.9480.9500.9420.9490.004PseudoScore0.9500.9490.9580.9590.9570.9540.9480.9420.9520.005

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Meancoverageprobabilities(CP)fortwelve,averagedoversamplesizes(100,100)or(50,50),c=3and6,=0:5and0:8 ExistingHalperinetal.0.9510.9510.000NewcombeScore0.9620.9600.003 UnrestrictedWald0.9420.944-0.001logitWald0.9510.9520.000LRT0.9480.9490.000Score0.9500.9490.000PseudoScore0.9510.9500.000 ParametricWald0.9420.9330.009logitWald0.9500.9430.006LRT0.9480.9430.006Score0.9480.9420.007PseudoScore0.9490.9410.007 Overallperformancesummariesofcoverageprobability(CP)fromsimulationstudyfortwelvemethods,averagedoverseveralsamplesizes,c=3and6,=0:5and0:8,andwhetherornotacumulativelogitmodelholds ExistingHalperinetal.0.9430.0100.083NewcombeScore0.9630.0130.250 UnrestrictedWald0.9220.0280.500logitWald0.9430.0100.125LRT0.9420.0080.125Score0.9510.0030.000PseudoScore0.9520.0030.000

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Whenwehavetworandomvariablesmeasuredonordinalmeasurementscaleswithccategories,theycanbeeitherindependentordependent,dependingonthedatastructure.InChapters2and3,weconstructeda2ccontingencytableunderanassumptionthatthetworandomvariablesareindependent,anddevelopedcondenceintervalmethodsforanordinaleectsizemeasurethatcanbeusedtosummarizethedierencebetweenthetwostochasticallyordereddistributions. Inthischapter,insteadofassumingtheyareindependent,weconsiderthetwosamplesinwhicheachobservationinonesamplepairswithanobservationintheother.Then,thetwosamplesaredependent.Thiskindofdataiscalledmatched-pairdataandcommonlyoccurinlongitudinalstudies.Sinceeachsamplehasccategories,thematched-pairsdataconstructsthefollowingccsquaretableinTable5-1.Here,nijisthenumberofobservationsthathavethecategoryifortherstobservationandthecategoryjforthesecondobservation,withithrowtotalni+=Pcj=1nijandjthcolumntotaln+j=Pci=1nij.n=Pci=1Pcj=1nijrepresentstotalsamplesize. Table5-1. Matched-pairsdatawithccategories 12c 1 ............ ...c

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Jointcellprobabilitiesofthematched-pairsdata 12c 1 ............ ...c andcellprobabilities.Inthischapter,weonlyconsideranunrestrictedmodel,whichmeansthatweusec21non-redundantcellprobabilities,becauseasimulationstudyinChapter4showedthatcondenceintervalmethodsunderanunrestrictedmodelaresaferthanthoseunderacumulativelogitmodel.Ourgoalinthischapteristondcondenceintervalsforsomemeasuresthatcanbeusedtocomparethetwodependentordereddistributions. Forthematched-pairsdata,Agresti(1980)discussedameasureP(Y1Y2),andAgresti(1983)consideredanothermeasureP(Y1Y2)forcomparingtwoordinalcategoricalvariables,Y1andY2,usingamultinomialdistributionwithforapair(Y1i;Y2i),i=1;;n.VarghaandDelaney(2000)usedP(Y1i
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Table5-3. 2ctablewithmarginalrowtotalsandcolumntotalsfromTable5-1 12c1cTotal Y2n+1n+2n+(c1)n+cn 71

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00:511 2[1+1221]. 2. 2,butMP2doesnotneedtobe1 2. 2,butMP1doesnotneedtobe1 2.

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73

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4[Xi;j2ijij(Xi;jijij)2]; Applyingthedeltamethodtotheasymptoticnormalityof^,wehavep

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Lettingl0beavalueoflevaluatedat~(0),whicharerestrictedMLestimatesofunderH0:MP1=0,andlettinglAbeavalueoflevaluatedattheoverallMLestimates^,the100(1)%LRT-basedcondenceintervalforMP1isf0:2[l0(0)lA]<2(1;1)g: 75

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Table5-4. Opinionaboutpremaritalandextramaritalsex ExtramaritalSex1234Total Premarital1144200146Sex233420393841461105412629255185 Total38749336475 opinionaboutpremaritalsexandextramaritalsex.Thecategoriesare1=alwayswrong,2=almostalwayswrong,3=wrongonlysometimes,and4=notwrongatall.Themarginalproportionsforpremaritalsexare(0.307,0.082,0.221,0.389),andthemarginalproportionsforextramaritalsexare(0.815,0.103,0.069,0.013).Thissuggeststhat 76

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Table5-5. 95%CondenceIntervalsforMP1 Lowerendpoints0.1830.1840.1840.1840.184Upperendpoints0.2290.2300.2290.2290.229 Table5-6. Jointcellprobabilitieswithc=6,=0:5,and=0:8 123456 1 0.1040.0410.0160.0050.0010.000 2 0.0410.0550.0400.0220.0080.001 3 0.0160.0400.0460.0380.0220.005 4 0.0050.0220.0380.0460.0400.016 5 0.0010.0080.0220.0400.0550.041 6 0.0000.0010.0050.0160.0410.104 77

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Table5-7. Overallperformancesformatched-pairsdataoverallconditionsconsidered MeanofCPMeanofjCP0:95jProportionof(jCP0:95j>0:02) Wald0.9340.0160.292LogitWald0.9490.0040.000LRT0.9510.0050.042Pseudo.Score0.9610.0110.194Score0.9600.0110.208 78

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Coverageprobabilitiesformatched-pairsdatawithc=6andsamplesizes=25 Halperin0.9510.9840.9990.9530.9830.9980.9500.9710.9870.9220.9490.962NewcombeScore0.9550.9870.9990.9580.9860.9990.9610.9810.9930.9620.9810.989 Wald0.9340.9390.9520.9320.9370.9440.9240.9190.9260.8890.8970.893LogitWald0.9490.9490.9560.9510.9490.9510.9500.9470.9460.9320.9360.932LRT0.9450.9450.9720.9470.9500.9670.9460.9570.9710.9370.9500.955Pseudo.Score0.9550.9650.9960.9630.9720.9900.9610.9700.9820.9360.9490.954Score0.9500.9620.9980.9750.9720.9860.9620.9690.9760.9370.9490.954 Coverageprobabilitiesformatched-pairsdatawithc=3andsamplesizes=25 Halperin0.9510.9820.9990.9520.9810.9980.9550.9650.9690.9610.9630.966NewcombeScore0.9640.9890.9990.9680.9890.9990.9780.9860.9900.9710.9740.977 Wald0.9300.9330.9420.9270.9330.9320.9240.9290.9270.9040.9090.908LogitWald0.9460.9440.9480.9460.9450.9380.9470.9460.9450.9600.9570.959LRT0.9440.9400.9480.9410.9430.9510.9410.9520.9540.9550.9610.967Pseudo.Score0.9500.9510.9790.9510.9590.9760.9630.9690.9710.9560.9590.965Score0.9490.9470.9790.9510.9580.9730.9640.9700.9710.9560.9590.965 Coverageprobabilitiesformatched-pairsdatawithc=6andsamplesizes=50 Halperin0.9530.9850.9990.9500.9840.9990.9530.9720.9870.9420.9600.971NewcombeScore0.9560.9870.9990.9550.9850.9990.9620.9800.9910.9600.9750.984 Wald0.9440.9420.9520.9390.9450.9510.9350.9340.9410.9170.9220.917LogitWald0.9510.9470.9540.9500.9510.9540.9520.9450.9500.9450.9410.945LRT0.9500.9450.9530.9470.9490.9580.9480.9470.9560.9500.9630.970Pseudo.Score0.9540.9510.9760.9520.9560.9760.9620.9680.9770.9620.9670.976Score0.9540.9480.9760.9510.9540.9730.9630.9690.9750.9650.9670.973

