<%BANNER%>

Design Considerations and Optimization for Univariate and Multivariate Generalized Linear Models

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Iwouldrstliketogivethankstomyadvisor,Dr.AndreI.Khuri,forhisenthusiasticsupervisionandencouragementduringmydoctoralresearch.Thisdissertationwouldnothavebeenpossiblewithouthisguidanceandtremendouspatience.Iwouldalsoliketothankallmycommitteemembers:Dr.MalayGhosh,Dr.RonaldRandles,Dr.TrevorParkandDr.RickSmith.SpecialthanksgotothefacultyandstaattheDepartmentofStatistics,UniversityofFlorida.Inaddition,IgivethankstomyundergraduateprofessorDr.AtindraMohanGoon.Iamindebtedtoallmyfriendsfortheirhelpandconstantencouragement,especiallyAnanyaRoy,AnalesDeBhaumik,VivekanandaRoy,DamarisSantanaMorantandLudwigHeigenhauser.Mostimportantly,Iwishtothankmyentirefamily,especiallyDebraj,Hanshu,OdaandTiti. iv

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page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. viii LISTOFFIGURES ................................ ix ABSTRACT .................................... x CHAPTER 1INTRODUCTION .............................. 1 2LITERATUREREVIEW .......................... 4 2.1ResponseSurfaceMethodology .................... 4 2.2GeneralizedLinearModels ....................... 6 2.2.1LocallyOptimalDesigns .................... 7 2.2.2SequentialDesigns ........................ 10 2.2.3BayesianOptimalDesigns ................... 11 2.2.4QuantileDispersionGraphs(QDGs)Approach ........ 12 2.3MultiresponseExperiments ....................... 13 2.4MultivariateGLMs ........................... 14 2.4.1OptimalDesignsforBivariateLogisticModels ........ 15 2.4.2OptimalDesignsforMultivariateLogisticModels ...... 17 2.5MultiresponseOptimization ...................... 18 3DESIGNCOMPARISONSFORUNIVARIATEGENERALIZEDLINEARMODELS ................................... 21 3.1Introduction ............................... 21 3.2GeneralizedLinearModels ....................... 22 3.2.1EstimationoftheMeanResponse ............... 23 3.2.2ThePredictionBias ....................... 25 3.2.3TheMean-SquaredErrorofPrediction ............ 26 3.3ChoiceofDesign ............................ 26 3.4QuantileDispersionGraphs ...................... 26 3.5Examples ................................ 28 3.5.1PoissonResponsewithaLogarithmicLink .......... 29 3.5.2PoissonResponsewithaSquareRootLink .......... 30 3.5.3ANumericalExample ...................... 31 v

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.... 41 4.1Introduction ............................... 41 4.2BiasinaGeneralMultivariateSetting ................ 41 4.3MultivariateGLM ........................... 42 4.4BiasinaMultivariateGLMSetting .................. 44 4.5FinalExpressionfortheBias ..................... 49 4.6Example:BivariateBinaryDistribution ............... 51 4.6.1BiasforaBivariateBinaryDistribution ............ 53 4.6.2NumericalExample ....................... 54 5COMPARISONOFDESIGNSFORMULTIVARIATEGENERALIZEDLINEARMODELS .............................. 57 5.1Introduction ............................... 57 5.2MultivariateGeneralizedLinearModels ................ 57 5.2.1EstimatingtheMeanResponse ................. 58 5.2.2PredictionVariance ....................... 59 5.2.3PredictionBias ......................... 60 5.2.4Mean-squaredErrorofPrediction ............... 62 5.3CriterionforComparingDesigns .................... 62 5.4QuantileDispersionGraphs ...................... 63 5.5BivariateBinaryDistribution ..................... 64 5.5.1BivariateBinaryDistributionwithaLogitLink ....... 65 5.5.2NumericalExample ....................... 67 6OPTIMIZATIONINMULTIVARIATEGENERALIZEDLINEARMODELS 75 6.1Introduction ............................... 75 6.2MultivariateGeneralizedLinearModels ................ 75 6.3SimultaneousOptimizationofaMultiresponseFunction ....... 76 6.3.1GeneralizedDistanceApproachinMultivariateGLM .... 77 6.3.2Constructionofcondenceintervalonthetruemeanresponse 78 6.4Example:BivariateBinaryDistribution ............... 79 6.4.1ANumericalExample ...................... 80 6.4.2ComputerProgramsUsedfortheSelectionProcess ..... 85 7SUMMARYANDFUTURERESEARCHTOPICS ............ 87 7.1Summary ................................ 87 7.2FutureResearch ............................. 88 APPENDIX APROOFOFRESULTSUSEDINCHAPTER 3 .............. 90 A.1DerivationofFormula( 3{30 ) ..................... 90 A.2DerivationofFormula( 3{33 ) ..................... 92 vi

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4 .............. 94 B.1DerivationofFormula( 4{15 ) ..................... 94 B.2DerivationofFormula( 4{20 ) ..................... 95 B.3DerivationofFormula( 4{21 ) ..................... 96 B.4DerivationofFormula( 4{23 ) ..................... 97 B.5DerivationofFormula( 4{25 ) ..................... 98 B.6DerivationofFormula( 4{27 ) ..................... 100 B.7DerivationofFormula( 4{29 ) ..................... 101 CPROOFOFRESULTSUSEDINCHAPTER 5 .............. 103 DRCODE:CONSTRUCTIONOFQUANTILEDISPERSIONGRAPHSINCHAPTER 3 ............................... 105 REFERENCES ................................... 115 BIOGRAPHICALSKETCH ............................ 120 vii

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Table page 3{1Experimentaldesignandresponsevalues. .................. 31 3{2Parameterestimatesandmodelanalysisusingloglink. .......... 32 3{3DesignD2. ................................... 33 3{4Parameterestimatesandmodelanalysisusingsquarerootlink. ..... 35 4{1Experimentaldesignandresponsevalues. .................. 55 4{2Maximumlikelihoodestimatesandstandarderrors. ............ 55 4{3Estimatedmeanresponsevalues. ...................... 56 5{1ExperimentaldesignD1(57factorial)andresponsevalues. ...... 70 5{2Maximumlikelihoodestimatesandstandarderrors. ............ 72 5{3DesignsD1(57factorial)andD2(32factorial). ............. 72 6{1Experimentaldesignandresponsevalues. .................. 81 6{2Maximumlikelihoodestimatesandstandarderrors. ............ 82 6{3Estimatedvariance-covariancematrixof^. ................. 83 6{4TheindividualoptimaandtheregionD. ................. 84 6{5Simultaneousoptima. ............................. 85 viii

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Figure page 3{1ConcentricsquareswithintheregionR. .................. 36 3{2QDGsfordesignsD1andD2usingPoissondistribution(=1)withaloglink. .................................... 37 3{3ComparisonoftheQDGsfordesignD1usingPoissondistribution(=1)withlogandsquarerootlinks. ...................... 38 3{4ComparisonoftheQDGsfordesignD2usingPoissondistribution(=1)withlogandsquarerootlinks. ...................... 39 3{5ComparisonoftheQDGsfordesignsD1andD2usingfourdierentdistributions,allwithloglink,using=1. ........................ 40 5{1DesignsD1(57factorial)andD2(32factorial) ............. 73 5{2ComparisonoftheQDGsfordesignsD1(57factorial)andD2(32factorial)giveninTable 5{3 ......................... 74 ix

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x

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xi

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NelderandWedderburn ( 1972 )asanextensionoftheclassoflinearmodels.UndertheframeworkofGLMs,discreteaswellascontinuousresponsescanbeaccommodated,andnormalityandconstantvariancesarenolongerarequirementfortheresponse.IntheareaofGLMs,oneofthemainconsiderationsisthechoiceofanexperimentaldesign.Unfortunately,littleworkhasbeendoneinthisarea.ThisismainlyduetoaseriousproblemcausedbythedependenceofadesignforaGLMontheunknownparametersofthettedmodel.ThisdependenceproblemcausesgreatdicultyintheconstructionandevaluationsofdesignsforGLMs.Forthemostpart,designcriteriaforlinearmodelsandGLMspertaintomodelsforasingleresponse.Thereare,however,manyexperimentalsituationswhereseveralresponsescanbeobservedforeachsettingofagroupofcontrol 1

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variables.Forexample,indoseresponsestudies,wemaycomeacrosssituationswhereseveralresponsescanbeobservedforthesamepatient.Forexample,inadditiontothestandardresponseconcerningtheecacyofadrug,wemayalsomeasureitstoxicity,thatis,anysideeectsexperiencedbythepatient.Thus,oneimportantareathatneedsinvestigationisthechoiceofdesignsformultivariateGLMs.ItiswellknownthatinsmallsamplestheparameterestimatesforamultivariateGLMarebiased.Thusthemean-squarederrorofprediction(MSEP)shouldbeusedtoassessthequalityofpredictionassociatedwithagivendesign.TheMSEPincorporatesboththepredictionvarianceandthepredictionbias,whichresultsfromusingmaximumlikelihoodestimatesoftheparametersofthettedlinearpredictor.Searchingtheliterature,wecameupwithanexpressionforthebiasofthemaximumlikelihoodestimates(MLEs)inageneralmultiparametersettinggivenby CoxandSnell ( 1968 ),andanapproximateexpressionofthebiasofMLEsinthecaseofunivariateGLMswasgivenby CordeiroandMcCullagh ( 1991 ).Hence,weneedanexpressionforthebiasinthemultivariateGLMcase.OneimportantobjectiveofmultiresponseexperimentationintheGLMsetupisthedeterminationofoptimumoperatingconditionsontheinput(control)variablesthatwillleadtothesimultaneousoptimizationofthepredictedvaluesoftheresponseswithinacertainregionofexperiment.Forexample,inadose-responseexperiment,wherebothecacyandtoxicityresponsesaremeasuredateachdose,theexperimentermaywishtondthedoselevelofthedrug(s)whichsimultaneouslymaximizesecacyandminimizestoxicity.Severalapproachesdealingwithmultiresponseoptimizationinthecaseoflinearmodelsareavailableintheliterature.MultiresponseoptimizationinaGLMsituationisstillinitsnascentstage.

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Inthisdissertation,asolutiontothedesigndependenceprobleminGLMsisproposedusingagraphicaltechniquecalledquantiledispersiongraphs(QDGs).Thisgraphicaltechniquehasbeenusedtocomparedesignsforbothunivariateandmultivariategeneralizedlinearmodels.AnapproximateexpressionforthebiasinthemultivariateGLMcase,andmultiresponseoptimizationinaGLMsituationhavealsobeendiscussed.Thetopicsdiscussedinthisdissertationareorganizedasfollows:Chapter2isareviewofliteratureconcerningGLMdesignsandoptimizationinthecaseofbothunivariateandmultivariateresponses.ComparisonofdesignsforPoissonregressionmodelsusingQDGsisdiscussedinChapter3.AdetailedderivationoftheexpressionforbiasinamultivariateGLMsetupisgiveninChapter4.Chapter5dealswiththecomparisonofdesignsusingQDGsformultivariateGLMs.DeterminationofoperatingconditionsonasetofcontrolvariablesthatsimultaneouslyoptimizetheresponsesinamultivariateGLMsituationisdiscussedinChapter6.Finally,asummaryandalistoffutureresearchtopicsaregiveninChapter7.

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BoxandWilson ( 1951 )laidthefoundationsforresponsesurfacemethodology.Itwasinitiallydevelopedtodetermineoptimumoperatingconditionsinthechemicalindustry.Itisnowusedinavarietyofeldsandapplications,notonlyinthephysicalandengineeringsciences,butalsointhebiological,clinicalandsocialsciences,tonamejustafew.Overthepastftyyears,therehavebeensomeextensivereviewsofRSM.The HillandHunter ( 1966 )paperfeaturedanextensivebibliographyandpresentedapplicationsinthechemicalandprocessindustries. Myersetal. ( 1989 )emphasizedthechangesthathadtakenplaceintheRSMtheoryandpracticeduringthe1970'sand1980's.TheirpaperdiscussedapplicationsofRSMinthephysicalandengineeringsciences,foodscience,socialscience,andthebiologicalsciences.Themostrecentreviewpaperwasthatof Myersetal. ( 2004 )whichfocussedonRSMactivitiessince1989.TheydiscusscurrentresearchareasinRSM,suchasrobustparameterdesignintheframeworkofRSM,RSMandgeneralizedlinearmodels,parametricandsemi-parametricRSM.Inadditiontothesereviewpapers,thebooksby BoxandDraper ( 1987 ), MyersandMontgomery ( 2002 ), KhuriandCornell ( 1996 ),and Khuri ( 2006 )provideacomprehensivecoverageofthevarioustechniquesinRSM. 4

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ThekeyideasinRSMwereinitiallydevelopedusingclassicallinearmodelswithcontinuousresponses(quiteoften,normallydistributed)withuncorrelatederrorsandhomogeneousvariances.However,inmanyexperimentalsituationsinvolving,forexample,clinicalandepidemiologicaldata,suchassumptionsareinvalid.Forexample,dataonhumanresponsestendtobemorevariablethanisexpectedunderthehomogeneouserrorvarianceassumption,biologicaldatamaybecorrelatedduetogeneticrelationships,anddose-responseexperimentsyielddiscretedata.Duetothenatureofthedataasdescribed,doingstatisticalanalysisusingstandardlinearmodelswillgiveinadequateresults.Forsuchdata,generalizedlinearmodels(GLMs)wouldbemoreappropriate.GLMshaveprovedveryusefulinseveralareasofapplicationsasinappliedbiomedicalelds,entomology,climatology,etc.However,touseGLMseectivelyintheabovementionedscenarios,anecientdesignwithdesirablepropertiesisneeded.ByachoiceofdesigninRSM,itismeantthedeterminationofthesettingsofthecontrolvariablesthatyieldadequatepredictionsoftheresponseofinterestthroughouttheexperimentalregion.Optimaldesigntheoryforlinearmodelshasbeenwidelystudied,butthesamecannotbesaidwithregardtoGLMs.ThisismainlybecauseoptimaldesignsforGLMsdependontheunknownparametersofthettedmodel.Thustheconstructionofadesignrequiressomepriorknowledgeoftheseparameters.Theexperimenteristhereforepresentedwithadilemmasincethepurposeofadesignistoestimatetheunknownparametersofthemodelusingthedatageneratedbythedesign.Commonapproachestosolvethisdesigndependenceproblemincludethefollowing:1.thespecicationofinitialvalues,orbest\guesses,"oftheparametersinvolved,andthesubsequentdeterminationoftheso-calledlocally-optimaldesignusinganappropriatedesigncriterionsuchasD-optimalityorG-optimality.

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2.thesequentialapproachwhichallowstheusertoobtainupdatedestimatesoftheparametersinsuccessivestages,startingwiththeinitialvaluesusedintherststage.3.theBayesianapproach,whereapriordistributionisassumedontheparameters,whichisthenincorporatedintoanappropriatedesigncriterionbyintegratingitoverthepriordistribution.4.theuseoftheso-calledquantiledispersiongraphs(QDGs)approach,whichallowstheusertocomparedierentdesignsbasedontheirquantiledispersionproles. RobinsonandKhuri ( 2003 )introducedthisapproachtodiscriminateamongdesignsforlogisticregressionmodels.InChapter3,weshalladdressthedesigndependenceproblemforgeneralizedlinearmodels.Themainfocusofattentionwillbeontheuseoftheaforementionedfourthapproach. NelderandWedderburn ( 1972 )asanextensionoftheclassoflinearmodels.Theycanbeusedtotdiscreteaswellascontinuousdatahavingavarietyofparentdistributions.Thetraditionalassumptionsofnormalityandhomogeneousvariancesoftheresponsedata,usuallymadeinananalysisofvariance(orregression)situation,arenolongerneeded.Manystatisticaldevelopmentsintermsofmodellingandmethodology,inthepasttwentyyears,maybeviewedasspecialcasesofGLMs.Examplesincludelogisticregressionforbinaryresponses,linearregressionforcontinuousresponses,andlog-linearmodelsforcounts.AclassicbookonGLMsistheoneby McCullaghandNelder ( 1989 ).Inaddition,themorerecentbooksby Dobson ( 2001 ), Lindsey ( 1997 ), McCullochandSearle ( 2001 ),and MyersandMontgomery ( 2002 )provideaddedinsightintotheapplicationandusefulnessofGLMs.IntheGLMsituationtheresponsevariabley,isassumedtofollowadistributionfromtheexponentialfamily.Thisincludesthenormalaswellas

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thebinomial,Poissonandgammadistributions.Themeanresponseismodelledasafunctionoftheform,E[y(x)]=(x)=h[f0(x)],wherex=(x1;:::;xk)0,f(x)isaknownvectorfunctionoforderp1andisavectorofpunknownparameters.Thefunctionf0(x)iscalledthelinearpredictor,andisusuallydenotedby(x).Itisassumedthath()isastrictlymonotonefunction.Usingtheinverseofthefunctionh()wehaveg[(x)]=(x).Thefunctiong()iscalledthelinkfunction.Estimationofisbasedonthemethodofmaximumlikelihoodusinganiterativeweightedleast-squaresprocedure[see McCullaghandNelder ( 1989 ,pp.40-43)].Thevariance-covariancematrix,Var(^),of^isapproximatedby1 RobinsonandKhuri ( 2003 )],whereisthedispersionparameter,Xisannpmatrixwhosejthrowisoftheformf0(xj),xjisthevalueofxatthejthexperimentalrun(j=1;:::;n),andWisadiagonalmatrixwhosejthdiagonalelementisw(xj;).Anexpressionforw(xj;)isgiveninChapter3.ItfollowsthatVar(^)dependsontheunknownparametervectorthroughw(xj;).Thisleadstotheaforementioneddesigndependenceproblem. MathewandSinha ( 2001 )concerningdesignsforalogisticregressionmodel.Otherrelatedworkincludesthatof AbdelbasitandPlackett ( 1983 ), Minkin ( 1987 ), KhanandYazdi ( 1988 ), Wu ( 1988 )and SitterandWu ( 1993 ).