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Coverageprobabilitiesformatched-pairsdatawithc=3andsamplesizes=50 Halperin0.9490.9820.9450.9490.9810.9970.9520.9690.9700.9480.9570.959NewcombeScore0.9620.9880.9990.9650.9890.9990.9790.9860.9890.9680.9750.976 Wald0.9380.9410.9470.9370.9430.9450.9380.9430.9420.9210.9310.929LogitWald0.9440.9450.9480.9470.9500.9470.9480.9570.9520.9480.9530.951LRT0.9430.9450.9450.9460.9480.9510.9440.9520.9610.9430.9540.954Pseudo.Score0.9480.9480.9520.9490.9510.9580.9530.9640.9610.9610.9690.968Score0.9470.9480.9500.9480.9510.9570.9540.9640.9610.9620.9690.969 Coverageprobabilitiesformatched-pairsdatawithc=6andsamplesizes=75 Halperin0.9500.9850.9990.9500.9850.9990.9480.9740.9880.94550.96710.9732NewcombeScore0.9540.9870.9990.9550.9860.9990.9560.9780.9910.95850.97670.9840 Wald0.9450.9450.9530.9420.9490.9480.9370.9390.9400.92930.93320.9287LogitWald0.9500.9480.9540.9480.9530.9480.9490.9480.9480.94730.94990.9455LRT0.9490.9470.9520.9470.9530.9500.9460.9460.9470.94990.95790.9594Pseudo.Score0.9520.9510.9590.9500.9550.9600.9510.9540.9630.96410.97210.9743Score0.9520.9500.9570.9500.9540.9600.9500.9540.9660.96650.97240.9719 Coverageprobabilitiesformatched-pairsdatawithc=3andsamplesizes=75 Halperin0.9510.9800.9990.9500.9790.9970.9480.9650.9760.9490.9530.959NewcombeScore0.9630.9870.9990.9650.9860.9990.9760.9860.9890.9680.9680.976 Wald0.9460.9440.9460.9440.9430.9430.9390.9440.9440.9280.9320.933LogitWald0.9500.9480.9470.9490.9480.9450.9480.9510.9500.9500.9470.953LRT0.9500.9480.9450.9480.9470.9460.9450.9490.9490.9420.9430.950Pseudo.Score0.9520.9520.9480.9520.9510.9490.9470.9500.9510.9620.9590.966Score0.9520.9510.9480.9510.9500.9490.9470.9500.9510.9630.9600.966

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ThecondenceintervaldevelopedbyHalperinetal.(1989)andthemethodsdiscussedinChapter2weredesignedfororderedcategoricaldata.Ontheotherhand,Newcombe'sscorecondenceintervalwasconstructedbyassumingthedatacamefromcontinuousdistributions,inparticularexponentialdistributions,althoughheusedhismethodfororderedcategoricaldataandshowedthatitperformedwellinasimulationstudy.Forcondenceintervalmethodsforwithcontinuousdistributions,seeChurchandHarris(1970),EnisandGeisser(1972),Govindarajulu(1967,1968),Owen,Craswell,andHanson(1964),andReiserandGuttman(1986).InthischapterweanalyzehowthemethodsdevelopedinChapter2workfordatafromcontinuousdistributions. Supposethatindependentsamplesofsizen1andn2,say(X11;;Xn1)and(X21;;X2n2),comefromunknowncontinuousdistributionsF1andF2.Toapply^,werankthedatafromthesmallesttothelargestandconstructa2ctablewithc=n1+n2,assumingtherearenoties.Asanexample,supposethatwegenerateasampleofsize2fromF1andgenerateasampleofsize3fromF2,andlettheorderofthese2samplesbeasfollows:X1(1)
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Anasymptoticnormalityof^inthiscaseiswellknownbasedonpropertiesofU-statistics,andsoWaldcondenceintervalsapplydirectly.Theestimatedasymptoticvarianceof^,say^var(^),isobtainedbysubstituting^and^intotheformin(2-2)with^i^i=0foreachi.TheresultingcondenceintervalisthesameastheHanley-McNeilWaldmethoddiscussedbyNewcombe(2006b). BeforewediscusstheLRT-based,thescore,andpseudoscore-typecondenceintervals,itshouldbenotedthatthenumberofcolumns,c,inthe2ctablesisincreasedasthesamplesizegrows,becausecisthesumoftwosamplesizes.Becauseofthis,regularityconditionstoapplymultinomialMLestimatesareviolated.Althoughthereisnotheoreticaljusticationtousethethreemultinomial-basedcondenceintervalsforfully-rankeddata,wewillchecktheperformanceofthethreecondenceintervalmethodsusingasimulationstudy. 82

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Weconductedsimulationevaluations,generatingdatafromnormaldistributionswithidenticalvariancesbutpossiblydierentmeans.Withoutlossofgenerality,weassumedX1N(0;1)andX2N(;1).Thentheparameterofinterestis=P(X1
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Coverageprobabilities(CP)andoverallsummariesforfully-rankeddatawithsamplesizes(10,10)and(20,30) Halperin0.9460.9220.9460.9450.9400.250NewcombeScore0.9480.9520.9480.9600.9520.000 Wald0.9200.8860.9380.9250.9170.725LogitWald0.9660.9730.9530.9550.9620.250LRT0.9470.9160.9460.9460.9390.250Score0.9500.9450.9490.9560.9500.000PseudoScore0.9630.9430.9520.9720.9580.250 Forexample,supposetheparametricmodelX1N(1;2)andX2N(2;2)trulyholds.Then,=(21 Table6-3. Relationshipbetweenand 84

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Withequalsamplesizesn,anaturalparametricestimateofis^=(X2X1 Theasymptoticeciencyoftheordinalestimaterelativetotheparametricestimateofisthelimitasnincreasesofeff=var(^) Forinstance,when=0.0,0.5,1.0,1.5,2.0,2.5,and3.0,theasymptoticrelativeecienciesare0.955,0.961,0.974,0.979,0.957,0.892,and0.782.Therefore,undernormality,theparametricestimatecanbemuchbetterthantheordinalestimatewhentheeectisverylarge.Otherwise,theordinalestimateholdsupwell,muchasthecorrespondingMann-Whitneytestdoesintermsoftheclassicresultaboutitslocaleciencycomparedtothettestfornormaldistributions. 85

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PlotofasymptoticeciencyofMann-Whitneyestimaterelativetoparametricestimate Whenaparametricmodelisplausibleinthefully-rankedcaseandtheeectisverylarge,itmightbepreferabletoestimateusingthatmodel.Inpractice,eventhenthismustbeweighedagainstthepossibilityofactualcoverageprobabilitiesforcorrespondingcondenceintervalspossiblybeingfarfromnominallevelswhenthereismodelmisspecication. 86

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InChapters1through4,wefocusedonndinggoodcondenceintervalsforanordinaleectsizemeasureinasingle2ctable.Practically,however,itismorerealistictoassumethatthemeasureisaectedbysomeexplanatoryvariables.Inotherwords,wemightneedtoconsidermorethanone2ctablesthatareconstitutedbycombinationsofalllevelsofthevariables.Themeasurescalculatedfromthevarioustablesmaysharecertaincharacteristicsthatcanbedescribedbyamodel. Asanexample,weillustratewithTable7-1,whichshowstheshouldertippainscoresafterlaparoscopicsurgery,nowstratiedbygenderandage,bybreakingageintotwocategories.Aprimaryinterestisagaintomeasurethedierencebetweenthetwotreatmentstoreducepatients'pain.Inthischapter,however,wearealsointerestedinknowingwhetherornottheirpainscorescanbeaectedbygenderandage. Table7-1. Shouldertippainscoresstratiedbygenderandage AgeGenderTreatment12345 20-70FemaleActive121000Control32302MaleActive51100Control10130 71+FemaleActive10000Control10000MaleActive21000Control10000 Inthischapter,weconsiderK(>1)2ctables.Thenumberoftables,K,istheproductofthenumberoflevelsofallcovariatesconsideredtomeasureaneectsize.Foreachk,k=1;;K,letYk1andYk2denoterandomvariableshavingcordinalcategoriesintherstandsecondrowsofthekth2ctable,respectively,andletyk1=(nk11;nk12;;nk1c)Tandyk2=(nk21;nk22;;nk2c)Tdenotecountsintherowsforthekthtable.AssumethatYk1hasamultinomialdistributionwithnk1=Pcjnk1jtrialsandcellprobabilitiesk1=(k11;k12;;k1c)T,andYk2hasamultinomialdistributionwith 87

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2P(Yk1=Yk2)=Tk2Ak1; Toestimatethemeasures,wedistinguishcovariateswithxedeectsfromoneswithrandomeects.Amongcovariateswithxedeects,wealsodistinguishcasesinwhichallcovariatesarecategoricalcovariatesfromcaseswithatleastonecontinuouscovariate.Ourprimaryinteresthereishowtouselogitmodelsthatconnectthemeasureswithcovariatesdirectly,insteadofusingacumulativelogitmodel,andhowtoestimatethemeasuresandmodelparametersusingthescorecondenceinterval,whichwasconsideredasthebestbasedonthesimulationstudywithindependentsamples. Inthischapter,weonlyconsiderthelogitmodelswiththecasesinwhichallcovariatesarecategorical,becauseofcomputationaldicultieswithcasesinwhichatleastoneofthecovariatesiscontinuousorhasrandomeects.Forthelattertwocases,wewilluseacumulativelogitmodeltoestimatetheordinaleectsizemeasureandmodelparameters.Whentherearecontinuouscovariates,itmaynotbepossibletodenestrata.Inthiscase,Brumback,Pepe,andAlonzo(2006)suggestedtoincludepairswithdierentvaluesforthecovariateintheanalysisaslongasthosecovariatesareincludedappropriatelyinthemodel.Forexample,ifXisacontinuouscovariate,wemighttamodeloftheformlogit[P(Y1(x1))
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Asanotherapproach,wecanmodellogit()directlywiththecovariates.LetXbeadesignmatrixcorrespondingtotheexplanatoryvariables,andletbeavectorofunknownparametersthatexplaineectsoftheexplanatoryvariables.Withalogitlink,weassume logit()=X;(7{1) Ourinteresthereistoestimatebothandusingscorecondenceintervals.Unlikeinacumulativelogitmodel,cellprobabilitiesinthemodelcannotbeexpressedintermsofthemodelparameters.Inthefollowingsubsection,wewilldiscusshowtotthemodelsafterreexpressingthemodelasaformofaconstraintfunction.OncewendtheMLestimatesofthecellprobabilities,say^,theMLestimatesofk'sareobtainedby^k=^Tk2A^k1;k=1;;K;

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Foreachi;j=0;1,denoteij=P(Y1
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Foramoregeneralcase,supposethatthereareIbinarycovariates.Forthemaineectsmodellogit()=0+1z1++IzI; 2I11Xi1=01Xij1=01Xij+1=01XiI=0[logit(^i1ij11ij+1iI)logit(^i1ij10ij+1iI)]: Lang(2005)developedmaximumlikelihoodinferencemethodsfortheclassofhomogeneouslinearpredictor(HLP)modelsforcontingencytables,whichisaspecialcaseoftheMPHmodeldiscussedinChapter2.AnHLPmodelcanbereexpressedasaformofanMPHmodel.TheonlydierencebetweenanHLPmodelandanMPHmodelisthatanHLPmodelincludesalinearpredictorsuchasXin(7-1),butan 91

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Letmk1andmk2beexpectedcellcountsintherstrowandthesecondrowinthekth2ctable.Sincemultinomialdistributionsareassumedfordataintherows,theexpectedcellcountsaremk1=nk1k1andmk2=nk2k2,andwehavethemeasurek=mTk2Amk1 whereL(m)=(L(m1);;L(mK))TandL(mk)=logmTk2Amk1 92

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Asmentionedearlier,anHLPmodelisaspecialcaseofanMPHmodelthathasaconstraintfunction.LetUbeanorthogonalcomplementmatrixofthedesignmatrixX.Thatis,UisafullcolumnrankmatrixthatspansthespaceorthogonaltothecolumnspaceofX.SinceUTX=0bythedenitionoftheorthogonalcomplement,wehavethefollowingconstraintformh(m)=UTL(m)=UTX=0:

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ItcanbeshownthattheconstraintfunctionhisZ-homogeneousoforder0andsatisesthefollowingconditions: (1)fm:m>0;h(m)=0g6=? (3)ThepartialderivativematrixH(m)=@hT(m)=@mhasfullcolumnrankwithrank=numberofcomponentsinL(m)minusrank(X).Theproofsareintheappendix.Combiningwiththerstcondition,weseethatourmodelisanHLPmodel.Therefore,wecanuseLang'sresulttondtheMLestimatesoftheexpectedcellcounts. AssumingtheMLestimates^mexistandareunique,theysolvetherestrictedlikelihoodequations,0B@ym+D(m)H(m)h(m)1CA=0; wherelogit(^)=B0B0U(UTB0U)1UTB0A0^=RX[B0B0U(UTB0U)1UTB0]RTXRXA0RTXA0=@L(vec(;)) 94

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ThePearsonstatistictotestH0:k=0assuming(7-2)isS2k(0)=KXi=12Xj=1cXr=1(nij^ijrnij~ijr(k))2 95

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Table7-2. MLestimatesofkandjparameters,withtheir95%scorecondenceintervals 96

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392795 291213 and 0295 322410 Apointestimateofis0.468forthersttable,and0.442forthesecondtable.Thus,thetwoseparatetablesshowthatobservationsintherstrowtendtobelargerthanthoseinthesecondrow.Ontheotherhand,thecombinedtable 3930810 3234513 has0.516asanestimateofthevalue,whichimpliesthatobservationsinthesecondrowtendtobelargerthanintherstrow.Thisistheoppositeresultfromthetwoseparatetables,sosatisesSimpson'sparadox. logit[P(YkjjX=x)]=jx0(k1);j=1;;c1;k=1;2:(7{4) Forthismodel,thenumberofparameterstobeestimatedisc+1,whichisxedassamplesizesgrow.Assumingcounts,(yk1(x);;ykc(x)),k=1;2,ineachrowofthe2ctable 97

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1+exp(j(k1)0x)exp(j1(k1)0x) 1+exp(j1(k1)0x);where=(1;;c1)T,0=,andc=1. Letdenotethecdfofastandardlogisticdistributionanddenotethepdfofthedistribution.Then,MLestimatesof(1;;c1;0;)aresolutionsofthefollowingequations:@l @=Xx2Xk=1cXj=1ykj(x)x(j(k1)0x)(j1(k1)0x) (j(k1)0x)(j1(k1)0x)=0;@l @0=XxcXj=1y2j(x)(j0x)(j10x) (j0x)(j10x)=0;Foreachj=1;;c1;@l @j=Xx2Xk=1(j(k1)0x)[ykj(x) (j(k1)0x)(j1(k1)0x)yk(j+1)(x) (j+1(k1)0x)(j(k1)0x)]=0:TheMLestimatesoftheparameterscanbeobtainedbyusingeither\procgenmod"or\proclogistic"inSAS.Since theMLestimateof(x)iscalculateddirectlybysubstitutingtheMLestimatesoftheparametersinto(7-5). Therstpartialderivativesof(x)withrespecttoeachparameterare@(x)

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dvar(^(x)) ^(x)(1^(x)); 99

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Table7-3. Opinionaboutsurrogatemotherhoodandthelikelihoodofsellingakidneyatage87 SurrogateKidneyforsalemotherhoodDenitelyNotProbablyNotPerhapsProbablyDenitely Forbidit20001Permitit00000 Ifweuseacumulativelogitmodelin(7-4)withabinarycovariatethatrepresentsopinionaboutsurrogatemotherhoodandacontinuouscovariatethatrepresentsage,thenwecanobtaintheMLestimatesfor,,and0byusing\proclogistic"inSAS:^=(0:838;0:242;0:538;1:424)withstandarderrors(s.e.)=0.172,0.171,0.171,and0.176,and^0=0:712withs.e.=0.003,and^=0:0118withs.e.=0.108.TheMLestimateofvalueatage87,obtainedbysubstitutingtheMLestimatesinto(7-5),is^(87)=0:604withs.e.=0.016,anda95%Wald-typecondenceintervalusingalogittransformationfor(87)is(0.574,0.635).Pointestimatesandcondenceintervalsof(x)atdierentxvaluesforthemaineectsmodelaredisplayedinTable7-4.Itshowsthatthepointestimateandthecondenceintervalof(x)areverysimilaraslevelsofxarechanged,althoughthepointestimatesslightlydecreaseasageincreases.Hence,whatevertherespondent'sageis,ifpeoplethinksurrogatemotherhoodshouldbepermitted,thentheytendtothinkthatoeringakidneyforsaleshouldbelegaltoo. Testingthemaineectsmodelisthesameastestingwhetherornottheeectsofthetwocovariatesarethesameforeachcumulativelogit.ThescoreteststatistictakenfromSASis10.8withdf=6,indicatingthatthemodelisadequateforthedata.Thedegrees 100

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Pointestimatesandcondenceintervalsof(x)asage(x)varies,undermain-eectsmodel x=age^(x)LogitWaldCondenceIntervals 200.611(0.579,0.643)300.611(0.579,0.642)400.610(0.578,0.642)500.610(0.578,0.641)600.609(0.577,0.639)700.607(0.576,0.638)800.606(0.575,0.636) offreedomis6becausethemodeltobetestedhas6parametersandamodelthatallowsseparateeectshas12parameters. SupposethatthereareKclustersandresponsesineachclusterhavecordinalcategories.Ifweassumethattheclustersarechosenrandomlyfromapopulation,thenavariablethatrepresentstheclustersisarandomeect.Usuallyobservationsinthesameclusterarenotindependentsincetheysharethesamerandomeect.However,ifaclusterisgiven,i.e.,ifarandomeectfortheclusterisgiven,thenitisassumedthattheobservationsintheclusterareindependent.Inotherwords,theobservationsintheK2ctablesareconditionallyindependent. 101

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Clinicaltrialrelatingtreatmenttoresponseforeightcenters CenterTreatmentResponsesUnchanged/worseBetterMuchbetter 1Placebo1011Drug6713 2Placebo122Drug1052 3Placebo282Drug72311 4Placebo230Drug8117 5Placebo511Drug5315 6Placebo104Drug5513 7Placebo1111Drug1347 8Placebo223Drug2915 LetYkdenotearandomvariableforanordinalresponseinclusterk,andletZdenoteabinarycovariaterepresentingtwogroupstobecompared,i.e,Z=I(i=2),wherei=1fortherstrowandi=2forthesecondrow.Attheclusterk,observationsgatheredfromtwogroupsforma2ctable.Letnijjkdenotethecountoftheithrowandthejthcolumnattheclusterk.Assumeavectorofthecounts(ni1jk;;nicjk)Thasamultinomialdistributionwithnijk=Pcj=1nijjktrialsandconditionalcellprobabilitiesijk,givenZ=zandtheclusterk,andobservationsindierentrowsareindependent.Fora2ctableintheclusterk,asinprevioussections,weconsideranordinaleectsizemeasurektocomparetherows:k=T2jkA1jk;k=1;;K: 102