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Considerabinaryresponsetakingthevalues0and1resultingfromanon-stochasticdoselevelx.Theresponsetakesthevalue1withprobability 1+e(+x);(2{1)whereandareunknownparametersand>0.Therearendistinctdoselevelsx1;:::;xn,andmjobservationsaretakenontheresponseatdoselevelxj(j=1;:::;n).LetPnj=1mj=m.Letyjdenotethenumberofonesobtainedatx=xj(j=1;:::;n).Then,y1;:::;ynaremutuallyindependentandyjhasabinomialdistribution(mj;p(xj));j=1;:::;n.Fortheestimationoftheunknownparametersand,orsomefunctionthereof,theoptimaldesignproblemistoselectthexj's(inagivenregion)andthemj's,withrespecttosomeoptimalitycriterion,foraxedm.Thisisdoneonthebasisofsomemeasureoftheinformationabout(;)yieldedbythedata.Sincethisisadicultandoftenintractableoptimizationproblem,acontinuousdesignmeasureisconsideredinwhichmj=misreplacedbyarealnumberjwith0j1andPnj=1j=1.Thisdesignmeasuredependsontheunknownvaluesofand,andthusgoodinitialestimatesofandareneeded.From( 2{1 )weseethatthedependenceentersthroughz=+xonlyandsotheproblemreducestoobtainingoptimalvaluesofzjandj.Theestimationproblemsthatareusuallyofinterestreferto(a)theestimationof,or=,orsomequantilesofp(x)in( 2{1 ),or(b)thejointestimationofapairofparameterssuchas(i)and,(ii)and=,(iii)andaquantileofp(x),and(iv)twoquantilesofp(x).Forestimatingtwoparametersjointly, MathewandSinha ( 2001 )consideredtheinformationmatrix,I();0=(;),oftheparametersandchosethexj'sandmj'stomaximizesomesuitablescalar-valuedfunctionofthismatrix.TheinformationmatrixI(),asymmetricmatrixoforderpp,pisthenumberofunknownparameters,isdenedas[see Lehmann ( 1983

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p.125)]I()=E@2logL(jx) 1p(xj)+mjlogf1p(xj)g375=nXj=1264log0B@mjyj1CA+yj(+xj)mjlogf1+exp(+xj)g375:Followingtheapproximatedesigntheory,adesignisdenotedbyD=f(xj;j);j=1;:::;ng.Weassumethedoseregion,R,tobe0
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estimatorsofand.TheyshowedthatD-optimaldesignsare2-pointsymmetricdesigns. Wu ( 1985 ), SitterandForbes ( 1997 ), SitterandWu ( 1999 ),amongothers.Inbinaryresponsemodels,theinformationmatrixdependsonboththedesignandthemodel'sparameters.Thustoobtainanoptimaldesign,\good"initialestimatesoftheparametersareneeded.Quiteoften,theseestimatesarepoor.Asolutiongivenby SitterandForbes ( 1997 )inthesecircumstanceswastoperformtheexperimentintwo(ormore)stages.Theinitialdesignisusedtogettheparameterestimatesandthentreatingtheseasthetrueparametervalues,asecond-stagedesignischosen.Thecombinationofrstandsecond-stagedesignsisoptimalinsomesense. SitterandForbes ( 1997 )performedaquantalresponsestudy,whereasubjectwasgivenastimulusxthatexhibitedaresponsewithprobabilityp(x). SitterandForbes ( 1997 )assumedaparametricmodelp(x)=H[(x)]forthequantalresponsecurvewhereandareunknownparametersandH()isaspecieddistributionfunction.Anumber,mj,ofindependentobservationsaretakenatthejthofnstimuluslevelsx1;:::;xn,andthecorrespondingresponsesareobserved.Theresponsesaremutuallyindependentbinomialrandomvariables.

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ThecorrespondingFisherinformationmatrixis[see SitterandForbes ( 1997 ,p.942)] SitterandForbes ( 1997 )reducedtheproblemofndinganoptimalsecond-stagedesigntoatwo-variable(inmanycasesaonevariable)maximizationproblemforalmostanyoptimalitycriterion.Thesearchfortheoptimalsecond-stagedesigncanberestrictedtotwo-pointdesignswhicharesymmetricaboutwithpossiblydierentweightsateachpoint. Chaloner ( 1987 ), ChalonerandLarntz ( 1989 1991 ), MukhopadhyayandHaines ( 1995 ), ChalonerandVerdinelli ( 1995 )and Khurietal. ( 2006 ).MostoftheBayesiancriteriaarebasedonnormalapproximationstotheposteriordistributionof,ascomputationsinvolvingtheexactposteriordistributionareoftenintractable.Weshallassume,asusual,thatthedesignmeasureputsrelativeweights(1;:::;n)atndistinctpoints(x1;:::;xn),respectively,withPnj=1j=1. Chaloner ( 1987 )and ChalonerandLarntz ( 1989 1991 )usedtwocriteria1()=E[logjI(;)j]and2()=E[trB()I(;)1];

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whereB()isappsymmetricmatrix.Iflinearcombinationsoftheparametersareofinterest,thenB()doesnotdependonandisavectorofknownweights.Incaseofnon-linearcombinationsoftheparameters,B()hasentriesthatdependon.The1and2,criteriaare,respectively,equivalenttoD-optimalityandA-optimalityinlinearmodels.AlloftheabovementionedBayesiancriteriaareconcaveoverthespaceofallprobabilitymeasuresonthedesignspaceR.TheEquivalenceTheoremforoptimalityoflinearmodels Whittle ( 1973 )hasbeenextendedtothenon-linearcaseby Silvey ( 1980 ), ChalonerandLarntz ( 1989 ),and Chaloner ( 1993 ). ChalonerandLarntz ( 1989 )consideralogisticregressionmodelsuchthatatdose-levelx,aBernoulliresponseisobserved.Therearetwoparameters0=(;)andtheprobabilityofsuccessis 1+exp((x)):(2{5)TheFisherinformationmatrixhereisthesameastheexpressiongivenin( 2{4 )usingH[(x)]=1 1+exp((x)): ( 1989 )investigatetheBayesianD-optimalityandA-optimalitycriteriainthecontextofabinaryresponselogisticregressionmodelwithasinglevariablexassumingindependentuniformpriorsontheparametersand.Theoptimaldesignsareobtainedthroughimplementingthesimplexalgorithmof NelderandMead ( 1965 ).Itwasnotedthatthenumberofsupportpointsoftheoptimaldesigngrewasthesupportofthepriorbecamelarger.Theoptimaldesignsingeneralwerenotequispacedandwereunequallyweighted. RobinsonandKhuri ( 2003 )inaGLMsituation.Inthisgraphicaltechnique,designsarecomparedonthebasisof

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theirquantiledispersionproles.Sinceinsmallsamplestheparameterestimatesareoftenbiased, RobinsonandKhuri ( 2003 )considerthemean-squarederrorofprediction(MSEP)asacriterionforcomparingdesigns.TheQDGsarebasedonstudyingthedistributionoftheMSEPthroughouttheexperimentalregion,R.ThedistributionoftheMSEPisstudiedintermsofitsquantiles.ThesegraphsprovideanassessmentoftheoverallpredictioncapabilityofagivendesignovertheentireexperimentalregionthroughavisualdisplayoftheMSEP.Theyalsogiveacleardepictionofthedependenceofthedesignonthemodel'sunknownparameters. KhuriandLee ( 1998 )usedagraphicalprocedureforcomparingdesignsfornonlinearmodels.TheQDGapproachof RobinsonandKhuri ( 2003 )generalizestheworkof KhuriandLee ( 1998 )byaddressingnon-normalityandnonconstanterrorvariance.Theyconsidertheproblemofdiscriminatingamongdesignsforlogisticregressionmodels.InChapters3and5,QDGswillbediscussedandusedtocomparedesignsforunivariateandmultivariateGLMs,respectively. HillandHunter ( 1966 ), Myersetal. ( 1989 )and Khuri ( 1996 )citedseveralpapersinwhichmultiresponseswereinvestigated.Inselectingadesignoptimalitycriterionformultiresponseexperiments,oneneedstoconsideralltheresponsessimultaneously.Onesuchcriterionwas

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developedby Fedorov ( 1972 ).Hisprocedure,however,requiredknowledgeofthevariance-covariancematrixassociatedwiththeseveralresponsevariables. WijesinhaandKhuri ( 1987 )modiedFedorov's( 1972 )procedurebyusinganestimateofthevariance-covariancematrix.VerylittleworkhasbeendoneonmultiresponseormultivariateGLMs,particularlyinthedesignarea.Suchmodelsareconsideredwheneverseveralresponsevariablescanbemeasuredforeachsettingofagroupofcontrolvariables,andtheresponsevariablesareadequatelyrepresentedbyGLMs. FahrmeirandTutz ( 2001 )and McCullaghandNelder ( 1989 )discusstheanalysisofmultivariateGLMs.Letqdenotethenumberofresponsevariables.Inmultivariategeneralizedlinearmodels(GLMs),theq-dimensionalvectorofresponses,y,isassumedtofollowadistributionfromtheexponentialfamily.Themeanresponse(x)=[1(x);:::;q(x)]0atagivenpointxintheregionofinterest,R,isrelatedtothelinearpredictor(x)=[1(x);:::;q(x)]0bythelinkfunctiong:Rq!Rq,

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whereh:Rq!Rq,then FahrmeirandTutz ( 2001 ,p.106)].Thevariance-covariancematrix,Var(^),of^isdependentontheunknownparametervector.ThiscausesthedesigndependenceprobleminmultivariateGLMs.Detailsconcerningtheestimationof^andVar(^)canbefoundinChapters4and5.ThekeyreferencesforoptimaldesignsinmultivariateGLMsare HeiseandMyers ( 1996 )and ZocchiandAtkinson ( 1999 ). HeiseandMyers ( 1996 )studiedoptimaldesignsforbivariatelogisticregressionwhileZocchiandAtkinson's( 1999 )workwasbasedonoptimaldesignsformultinomiallogisticmodels. HeiseandMyers ( 1996 )chosethe\Gumbelmodel"basedonthebivariatelogisticcdfgivenby Gumbel ( 1961 ).ThestandardGumbelcdfisgivenby 1+eu1 1+ev1+euv 2{8 )wecanwrite

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[see HeiseandMyers ( 1996 ,p.615)]p(1;1;x)=P(Y1=1;Y2=1jX=x)=F(x1;x2)=E1E2+A;p(1;0;x)=P(Y1=1;Y2=0jX=x)=F(x1;1)F(x1;x2)=E1E1E2A;p(0;1;x)=P(Y1=0;Y2=1jX=x)=F(1;x2)F(x1;x2)=E2E1E2A;p(0;0;x)=P(Y1=0;Y2=0jX=x)=1F(x1;1)F(1;x2)+F(x1;x2)=1E1E2+E1E2Awherex1=1+1x,x2=2+2x,1;1;2;and2aretheparameters,andE1,E2andAaredenedbelowas:E1=1 1+ex1;E2=1 1+ex2;A=ex1x2 HeiseandMyers ( 1996 ,p.616)] HeiseandMyers ( 1996 )consideredtheinformationmatrix,I(),oftheparametersandchosethexj'sandmj'stomaximizesomesuitablescalar-valuedfunctionofthismatrix.The55informationmatrixisgivenin HeiseandMyers ( 1996 ,p.617,formula4). HeiseandMyers ( 1996 )developedD-optimalbymaximizingjI()jandQ-optimaldesignbyminimizingtheaverageasymptoticpredictionvarianceofanestimateofp(1;0;x),theprobabilityofecacywithouttoxicityatdosex,over

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adesiredrangeofdoses.Theoptimaltwoandthreepointdesignscanbefoundtabulatedintheirearliertechnicalreport[ HeiseandMyers ( 1995 )]. ZocchiandAtkinson ( 1999 )studiedtheinuenceofgammaradiationonthreedisjointedoutcomesrelatedtotheemergenceofhouseies.Theoutcomesweredeathsbeforethepupaeopened,deathduringemergence,andlifeafteremergence.Thissortofexperimentisingeneralveryexpensivesinceitusesagammaradiationsource.Experimentsthereforeneedtobedesignedtoinvolveaminimumnumberofdierentdoses. ZocchiandAtkinson ( 1999 )useoptimaldesignstoecientlyestimatetheparametersofthemodelandappreciablyreduceexperimentalcosts.Theirfocusisondesignsformultinomiallogisticmodelsthatbelongtotheclassofmultivariatelogisticmodelsdescribedby GlonekandMcCullagh ( 1995 ).Letybearesponsevariablewithqresponsecategoriesandi(x)betheprobabilitythattheindividualhasresponseiatdoselevelx,with1(x)+:::+q(x)=1.Thustheresponsevector,y=(y1;:::;yq)0,followsamultinomialdistributionwithparametersmand(x)where,(x)=[1(x);:::;q(x)]0,andPqi=1yi=m. ZocchiandAtkinson ( 1999 )studiedthedependencyof(x)ondosexthroughthemultivariatelogisticmodelsof GlonekandMcCullagh ( 1995 ).Fromthemultivariatelogisticregressionmodelsin ZocchiandAtkinson ( 1999 ,p.439,formula5),wehave

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Thelog-likelihoodfunctionofyis[see ZocchiandAtkinson ( 1999 ,p.439)] ZocchiandAtkinson ( 1999 ,p.439,formula6). ZocchiandAtkinson ( 1999 )derivedD-optimaldesignsforthemultivariatelogisticmodelsbymaximizinglogjI()j.TheyalsodealtwithBayesianD-optimaldesignsandmaximizedElogjI()jbyassumingapriordistributionon.Itwasnotedthatthesupportpointsoftheoptimaldesigngrewastheyincreasedtheprioruncertainty. Harrington ( 1965 ), DerringerandSuich ( 1980 ),and KhuriandConlon ( 1981 ).Multiresponseoptimizationusingtheso-calledgeneralizeddistanceapproachwasintroducedby KhuriandConlon ( 1981 ).Inthisprocedure,allresponsesdependonthesamesetofcontrolvariablesandcanberepresentedbypolynomialregressionmodelsofthesamedegreeandformwithintheexperimentalregion.Letnbethenumberofexperimentalrunsandqbethenumberofresponses.Thenthe

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modelfortheithresponseis 2{12 )canbeexpressedintheformY=X0+;whereY=[y1;:::;yq],=[1;:::;q],and=[1;:::;q].Theassumptionsarethattherowsofaremutuallyindependent,eachhavingazeromeanvectorandacommonvariance-covariancematrix.SinceallresponsemodelshavethesameX0matrix,thenanestimateofisgivenby^=Y0[InX0(X00X0)1X00]Y ^yi(x)=z0(x)^i;i=1;:::;q;(2{13)wherex=(x1;:::;xk)0,z0(x)isavectorofthesameformasarowofthematrixX0evaluatedatthepointx,^iistheleastsquaresestimatorofi.Thevariance-covariancematrixof^y(x)=[^y1(x);:::;^yq(x)]0isz0(x)(X00X0)1z(x).AnunbiasedestimateofVar[^y(x)]namely,dVar[^y(x)],canbeobtainedbyreplacingby^.Let^i(i=1;:::;q)betheoptimumvalueof^yi(x)optimizedindividuallyovertheexperimentalregionandlet^=(^1;:::;^q)0bethevectorofalltheqoptima.Iftheseindividualoptimaareattainedatthesamex,thenanidealoptimumissaidtobeachieved,butthisrarelyoccurs.Instead,wetrytond

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conditionsonxthatminimizethedistancebetweentheestimatedresponsesandtheircorrespondingoptimumvalues.Let[^y(x);^]beadistancefunctionwhichmeasuresthedistanceof^y(x)from^.Severalchoicesofarepossible. KhuriandConlon ( 1981 )consideredthefollowingmetric PaulandKhuri ( 2000 )usedmodiedridgeanalysistocarryoutoptimizationofthemeanresponse.Ridgeanalysisisawell-knownprocedurefortheoptimizationofasecond-degreeresponsemodeloverasphericalregionofinterest. KhuriandMyers ( 1979 )introducedamodicationofthisprocedurebyincorporatingacertainconstraintonthepredictionvarianceandcalleditmodiedridgeanalysis.Insteadofoptimizingthemeanresponsedirectly, PaulandKhuri ( 2000 )optimizedthelinearpredictor.Chapter6dealswithoptimizationinmultivariateGLMs.Thegeneralizeddistanceapproachof KhuriandConlon ( 1981 )isusedtocarryoutthesimultaneousoptimizationofthemeanresponses.

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21

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NelderandWedderburn ( 1972 ).Theycanbeusedtotdiscreteaswellascontinuousdatahavingavarietyofparentdistributions.Thusthetraditionalassumptionsofnormalityandhomogeneousvariances,usuallymadeinananalysisofvariance(orregression)situation,arenolongerneeded.TheuseofGLMsrequiresthespecicationofthefollowingthreecomponents:1.thedatasetunderconsiderationconsistsofindependentrandomvariables,y1;y2;:::;yn,suchthatyjhasadistributionintheexponentialfamilywithaprobabilitymassfunction(oradensityfunctionforacontinuousdatadistribution)givenby McCullaghandNelder ( 1989 ,pp.28-29).2.alinearregressionfunction,,calledalinearpredictor,inkcontrolvariables,x1;:::;xk,oftheform 3{5 )indicatesthatthemeanresponseis,ingeneral,representedbyanonlinearmodel.