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logit(k)=+uk;(7{6) whereukisthevalueofarandomeectfortheclusterkandisassumedtofollowanormaldistributionwithmean=0andvariance=2.Itshouldbenoticedthatcellprobabilitiesinthe2ctablescannotbereexpressedintermsoftheparametersofinterest,and,inthismodel,whichmightcauseaproblemtondtheMLestimatesoftheparameters.ThesamesituationshappenedinSection7.1,inwhichallcovariateswerecategorical,buttheparametricmodelinthesectionwasexpressedinaformofanHLPmodelandsoMLestimatesofthecellprobabilitiesandmodelparameterscouldbeobtainedbyusingLang's(2004)algorithm.Ontheotherhand,inthemodelin(7-6),wecannotexpressthemodelinaformofaconstraintfunctionbecauseukinthemodelisarandomeect.Becauseofthisdiculty,inthissection,weonlyconsidercumulativelogitmodelswithrandomeects. logit[P(Ykjjz;uk)]=jzuk;j=1;;c1;(7{7) where<1<
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Underthemodel,thecumulativelogits,ijk=(i1jk;;i(c1)jk)T,fortheithrowatkthclusterareijjk=P(Ykjjz;uk)=exp(jzuk) 1+exp(jzuk);j=1;;c1; Toseeaneectofonk,givenanduk,weregardkasafunctionof.Wheniszero,k=0:5because1issameas2,regardlessofvaluesofrandomeectsuk.AsmentionedinChapter3,thecumulativelogitmodelcanbeexplainedbyusingunderlyinglogisticdistributionswithscale=1.Alocationparameterfortherstrowis0andthatforthesecondrowis.Thus,ifispositive,thenadistributioncorrespondingtothesecondrowisstochasticallylargerthanonefortherstrow,whichimplieskisgreaterthan0.5regardlessoftherandomeectsuk.Similarly,ifisnegative,thenkislessthan0.5.Therefore,thesignofk0:5issameasthesignof,andisnotaectedbyuk. Aproblemwiththemodelin(7-7)isthatitassumestheroweectisthesameinallclusters,whichisoftennotrealistic.Torelaxthisassumption,amodelthatallowsaheterogeneousgroupeectforeachclusterisneeded.Apossiblemodelisacumulativelogitmodelwitharandominterceptaswellasarandomslope: logit[P(Ykjjz;uk)]=jbkzuk;j=1;;c1:(7{8) 104

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1+exp(jbkzuk);j=1;;c1; whereyijjk=nijjk=nijk;k=1;;K;i=1;2;j=1;;c1ljk=exp(luk) 1+exp(luk)exp(l1uk) 1+exp(l1uk)2ljk=exp(luk) 1+exp(luk)exp(l1uk) 1+exp(l1uk);l=1;;c:0=;c=1;

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2p b)22uk(bk) where1ljk,l=1;;c,aresameas(7-9)and2ljk=exp(lbkuk) 1+exp(lbkuk)exp(l1bkuk) 1+exp(l1bkuk);l=1;;c. TondtheMLestimatesoftheparametersinequations(7-9)and(7-10),weneedtomaximizethelikelihoodfunctionswithrespecttotheparameters.Sincetheyhaveintegralsinsidethelikelihoodfunctions,werstneedtoapproximatetheintegrals.ThereareseveralmethodstoapproximatetheintegralsincludingLaplaceapproximation,binomialapproximation(TenHave,1999),andadaptiveGauss-Hermitequadrature(LiuandPierce,1994).Inthissection,wewillusetheadaptiveGauss-HermitequadraturemethodsinceitconvergestotheMLsolutionasthenumberofquadraturepointsincreases(CoullandAgresti,2003).ThismethodusesthesameweightsasGauss-Hermitequadrature,butitusesnodesthatarecenteredbythemodeofanintegrandandscaledaccordingtotheestimatedcurvatureoftheintegrandatthemode. Forsimplicity,weusethelikelihoodfunctionin(7-9)toillustratetheadaptiveGauss-Hermitemethod.Leth(uk;;;)="cYl=1(1ljky1ljk)(2ljkyk2l)#1 106

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where1ljk=expfl(p Forclusterk,denotenodesthatarezerosofthesth-orderHermitepolynomialbyfqjk:q=1;;sg,andthecorrespondingweightsbyfwqjk:q=1;;sg(Liuetal.1994).Letqjk=p where1ljk=expflqjkg

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Nowtheintegralin(7-9)isapproximatedbyanitesummationwithsquadraturepoints,andsowecanmaximizeitusingaquasi-Newtonalgorithm.LetLqjk=wqjkexp(qjk2)cYl=1(1ljky1ljk)(2ljky2ljk)!1 @j=XkPsq=1Lqjk@ @jhP2i=1Pcj=1yiljklog(iljk)i Psq=1Lqjk;@l @=XkPsq=1Lqjk@ @hP2i=1Pcj=1yiljklog(iljk)i Psq=1Lqjk;@l @=XkPsq=1Lqjk(qjk2

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@j[2Xi=1cXl=1yiljklog(iljk)]=2Xi=1(jI(i=2)kq)[yijjk @[2Xi=1cXl=1yiljklog(iljk)]=cXl=1yiljk(jqjk)(j1qjk) (jqjk)(j1qjk): Oneofthemaininterestsinthissectionistopredictkwhenitisafunctionofarandomeect.AnempiricalBayes(EB)estimatecanbeusedforthispurpose,whichisaconditionalexpectationofk,giventheobservations.GiventheMLestimates(^1;;^c1;^,and^),theparameterofinterestkisafunctionofukandtheconditionaldensityofukisf(ukjyk;^;^;^)=f(ykjuk;^;^;^)f(ukj^)=Rf(ykjuk;^;^;^)f(ukj^)duk:Thus,anEBestimateofkis ^k=E[(T2jkA1jk)jyk;^;^;^]=Z1(~T2jkA~1jk)f(ukjyk;^;^;^)duk=Z1(~T2jkA~1jk)h(uk;^;^;^)duk=Zh(uk;^;^;^)duk;(7{13) where~ijkarevaluesofijksubstitutedwith^,^,and^.ThedenominatorRh(uk;^;^;^)dukisapproximatedbyp 109