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Forexample,theresponseymaybebinarywithaprobabilitymassfunctionoftheform McCullaghandNelder ( 1989 ,p.31)]namely, 1(x);(3{7)andthelinearpredictorisoftheform log(x) 1(x)=f0(x);(3{8)where(x)denotestheprobabilityofsuccess(thatis,achievingthevalue1)atx.Themeanresponse,(x),atxisthengivenby 1+exp[f0(x)]:(3{9)Model( 3{9 )iscalledthelogisticregressionmodel.Wenotethatthelinkfunctionin( 3{7 )wasobtainedbyequating(x)tothecanonicalparameter.Thisparticularlinkfunctioniscalledthecanonicallink,anditsuseleadstodesirablestatisticalpropertiesofthemodel,particularlyinsmallsamples[see McCullaghandNelder ( 1989 ,p.32)]. 3{3 )isbasedonthemethodofmaximumlikelihood,whichiscarriedoutusinganiterativeweightedleast-squaresprocedure.Anestimateof

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3{3 )isthengivenby ^(x)=f0(x)^;(3{10)andthecorrespondingestimateofthemeanresponse(x)in( 3{5 )is ^(x)=h[f0(x)^];(3{11)where^isthemaximumlikelihoodestimateof.Thevariance-covariancematrixof^isapproximatedby[see RobinsonandKhuri ( 2003 )] Var(^):=1 3{14 ),dj RobinsonandKhuri ( 2003 )] Var[^(x)]:=1 Var[^(x)]:=1

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CordeiroandMcCullagh ( 1991 )developedanexpressionforthebiasof^forGLMsusingthetensormethodology. Cadigan ( 1994 )presentedamethodforthecomputationofthebiaswhichdidnotrequireusingthetensormethodology.Hisexpressionforthebiasisapproximatedby Bias(^):=1 2(X0WX)1X0ZdF1n;(3{17)whereXandWarethesameasinSection3.2.1,Zd=diag(z11;:::;znn),wherezjjisthejthdiagonalelementofZ=X(X0WX)1X0,F=diag(f11;:::;fnn),whereforj=1;:::;n,fjjisgivenby 3{18 ),d2j CordeiroandMcCullagh ( 1991 ,formula6.3)showedthatthebiasof^canalsobederivedapproximatelythroughasimpleweightedlinearregressioncomputationoftheform Bias(^):=(X0WX)1X0W;(3{19)where=1 2W1ZdF1nandWplaystheroleoftheweightmatrix.Using( 3{19 ),thebiasof^(x)in( 3{10 )isapproximatedby Bias[^(x)]:=f0(x)(X0WX)1X0W:(3{20)Furthermorethepredictionbias,whichisthebiasof^(x)in( 3{11 ),isapproximatedby Bias[^(x)]:=Bias[^(x)]d(x) 2Var[^(x)]d2(x)

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whered2(x) 3{15 )[see RobinsonandKhuri ( 2003 )]. MSE[^(x)]=E[^(x)(x)]2:(3{22)Theright-handsideof( 3{22 )canbepartitionedintotwocomponents,namely, MSE[^(x)]=Var[^(x)]+fBias[^(x)]g2;(3{23)sinceBias[^(x)]=E[^(x)](x).Bycombiningformulas( 3{16 )and( 3{21 )weobtain MSE[^(x)]:=1 2Var[^(x)]d2(x) whereVar[^(x)]andBias[^(x)]areapproximatedbyformulas( 3{15 )and( 3{20 ),respectively. 3{24 ),overtheregionR.OnemajorproblemindoingsoisthedependenceoftheMSEPontheunknownparametervectorinmodel( 3{3 ),since(x),andhence(x),dependson.ThisisanalogoustothedesigndependenceproblemmentionedearlierinSection 2.1 withregardtoGLMs.

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chosendesign,denotedbyD,andthevectorofunknownparameters,.LetusthereforedenotesuchavaluebyD(x;).WeassumethatbelongstoaspeciedparameterspacedenotedbyC.InordertostudytheeectofthedesignDonMSEP,weexaminethevaluesofD(x;)withintheregionRforagiveninC.Inparticular,weconsideritsvaluesonconcentricsurfaces,denotedbyR,whichareobtainedbyreducingtheboundaryofRusingashrinkagefactor,.ForagivendesignDandagiveninC,thevaluesofD(x;),forallxonR,formadataset.LetQD(p;;)denotethepthquantileofsuchadataset,0p1.ThesequantilesprovideadescriptionofthedistributionofD(x;)forvaluesofxonR.ByvaryingwecancovertheentireregionR.SmallvaluesofD(x;)throughoutRareobviouslydesirable.RecallthatanydesigncriterionforGLMssuersfromtheproblemofthedependenceofthedesignon.Toaddressthisproblem,wecanvarywithinthesetCandthenexaminethecorrespondingvaluesofQD(p;;)foragivendesignDandgivenpand.An\initial"datasetthatmaybeavailableontheresponsescanbeusedtoconstructtheparameterspaceC.ThedependenceofthequantilesofD(x;)on,forxonR,canbeinvestigatedbycomputingQD(p;;)forseveralvaluesofthatformagrid,C,insideC.Subsequently,theminimumandmaximumvaluesofQD(p;;)overthevaluesofinCareobtained.ThisresultsinthefollowingextremaofQD(p;;)foreachandagivenp:

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capabilityassociatedwiththedesignDthroughouttheregionR.Suchplotscanbeconstructedforeachofseveralcandidatedesignsforthemodel.Itshouldbenotedthatforagiven,adesirablefeatureofadesignistohavecloseandsmallvaluesofQminDandQmaxDovertherangeofp(0p1).ThesmallnessofQmaxDindicatessmallMSEPvaluesonR,andtheclosenessofQminDandQmaxDindicatesrobustnesstochangesinthevaluesofthatisinducedbythedesignD.Thereareseveraladvantagestothisapproach,namely,1.theperformanceofadesigncanbeevaluatedthroughouttheregionR.Otherdesigncriteriabasetheevaluationofadesignonasinglemeasure,suchasD-eciency,butdonotconsiderthequalityofpredictioninsideR.2.unlikeothervariance-baseddesigncriteria,suchasD-optimalityorG-optimality,estimationbiasistakenintoaccountintheevaluationofagivendesign.3.thedependenceofadesignonthemodel'sunknownparametersisclearlydepictedbytheQDGsthroughouttheregionR.4.plottingthequantilesoftheMSEPagainstppermitstheconsiderationofmodelswithseveralcontrolvariables,insteadofjustone,asisusuallythecasewithotherdesigncriteria,suchasD-optimality. FahrmeirandTutz ( 2001 ,p.36)].

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3{27 )with( 3{1 )wenotethatthedistributionofyjbelongstotheexponentialfamilywithj=logj;b(j)=j;c(yj)=log(yj!),andd(yj;)=0.ThemeanandvarianceofyjareE(yj)=jandVar(yj)=j NelderandWedderburn ( 1972 )suggestedtaking^=(np)=X2,whereX2isaPearson-typestatisticandpisthenumberofelementsintheparametervector[see Agresti ( 2002 ,p.150)].Letxbeavectorofdesignsettingsintheregionofinterest,R.Atthejthexperimentalrun,x=xj,thecorrespondingresponsevalueisyj(j=1;:::;n).Themeanresponseatxisdenotedby(x).Thecanonicallinkfunctionisthelogarithmic(log)link[see McCullaghandNelder ( 1989 ,p.30)] ^(x)=^(x)=exp[f0(x)^];(3{29)where^isthemaximumlikelihoodestimator(MLE)of.Using( 3{16 )and( 3{21 )itcanbeshown[seeAppendix A.1 ]thatthemean-squarederrorofprediction(MSEP)atxisapproximatedby MSE[^(x)]:=[(x)]2f0(x)(X0WX)1 (3{30) +ff0(x)Bias(^)(x)+1 2f0(x)(X0WX)1

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whereBias(^):=(X0WX)1X0W;W=diag(w1;:::;wn)withwj=j,andisann1vectorwhosejthelementiszjj ^(x)=^(x)=[f0(x)^]2:(3{32)Equations( 3{16 )and( 3{21 )show[seeAppendix A.2 ]thatthemean-squarederrorofprediction(MSEP)atxisapproximatedby MSE[^(x)]:=4(x)f0(x)(X0WX)1 (3{33) +f2f0(x)Bias(^)p 3{19 ),W=diag(w1;:::;wn)withwj=4,andthejthelementofiszjj

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Piegorschetal. ( 1988 ).Inabiomedicalstudyoftheimmuno-activatingabilityoftwoagents,TNF(tumornecrosisfactor)andIFN(interferon),toinducecelldierentiation,thenumberofcellsthatexhibiteddierentiationafterexposuretoTNFand/orIFNwasrecordedusinga44factorialdesign.Ateachofthe16dosecombinationsofTNF/IFN,200cellswereexamined.Thenumberyofcellsdierentiatinginonetrial,andthecorrespondingdoselevelsofthetwofactorsaregiveninTable 3{1 Table3{1. Experimentaldesignandresponsevalues.

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Model( 3{28 )wasttedtothedatasetinTable 3{1 usingthefollowingmodelforthelinearpredictor 3{34 )areshowninTable 3{2 Table3{2. Parameterestimatesandmodelanalysisusingloglink. ParameterEstimateStd.errorP-value ScaledDeviance=54:1883;DF=10. WerefertothedesigninTable 3{1 asD1.Wealsoconsideranotherdesign,D2,namely,aface-centeredcubewithreplicatedcenterandface-centerpointsgiveninTable 3{3 .DesignsD1andD2havedierentdesignsettings.Foreachofthetwodesigns,weconsiderthedistributionofD(x;)oneachoftheseveralconcentricsquares,R,whichareobtainedbyareductionoftheboundaryofRusingashrinkagefactor,0:5<1.Thus,Risdeterminedbytheinequalitiesai+(1)(biai)xibi(1)(biai);i=1;2;

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Table3{3. DesignD2. whereaiandbiaretheboundsonxiinR(i=1,2),thatis,a1=1;a2=1;b1=1;b2=1(seeFigure 3{1 ).ToinvestigatethedependenceofD(x;)on,aparameterspacewasestablished.Foreachparameter,arangeconsistingoftheparameter'spointestimateplus/minusfourstandarderrors(fromTable 3{2 )wasconsidered.Asubset,C,ofthisparameterspacewasobtainedbyselectingthreepointswithineachparameterrange,namely,thepointestimateandthetwoendpoints.Thus,thenumberofpointsinCis36=729.Thesameparameterspacewasusedforthetwodesigns.ForeachdesignandaselectedvalueofinC,quantilesofthedistributionofD(x;)areobtainedforx2R,whereisoneofseveralvalueschosenfromtheinterval(0:5;1].ThenumberofpointschosenoneachRwas1000,consistingof250pointsoneachside.Thequantilesarecalculatedforp=0(0:05)1.TheprocedureisrepeatedforothervaluesofinthesubsetC.Then,QmaxD(p;)and

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3{25 )and( 3{26 ).PROCGENMODinSAS[ SASInstituteInc. ( 2000 )]wasusedtocalculatetheparameterestimates.AllothercomputationsweredoneusingtheRsoftware(Version1.6).Forbetterrepresentation,wetakethenaturallogarithm(log)ofthequantilesoftheMSEPwhenplottingtheQDGs.Tocomparethetwodesigns,weexaminethecorrespondingQDGsshowninFigure 3{2 .WenotethatthemaximumquantilesofD1areabovethoseofD2formostvaluesofp,indicatingthatD2hasbetterpredictioncapabilitythanD1.ThevaluesofQmaxD(p;)andQminD(p;)forbothdesignsareatsomedistancefromeachother,indicatingsensitivitytotheparametervalues.Asdecreases(thatis,aswegetclosertothecenterofR),thereisaslightdecreaseinthedistancebetweenQmaxD(p;)andQminD(p;)forD2,whilethequantilesofD1donotchangemuch.Thus,D2becomesslightlymorerobusttotheparametervaluesnearthedesigncenter,butthequantilesforD1remainthesame.ToassessthesensitivityoftheQDGstotheformofthelinkfunction,weusethesamedataandassumethesamePoissondistribution,asbefore,butemployadierentlink,namely,thesquareroot.Westartwithttingmodel( 3{31 )tothedatasetinTable 3{1 usingmodel( 3{34 )forthelinearpredictor.Thedispersionparameterwastakentobe1.ThesametwodesignsD1andD2areusedasbefore.Ourregionofinterest,R,andtheseveralconcentricsquares,R,remainunchanged.TheparameterestimatesandtheirstandarderrorsinTable 3{4 usingthedatasetbasedondesignD1,however,change.Thechangeoccursduetothedierentlinkfunctionused.Thecorrespondingquantileswerecomputedinasimilarfashion.TheQDGsforbothdesignsarecomparedusingthetwodierentlinks.Figure 3{3 givesacomparisonoftheQDGsforD1usingthePoissondistributionwithaloglinkandasquarerootlink,andFigure 3{4 showsthesameforD2.

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Table3{4. Parameterestimatesandmodelanalysisusingsquarerootlink. ParameterEstimateStd.errorP-value ScaledDeviance=58:1596;DF=10. ThecanonicallinkforthePoissondistributionistheloglink.Misspecifyingthelinkassquarerootchangestherobustnessofthedesignstotheparametervalues.ThedistancebetweenthemaximumandminimumquantilesfordesignD1increasesslightlyaswechangethelinkfromlogtosquareroot.Thisindicatesaslightincreaseinsensitivitytotheparametervaluesforthesquarerootlink.ThemaximumandminimumquantilesforD2forthesquarerootlinkarefurtherapartfromeachotherthantheloglinkcase.Thus,D2losesitsrobustnesstotheparametervaluesaswealterthelink.NextwetrytostudytherobustnessoftheQDGstodierentdistributions.Weusefourdistributions,namely,Poissondistributionwith=1,Poissondistributionwith=0:1824(isestimatedbyusingPROCGENMODinSAS[options:scale=Pearson]),negativebinomialdistribution,andgammadistribution,eachwithloglink.ThemodelforthelinearpredictorremainsthesameandtheexperimentalregionRalsoremainsunchanged.Theparameterestimatesandtheirrespectivestandarderrors,however,change.Figure 3{5 givesthecomparisonoftheQDGswiththesefourdistributions.ThedistancebetweenthemaximumandtheminimumquantilesforbothdesignsincreasesalittleforthePoissondistributionwithanestimated,indicatingaslightlossinrobustnesstotheparametervalues.Forthenegativebinomial

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distribution,thedesignsbecomemoresensitivetotheparameters.Inthecaseofthegammadistribution,thequantileplotsforbothD1andD2areveryclosetooneanother.WealsonoticethatthedistancebetweenthemaximumandminimumissmallerforthegammadistributionbycomparisontotheestimatedPoissonorthenegativebinomialdistributions.Bothdesignsshowsimilarpredictioncapabilitiesforallfourdistributions.Overall,theshapeoftheQDGshasbeenaectedbythedistribution,butthesuperiorityofD2overD1,inthisexample,appearstobeunaectedbythedistribution. ConcentricsquareswithintheregionR.

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QDGsfordesignsD1andD2usingPoissondistribution(=1)withaloglink.

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ComparisonoftheQDGsfordesignD1usingPoissondistribution(=1)withlogandsquarerootlinks.

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ComparisonoftheQDGsfordesignD2usingPoissondistribution(=1)withlogandsquarerootlinks.

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ComparisonoftheQDGsfordesignsD1andD2usingfourdierentdistributions,allwithloglink,using=1.

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Bartlett ( 1953 ).Ageneralizationofthebiasformulaforalikelihoodfunctionwithseveralparameterswasestablishedby CoxandSnell ( 1968 ).UsingCoxandSnell's( 1968 )results, CordeiroandMcCullagh ( 1991 )developedarst-orderapproximationofthebiasinthespecialcaseofgeneralizedlinearmodels(GLMs).Inthischapter,weprovideafurtherextensionofthebiasformulathatappliestomultivariateGLMswhereseveralresponsevariablesareunderconsiderationinagivenexperimentalsituation.AnexpressionforthebiasofMLEsoftheparametersinamultivariateGLMsetupisveryimportantformanyreasons.Oneprimaryreasonistheneedtoobtainanexpressionfortheso-calledmean-squarederrorofprediction(MSEP)thatcanbeutilizedasacriterionforthecomparisonofdesignsformultivariateGLMs[see MukhopadhyayandKhuri ( 2005 )].TheMSEPincorporatesboththevarianceandbiasassociatedwithestimatingthemeanresponsesunderconsideration.ThustheMSEPcanprovideusefulinformationconcerningthequalityofestimationinamultivariateGLMsituation. 41

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becorrelated.Letbeap1vectorofunknownparameterssuchthat=(1;:::;p)0.Thelikelihoodandlog-likelihoodfunctionsare,respectively, (4{1) wherejisthedensityfunctionofyj. CordeiroandMcCullagh ( 1991 ,p.634,formula4.1)givethefollowingexpressionforthebias,correctuptoorder1 Bias(^a):=pXr=1pXs=1pXt=1Iar[Its(Jt;rs+1 2Krst)];(4{3)whereIuvisthe(u;v)th(u;v=1;:::;p)elementoftheinverseofFisherinformationmatrix,I()=E@2l @@0;Jt;rs=E@l @t@2l @r@sandKrst=E@3l @r@s@t;r;s;t=1;:::;p:Stackinguptheseexpressions,wegetthebiascorrectuptoorder1 Bias[^]=0BBBB@Bias[^1]...Bias[^p]1CCCCA:=0BBBB@Ppr=1Pps=1Ppt=1I1r[Its(Jt;rs+1 2Krst)]...Ppr=1Pps=1Ppt=1Ipr[Its(Jt;rs+1 2Krst)]1CCCCA:(4{4)InlatersectionsweshalldeveloptheformsofKrstandJt;rs(r;s;t=1;:::;p)foramultivariateGLM.Forsimplicity,weshallrefertoj(yj;)informulae( 4{1 )and( 4{2 )asj(yj).

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4{5 ),theloglikelihoodfunctionis 4{3 ).

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^j=Z0(xj)^;j=1;:::;n;(4{10)andthecorrespondingestimateofthemean,j,is ^j=h[Z0(xj)^];j=1;:::;n;(4{11)where^istheMLEof.Thevariance-covariancematrixof^isapproximatelygivenby[see FahrmeirandTutz ( 2001 ,p.106)], Var(^):=(X0WX)1;(4{12)whereX=[Z(x1);:::;Z(xn)]0andWisablock-diagonalmatrixoftheform,W=diag[W1;:::;Wn],Wj=@j 4{3 )foramultivariateGLM.Sincethisderivationisquiteinvolvedandlengthy,weshallbreakitupusingtwolemmasandseveralappendices.Forthispurpose,expressionsfor@l @rand@2l @r@s(r;s=1;:::;p)willbedetermined,asbothareneededtodevelopKrstandJt;rs.Startingwith@l @r,weusethechainruletowriteitas @r=@l @0j@j

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Dierentiatingthelog-likelihoodlwithrespectto0j(j=1;:::;n),weget@l @0j=nXj=1(y0j0j);since@b(j) 4{13 )givesthefollowingformof@l @r(r=1;:::;p): @r=nXj=1(y0j0j)@j @rwithrespecttos(s=1;:::;p)toobtain@2l @r@s(intermediatestepsaregiveninAppendix B.1 ), @r@s=nXj=10BBBB@f01(xj)@1 @s@j @s@j WenowhavethenecessarytoolstoproceedwiththedevelopmentofKrstandJt;rs.Lemma 4.4.1 givesdetailsofthederivationofKrst.

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@t@0j @t@j @t@j @s@j @s@j @t@2l @r@s,wersttakethepartialderivativeof@2l @r@swithrespecttot(t=1;:::;p)andthenndexpectationoftheresultingquantity.Thisgives, @tnXj=10BBBB@f01(xj)@1 +E@ @tnXj=1(y0j0j)@ @s@j +E@ @tnXj=1(y0j0j)@j @s@j

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Now,eachcomponentin( 4{16 )-( 4{18 )isdevelopedfurtherasfollows:Theexpression( 4{16 )canbeexpandedas Expression( 4{16 )=nXj=10BBBB@f01(xj)@1 @t@0j @t@j @t@j ItisshowninAppendix B.2 thatexpression( 4{17 )canbesimpliedfurtheras Expression( 4{17 )=nXj=10BBBB@f01(xj)@1 @s@j 4{18 )canbereducedto(seeAppendix B.3 ) Expression( 4{18 )=nXj=10BBBB@f01(xj)@1 @s@j 4{19 )-( 4{21 )wegetthedesiredexpressionforKrst.ThederivationofJt;rsisgiveninLemma 4.4.2 .Theexpressionsin( 4{14 )and( 4{15 )areusedtoconstructtheproofofthislemma.

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@s@j @s@j @t@2l @r@s.Hence,toobtainanexpressionforJt;rs,werstmultiply@l @t,using( 4{14 ),with@2l @r@sin( 4{15 ),andthentaketheexpectationoftheproduct.CarryingouttheabovecalculationsresultsinthefollowingthreequantitieswhosesumisJt;rs:ThersttermofJt;rsis B.4 ) Expression( 4{22 )=0;(4{23)ThesecondtermofJt;rsis EnXj=1(y0j0j)@j @s@j

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Thiscanbereducedto(seeAppendix B.5 ) Expression( 4{24 )=nXj=10BBBB@f01(xj)@1 @s@j Finally,thethirdtermofJt;rsis EnXj=1(y0j0j)@j @s@j B.6 ) Expression( 4{26 )=nXj=10BBBB@f01(xj)@1 @s@j Addingupthetermsontheright-handsidesofequations( 4{23 ),( 4{25 ),and( 4{27 )givesthedesiredexpressionforJt;rs. 4.4.1 and 4.4.2 ,respectively,wecannowusetheseexpressionsinformula( 4{4 )togetanapproximateformulaforBias(^)inamultivariateGLMsituation.AsinSection 4.2 ,webeginwiththebiasofasingleelementof^usingformula( 4{3 ), Bias(^a):=pXr;s;t=1Iar[Its(Jt;rs+1 2Krst)];a=1;:::;p;(4{28)

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recallthatIuvisthe(u;v)th(u;v=1;:::;p)elementoftheinverseofFisherinformationmatrix.UsingtheresultsofLemmas 4.4.1 and 4.4.2 informula( 4{28 )gives(detailsaregiveninAppendix B.7 ), Bias(^a):=pXr;s;t=1IarItsnXj=1 @t@0j @t@j @t@j @s@j @s@j Tosimplifythenotation,letusdenotebyAjtandBjs(j=1;:::;n)thefollowingexpressions:Ajt=@ @t@0j @t@j @t@j @s@j @s@j 4{29 )canthenbewrittenas

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= 4{30 ),westackthemuptogetBias(^)correctuptoorder1

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Cox ( 1972 ,p.115)and McCullaghandNelder ( 1989 ,p.226),wecanwrite( 4{32 )as 4{33 )[see CasellaandBerger ( 2002 ,p.180)and FahrmeirandTutz ( 2001 ,p.70,formula3.12)]thattheresponseszj=(zj1;zj2;zj3)0followthemultinomialdistributionwithparametersmjandj=(j1;j2;j3)0,j=1;:::;n.Letuswrite( 4{33 )as 4{34 )with( 4{5 )wenotethat=1,ji=logji 4{8 )givesg(j)=j=Z0(xj);j=1;:::;n;

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where,xj=(xj1;:::;xjk)0,Z0(xj)=3i=1f0i(xj),fi(x)isaknownvectorfunction,and=[01;02;03]0isap-dimensionalvectorofunknownparameters.Thei's;(i=1;2;3)correspondtothethreeresponsesandareoforderpi1,whereP3i=1pi=p.Thecorrespondinglinkfunctionusedhereis[see Agresti ( 2002 ,pp.267-274)and FahrmeirandTutz ( 2001 ,p.73)] ^j=h[Z0(xj)^];(4{38)where^istheMLEof. 4{31 ),forabivariatebinarydistribution.Tond@j 4{35 ).Thenextstepistosearchforanexpressionfor@j 4{31 )are@ @d@j @d@j

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rst-orderpartialderivativesof@j @d@j @d@j @d@j @d@j 4{31 ).ThisgivesanapproximateexpressionforBias(^)forthebivariatebinarydistribution. ( 1970 )publishedastudyconcerningtworespiratoryailmentsofworkingcoalminerswhowereclassiedassmokerswithoutradiologicalevidenceofpneumoconiosis,agedbetween20and64(inclusive)atthetimeoftheirexamination.Eachrespondentwasclassiedaccordingtowhetherornotheissueringfrombreathlessness(B)orwheeze(W).Eachoftheresponsevariableshastwolevelsandallfourcombinationsarepossible.Oneaimoftheinvestigationwastostudyhowtheassociatedprobabilitiesconcerningbreathlessnessandwheezearerelatedtoage.ThedesignandresultingdataarepresentedinTable 4{1 .Thettedmodelis( 4{6 )with 4{36 ).Estimatesoftheelementsof=(1;2;:::;6)0andthecorrespondingstandarderrors,obtainedbyusingPROCLOGISTICinSAS[ SASInstituteInc.

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Table4{1. Experimentaldesignandresponsevalues. Age-groupBreathlessnessNobreathlessnessTotalinyearsWheezeNowheezeWheezeNowheeze 20-2497951841195225-292391051654179130-3454191771863211335-39121482572357278340-44169542731778227445-49269883241712239350-544041172451324209055-59406152225967175060-643721061325261136 Total182760018331402218282 ( 2000 )],areshowninTable 4{2 .Theestimatedmeanresponsevaluesatthevariousexperimentalrunsareobtainedusingthevalueof^(fromTable 4{2 )informula( 4{38 ).Thesevalues,whicharedisplayedinTable 4{3 ,arethenusedinformula( 4{31 )toobtainanestimateofthebiasvector,whichisBias(^)=(0:0008;0:0003;0:0023;0:0005;0:0005;0:0001)0: Maximumlikelihoodestimatesandstandarderrors. ParameterEstimateStd.errorP-value Scaleddeviance=41:4646;DF=21.

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Table4{3. Estimatedmeanresponsevalues. Age-group^j1^j2^j3inyears 20-2417.79417.1418109.286225-2927.694510.7396119.957130-3454.957720.5912167.882335-39120.230643.5240259.021740-44160.170256.0216243.358945-49267.475390.3894286.611450-54356.9859116.5588269.777155-59434.7472137.1485231.704960-64386.8044117.8979145.3899

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4.3 showsthatinanalogytotheunivariatecase,multivariategeneralizedlinearmodels(GLMs)requirethespecicationofthefollowingthreecomponents:1.thedatasetunderconsiderationconsistsofnindependentq-dimensionalrandomvariables,y1;:::;yn.Thedistributionofyj(j=1;:::;n)belongsto 57

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theexponentialfamilywiththedensityfunction FahrmeirandTutz ( 2001 ,p.106)].Anestimateof(x),isthengivenby ^(x)=Z0(x)^;j=1;:::;n;(5{6)

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andthecorrespondingestimateofthemean,(x),is ^(x)=h[Z0(x)^];j=1;:::;n;(5{7)where^istheMLEof. FahrmeirandTutz ( 2001 ,p.106)] Var(^):=(X0WX)1;(5{8)whereX=[Z(x1);:::;Z(xn)]0andWisablock-diagonalmatrixoftheform,W=diag[W1;:::;Wn],Wj=@j FahrmeirandTutz ( 2001 ,p.106)]forlargen.BasedonWald's( 1943 )results,weapproximatelyhave (^)0[dVar(^)]1(^)2p;(5{9)wherepisthetotalnumberofparametersand^Var(^):=(X0^WX)1;^WisanestimateofWusingmaximumlikelihoodestimation.Thusanapproximate100(1)%condenceregionforisgivenby 5{6 )and( 5{8 )weapproximatelyget Var[^(x)]=Z0(x)Var(^)Z(x):=Z0(x)(X0WX)1Z(x):(5{11)

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Arst-ordermultivariableTaylorseriesexpansionof^(x)=h[^(x)]around(x)gives ^(x):=h[(x)]+@h[(x)] ^(x):=(x)+@(x) 5{11 ),Var[^(x)]canbeapproximatelywrittenas Var[^(x)]:=@(x) 4{31 )inChapter 4 tondanexpressionforthebiasassociatedwithestimating(x).Formula( 4{31 )gavethefollowingexpressionforBias(^)correctuptoorder1

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whereIuvisthe(u;v)th(u;v=1;:::;p)elementoftheinverseofFisherinformationmatrix,I()=E@2l @@0:Here,listhelog-likelihoodfunctionobtainedusing( 5{1 ),thatis,l=nXj=1[fy0jjb(j)g+c(yj;)];@i @t@0j @t@j @t@j @s@j @s@j @d@j @d@j 5{6 )and( 5{15 )jointly,thebiasof^(x)isgivenby Bias[^(x)]=Z0(x)Bias[^]:(5{16)Fromtheexpansionof^(x)=h[^(x)]inaneighborhoodof(x)byarst-orderapproximationofTaylorseriesasin( 5{13 ),wehave^(x)(x):=@(x)

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Takingexpectationonbothsidesoftheaboveequation,wegetanapproximateexpressionforthepredictionbias,thatis, Bias[^(x)]:=@(x) MSE[^(x)]=E[(^(x)(x))(^(x)(x))0]=Var[^(x)]+fBias[^(x)]gfBias[^(x)]g0: Usingformulae( 5{14 )and( 5{17 )wegetanapproximateexpressionfortheMSEPinthemultivariatecase.

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constructthebestdesign,however,oneneedstoknowthetruevaluesoftheparameters.OtherdesignoptimalitycriteriasuchasA-,D-,E-optimality,whicharevariance-based,alsosuerfromthesameproblem.CommonapproachestosolvingthisdesigndependenceproblemweresummarizedinSection 2.1 .Inthischapter,weextendtheuseofthequantiledispersiongraphsapproach,mentionedearlierinSection 5.1 ,tomultivariateGLMs. 5{3 ).ByvaryingthevalueofwecancovertheentireregionR.RecallthatanydesigncomparisoncriterionforaGLMsuersfromtheproblemofthedependenceofthedesignon.Toaddressthisproblem,wecanvarywithinaspeciedparameterspace,C,andthenexaminethecorrespondingvaluesofQD(p;;)foragivendesignDandgivenpand.ThesetCischosentobethe(1)100%condenceregionongivenbyformula( 5{10 ).An\initial"datasetthatmaybeavailableontheresponsescanbeusedtoconstructsuchacondenceregion.ThusthedependenceofthequantilesofD(x;)on,forxonR,canbeinvestigatedbycomputingQD(p;;)forseveralvaluesofthatform

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agrid,C,ofpointsrandomlychosenfromC.TheminimumandmaximumvaluesofQD(p;;)aresubsequentlyobtainedoverthevaluesofinC,forxedpand,andforagivendesignD.Wecanthereforecomputethefollowingfunctions: Foraxed,plottingthesevaluesagainstpresultsinthequantiledisper-siongraphs(QDGs)oftheEMSEPovertheregionR.ByrepeatingthesameprocessforseveralselectedvaluesofweobtainplotsthatportraythepredictioncapabilityassociatedwiththedesignDthroughouttheregionR.Smallandclosevaluesoftheminimumandmaximumquantilesin( 5{19 )and( 5{20 )ofadesignDovertherangeofp(0p1)arehighlydesirable.ThesmallnessofQmaxDindicatessmallEMSEPvaluesonR,andtheclosenessofQminDandQmaxDindicatesrobustnesstochangesinthevaluesoftheparametervector,. PerucaandPisani ( 1989 ),withtheassumptionthattheyareuncorrelated.However,sincethetworesponsescomefromthesamesubject,itisprudenttoallowforcorrelationintheresponses,thatis,toconsiderthemasabivariateresponse.

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4.6 .Formula( 4{33 )inChapter 4 showsthattheresponsezj=(zj1;zj2;zj3)0followsthemultinomialdistributionwithparametersmjandj=(j1;j2;j3)0;j=1;:::;n.ReferbacktoSection 4.6 forthedenitionsofzj,jandmj.Themeanofzjisj=(j1;j2;j3)0,wherej1=mjj1,j2=mjj2,j3=mjj3,andthevariance-covariancematrixofzjisj,where 5.2 ,x=(x1;:::;xk)0isvectorofcontrolvariables,Z0(x)=3i=1f0i(x),fi(x)isaknownvectorfunction,and=[01;02;03]0isap-dimensionalvectorofunknownparameters.Thei's(i=1;2;3)correspondtothethreeresponses(outcomesinthemultinomialdistribution)andareoforderpi1,whereP3i=1pi=p.Thecorrespondinglinkfunctionusedhereis[see FahrmeirandTutz ( 2001 ,p.73)and Agresti ( 2002 ,pp.267-274)] 1P3l=1l(x)=f0i(x)i;i=1;2;3:(5{22)

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Notethati(xj)=ji=ji(i=1;2;3;j=1;:::;n):Hence,^i(x)=exp[f0i(x)^i] 1+P3l=1exp[f0l(x)^l];i=1;2;3:where^iistheMLEofifori=1;2;3.Usingformulas( 5{14 )and( 5{17 ),itcanbeshown(seeAppendix C )thatthemean-squarederrorofprediction(MSEP)atxisapproximatelygivenby MSE[^(x)]:=@(x) whereX,Z,andBias(^)areasdenedinSection 5.2.2 ,andtherst-orderpartialderivativeofwithrespectto0evaluatedatxis 1+P3l=1exp(jl)=exp[f0i(xj)i] 1+P3l=1exp[f0l(xj)l];i=1;2;3;j=1;:::;n:ItshouldalsobenotedherethattheMSEPisa33matrixanddependson.

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Genningsetal. ( 1994 ,pp.429-451).Inacombinationdrugtherapystudyonmalemice,thepainrelieving(analgesic)abilityoftwodrugs,namely,49-tetrahydro-cannabinol(49-THC)andmorphinesulfatearestudied.Thoughbothdrugsareanalgesic(i.e.providepainrelief),theyarealsoassociatedwithadverseside-eects.Thetworesponsesarey1,painrelief,andy2,side-eect.Theresponsey1takesthevalue1ifamousetakesmorethan8secondstoickitstailwhenplacedunderaheatlamp,andy1=0,otherwise.Responsey1=1isconsideredgoodbecausethemousedoesnotfeelpainwhenplacedunderaheatlampforatleast8secondsduetothepainrelieving(analgesic)abilityofthedrugs.Thesideeectresponse,y2,wasdeterminedbyrecordingtherectaltemperatureofthemouseafter60minutesfollowingdrugadministration.Thisresponseisequalto1whentherectaltemperatureofthemousedropsbelow35C(resultinginhypothermia)afterthedrugadministration,andisequaltozero,otherwise.Hypothermia(y2=1)isatoxicorharmfulside-eect.Thuseachoftheresponsevariableshastwolevelsandallfourcombinationsarepossible.Oneaimoftheinvestigationwastostudyhowtheassociatedprobabilitiesconcerningpainreliefandhypothermiaarerelatedtodoselevelsofthetwodrugs.Forthepainreliefandside-eectresponses,35groupsofmice(sixanimalspergroup)froma57factorialexperimentwererandomlyassignedtoreceivethetreatments,whereatreatmentconsistsofasingleinjectionofoneofthepossiblecombinationsofmorphinesulfate(0,2,4,6,8mg/kg)and49-THC(0,0.5,1.0,2.5,5.0,10.0,15.0mg/kg).Thuswehave35runswith6experimentalunits(mice)ineachrun.ThedesignandtheresultingdataarepresentedinTable 5{1

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Model( 5{3 )wasttedtothedatainTable 5{1 usingthefollowingrst-degreemodelsforthelinearpredictors Here,x1andx2representthedoselevelsofthedrugs,morphinesulfateand49-THC,respectively.Thelinkfunctionsarethesameasthosedescribedinformula( 5{22 ).Theexperimentalregion,R,isrectangularinshapewithR:f0x18;0x215g.Theparameterestimatesandtheirstandarderrorsfortheabovemodels,obtainedbyusingPROCLOGISTICinSAS[ SASInstituteInc. ( 2000 )],areshowninTable 5{2 .LetusnowrefertothedesigninTable 5{1 asD1.Wecomparethisdesignwithanotherdesign,D2.TheadditionaldesigndiersindesignsettingsfromD1,buthasthesamenumberofexperimentalruns(=35)andthesamenumberofmiceperrun(=6).DesignD2isa32factorialwiththecenterpoint(4;7:5)replicatedthreetimesandallotherpointsreplicatedfourtimes.ThelistingofthetwodesignsisgiveninTable 5{3 .Figure 5{1 showsthedesignpointsofD1andD2.ForeachdesigninTable 5{3 westudythedistributionoftheEMSEPoneachofseveralconcentricrectangles,R,whichareobtainedbyareductionoftheboundaryofRusingashrinkagefactor,0:5<1.Thus,Risdeterminedbytheinequalitiesai+(1)(biai)xibi(1)(biai);i=1;2;

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whereaiandbiaretheboundsonxiinR(i=1,2),thatis,a1=0;a2=0;b1=8;b2=15.Fourorvevaluesofaretypicallyusedinthecalculationofthequantiles.ToinvestigatethedependenceofEMSEPonweconsiderCtobethe95%condenceregionon[seeformula( 5{10 )]andCtobeasetof500pointsrandomlychosenfromC.ForeachdesignandaselectedvalueofinC,quantilesofthedistributionoftheEMSEPareobtainedforx2R.ThenumberofpointschosenoneachRwas500,consistingof125pointsoneachside.Thequantilesarecalculatedforp=0(0:05)1(pfrom0to1instepsof0.05).TheprocedureisrepeatedforallthevaluesofinC.Then,QminD(p;)andQmaxD(p;)arecalculatedusingformulas( 5{19 )and( 5{20 ).TheRsoftware(Version2.0.1)wasusedtocarryoutthenumericalinvestigationsanddrawtheQDGs.Tocomparethetwodesigns,weexaminethecorrespondingQDGsshowninFigure 5{2 .For=1,wenotethatthemaximumquantilesofD1areabovethoseofD2formostvaluesofp,indicatingthatD2hasbetterpredictioncapabilitythanD1.ThevaluesofQmaxD(p;)andQminD(p;)forbothdesignsareclosetoeachotherformostofthevaluesofpindicatingthattheyarerobusttochangesintheparametervalues.Asdecreases(thatis,aswegetclosertothecenterofR),thereisadecreaseinthedistancebetweenthemaximumquantilesofD1andD2indicatingthattheyhavesimilarpredictioncapabilitiesnearthecenterofR.So,overall,D2appearstobethebetterdesign,intermsofpredictioncapability,foralmostallvaluesofpand.