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Avarianceof^kcanalsobecalculatedconditionally.AnaivevarianceisdVar(^k)=E[(T2jkA1jk)2jyk;^;^;^](E[T2jkA1jkjyk;^;^;^])2=Z1(~T2jkA~1jk)2h(uk;^;^;^)duk=Zh(uk;^;^;^)duk^2k; TheCMSEPisestimatedby where^ak=akj(;;)=(^;^;^)and^Iistheinformationmatrixevaluatedat(^;^;^).AlthoughanEBestimateofkin(7-13)andthepredictionvarianceof^kin(7-14)canbeobtainedbyusingtheadaptiveGauss-Hermitemethoddirectly,theyarealsoobtainedbyusing\procnlmixed"inSAS.InSAS,inordertondthepredictionvarianceof^k,rstitcalculatesthepredictionvariancematrixfor(^;^;^;^uk),andthenusesthedeltamethodtondthepredictionvariancefor^k.Lettinguk=E[ukjyk;;;]and 110

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Table7-6. MLestimatesofparametersandtheirstandarderrors RandomInterceptRandomInterceptandSlopeParametersMLestimatesStandarderrorsMLestimatesStandarderrors Table7-7showsEBestimatesofthemeasuresfortheeightclinics.Thesevaluesareverysimilarandaregreaterthan0.5,whichisexpectedsince^>0andsincethecumulativeoddsratioeectmeasureisconstantforthismodel. Thecumulativelogitmodelwithbotharandominterceptandarandomslope,whichallowsheterogeneoustreatmenteectsacrosstheclinics,gives^=0:923withs.e.=0.526.Theoveralltreatmenteect^issimilarasoneunderthepreviousmodel,butitsstandarderrorisalmostdouble.Thisisnotsurprising,sincethismodelallowsextraheterogeneity,andtheestimatedheterogeneityissubstantial(^b=1.22).Sincethe95%Waldcondenceintervalfortheparameterincludes0underthismodel,wecannotreject 111

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Table7.7alsoshowsEBestimatesofthe'sunderthismodel.Unlikethepreviousmodel,thereisconsiderableheterogeneityintheestimates,whichiscausedbyanassumptionthateachclinicmighthaveadierenttreatmenteect.TheEBestimatesforclinics2and6arelessthan0.5,andthosefortheotherclinicsaregreaterthan0.5.Intermsoftraditionalstatisticalsignicance,theonlycentersthatwecanconcludehaveabettereectfordrugsthanplaceboare1,5,and7.TheinterpretationoftheEBestimateforclinic1,forinstance,isasfollows:Supposethatweselectrandomlyonepatientwhoreceivesaplaceboandanotherpatientwhoreceivesanactivedrugforasthmarandomlyandindependentlyinclinic1.Thentheestimatedprobabilitythatthepatientreceivingthedrughasabetterresponsethanthepatientreceivingtheplacebois0.793. Thesecondmodel,whichisthecumulativelogitmodelwithbotharandominterceptandarandomslopebecomestherstmodel,whichhasarandominterceptifb=ub=0.ThelikelihoodratioteststatisticfortestingH0:b=ub=0againstHa:b>0isthedierencebetweendeviancesofthetwomodels.Thenulldistributionofthestatisticisamixtureof21and22withthesameprobability(seeMolenberghsandVerbeke(2007),andStramandLee(1994,1995)).Sincethestatisticis6:9(=571:2564:3)forthesedata,thecorrespondingp-valueis0.021.Hencewecaninferthatthereisstrongevidenceofheterogeneousgroupeects. 112

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EBestimatesof'sandtheirstandarderrorsofprediction UnrestrictedModelRandomInterceptModelRandomInterceptandSlopeModelParameterssample^s.e.of^EBestimatess.eofpredictionEBestimatess.eofprediction

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Toestimatethemeasure,wedevelopedveasymptoticcondenceintervals:theWaldcondenceinterval,theWaldcondenceintervalusingalogittransformation,thelikelihoodratiotest-basedcondenceinterval,thescorecondenceinterval,andapseudoscore-typecondenceinterval.Sincewedealtwithadatasetthatwassummarizedinaformofa2ctable,withcoutcomecategories,weassumedobservationsinthetablewerefromtwomultinomialdistributions.Todeterminecellprobabilitiesofthemultinomialdistributions,weusedtwodierentmodels:anunrestrictedmodelandacumulativelogitmodel. InChapter2wedevelopedthevecondenceintervalmethodsoftheordinaleectsizemeasureunderanunrestrictedmodel.ThemaximumlikelihoodestimateofthemeasurewastheMann-WhitneyU-statisticallowingties,dividedbytheproductofthesamplesizes,anditsvariancewasexpressedintermsofmultinomialcellprobabilities.Waldcondenceintervalswereeasilycalculatedusingthefactsthatthemaximumlikelihoodestimateofthemeasurehasanasymptoticnormalityandthemaximum 114

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Inanalternativeapproach,inChapter3weconsideredacumulativelogitmodel.Unlikeanunrestrictedmodelthathad2(c1)parameters,thismodelhasonlycparameters.Thenumberofparameterswasreducedsubstantiallyforlargec.Underthismodel,wedevelopedveasymptoticcondenceintervalmethodsforthemeasure,usingsimilarargumentsasinChapter2.SimulationstudiesinChapter4showedthatthemethodsdevelopedunderanunrestrictedmodelweresafertouseinpractice,andthescorecondenceintervalandapseudoscore-typecondenceintervalsperformedbest. Chapter5andChapter6focusedonperformanceofthemethodsforotherdatastructuresincludingmatched-pairsdataandfully-rankeddata.Formatched-pairsdata,theWaldcondenceintervalusingalogittransformationperformedwell.Forfully-rankeddata,ontheotherhand,thescorecondenceintervalperformedwell,alongwithapseudoscore-typecondenceintervaldevelopedbyNewcombe(2006b). InChapter7,wefocusedonmodelingtheordinaleectsizemeasurewhenexplanatoryvariableswereavailable.Whenallcovariateswerecategorical,weconsideredalogitmodelthatdescribedthecovariatesdirectly,insteadofusingacumulativelogitmodel.Wedevelopedthescorecondenceintervalforthemeasure,aftershowingthatourcasewasaspecialcaseofthehomogeneouslinearpredictor(HLP)model.Whentherewereeither 115

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Oneoftheunsolvedproblemsinthisdissertationistheexistenceanduniquenessoftherestrictedmaximumlikelihoodestimatesofcellprobabilities.Inthisdissertation,weassumedtheywereuniquelyexistent.Infuture,wewillinvestigatethisproblemforourmodel,sincethereisnogeneraltheoryaboutit. Themeasureweusedinthisdissertationisdesignedforordinalcategoricalresponses.Forfuturework,wewillinvestigateeectsizemeasuresfornominalresponses.Onepossiblemeasureistheindexofdissimilarity,whichisdenedasPci=1jiij=2ina2ctable.Weplantodevelopgoodcondenceintervalsforthisandothermeasures. 116