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Table5{1. ExperimentaldesignD1(57factorial)andresponsevalues. Morphinesulfate49-THCPainreliefNopainreliefx1x2Side-eectNosideeectSide-eectNoside-eectm

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Table5-1(continued) Morphinesulfate49-THCPainreliefNopainreliefx1x2Side-eectNosideeectSide-eectNoside-eectm

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Table5{2. Maximumlikelihoodestimatesandstandarderrors. ParameterEstimateStd.errorP-value ScaledDeviance=78:0672,DF=96. Table5{3. DesignsD1(57factorial)andD2(32factorial). 410 00 81500.5 415 00 41501 60 00 41502.5 60.5 00 41505 61 40 415010 62.5 40 015015 65 40 01520 610 40 01520.5 615 80 01521 80 80 07.522.5 80.5 80 07.525 81 80 07.5210 82.5 87.5 07.5215 85 87.5 47.540 810 87.5 47.540.5 815 87.5 47.541 815 42.5 815 45 815

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DesignsD1(57factorial)andD2(32factorial)

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ComparisonoftheQDGsfordesignsD1(57factorial)andD2(32factorial)giveninTable 5{3

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KhuriandConlon ( 1981 ).Inthisalgorithm,werstcomputetheestimatedmeanresponsesusingthecorrespondingmultiresponsedata.Thenextstepistoobtainoptimafortheindividualestimatedmeanresponses.Ifalltheindividualoptimaareattainedatthesamesetofconditionsontheinput(control)variablesorfactors,thenwesaythatan\idealoptimum"hasbeenreached.Unfortunately,thisisrarelythecase.Optimalconditionsforonemeanresponsemaybefarfromoptimalorbeevenphysicallyimpracticalfortheothers.Thusweresorttondingcompromiseconditionsontheinputvariablesthatare\favorable"toallthemeanresponses.Thedeviationfromtheidealoptimumismeasuredbyadistancefunctionexpressedintermsoftheestimatedmeanresponsesalongwiththeirvariance-covariancematrix.Byminimizingsuchadistancefunctionwearriveatasetofconditionsfora\compromiseoptimum". 5.2 weknowthattheq-dimensionalresponsevectoryj(j=1;:::;n)isassumedtobelongtoanexponentialfamilywithaprobabilitydistribution(yjjj;)[seeformula( 5{1 )].Thelinearpredictor,(x)=Z0(x),isrelatedtothemeanresponse(x)byalinkfunction(x)=g[(x)][refertoequations( 5{2 )and( 5{3 )],wherex=(x1;:::;xk)0isthevectorofcontrol 75

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variables,=(01;::;0q)0isap-dimensionalvectorofunknownparameters,andg:Rq!Rq.Usingthemaximumlikelihoodestimate(MLE),^,of,wecanobtainanestimateofthemeanresponse(x)givenby ^(x)=h[Z0(x)^]:(6{1)Thevariance-covariancematrixof^isapproximatelygivenby[seeformula( 5{8 )] Var(^):=(X0WX)1;(6{2)whereXandWarethesameasin( 5{8 ).Anapproximate100(1)%condenceregionforis[seeformula( 5{10 )] 5{14 )] Var[^(x)]:=@(x) KhuriandConlon ( 1981 ).

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6{5 ).But,beforeminimizing,wehavetotakeintoaccountthevariabilityassociatedwith^.Todoso,weapplythefollowingprocedure,givenby KhuriandConlon ( 1981 ):Leti()bethetrueoptimumvalueofthei-th(i=1;:::;q)meanresponseoptimizedindividuallyoverR,andlet()=[1();:::;q()]0.Ourobjectiveistondanx2Rsuchthat[^(x);()]isminimizedoverR.As()isunknown,tryingtominimize[^(x);()],whichisafunctionof(),isimpossible.Instead,weminimizeanupperboundon[^(x);()]asfollows:LetDbeacondenceregionfor().Then,whenever()2D, 6{7 )servesasanoverestimateof[^(x);()].Wethereforeminimizetheright-handsideof( 6{7 )overtheregionR,thusadoptingaconservativedistanceapproachtoourminimizationproblem.

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( 1973 ,p.240)showsthatforanycontinuousfunctionl(),denedonasubsetofthep-dimensionalEuclideanspacethatcontainsC, Pfmin2Cl()l()max2Cl()g1;(6{8)whereCisthe100(1)%condenceregionongivenin( 6{3 ).Theinterval[min2Cl();max2Cl()]denesaconservativecondenceintervalonl().Sincetheindividualoptimum,i()(i=1;:::;q)oftheithtruemeanresponse,i(x),canbeexpressedasacontinuousfunctionoftheparametervector,theaboveresultin( 6{8 )canbeusedtoobtainaconservativecondenceintervaloni()usingthecondenceregionin( 6{3 ).Moreover,sincewhenever2C,i()2[min2Ci();max2Ci()];i=1;:::;q;thenP[min2Ci()i()max2Ci()ji=1;:::;q]P[2C]1:Itfollowsthat[min2Ci();max2Ci()],i=1;:::;q,formconservativesimultaneouscondenceintervalsonthei()withajointcoverageprobabilityapproximatelygreaterthanorequalto1.Now,letusdenotetheinterval[min2Ci();max2Ci()]byDi(C),i=1;:::;q.Sincei()2Di(C)fori=1;:::;qifandonlyif()2qi=1Di(C),whereqi=1Di(C)denotestheCartesianproductoftheDi(C),then P[()2qi=1Di(C)]=P[i()2Di(C)ji=1;:::;q]P[2C]1:(6{9)Consequently,qi=1Di(C)formsarectangularconservativecondenceregionon().WethereforechoosethisregiontobetheonedescribedinSection 6.3.1 ,thatis,D.

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4.6 and 5.5.1 .Theprobabilitymassfunctionoftheresponsezjis[seeformula( 4{33 )] 6{10 )thattheresponseszj=(zj1;zj2;zj3)0followthemultinomialdistributionwithparametersmjandj=(j1;j2;j3)0;j=1;:::;n.Thenotationsyjw1,yjw2,zj,jandmjarethesameasdenedinSection 4.6 .Themeanofzjisj=(j1;j2;j3)0,wherej1=mjj1,j2=mjj2,j3=mjj3,andthevariance-covariancematrixofzjisj,where

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Thecorrespondinglinkfunctionusedhereis 1P3l=1l(x)=f0i(x)i;i=1;2;3;(6{12)wherefi(x)isaknownvectorfunctionofxdenedinformula( 5{3 ).Hencewecanwrite,^i(x)=exp[f0i(x)^i] 1+P3l=1exp[f0l(x)^l];i=1;2;3:where^iistheMLEofifori=1;2;3.Therst-orderpartialderivativeofwithrespectto0evaluatedatxis[seeformula( 5{24 )]@(x) 6{1 .Notethatthecenterofeachagegroupwasusedasthevalueoftheagevariable.

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Table6{1. Experimentaldesignandresponsevalues. (a)(b)(b)AgegroupAgeDosex1x2z1z2z3z4m (a)Thecenterofeachagegroupisusedasthevalueoftheagevariable.(b)x1andx2arethecodedlevelsofageanddoserespectively. AsinSection 6.4 ,wedenotebyzj1(j=1;:::;15)thenumberofpatientsinthethej-thrunwhosediastolicbloodpressurewasreducedbyatleast15mmHgbutexperiencedsomeunwantedside-eects,zj2thenumberofpatientsinthethej-thrunwhosediastolicbloodpressurewasreducedbyatleast15mmHgbuthadnounwantedside-eectsafterbeinggiventhedrug,zj3isthenumberpatientsinthej-thrunwhoexperiencedonlyside-eectsbutnoreductionintheirdiastolicbloodpressure,andzj4isthenumberofpatientsinthethej-thrunwhoexperienceneitherside-eectsnoranyreductionintheirdiastolicbloodpressure.Weinvestigatetheeectofthetwofactors,ageanddoselevelsofthevasodilator,onthemeansoftheresponses,namely,1(x),2(x)and3(x),x=(x1;x2)0andx1andx2arethecodedlevelsofageanddose,respectively,givenbyx1=(age60)=11andx2=(dose10)=10.Itisofinteresttosimultaneouslyminimize1(x)and3(x),andmaximizethemeanresponse2(x).

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Herei(x);i=1;2;3denotesthemeanofthei-thresponseatagivenpointxinthecodedexperimentalregion,R:1x11;1x21.Model( 5{2 )wasttedtothedatainTable 6{1 usingthefollowingmodelsforthelinearpredictors1(x)=1+2x1+3x2;2(x)=4+5x1+6x2+7x22;and3(x)=8+9x1+10x2:Thelinkfunctionsarethesameasthosedescribedinformula( 6{12 ).Theparameterestimatesandtheirstandarderrorsfortheabovemodels,obtainedbyusingtheRsoftware(Version2.0.1),areshowninTable 6{2 .Theseparameterestimatesandtheestimatedvariancecovariancematrixof^inTable 6{3 areusedjointlytondC,theapproximate95%condenceregionon. Table6{2. Maximumlikelihoodestimatesandstandarderrors. ParameterEstimateStd.errorP-value ScaledDeviance=113:3755,DF=35.

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Table6{3. Estimatedvariance-covariancematrixof^. 0.0350-0.00760.02300.01840.00090.02850.00980.0232-0.00110.0286-0.00760.03960.0122-0.00070.0140-0.0042-0.00120.00020.01800.00390.02300.01220.08170.01560.00450.04840.02720.03140.00430.05790.0184-0.00070.01560.04480.00220.0358-0.03980.0182-0.00060.01720.00090.01400.00450.00220.03120.00120.01240.00120.01290.00370.0285-0.00420.04840.03580.00120.0938-0.00010.0281-0.00230.04740.0098-0.00120.0272-0.03980.0124-0.00010.14190.00990.00020.02250.02320.00020.03140.01820.00120.02810.00990.0545-0.00760.0330-0.00110.01800.0043-0.00060.0129-0.00230.0002-0.00760.06280.01200.02860.00390.05790.01720.00370.04740.02250.03300.01200.1118

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Usingformula( 6{9 ),thecondenceregionDistheCartesianproductoftheindividualcondenceintervals,Di(C),oni();i=1;2;3.TondDi(C),thefollowingstepsaretaken:Foreachof1000valuesofrandomlyselectedfromC(selectionprocessisdiscussedindetailbelow),wecomputetheoptimumofi(x);i=1;2;3,overtheexperimentalregionR.Wedenotetheoptimumofi(x)byi().Next,wendtheminimumandmaximumofthesei;i=1;2;3,overthevaluesof2CtogetthelowerandupperboundsofDi(C),respectively.Theintervals,Di(C),fori=1;2;3aregiveninTable 6{4 .Thistablealsoincludestheindividualoptima(^i)ofthethreeestimatedmeanresponses,^i(x);i=1;2;3,andthelocationsoftheseoptima.ThenextstepistousethecondenceintervalsDi(C)togetthecondenceregionD. Table6{4. TheindividualoptimaandtheregionD. LocationofindividualoptimaDi(C) ^10.0362-1-10.00910.1113^20.7638-10.07500.56860.8748^30.0350-1-0.10700.01010.0826 Foreachof1380valuesofxselectedfromtheregionRbyagridsearch(selectionprocessisdiscussedindetailbelow),wecomputethemaximumofthedistancefunction,[^(x);]withrespectto2D.Letusdenotethemaximumof[^(x);]bymax(x).Minimizingmax(x)overthevaluesofxinR,wearriveattheminimaxdistance.Theminimaxdistanceandthecorrespondingsimultaneousoptimaof^1(x),^2(x),and^3(x)andtheirlocationsaregiveninTable 6{5 .Notethatincreasingthenumberofselectedvaluesofandxabove1000and1380,respectively,doesnotcauseanyappreciablechangeintheresults.

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Table6{5. Simultaneousoptima. Minmax9.6826 Simultaneous^10.0723optima^20.7056^30.0387 Locationofx1-0.8823simultaneousoptimax2-0.1849 FromTable 6{5 weseethatthelocationofxcorrespondingtotheminimaxdistanceis(0:8823;0:1849).Usingtheoriginal(uncoded)settingsofthevariables,ageanddose,thecoordinatesoftheminimaxpointareage=50anddose=8:1508mg.Thisresulttellsusthatecacyofthevasodilatorismaximizedanditstoxicityeectsareminimizedforpatientsneartheagegroup[4454]receivingamoderatedoselevelof8.1508mgofthedrug. DATAisasetofMATLABprogramsusedtogeneratedatapointsrandomlyfromagivenregion.WeusedRANDOM DATAtogenerate'srandomlyfromanellipsoidalregion,C.TherequiredMATLABlescanbedownloadedfromhttp://www.csit.fsu.edu/burkardt/m src/random data/random data.html.AsetofMATLABroutinescalledHEX GRIDwasusedtoselectx'sfromtheexperimentalregionR.HEX GRIDcomputespointsonahexagonalgriddenedonarectangularregion.TherequiredMATLABlescanbedownloadedfromhttp://www.csit.fsu.edu/burkardt/m src/hex grid/hex grid.html.AcomputerprogramcalledMCS[ HuyerandNeumaier ( 1999 )],wasusedinalltheaboveoptimizations.MCSisaMATLABprogramforconstrainedglobaloptimizationusingfunctionvaluesonly.Itisbasedonamultilevelcoordinatesearchthatbalancesglobalandlocalsearches.Thelocalsearchfortheoptimumis

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doneviaasequentialquadraticprogrammingprocedure.TherequiredMATLABlescanbedownloadedfromhttp://www.mat.univie.ac.at/neum/software/mcs/.Toruntheprograms,thetwoMATLAB5programsMINQ(boundconstrainedquadraticprogramsolver)andGLS(globallinesearch)arealsorequired.BothMINQandGLSarefoundinthesamewebsite.

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3 demonstratesthatthegraphicalapproachofquantiledispersiongraphs(QDGs)isaconvenienttechniqueforevaluatingandcomparingdesignsforGLMs.Thesegraphsalsoprovideinformationonthepredictioncapabilityofadesignthroughouttheexperimentalregion,andonitsdependenceonthemodel'sparameters.TheproposedmethodologyisillustratedwithanexamplebasedondesignsforPoissonregressionmodels.TheexamplealsoshowsthatthechoiceofthelinkfunctioncaninuencetheshapeoftheQDGsforagivendistribution.Foragivenlinkfunction,theexampleshowsthattheQDGsmayalsobeinuencedbythedistributionoftheresponsevariable.ThemaincontributionofChapter 4 isthederivationofformula( 4{31 )representingthebiasofMLEsoftheparametersinamultivariateGLMsetting.AnapplicationoftheuseofthisformulainthespecialcaseofabivariatebinarydistributionwasdiscussedinSection 4.6 ofthischapter. 87

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TherstportionofChapter 5 dealswiththedevelopmentofanexpressionforthemean-squarederrorofprediction(MSEP)concerningdesignsformultivariateGLMs.TheremainingportioncomparesthesedesignsformultivariateGLMsbasedontheirMSEP.AnexampleofabivariatebinarysituationisusedtoillustratethecomparisonofdesignsinamultivariateGLMsetup.InChapter 6 ,thegeneralizeddistanceapproachof KhuriandConlon ( 1981 )wasusedtodetermineoptimumoperatingconditionsontheinputvariablesthatresultinasimultaneousoptimizationofseveralpredictedmeanresponsesinamultivariateGLMsituation.AnexampleispresentedtoillustrateoptimizationinamultivariateGLMsetup. 3 through 6 ,inadditiontocertainproblemsinthecontextoflinearmixedeectsmodels.Alistingofthesetopicsfollows:1.ThegraphicalapproachofQDGsallowsforthecomparisonofseveralcandidatedesignsbutisnotcurrentlyusedtoconstruct\optimal"designs.Itwouldbedesirabletodevelopthisapproachsothatitcangeneratea\better"designsequentiallystartingwithaninitialdesign.2.ComparisonofdesignsforGLMsinChapter 3 and 5 isbasedonadistributionalformforthedataunderconsideration.Theremayarisesituationswherewelackawell-denedlikelihoodfunctionforthemultiresponsedata.Thus,apossibleextensionwillbetocomparedesignsforbothunivariateandmultivariateGLMsinthesesituationsbasedonaquasi-likelihoodfunction.3.InChapter 6 webaseouroptimizationalgorithminamultivariateGLMsetuponaspecicdistributionalform.Optimizationtechniquesthatcanbedevelopedwithoutassumingadistributionalformarethereforeneeded.Forexample,theremaybeaneedtodeterminethesettingsoftheinputvariableswhichsimultaneouslyoptimizetworesponses,bothfollowingaPoissondistribution.Inthiscase,toapplythesamealgorithmasinChapter 6 ,weshallrstneedtocomeupwithamultivariatePoissondistributionto

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estimatethemeanresponses.Unfortunately,aclosed-expressionformforthemultivariatePoissondistributionisnotavailableinthestatisticalliterature.4.Thereisaneedforresearchondesignissuesforgeneralizedlinearmixedmodels(GLMMs)andnon-linearmixed-eectsmodels(NLMEs).ItshouldbepossibletouseQDGsforthecomparisonandevaluationofdesignsforGLMMsandNLMEs.5.Myfutureplanalsoextendstolinearmixedeectsmodelswithheterogeneousvariances.Theexperimentalerrorvariance,inaresponsesurfacemodelwithablockeect,hastraditionallybeenassumedtobeconstant.Insomeexperimentalsituations,however,thisvariancemaybedierentforthedierentblocksthatmakeuptheassociateddesign.Astudyonthechoiceofdesignsfortheselinearmixedeectsmodelswithheterogeneousvariancesisapossibleextension.SomeoftheproceduresfortestingtherandomeectsinthesemodelsarelikelihoodratioandANOVA-basedtests.Acomparisonofthepowersofthesetwotestsfortherandomeectsisalsoafuturetarget.6.Generalizedleastsquares(GLS)estimatesofxedeectsinmixedlinearmodelsdependonthevaluesofthevariancecomponents.However,thesevaluesaretypicallyunknown,anditisacommonpracticetoreplacethemwithestimatedvalues.TheprocedureofrstestimatingthevariancecomponentsandthenusingthoseestimatesinGLScomputationisoftenreferredtoasatwo-stagemixedmodelanalysis,orestimatedgeneralizedleastsquares(EGLS).Withfewexceptions,propertiesofEGLSxedeectestimatesareunknown.ThevarianceoftheEGLSestimatedependsonthedesign.ItisdesirabletoperformadiagnosistocheckifitispossibletoreducethedierencebetweenthevarianceoftheGLSestimateandthatoftheEGLSestimatebychoosinganappropriatedesign.

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3 3{30 )Forlog-linearPoissonregression,(x)=(x),=log,and 3{1 ).Hence,from( A{1 ), 3{13 ),W=diag(w1;w2;:::;wn),wherefrom( 3{14 ), 3{28 ) =j;j=1;:::;n;and 90

PAGE 102

Here, =j;j=1;:::;n:Hence,from( A{4 ),( A{5 )and( A{7 ), 3{19 )isoftheform 2zjjj =zjj MSE[^(x)]=Var[^(x)]+fBias[^(x)]g2:(A{10)From,( 3{16 )and( A{4 ),wehavetheapproximation Var[^(x)]:=d(x) (A{11) =[(x)]2f0(x)(X0WX)1 3{15 ),( 3{21 ),( A{4 )and( A{7 ), Bias[^(x)]:=Bias[^(x)]d(x) 2Var[^(x)]d2(x) (A{12) =Bias[^(x)](x)+1 2f0(x)(X0WX)1 3{15 ),Var[^(x)]isapproximatelyequaltof0(x)(X0WX)1f(x) 3{20 ),thebiasof^(x)in( A{12 )isapproximatedbyBias[^(x)]:=f0(x)(X0WX)1X0W:Using( A{11 )and( A{12 )in( A{10 ),weobtainformula( 3{30 ).

PAGE 103

3{33 )ForPoissonregressionwithasquarerootlink,(x)=(x),=p 3{1 ).Hence,from( A{13 ), 3{13 ),W=diag(w1;w2;:::;wn),wherefrom( 3{14 ), 3{31 ) =2p A{16 ),( A{17 )and( A{19 ),

PAGE 104

Itfollowsthatthejthelementofin( 3{19 )isoftheform 21 4zjj4 =zjj MSE[^(x)]=Var[^(x)]+fBias[^(x)]g2:(A{22)From,( 3{16 )and( A{16 ),wehave Var[^(x)]:=d(x) (A{23) =[4(x)]f0(x)(X0WX)1 3{15 ),( 3{21 ),( A{16 )and( A{19 ), Bias[^(x)]:=Bias[^(x)]d(x) 2Var[^(x)]d2(x) (A{24) =2Bias[^(x)]p 3{15 ),Var[^(x)]isapproximatedbyf0(x)(X0WX)1f(x) 3{20 ),thebiasof^(x)in( A{24 )isapproximatelygivenbyBias[^(x)]:=f0(x)(X0WX)1X0W:Using( A{23 )and( A{24 )in( A{22 ),weobtainformula( 3{33 ).

PAGE 105

4 4{15 )Dierentiating@l @rfrom( 4{14 )withrespecttosgives @r@s=@ @s@l @r=nXj=1@ @s(y0j0j)@j +nXj=1(y0j0j)@ @s@j +nXj=1(y0j0j)@j @s@j Notethat@ @s(y0j0j)is @s(y0j0j)=@j Replacing@ @s(y0j0j)in( B{1 )withtheexpressionontheright-handsideof( B{4 )givesthedesiredformulain( 4{15 ). 94

PAGE 106

4{20 )Considertheexpressionin( 4{17 ),whichcanbewrittenas E@ @tnXj=1(y0j0j)@ @s@j @t(y0j0j)@ @s@j +EnXj=1(y0j0j)@ @t@ @s@j +EnXj=1(y0j0j)@ @s@j @t@j Using( B{4 ),wecanreplace@ @t(y0j0j)in( B{5 )with0BBBB@f01(xj)@1 B{6 )and( B{7 )reduceto0,and( 4{17 )becomesequaltotheright-handsideofformula( 4{20 ),thatis,E@ @tnXj=1(y0j0j)@ @s@j @s@j

PAGE 107

4{21 )Theexpressionin( 4{18 )canbeformulatedas E@ @tnXj=1(y0j0j)@j @s@j @t(y0j0j)@j @s@j +EnXj=1(y0j0j)@ @t@j @s@j +EnXj=1(y0j0j)@j @t@ @s@j Using( B{4 )andthefactthatE(yj)=j;j=1;:::;n,weareabletoreduce( 4{18 )toE@ @tnXj=1(y0j0j)@j @s@j @s@j 4{21 ).

PAGE 108

4{23 )Theexpressionin( 4{22 )issimpliedasfollows,EnXj=1(y0j0j)@j 4{23 )followsfromtheaboveequality.

PAGE 109

4{25 )Formula( 4{24 )canbewrittenas EnXj=1(y0j0j)@j @s@j @s@j @s@j0 UsingE(yj)=jandtheindependenceofyj's,( B{11 )isreducedto EnXj=12(y0j0j)@j @s@j

PAGE 110

Replacing(y0j0j)@j B{12 ),weget EnXj=120BBBB@f01(xj)@1 @s@j B{13 )with1 @s@j B{14 )becomes @s@j

PAGE 111

Fromawell-knownpropertyofJacobianmatrices,wehave@j B{15 )isnallywrittenasnXj=10BBBB@f01(xj)@1 @s@j 4{25 ). 4{27 )Theexpressionin( 4{26 )canbewrittenas EnXj=1(y0j0j)@j @s@j @s@j @s@j0 Asbefore,usingE(yj)=jandtheindependenceofyj's,( B{16 )simpliesto EnXj=12(y0j0j)@j @s@j

PAGE 112

UsingagainthesametechniquesasinAppendix( B{11 ),weareabletoreduce( B{17 )tonXj=10BBBB@f01(xj)@1 @s@j 4{27 ). 4{29 )ThisformulafollowsfromusingtheresultsofLemmas4.4.1and4.4.2in( 4{28 ),whereJt;rs+1 2Krst=nXj=10BBBB@f01(xj)@1 @s@j @s@j 2nXj=10BBBB@f01(xj)@1 @t@0j @t@j @t@j 2nXj=10BBBB@f01(xj)@1 @s@j @s@j

PAGE 113

=nXj=1 @t@0j @t@j @t@j @s@j @s@j

PAGE 114

5 5{23 ) 5.5.1 ,(x)=(x),=1, 5{21 ),wecanwrite 5.2.2 thatW=diag[W1;:::;Wn],whereWj=@j C{3 ).Thus, 103

PAGE 115

Usingjfromformula( 5{21 )and@j C{4 ),weget, MSE[^(x)]=Var[^(x)]+fBias[^(x)]gfBias[^(x)]g0:(C{6)From( 5{14 )and( C{5 )wehave Var[^(x)]:=@(x) C{3 )@ 4.6.1 ofChapter 4 .From( 5{17 )wehave Bias[^(x)]:=@(x) C{8 ),andtheoneforBias(^),wegetthepredictionbias.Substituting( C{7 )and( C{9 )in( C{6 ),weobtainformula( 5{23 ).

PAGE 116

3 #############DATAENTRY############n<16X1
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###################################PARAMETERESTIMATESANDSTANDARDERROR##################################coef.stder
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var.lin.pred
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rect0.9
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################QUANTILESOFD1###############prob.cuto
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q.D1.rect0.7.min.l
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mse.pred.out.D2.rect1[,i]
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#####################LOGOFQUANTILESOFD2####################q.D2.rect1.max.l
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lines(spline(prob.cuto,q.D1.rect1.max.l,n=500),lty=1)points(prob.cuto,q.D1.rect1.min.l,pch=16)lines(spline(prob.cuto,q.D1.rect1.min.l,n=500),lty=1)points(prob.cuto,q.D2.rect1.max.l,pch=2)lines(spline(prob.cuto,q.D2.rect1.max.l,n=500),lty=3)points(prob.cuto,q.D2.rect1.min.l,pch=2)lines(spline(prob.cuto,q.D2.rect1.min.l,n=500),lty=3)plot(prob.cuto,q.D1.rect0.9.max.l,xlab="",ylab="Quantileoflog(MSEP)",type="n",ylim=c(yminlimit,ymaxlimit))box()mtext("p",side=1,line=1)mtext("nu=0.9",side=3)points(prob.cuto,q.D1.rect0.9.max.l,pch=16)lines(spline(prob.cuto,q.D1.rect0.9.max.l,n=500),lty=1)points(prob.cuto,q.D1.rect0.9.min.l,pch=16)lines(spline(prob.cuto,q.D1.rect0.9.min.l,n=500),lty=1)points(prob.cuto,q.D2.rect0.9.max.l,pch=2)lines(spline(prob.cuto,q.D2.rect0.9.max.l,n=500),lty=3)points(prob.cuto,q.D2.rect0.9.min.l,pch=2)lines(spline(prob.cuto,q.D2.rect0.9.min.l,n=500),lty=3)plot(prob.cuto,q.D1.rect0.7.max.l,xlab="",ylab="Quantileoflog(MSEP)",type="n",ylim=c(yminlimit,ymaxlimit))box()mtext("p",side=1,line=1)mtext("nu=0.7",side=3)points(prob.cuto,q.D1.rect0.7.max.l,pch=16)lines(spline(prob.cuto,q.D1.rect0.7.max.l,n=500),lty=1)

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points(prob.cuto,q.D1.rect0.7.min.l,pch=16)lines(spline(prob.cuto,q.D1.rect0.7.min.l,n=500),lty=1)points(prob.cuto,q.D2.rect0.7.max.l,pch=2)lines(spline(prob.cuto,q.D2.rect0.7.max.l,n=500),lty=3)points(prob.cuto,q.D2.rect0.7.min.l,pch=2)lines(spline(prob.cuto,q.D2.rect0.7.min.l,n=500),lty=3)plot(prob.cuto,q.D1.rect0.3.max.l,xlab="",ylab="Quantileoflog(MSEP)",type="n",ylim=c(yminlimit,ymaxlimit))box()mtext("p",side=1,line=1)mtext("nu=0.6",side=3)points(prob.cuto,q.D1.rect0.6.max.l,pch=16)lines(spline(prob.cuto,q.D1.rect0.6.max.l,n=500),lty=1)points(prob.cuto,q.D1.rect0.6.min.l,pch=16)lines(spline(prob.cuto,q.D1.rect0.6.min.l,n=500),lty=1)points(prob.cuto,q.D2.rect0.6.max.l,pch=2)lines(spline(prob.cuto,q.D2.rect0.6.max.l,n=500),lty=3)points(prob.cuto,q.D2.rect0.6.min.l,pch=2)lines(spline(prob.cuto,q.D2.rect0.6.min.l,n=500),lty=3)legend(0.0,0.0,c(\D1",\D2"),lty=c(1,3),pch=c(16,2))

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115

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SiuliMukhopadhyaywasborninCalcutta,India,onAugust14,1978.Sheearnedherbachelor'sdegreeinstatisticsfromSt.Xavier'sCollege,Calcutta,in1999.Aftercompletinghermaster'sdegreeinstatisticsfromtheDepartmentofstatistics,CalcuttaUniversity,Calcutta,in2001,shecametotheUnitedStates.ShejoinedtheDepartmentofstatisticsattheUniversityofFloridatopursueaPh.D.instatisticsundertheguidanceofDr.AndreI.Khuri.From2001-2005shewasappointedasanAlumniFellowattheDepartmentofStatistics.InthepastyearsheworkedasateachingassistantattheDepartmentofStatistics.Shereceivedthe2006DepartmentofStatisticsSeniorStudentGraduateAward.SheexpectsaPh.D.inAugust,2006.AftercompletingherdoctoralstudiesshewillworkasanAssistantProfessorattheDepartmentofBiostatisticsattheMedicalCollegeofGeorgiainAugusta,Georgia. 120


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Title: Design Considerations and Optimization for Univariate and Multivariate Generalized Linear Models
Physical Description: Mixed Material
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DESIGN CONSIDERATIONS AND OPTIMIZATION FOR UNIVARIATE AND
MULTIVARIATE GENERALIZED LINEAR MODELS
















By
SIULI MUK(HOPADHYAY


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006































Copyright 2006

by
Siuli Mukhopadhyay



































Dedicated to my family and friends.















ACKNOWLEDGMENTS

I would first like to give thanks to my advisor, Dr. Andrd I. K~huri, for his

enthusiastic supervision and encouragement during my doctoral research. This

dissertation would not have been possible without his guidance and tremendous

patience. I would also like to thank all my committee members: Dr. Malay Ghosh,

Dr. Ronald Randles, Dr. Trevor Park and Dr. Rick Smith. Special thanks go to

the faculty and staff at the Department of Statistics, University of Florida. In

addition, I give thanks to my undergraduate professor Dr. Atindra Mohan Goon.

I am indebted to all my friends for their help and constant encouragement,

especially Ananya Roy, Anales DeBhaumik, Vivekananda Roy, Damaris Santana

Morant and Ludwigf Heigfenhauser.

Most importantly, I wish to thank my entire family, especially Debraj, Hanshu,

Oda and Titi.


















TABLE OF CONTENTS

page

ACK(NOWLEDGMENTS ......... .. iv

LIST OF TABLES ......... .. .. viii

LIST OF FIGURES ......... ... ix

ABSTRACT ......... .... .. x

CHAPTER

1 INTRODUCTION . ...... ... .. 1

2 LITERATURE REVIEW ......... .. 4

2.1 Response Surface Methodology ...... .. 4
2.2 Generalized Linear Models . .... 6
2.2.1 Locally Optimal Designs .... ... 7
2.2.2 Sequential Designs . .... .. 10
2.2.3 B li-.- I m Optimal Designs ... ... .. 11
2.2.4 Quantile Dispersion Graphs (QDGs) Approach .. .. .. 12
2.3 Multiresponse Experiments . ..... .. 1:3
2.4 Multivariate GLMs . ..... .... 14
2.4. 1 Optimal Designs for Bivariate Logistic Models .. .. .. 15
2.4.2 Optimal Designs for Multivariate Logistic Models .. .. 17
2.5 Multiresponse Optimization . ..... .. 18

:3 DESIGN COMPARISONS FOR ITNIVARIATE GENERALIZED LINEAR
MODELS ......... .... .. 21

:3.1 Introduction ......... .. .. 21
:3.2 Generalized Linear Models . .... 22
:3.2.1 Estimation of the Mean Response ... ... .. 2:3
:3.2.2 The Prediction Bias . ... 25
:3.2.3 The Mean-Squared Error of Prediction .. .. 26
:3.3 Clon .! ..of Design ......... ... 26
:3.4 Quantile Dispersion Graphs . ..... .. 26
:3.5 Examples ... . . . 28
:3.5.1 Poisson Response with a Logfarithmic Link .. .. .. 29
:3.5.2 Poisson Response with a Square Root Link .. .. .. .. :30
:3.5.3 A Numerical Example . .... .. :31










4 BIAS IN MULTIVARIATE GENERALIZED LINEAR MODELS

4. 1 Introduction.
4.2 Bias in a General Multivariate Settingf
4.3 Multivariate GLM
4.4 Bias in a Multivariate GLM Setting .
4.5 Final Expression for the Bias
4.6 Example: Bivariate Binary Distribution
4.6.1 Bias for a Bivariate Binary Distribution.
4.6.2 Numerical Example.


5 COMPARISON OF DESIGNS FOR MULTIVARIATE
LINEAR MODELS.

5.1 Introduction.
5.2 Multivariate Generalized Linear Models.
5.2.1 Estimating the Mean Response .
5.2.2 Prediction Variance.
5.2.3 Prediction Bias
5.2.4 Mean-squared Error of Prediction
5.3 Criterion for Comparing Designs .
5.4 Quantile Dispersion Graphs
5.5 Bivariate Binary Distribution
5.5.1 Bivariate Binary Distribution with a Logit
5.5.2 Numerical Example.


GENERALIZED


Link


6 OPTIMIZATION IN MULTIVARIATE GENERALIZED LINEAR MODELS 75

6.1 Introduction ........ .. .. ... 75
6.2 Multivariate Generalized Linear Models .. .. .. 75
6.3 Simultaneous Optimization of a Multiresponse Function .. .. .. 76
6.3.1 Generalized Distance Approach in Multivariate GLM .. 77
6.3.2 Construction of confidence interval on the true mean response 78
6.4 Example: Bivariate Binary Distribution .. .. .. 79
6.4. 1 A Numerical Example . . .. .. 80
6.4.2 Computer Programs Used for the Selection Process .. .. 85

7 SITAINARY AND FITTIRE RESEARCH TOPICS .. .. 87

7. 1 Suninary ......... ... 87
7.2 Future Research ......... .. 88

APPENDIX

A PROOF OF RESITLTS ITSED IN CHAPTER :3 .. .. .. 90

A.1 Derivation of Formula (:3-30) ...... .. 90
A.2 Derivation of Formula (:3-33) ...... .. 92










B PROOF OF RESULTS USED IN CHAPTER 4 ........


B.1
B.2
B.:3
B.4
B.5
B.6
B.7


Derivation
Derivation
Derivation
Derivation
Derivation
Derivation
Derivation


of Formula
of Formula
of Formula
of Formula
of Formula
of Formula
of Formula


15)
20)
21)
2:3)
25)
27)
29)


C PROOF OF RESULTS USED IN CHAPTER 5 ........

D R CODE: CONSTRUCTION OF QUANTILE DISPERSION GRAPHS
IN CHAPTER :3

REFERENCES ...................

BIOGRAPHICAL SKETCH ..... ..... ....

