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(1)Let=P(Y1
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(1)LetY1andY2beordinalcategoricalvariableshavingccategories.Assumingtheyareidentical,wehave(1;2;;c)Tascellprobabilitiesforeachtable.Then,themeasureis=P(Y1aab+0:5cXa=12a=0:5(cXi=1i)2=0:5: 2,whichimplies bythesymmetry.Hencethemeasureis=P(Y1
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Proof: Bamber(1975)showedthatthemeasureisthesumoftheareaofalltrapezoidsunderstraight-linesegmentsbetweentheadjacentpointsoff(P(Y1>c);P(Y2>c));(P(Y1>c1);P(Y2>c1));;(P(Y1>1);P(Y2>1));(1;1)g: Proof: Let(T1;T1)Tand(T2;T2)Tbein.Then,1T1=1,1T2=1,1T1=1,and1T2=1.Let0c1.Then,1T(c1+(1c)2)=1and1T(c1+(1c)2)=1. 119

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Proof: Foreach(T1;T1)T;(T2;T2)T2and0c1,l(c(1;1)+(1c)(1;1))=l((c1+(1c)2;c1+(1c)2))=Xan1alog(c1a+(1c)2a)+Xbn2blog(c1a+(1c)2a)>Xan1a[clog1a+(1c)log2a)]+Xbn2b[clog1b+(1c)log2b)])=cl(1;1)+(1c)l(2;2): Proof1:UsingDeltaMethod Notethat,bythemultivariatecentrallimittheorem,p SinceX1andX2areindependent,^and^arealsoindependentandsop

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2!TB!; SinceBisasymmetricmatrix,@g @!=B!: Dene10=varX1[EX2((X1ij+1 2i)22=C2:

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Notethat0R2c: Toshowthat0isbounded,notethatalliandjarebetween0and1.Thus,jj(;)jj2=cXi=12i+cXj=12j2c: 122

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: Fory12Rc,y22Rc,and=0B@121CA,notethatDiag(Z)0B@y1y21CA=0B@1y12y21CA: Recallthat,underH0:=0,l0=sup(;)20l(;): 123

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and1T(a1+(1a)2)=1:

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2[1+1221]. Using22=1111221,MP1=+1+20B@0:5010:51CA0B@1+2+1CA=(0:5+1++2)1++0:5+22+=0:5(1+1221):

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(1)Supposethatthereexistsj02f1;;c1gsuchthatij=0forallelementsinf(i;j):ij0orjj0+1g.Then,(1+;;j0+)=0and(+(j0+1);;+c)=0.MP1isthemeasureappliedtothefollowing2ctable: 1j0 00 Thus,MP1=(0j(j0+1)+;;c+)A(+1;;+j0j0)T=0: 1j0 00 (1)Supposethatij=0forallij.Then,Pc1i=1Pcj=i+1ij=0andPci=1ii=0.Thus,MP1=0. (2)Supposethatij=0forallji.Then,Pc1i=1Pcj=i+1ij=1becauseofPci=1Pcj=1ij=1.Thus,MP1=1. 126

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2. Supposethatij=jiforalliandj.Then,i+=+iforalli,whichimpliesMP1=0:5.Ontheotherhand,since2Pc1i=1Pcj=i+1ij+Pci=1ii=1bythesymmetry,Pc1i=1Pcj=i+1ij=0:5(1Pci=1ii).Hence,MP2=c1Xi=1cXj=i+1ij+0:5cXi=1ii=0:5(1cXi=1ii)+0:5cXi=1ii=0:5: 2,butMP2doesnotneedtobe1 2. Ifthemarginalrowdistributionandthemarginalcolumndistributionarebothsymmetricwithsupportoverallccategories,thenbyProperty1.3,MP1=0:5.However,itisnottrueforMP2.Forinstance,considerthefollowingtable: 0.1670.1640.002 0.333 0.0310.2720.030 0.333 0.0020.1640.167 0.333 0.2000.6000.200 1 Thisjointcellprobabilitiessatisfytheassumptioninthisproperty,andsogiveMP1=0:5,butMP2=0:464. 2,butMP1doesnotneedtobe1 2.

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IfPc1i=1Pj>iij=Pc1j=1Pi>jij,thenPc1i=1Pj>iij=0:5(1Piii),whichgivesMP2=0:5(1Piii)+0:5Piii=0:5.Ontheotherhand,itisnottrueforMP1.Forinstance,considerthefollowingtable: 0.0500.2000.100 0.350 0.1000.0500.100 0.250 0.1500.1500.100 0.400 0.3000.4000.300 1 Thisjointcellprobabilitiessatisfytheassumptioninthisproperty,andsogiveMP2=0:5,butMP1=0:483. SupposethatY1andY2areindependent.Then,bythedenition,ij=i++jforalliandj.Thus,MP2=c1Xi=1cXj=i+1ij+0:5cXi=1ii=c1Xi=1cXj=i+1i++j+0:5cXi=1i++i=c1Xi=1i+(cXj=i+1+j)+0:5cXi=1i++i=MP1: (1)Ifcategoriesarereversed,thenwehave(c+;;1+)Tand(+1;;+c)asmarginalrowprobabilitiesandmarginalcolumnprobabilities,respectively.By 128

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(2)Ifrowsandcolumnsareinterchanged,wehave(+1;;+c)Tand(1+;;c+)asmarginalrowprobabilitiesandmarginalcolumnprobabilities.ByProperty1.2,MP1changesto1MP1.Ontheotherhand,MP2changesto(1MP2)usingthesameargumentusedforreversingcategories. 129

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y1
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n2reltol)&(iter
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pp1<-x1/pi12 pp12
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H1
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theta.check=1e-8)MLE.beta
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if(abs(theta.hat)>1e-8&abs(theta.hat-1)>1e-8)f L.1
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Score<1e10 rest0)cal2[j]<(x2[j]-n2*lambda.nal)2/(n1*lambda.nal) if(Score<=critic.Chisq)store.Score[k]0]) Score.CI
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critic.Chisq
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for(iin1:(c2))f var.pair1e-5&abs(theta.pair-1)>1e-5)f L.1
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TableC-1. Opinionaboutsurrogatemotherhoodandthelikelihoodofsellingakidneywithagebetween18and35 18Forbidit011002Permitit100001 19Forbidit000112Permitit0133613 20Forbidit303208Permitit302439 21Forbidit213016Permitit014328 22Forbidit110013Permitit4242514 23Forbidit521109Permitit1053716 24Forbidit014117Permitit5566729 25Forbidit401106Permitit2264519 26Forbidit010022Permitit1165922 27Forbidit210227Permitit3147520 28Forbidit311308Permitit6328322 29Forbidit221005Permitit5223517 30Forbidit213118Permitit8551726 31Forbidit322007Permitit2493624 32Forbidit5220312Permitit6373322 33Forbidit5411415Permitit5471524 34Forbidit102104Permitit7263523 35Forbidit332109Permitit5313618