LIST OF TABLES

Table page

3-1 Experimental design and response values. .... .. .. 31

3-2 Parameter estimates and model analysis using log link. .. .. .. 32

3-3 Design D2* *......... * * 33

3-4 Parameter estimates and model analysis using square root link. .. .. 35

4-1 Experimental design and response values. .... .. .. 55

4-2 Maximum likelihood estimates and standard errors .. . .. 55

4-3 Estimated mean response values. ...... .. 56

5-1 Experimental design D1 (5 x 7 factorial) and response values. .. .. 70

5-2 Maximum likelihood estimates and standard errors .. . .. 72

5-3 Designs D1 (5 x 7 factorial) and D2 (2 faCtOrial). ... .. .. 72

6-1 Experimental design and response values. .... .. .. 81

6-2 Maximum likelihood estimates and standard errors .. . .. 82

6-3 Estimated variance-covariance matrix of P. ... .. 83

6-4 The individual optima and the region DC. .... .. 84

6-5 Simultaneous optima. .. ... .. 85
















LIST OF FIGURES

Figure page

:31 Concentric squares within the region R. ... .. .. 36

:32 QDGs for designs D1 and D2 using Poisson distribution (4 = 1) with a
log link. ......... . . :37

:33 Comparison of the QDGs for design D1 using Poisson distribution (4
1) with log and square root links. ..... ... :38

:34 Comparison of the QDGs for design D2 using Poisson distribution (4
1) with log and square root links. ..... ... :39

:35 Comparison of the QDGs for designs D1 and D2 using four different distributions,
all with log link, using a = 1. . ..... .. 40

5-1 Designs D1 (5 x 7 factorial) and D2 (2 factOrial) ... .. .. 7 3

5-2 Comparison of the QDGs for designs D1 (5 x 7 factorial) and D2 (2
factorial) given in Table 5-3. . .... . 74















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

DESIGN CONSIDERATIONS AND OPTIMIZATION FOR UNIVARIATE AND
MULTIVARIATE GENERALIZED LINEAR MODELS

By

Siuli Mukhopadhyay

August 2006

C'I ny~: Andrd' I. K~huri
Major Department: Statistics

Most response surface designs are based on linear models in which the

responses are assumed to be continuous with uncorrelated errors and homogeneous

error variances. However, clinical or epidemiological data, for example, quite

frequently do not satisfy these assumptions. In such situations, analysis of the data

using generalized linear models (GLMs) would be more appropriate. One inl li r~

problem that handicaps the choice of design for a GLM is its dependence on the

unknown parameters of the fitted model. One approach to solving this problem

uses the so-called quantile dispersion graphs (QDGs) of the mean-squared error of

prediction associated with a given model. Quantiles of the mean-squared error of

prediction are obtained within a certain region of interest. These quantiles depend

on the unknown parameters of the linear predictor. Plots of the maxima and

minima of the quantiles over a subset of the parameter space produce the QDGs.

The plots assess the design's performance throughout the experimental region

and give a clear depiction of the dependence of the design on the parameters.

In this dissertation, we use the graphical technique of QDGs to compare and

evaluate designs for both univariate and multivariate GLMs. Multivariate GLMs










are considered whenever several response variables can he measured for each setting

of a group of control (input) variables, and the response variables are adequately

represented by GLMs.

Alaxiniun likelihood estimates are known to be biased when the sample size

is small. In this dissertation, a general approximate expression is derived for the

bias associated with nmaxiniun likelihood estimates of the parameters of the linear

predictor in a multivariate GLM.

One of the primary objectives in response surface methodology is the

determination of operating conditions on a set of control variables that result in an

optiniun response. Optimization is more complex in a nmultiresponse situation as

it requires finding the settings of the control variables that yield optimal, or near

optimal, values for all the responses under consideration simultaneously. The final

topic in this dissertation deals with optimization in a multivariate GLM situation.















CHAPTER 1
INTRODUCTION

Response surface methodology (RSM) is a set of techniques that encompasses

setting up a series of experiments that yield reliable and adequate measurements

of the response of interest, fittingf and evaluating a given model, and determining

the optimum settings of the factors that yield maximum (minimum) value of the

predicted response. Most traditional response surface techniques were developed

within the framework of linear models under the strong assumption of continuous

responses (quite often, normally distributed) with uncorrelated errors and equal

error variances. However, in some situations, the response data may be discrete,

possibly correlated, and/or exhibit heterogeneous variances. This is quite common

in, for example, clinical studies and dose-response experiments. For such data,

statistical analysis using generalized linear models (GLMs) would be more

appropriate. GLMs were introduced by Nelder and Wedderburn (1972) as an

extension of the class of linear models. Under the framework of GLMs, discrete as

well as continuous responses can be accommodated, and normality and constant

variances are no longer a requirement for the response.

In the area of GLMs, one of the main considerations is the choice of an

experimental design. Unfortunately, little work has been done in this area. This is

mainly due to a serious problem caused by the dependence of a design for a GLM

on the unknown parameters of the fitted model. This dependence problem causes

great difficulty in the construction and evaluations of designs for GLMs.

For the most part, design criteria for linear models and GLMs pertain to

models for a single response. There are, however, many experimental situations

where several responses can be observed for each setting of a group of control










variables. For example, in dose response studies, we may come across situations

where several responses can he observed for the same patient. For example, in

addition to the standard response concerning the efficacy of a drug, we may also

measure its toxicity, that is, any side effects experienced by the patient. Thus, one

important area that needs investigation is the choice of designs for multivariate

GLMs.

It is well known that in small samples the parameter estimates for a

multivariate GLM are biased. Thus the nican-squared error of prediction (jl!SEPl)

should be used to assess the quality of prediction associated with a given design.

The MSEP incorporates both the prediction variance and the prediction hias,

which results from using nmaxiniun likelihood estimates of the parameters of the

fitted linear predictor. Searching the literature, we came up with an expression for

the bias of the nmaxiniun likelihood estimates (AILEs) in a general multiparanleter

setting given by Cox and Snell (1968), and an approximate expression of the bias

of MLEs in the case of univariate GLMs was given by Cordeiro and Alc~ullagh

(1991). Hence, we need an expression for the bias in the multivariate GLM case.

One important objective of nmultiresponse experimentation in the GLM setup

is the determination of optiniun operating conditions on the input (control)

variables that will lead to the simultaneous optimization of the predicted

values of the responses within a certain region of experiment. For example, in a

dose-response experiment, where both efficacy and toxicity responses are measured

at each dose, the experimenter may wish to find the dose level of the drugs) which

simultaneously nmaxintizes efficacy and nxinintizes toxicity. Several approaches

dealing with nmultiresponse optimization in the case of linear models are available in

the literature. Multiresponse optimization in a GLM situation is still in its nascent

stage.










In this dissertation, a solution to the design dependence problem in GLMs is

proposed using a graphical technique called quantile dispersion graphs (QDGs).

This graphical technique has been used to compare designs for both univariate and

multivariate generalized linear models. An approximate expression for the bias in

the multivariate GLM case, and multiresponse optimization in a GLM situation

have also been discussed.

The topics discussed in this dissertation are organized as follows: Chapter

2 is a review of literature concerning GLM designs and optimization in the case

of both univariate and multivariate responses. Comparison of designs for Poisson

regression models using QDGs is discussed in OsI Ilpter 3. A detailed derivation

of the expression for hias in a multivariate GLM setup is given in C'!s Ilter 4.

('I Ilpter 5 deals with the comparison of designs using QDGs for multivariate

GLMs. Determination of operating conditions on a set of control variables that

simultaneously optimize the responses in a multivariate GLM situation is discussed

in C'!s Ilter 6. Finally, a summary and a list of future research topics are given in















CHAPTER 2
LITERATURE REVIEW

2.1 Response Surface Methodology

The tools needed for the adequate selection of a design and the subsequent

fitting and evaluation of a given model, using the data generated by the design,

have been developed in an area of experimental design known as response -;4;. .~~~

I,~I,~ I 111;...J., (RSM). Box and Wilson (1951) laid the foundations for response

surface methodology. It was initially developed to determine optimum operating

conditions in the chemical industry. It is now used in a variety of fields and

applications, not only in the physical and engineering sciences, but also in the

biological, clinical and social sciences, to name just a few.

Over the past fifty years, there have been some extensive reviews of R SM. The

Hill and Hunter (1966) paper featured an extensive bibliography and presented

applications in the chemical and process industries. Myers et al. (1989) emphasized

the changes that had taken place in the R SM theory and practice during the

1970's and 1980's. Their paper discussed applications of RSM in the physical

and engineering sciences, food science, social science, and the biological sciences.

The most recent review paper was that of Myers et al. (2004) which focused on

R SM activities since 1989. They discuss current research areas in R SM, such as

robust parameter design in the framework of R SM, R SM and generalized linear

models, parametric and semi-parametric R SM. In addition to these review papers,

the books by Box and Draper (1987), Myers and Montgomery (2002), K~huri and

Cornell (1996), and K~huri (2006) provide a comprehensive coverage of the various

techniques in R SM.










The key ideas in RSM were initially developed using classical linear models

with continuous responses (quite often, normally distributed) with uncorrelated

errors and homogeneous variances. However, in many experimental situations

involving, for example, clinical and epidemiological data, such assumptions are

invalid. For example, data on human responses tend to be more variable than is

expected under the homogeneous error variance assumption, biological data may be

correlated due to genetic relationships, and dose-response experiments yield discrete

data. Due to the nature of the data as described, doing statistical analysis using

standard linear models will give inadequate results. For such data, generalized

linear models (GLMs) would be more appropriate. GLMs have proved very

useful in several areas of applications as in applied biomedical fields, entomology,

climatology, etc.

However, to use GLMs effectively in the above mentioned scenarios, an efficient

design with desirable properties is needed. By a choice of design in RSM, it is

meant the determination of the settings of the control variables that yield adequate

predictions of the response of interest throughout the experimental region. Optimal

design theory for linear models has been widely studied, but the same cannot

be said with regard to GLMs. This is mainly because optimal designs for GLMs

depend on the unknown parameters of the fitted model. Thus the construction of

a design requires some prior knowledge of these parameters. The experimenter is

therefore presented with a dilemma since the purpose of a design is to estimate

the unknown parameters of the model using the data generated by the design.

Common approaches to solve this design dependence problem include the following:

1. the specification of initial values, or best "guesses," of the parameters
involved, and the subsequent determination of the so-called '7... ril;i-optimal
design using an appropriate design criterion such as D-oldl.:l,,al.:lt or G-










2. the .sequential approach which allows the user to obtain updated estimates of
the parameters in successive stages, starting with the initial values used in the
first stage.

3. the Ba .;,. .:r t approach, where a prior distribution is assumed on the
parameters, which is then incorporated into an appropriate design criterion by
integrating it over the prior distribution.

4. the use of the so-called quantile li~spersion Il'aft-, (QDGs) approach, which
allows the user to compare different designs based on their quantile dispersion
profiles. Robinson and K~huri (2003) introduced this approach to discriminate
among designs for logistic regression models.

In ChI Ilpter 3, we shall address the design dependence problem for generalized

linear models. The main focus of attention will be on the use of the aforementioned

fourth approach.

2.2 Generalized Linear Models

Generalized linear models (GLMs) were first introduced by Nelder and

Wedderburn (1972) as an extension of the class of linear models. They can he used

to fit discrete as well as continuous data having av 0-1I i. b of parent distributions.

The traditional assumptions of normality and homogeneous variances of the

response data, usually made in an analysis of variance (or regression) situation,

are no longer needed. Alany statistical developments in terms of modelling and

methodology, in the past twenty years, may be viewed as special cases of GLMs.

Examples include logistic regression for binary responses, linear regression for

continuous responses, and log-linear models for counts. A classic book on GLMs

is the one by McCullagh and Nelder (1989). In addition, the more recent books

by Dobson (2001), Lindsey (1997), McCulloch and Searle (2001), and Myers and

Montgomery (2002) provide added insight into the application and usefulness of

GLMs.

In the GLM situation the response variable y, is assumed to follow a

distribution from the exponential family. This includes the normal as well as









the hinomial, Poisson and ganina distributions. The mean response is modelled as

a function of the form, E[y(x)] = p(x) = b [f'(x)P], where x = (. ., .k) f@X) is

a known vector function of order p x 1 and P is a vector of p unknown parameters.

The function f'(x)P is called the linear predictor, and is usually denoted by ty(x).

It is assumed that h(-) is a strictly monotone function. Using the inverse of the

function h(-) we have g[p(x)] = ty(x). The function g(-) is called the link function.

Estimation of p is based on the method of nmaxiniun likelihood using an

iterative weighted least-squares procedure [see McCullagh and Nelder (1989,

pp. 40-4:3)]. The variance-covariance matrix, Var(P), of P is approxiniated by

~(X'WX)-1 [see R~obinson and Kihuri (200:3)], where is the dispersion parameter,
X is an n x p matrix whose jth row is of the form f'(xj), xj is the value of x at

the jth experimental run (j = 1,..., n), and W is a diagonal matrix whose jth

diagonal element is w(xj, p). An expression for w(xj, p) is given in ChI Ilpter :3. It

follows that Var(P) depends on the unknown parameter vector P through w(xj, P).

This leads to the aforementioned design dependence problem.

2.2.1 Locally Optimal Designs

Designs for GLMs depend on the unknown parameters of the fitted model.

Due to this dependence, the construction of a design requires some prior knowledge

of the parameters. If initial values of the parameters are assumed, then a

design obtained on the basis of an optinmality criterion, such as D-optinmality or

A-optinmality, is called '7... ril;i optimal. The adequacy of such a design depends on

how close the initial values are to the true values of the parameters.

A key reference in this area is the one by Mathew and Sinha (2001) concerning

designs for a logistic regression model. Other related work includes that of

Abdelhasit and Plackett (198:3), Minkin (1987), K~han and Yazdi (1988), Wu

(1988) and Sitter and Wu (199:3).









Consider a binary response taking the values 0 and 1 resulting from a

non-stochastic dose level x. The response takes the value 1 with probability


p(x) (2-1)
1 + e-("+pm)

where a~ and p are unknown parameters and p > 0. There are a distinct dose

levels xl,..., x,, and mj observations are taken on the response at dose level

xy (j = 1,..., n1). Let, CE~ mi = m Let yyi denote the number of ones obtained
at x = xj (j = 1,..., n). Then, yl,..., y, are mutually independent and yj

has a binomial distribution (mj,p(xj)), j = 1,..., n. For the estimation of

the unknown parameters a~ and p, or some function thereof, the optimal design

problem is to select the xj's (in a given region) and the mj's, with respect to some

optimality criterion, for a fixed m. This is done on the basis of some measure of

the information about (a~, p) yielded by the data. Since this is a difficult and often

intractable optimization problem, a continuous design measure is considered in

which myl/mr is replaced by a real nlumber (4 with 0 < (4 < 1 anld CE (4 = 1. Th'lis

design measure depends on the unknown values of a~ and P, and thus good initial

estimates of a~ and p are needed. From (2-1) we see that the dependence enters

through z = a~ + px only and so the problem reduces to obtaining optimal values of

zj and (y.

The estimation problems that are usually of interest refer to (a) the estimation

of p, or c0/#, or some quantiles of p(x) in (2-1), or (b) the joint estimation of a

pair of parameters such as (i) a~ and p, (ii) P and a gP, (iii) P and a quantile of

p(x), and (iv) two quantiles of p(x). For estimating two parameters jointly, Mathew

and Sinha (2001) considered the information matrix, I(0), 8' = (a~, P), of the

parameters and chose the xj's and mj's to maximize some suitable scalar-valued

function of this matrix. The information matrix I(8), a symmetric matrix of order

p x p, p is the number of unknown parameters, is defined as [see Lehmann (1983,









p. 125)]
82 log L(0|x)dee
I(0) = E

where L(0|x) is the likelihood function. This is equivalent to minimizing some
scalar-valued function of the variance-covariance matrix of the maximum likelihood

estimators of the parameters. The D-optimality and A-optimality criteria are

well-known examples that illustrate this process.

For the logistic regression model, the likelihood function takes the form





Thus the log-likelihood function is


1(a, px)= ~ lo my I+ yj log p~y + mj log {1 p(xj)}
1 p(xj)



= log + yj(a~ + Pxj) mj log {1 + exp(a~ + pxj)

Following the approximate design theory, a design is denoted by ~D=

{(xj, (4), j = 1,. .,,}. We assume the dose region, R, to be 0 < x < 00.
Thus the information matrix for the joint estimation of a~ and P underlying the

design ~D is given by [see Mathew and Sinha (2001, p. 297)]

n exp(zy) nexp(-zj)
I~a, 4 = n =1 1 (1+exp(-zy2 Cj=1 1 1 (1+exp(-zy))' 23
n2 ex(z) n 2 exp(-zj)
j=1 -3 3 (1+exp(-zy))'=1 1 J (1i+exp(--zy))

where zy = a~+ pxj, j = 1,. .,n.

Mathew and Sinha (2001) presented a unified approach for the derivation of

D- and A-optimality for binary data under the two-parameter logistic regression

model. To obtain a D-optimal design, they maximize |l(ca, P)|, and for A-optimal

designs, they minimize [Var & + Var P], where &i and p are the maximum likelihood










estimators of a~ and p. They showed that D-optimal designs are 2-point symmetric

designs .

2.2.2 Sequential Designs

In Section 2.2.1, a locally optimal design was developed using initial values

of the parameters as "best go~ --' Response values can then be obtained on

the basis of the generated design. In the sequential approach, experimentation is

not stopped at the initial stage. Instead, using the information obtained, initial

estimates of the parameters are updated and used to find additional design points

in the subsequent stages. This process is carried out till convergence is achieved

with respect to some optimality criterion, for example, D-optimality. Sequential

designs were proposed by Wu (1985), Sitter and Forbes (1997), Sitter and Wu

(1999), among others.

In binary response models, the information matrix depends on both the

design and the model's parameters. Thus to obtain an optimal design, t;ood"

initial estimates of the parameters are needed. Quite often, these estimates are

poor. A solution given by Sitter and Forbes (1997) in these circumstances was to

perform the experiment in two (or more) stages. The initial design is used to get

the parameter estimates and then treating these as the true parameter values, a

second-stage design is chosen. The combination of first and second-stage designs

is optimal in some sense. Sitter and Forbes (1997) performed a quantal response

study, where a subject was given a stimulus x that exhibited a response with

probability p(x).

Sitter and Forbes (1997) assumed a parametric model p(x) = H[P(x p)]

for the quantal response curve where p and P are unknown parameters and H(-) is

a specified distribution function. A number, mj, of independent observations are

taken at the jth of a stimulus levels xl, .., x,, and the corresponding responses

are observed. The responses are mutually independent binomial random variables.









The corresponding Fisher information matrix is [see Sitter and Forbes (1997,

p. 942)]


;- p 2s, S o S~- S ] (2 -4 )

where So = C" 9j (ywjz, S1 = CnOy~y, 2 =
p(xy p), (4 = mj/m, m = y ndwt [H'(t)]2/[H(t) {1 H(t)}]. Sitter

and Forbes (1997) reduced the problem of finding an optimal second-stage design

to a two-variable (in many cases a one variable) maximization problem for almost

any optimality criterion. The search for the optimal second-stage design can be

restricted to two-point designs which are symmetric about p with possibly different

weights at each point.