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Opinionaboutsurrogatemotherhoodandthelikelihoodofsellingakidneywithagebetween36and55 36Forbidit330107Permitit3423618 37Forbidit4243114Permitit5287729 38Forbidit5203111Permitit7354524 39Forbidit4123212Permitit6024618 40Forbidit7110211Permitit8252421 41Forbidit4222313Permitit3322515 42Forbidit8112012Permitit10256629 43Forbidit103127Permitit10419327 44Forbidit2321210Permitit5132516 45Forbidit400206Permitit7234218 46Forbidit001012Permitit4244014 47Forbidit401128Permitit4122413 48Forbidit5320111Permitit4545523 49Forbidit6322215Permitit5545221 50Forbidit201003Permitit0237214 51Forbidit402006Permitit4441215 52Forbidit210104Permitit9102113 53Forbidit501107Permitit3322313 54Forbidit300418Permitit3232212 55Forbidit601017Permitit500207

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Opinionaboutsurrogatemotherhoodandthelikelihoodofsellingakidneywithagebetween56and75 56Forbidit401218Permitit113117 57Forbidit511007Permitit010023 58Forbidit200002Permitit3313212 59Forbidit412029Permitit331018 60Forbidit400004Permitit113117 61Forbidit403209Permitit020114 62Forbidit500027Permitit400228 63Forbidit312006Permitit100214 64Forbidit320117Permitit210328 65Forbidit401005Permitit311027 66Forbidit120104Permitit101012 67Forbidit301004Permitit220318 68Forbidit100001Permitit200305 69Forbidit211015Permitit112026 70Forbidit330118Permitit101226 71Forbidit202004Permitit000011 72Forbidit300036Permitit223018 73Forbidit500005Permitit110013 74Forbidit400206Permitit011013 75Forbidit101125Permitit401106

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Opinionaboutsurrogatemotherhoodandthelikelihoodofsellingakidneywithagebetween76and89 76Forbidit200114Permitit200024 77Forbidit001001Permitit320207 78Forbidit010021Permitit401005 79Forbidit110103Permitit200024 80Forbidit201014Permitit010102 81Forbidit002002Permitit302106 82Forbidit002002Permitit302106 83Forbidit000011Permitit100102 84Forbidit110103Permitit000000 85Forbidit000000Permitit002002 86Forbidit000101Permitit100001 87Forbidit200013Permitit000000 88Forbidit000011Permitit120014 89Forbidit100023Permitit000000

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Lehmann,E.L.(1975)Nonparametrics:statisticalmethodsbasedonranks.California:Holden-Day. Liu,Q.,andPierce,D.A.(1994).AnoteonGauss-Hermitequadrature.Biometrika81,624{629. Lumley,T.(1996).Generalizedestimatingequationsforordinaldata:anoteonworkingcorrelationstructures.Biometrics52,354{361. Mann,H.B.,andWhitney,D.R.(1947).Onatestofwhetheroneoftworandomvariablesisstochasticallylargerthantheother.AnnalsofMathematicalStatistics18,50{60. McGraw,K.O.,andWong,S.P.(1992).Acommonlanguageeectsizestatistic.PsychologicalBulletin111,361{365. Molenberghs,G.,andVerbeke,G.(2007).Likelihoodratio,score,andWaldtestsinaconstrainedparameterspace.TheAmericanStatistician61,22{27. Newcombe,R.G.(1998).Intervalestimationforthedierencebetweenindependentproportions:comparisonofelevenmethods.StatisticsinMedicine17,873{890. Newcombe,R.G.(2006a).CondenceintervalsforaneectsizemeasurebasedontheMann-Whitneystatistic.Part1:generalissuesandtailareabasedmethods.StatisticsinMedicine25,543{557. Newcombe,R.G.(2006b).CondenceintervalsforaneectsizemeasurebasedontheMann-Whitneystatistic.Part2:asymptoticmethodsandevaluation.StatisticsinMedicine25,559{573. Newcombe,R.G.(2007).ArelativemeasureofeectsizeforpaireddatageneralisingtheWilcoxonmatched-pairssigned-ranksteststatistic.Statis-ticsinMedicine,submitted. Noether,G.E.(1967).Elementsofnonparametricstatistics.NewYork:Wiley. Owen,D.B.,Craswell,K.J.,andHanson,D.L.(1964).NonparametricuppercondenceboundsforPr(X
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Stram,D.O.,andLee,J.W.(1994).Variancecomponentstestinginthelongitudinalmixedeectsmodel.Biometrics50,1171{1177. Stram,D.O.,andLee,J.W.(1995).CorrectiontoVariancecomponentstestinginthelongitudinalmixedeectsmodel.Biometrics51,1196. TenHave,T.R.,Kunselman,A.R.,andTran,I.(1999).Acomparisonofmixedeectslogisticregressionmodelsforbinaryresponsedatawithtwonestedlevelsofclustering.StatisticsinMedicine18,947{960. Troendle,J.F.(2002),AlikelihoodratiotestforthenonparametricBeherens-Fisherproblem.BiometricalJournal44,813{824. Vargha,A.,andDelaney,H.D.(1998).TheKruskal-Wallistestandstochastichomogeneity.JournalofEducationalandBehavioralStatistics23,170{192. Vargha,A.,andDelaney,H.D.(2000).AcritiqueandimprovementoftheCLcommonlanguageeectsizestatisticsofMcGrawandWong.JournalofEducationalandBehavioralStatistics25,101{132. Ury,H.K.(1972).Ondistribution-freecondenceboundsforPr(Y
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EuijungRyuwasbornonApril23rd,1975,inYecheon,Kyungpookprovince,Korea.ShelivedinYecheonuntil1990andthenmovedtoDeaguwithhersistersandbrother.In1995,EuijungattendedKyungpookNationalUniversity,whereshestudiedintheDepartmentofMathematicalSciences.AftergraduatingwithaBachelorofScienceinStatistics,shecontinuedherstudyintheDepartmentofStatistics,andreceivedhermaster'sdegree.EuijungwasacceptedasaPh.D.studentintheDepartmentofStatisticsattheUniversityofFlorida.Inthesummerof2002,shemovedtoGainesville,Florida,andcontinuedhergraduatestudyintheU.S.Fortherstfouryears,shewasontheGraduateAlumniFellowship.Forherfthyear,sheworkedasateachingassistant.ShewillreceiveherPh.D.degreeinAugust2007.Aftergraduating,EuijungwillmovetoRochester,Minnesota,whereshewillbearesearchassociateintheMayoClinic. 147