2.2.3 Bayesian Optimal Designs

In the B li-o -1 .Is approach, a prior distribution is assumed on the parameter

vector, 8, which is then incorporated into an appropriate design criterion by

integrating it over the prior distribution. For example, one criterion maximizes the

average over the prior distribution of the logarithm of the determinant of Fisher

information matrix. This criterion is equivalent to D-optimality in linear models.

B li-. -1 Ia versions of other alphabetic optimality criteria can also be used such as

A-optimality. The B li-o -1 .Is approach was discussed by several authors including

Chaloner (1987), Chaloner and Larntz (1989, 1991), Mul:1su uI lllivay and Haines

(1995), C'll II..11. 1- and Verdinelli (1995) and K~huri et al. (2006).

Most of the B li-. -1 Ia criteria are based on normal approximations to the

posterior distribution of 8, as computations involving the exact posterior

distribution are often intractable. We shall assume, as usual, that the design

measure ( puts relative weights ((1,...,(s) at a distinct points (xl,..., x,),

respectively, with CE g = 1. ('I! IInin! 1- (1987) and ('I! IIh e.. 1- and Larntz (1989,

1991) used two criteria i(() = Ea[log |i(0, ()|] and #2(() = -E[trB(0)I(0, ()- ],









where B(0) is a p x p symmetric matrix. If linear combinations of the parameters

are of interest, then B(0) does not depend on 8 and is a vector of known weights.

In case of non-linear combinations of the parameters, B(0) has entries that depend

on 0. The 01 and 42, cTiterl8 are, Tespectively, equivalent to D-optimality and

A-optimality in linear models. All of the above mentioned B li-, -i Ia criteria are

concave over the space of all probability measures on the design space R. The

Equivalence Theorem for optimality of linear models Whittle (197:3) has been

extended to the non-linear case by Silvey (1980), Chaloner and Larntz (1989), and

Chaloner (199:3).

C'I IIh e.. 1- and Larntz (1989) consider a logistic regression model such that at

dose-level :r, a Bernoulli response is observed. There are two parameters 8/ = (ft, /3)

and the probability of success is


p(:r)= (2-5)
1 + exp(-73(: ft))'

The Fisher information matrix here is the same as the expression given in (2-4)

usmng

H[/3(: 1-1)] =.
1 + exp(-73(r ft))'

C'I IIh e.. 1- and Larntz (1989) investigate the B li-. -1 Ia D-optimality and

A-optimality criteria in the context of a binary response logistic regression model

with a single variable :r assuming independent uniform priors on the parameters

/3 and I-1. The optimal designs are obtained through implementing the simplex

algorithm of Nelder and Mead (1965). It was noted that the number of support

points of the optimal design grew as the support of the prior became larger. The

optimal designs in general were not equispaced and were unequally weighted.

2.2.4 Quantile Dispersion Graphs (QDGs) Approach

This approach was recently introduced by Robinson and K~huri (200:3) in a

GLM situation. In this graphical technique, designs are compared on the basis of










their quantile dispersion profiles. Since in small samples the parameter estimates

are often biased, Robinson and K~huri (2003) consider the mean-squared error of

prediction (jl!SEPl) as a criterion for comparing designs. The QDGs are based on

studying the distribution of the MSEP throughout the experimental region, R.

The distribution of the MSEP is studied in terms of its quantiles. These graphs

provide an assessment of the overall prediction capability of a given design over

the entire experimental region through a visual di pl .vi of the MSEP. They also

give a clear depiction of the dependence of the design on the model's unknown

parameters. K~huri and Lee (1998) used a graphical procedure for comparing

designs for nonlinear models. The QDG approach of Robinson and K~huri (2003)

generalizes the work of K~huri and Lee (1998) by addressing non-normality and

nonconstant error variance. They consider the problem of discriminating among

designs for logistic regression models.

In C'!s Ilters 3 and 5, QDGs will be discussed and used to compare designs for

univariate and multivariate GLMs, respectively.

2.3 Multiresponse Experiments

In the previous sections, we discussed experiments involving only single

response variables. Quite often, however, research workers obtain measurements

on several response variables. An experiment in which a number of responses are

measured simultaneously for each setting of a group of control variables is called a

multiresponse experiment. Examples of multiresponse experiments are numerous,

for example, a chemical engineer may be interested in maximizing the yield while

minimizing the cost of a certain chemical process. Hill and Hunter (1966), Myers

et al. (1989) and K~huri (1996) cited several papers in which multiresponses were

investigated.

In selecting a design optimality criterion for multiresponse experiments,

one needs to consider all the responses simultaneously. One such criterion was









developed by Fedorov (1972). His procedure, however, required knowledge of

the variance-covariance matrix associated with the several response variables.

Wijesinha and K~huri (1987) modified Fedorov's (1972) procedure by using an
estimate of the variance-covariance matrix.

Very little work has been done on multiresponse or multivariate GLMs,

particularly in the design area. Such models are considered whenever several

response variables can be measured for each setting of a group of control variables,

and the response variables are adequately represented by GLMs.

2.4 Multivariate GLMs

In dose-response studies, one may come across situations where several

responses can be observed for the same patient. For example, in addition to the

standard response concerning the efficacy of a drug, it may also be of interest to

measure its tolerability, that is, any side effects experienced by the patient. Thus,

one important area that needs investigation is the choice of designs for multivariate

GLMs. Books by Fahrmeir and Tutz (2001) and McCullagh and Nelder (1989)

discuss the analysis of multivariate GLMs.

Let q denote the number of response variables. In multivariate generalized

linear models (GLMs), the q-dimensional vector of responses, y, is assumed to

follow a distribution from the exponential family. The mean response pw(x)=

[pI-(x),..., p,(x)]' at a given point x in the region of interest, R, is related to the

linear predictor rl(x) = [rll(x),..., r,(x)]' by the link function g : RV i RV,


rl(x) = Z'(x)P = g [p(x)], (2-6)


where x = (xl,..., xk)', Z(x) = egzfi(x), fi(x) is a known vector function

of x, p = (B', .. ,)' is a p.-dimencsion~al vector of un~known? pa~rame??ters, and

Pi = (Pil, ipi,)' is a vector of unknown parameters for the ith response of order

pi x 1(i = 1, .. ,q), where C pi = p.. If the inverse of g, denoted by h, exists,









where h : RV RV, then


p,(x) = h[rl(x)] = h[Z'(x)P]. (2-7)


Estimation of p is based on the method of nmaxiniun likelihood using an iterative

weighted least-squares procedure [see Fahrnicir and Tutz (2001, p. 106)]. The

variance-covariance matrix, Var(P), of P is dependent on the unknown parameter

vector p. This causes the design dependence problem in multivariate GLMs.

Details concerning the estimation of P and Var(P) can he found in Chapters 4 and



The key references for optimal designs in multivariate GLMs are Heise and

Myers (1996) and Zocchi and Atkinson (1999). Heise and Myers (1996) studied

optimal designs for hivariate logistic regression while Zocchi and Atkinson's (1999)

work was based on optimal designs for niultinontial logistic models.

2.4.1 Optimal Designs for Bivariate Logistic Models

In a drug-testing experiment the primary responses of interest are efficacy

and toxicity. By efficacy we usually refer to the desired therapeutic effect of the

drugss, and by toxicity we refer to the undesirable effects of any of the drugss.

Since the two responses come front the same patient or subject, they are assumed

to be correlated. As in the case of a single binary response, the effect of the dose(s)

on the two responses is modelled using a hivariate logistic cumulative distribution

function (cdf). Heise and Myers (1996) chose the "Gumbel model" hased on the

hivariate logistic cdf given by Gumbel (1961).

The standard Gumbel cdf is given by

1 1 (e-"-"U
Fa y (u, v) =1+,- ,-0 1 +e-"1+ e-"'' (1 +e-'')(1 +e->'U)
(2-8)

Here, 4 is the correlation parameter. Thus if yl is a binary variable for efficacy, y2

for toxicity, and .r represents dose level, then using the cdf F in (2-8) we can write










[see Heise and Myers (1996, p. 615)]


p(1 1;x)= P(Yi = 1, Y2 = 1|X = x) = F(xl, x2) = E1E2 + A;

p(1, 0; x) = P(Yi = 1, Y2 = 0|X = x) = F(xl, 00) F(xl, x2) = El E1E2 A;

p(0, 1; x) = P(Yi = 0, Y2 = 1|X = x) = F(oo, x2 1 2~l,) = E2 E1E2 A;

p(0, 0; x) = P(Yi = 0, Y2 = 0|X = x) = 1 F(x 00) F(oo, x2 1 2~l ~

= 1- El-E2 +E1E2- A


where xl = atl + Pix, x2 = c02 2ZP, 01, P1, c82, and p2 are the parameters, and El,

E2 and A are defined below as:

1 1 ~e-" -"2
El = E A=.
1 +e-"] 1 +e-zi (1 + e-"1)2 1 + -"22 2

For the bivariate logistic regression model, the likelihood takes the form [see

Heise and Myers (1996, p. 616)]


L~(0|x)= p11: )"'p1O: ) -"(01: ) ""p0O: ) }
j=1
(2-9)

where 8' = (axl, pr, 82, 2a, ~), X = (x1, *, k~)', a = number of dosage levels, mj

number of observations at the jth dose, xj = jth dose, and yl, y2 E {0, 1}.

We are interested here in jointly estimating the parameters atl, pr, c82, 2, and

~. For estimating the five parameters jointly, Heise and Myers (1996) considered

the information matrix, I(0), of the parameters and chose the xj's and mj's to

maximize some suitable scalar-valued function of this matrix. The 5 x 5 information

matrix is given in Heise and Myers (1996, p. 617, formula 4 ).

Heise and Myers (1996) developed D-optimal by maximizing |i(0)| and

Q-optimal design by minimizing the average .-i-mptotic prediction variance of an

estimate of p(1, 0; x), the probability of efficacy without toxicity at dose x, over









a desired range of doses. The optimal two and three point designs can he found

tabulated in their earlier technical report [Heise and Myers (1995)].

2.4.2 Optimal Designs for Multivariate Logistic Models

Multinomial responses frequently occur in dose-level experiments. Zocchi

and Atkinson (1999) studied the influence of gamma radiation on three disjointed

outcomes related to the emergence of house flies. The outcomes were deaths

before the pupae opened, death during emergence, and life after emergence. This

sort of experiment is in general very expensive since it uses a gamma radiation

source. Experiments therefore need to be designed to involve a minimum number

of different doses. Zocchi and Atkinson (1999) use optimal designs to efficiently

estimate the parameters of the model and appreciably reduce experimental costs.

Their focus is on designs for multinomial logistic models that belong to the class of

multivariate logistic models described by Glonek and Alc~ullagh (1995).

Let y be a response variable with q response categories and ~ri(:) be the

probability that the individual has response i at dose level r, with xi(:r) + ... +

xr,(:r) = 1. Thus the response vector, y = (yl., y *U)', follows a multinomial

distribution with parameters m and ;r(:r) where, x(:r) = [xi(:r), ..., x,(:r)]', and

CI Ui = m. Zocchi and Atkinson (1999) studied the dependency of x(:r) on dose

r through the multivariate logistic models of Glonek and Alc~ullagh (1995).

From the multivariate logistic regression models in Zocchi and Atkinson (1999,

p. 439, formula 5), we have


Z'(:r)0 = C' log [Lw r(:r), (2-10)


where, Z(x) = elifi(x), fi(x) is a known vector function of x, p = (0',,.., ',)' is

a p.-dimensional vector of unknown parameters, L and C are matrices of zeros and

ones only, and log[Lwr(:r)] is a vector representing the logarithm of each element of

the vector Lwr(:r).









The log-likelihood function of y is [see Zocchi and Atkinson (1999, p. 439)]


L,(p|x) = Cy' log w(xy)] (2-11)
j= 1

where log [;r(xj)] = [log x1(xj),. .. log x,(xj)]'. The corresponding information

matrix is given in Zocchi and Atkinson (1999, p. 439, formula 6).

Zocchi and Atkinson (1999) derived D-optimal designs for the multivariate

logistic models by maximizing log |I(P)|i. They also dealt with B li- Im D-optimal

designs and maximized Ep log |I(0)|I by assuming a prior distribution on P. It was

noted that the support points of the optimal design grew as they increased the

prior uncertainty.

2.5 Multiresponse Optimization

M~ultiresponse optimization requires finding the settings of the factors or

control variables that yield optimal, or near optimal values for the responses

under consideration simultaneously. In multiresponse experiments, the meaning

of "optimum" is sometimes unclear as there is no unique way to order the

multiresponse data. Conditions that are optimal for one response may be far

from optimal or even physically impractical for the other responses from the

experimental point of view. The problem of simultaneous optimization for linear

multiresponse models was addressed in the works done by Harrington (1965),

Derringer and Suich (1980), and K~huri and Conlon (1981).

Multiresponse optimization using the so-called generalized distance approach

was introduced by K~huri and Conlon (1981). In this procedure, all responses

depend on the same set of control variables and can be represented by polynomial

regression models of the same degree and form within the experimental region. Let

a be the number of experimental runs and q be the number of responses. Then the









model for the ith response is


Yi = Xo i + Ei, i = 1, q (2-12)


where yi is an a x 1 vector of observations on the ith response, Pi is a vector of p

unknown regression parameters, Ei is a vector of random errors associated with the

ith response, and Xo is n x p of rank p. A multivariate formulation of the models

in (2-12) can be expressed in the form


Y = XoP + E,


where Y = [y,,...,y,], p = [P1,... pV], and E = [E1,...,Ey]. The assumptions are

that the rows of E: are mutually independent, each having a zero mean vector and a

common variance-covariance matrix E. Since all response models have the same Xo

matrix, then an estimate of E is given by

Y'[I, Xo(X'oXo)-1X'o]Y


The prediction equation for the ith response is


Di(x) = z'(x)Pi, i = 1, q, (2-13)


where x = (xl,..., xk) Z (X) is a vector of the same form as a row of the

matrix Xo evaluated at the point x, Pi is the least squares estimator of pi. The

variance-covariance matrix of y (x) = [ 1(x), y(x)]' is z'(x) (X'oXo)-1z (x) An

unbiased estimate of Var[yr(x)] namely, Var[yr(x)], can be obtained by replacing E

by E.

Let ki (i = 1,...,q) be the optimum value of 4i(x) optimized individually

over the experimental region and let it = (ky,..., &,)' be the vector of all the

q optima. If these individual optima are attained at the same x, then an ideal

optimum is said to be achieved, but this rarely occurs. Instead, we try to find









conditions on x that minimize the distance between the estimated responses and

their corresponding optimum values. Let p[yr(x), k] be a distance function which

measures the distance of jr(x) from it. Several choices of p are possible. K~huri and

Conlon (1981) considered the following metric


p~f(x) k]= ((x)- A'{Vr[9x)] (9x) A)1/2(2-14)


The problem of optimization in a GLM environment is not as well developed

as in the case of linear models. In single-response GLMs, Paul and K~huri (2000)

used modified ridge analysis to carry out optimization of the mean response. Ridge

analysis is a well-known procedure for the optimization of a second-degree response

model over a spherical region of interest. K~huri and Myers (1979) introduced

a modification of this procedure by incorporating a certain constraint on the

prediction variance and called it modified ridge analysis. Instead of optimizing the

mean response directly, Paul and K~huri (2000) optimized the linear predictor.

ChI Ilpter 6 deals with optimization in multivariate GLMs. The generalized

distance approach of K~huri and Conlon (1981) is used to carry out the simultaneous

optimization of the mean responses.















CHAPTER 3
DESIGN COMPARISONS FOR UNIVARIATE GENERALIZED LINEAR
MODELS

3.1 Introduction

The choice of design for a generalized linear model (GLM) depends on the

unknown parameters of the fitted model. This poses a difficult problem since the

purpose of a design is to provide efficient estimates of the model's parameters. In

this chapter, the so-called quantile dispersion Il g/t -, (QDGs) are used to compare

and evaluate designs for GLMs. The mean-squared error of prediction (jl!SEP i) is

used to assess the quality of prediction associated with a given design. The MSEP

incorporates both the prediction variance and the prediction bias, which results

from using maximum likelihood estimates (MLEs) of the parameters of the fitted

model. For a given design, quantiles of the MSEP are obtained within a region of

interest. These quantiles depend on the unknown parameters of the model. Plots

of the maxima and minima of the quantiles, over a subset of the parameter space,

produce the QDGs. The plots provide a comprehensive assessment of the quality

of prediction afforded by a given design. They also portray the dependence of the

design on the model's parameters.

In this chapter, the application of the QDGs is demonstrated using a model

with a logarithmic link function and a Poisson-distributed response variable.

Several variants of these conditions are considered, including a square root link

in conjunction with the Poisson distribution and several other combinations.

The results indicate that the choice of the link function and/or the nature of the

response distribution can have an effect on the shape of the QDGs for a given

design.









3.2 Generalized Linear Models

Generalized linear models (GLMs) were first introduced by Nelder and

Wedderburn (1972). They can be used to fit discrete as well as continuous data

having a variety of parent distributions. Thus the traditional assumptions of

normality and homogeneous variances, usually made in an an~ ll-h- of variance (or

regression) situation, are no longer needed.

The use of GLMs requires the specification of the following three components:

1. the data set under consideration consists of independent random variables,
Y1, Y2, > Un, Such that yj has a distribution in the exponential family with
a probability mass function (or a density function for a continuous data
distribution) given by

6(yj, 0s, 4) = exp[4{yj0; b(04) + c(y>)} + d(yj,4)], j = 1,..., n, (3-1)

where b(-), c(-) and d(-, -) are known functions, 8j is a canonical parameter,
and is a dispersion parameter, possibly unknown. The mean of yj is
py = ), and its variance is

S1 d2b(04)
of=, a j= 1,..n. (3-2)

See McCullagh and Nelder (1989, pp. 28-29).

2. a linear regression function, rl, called a linear predictor, in k control variables,
X1,... Xk, of the form
rl(x) = f'(x)P, (3-3)
where f(x) is a known p-component vector-valued function of x = (xl,..., xk)'
p is a vector of p unknown parameters, and f'(x) denotes the transpose of
f(x).

3. a link function g(p) which relates rl(x) to the mean response p(x) at a point x
in a region of interest, R,
q~x) g [px)],(3-4)
where g(-) is a monotone differentiable function whose inverse is denoted by
h(-). Thus
p(x) = h[rl(x)]. (3-5)
Formula (3-5) indicates that the mean response is, in general, represented by
a nonlinear model.









For example, the response y may be binary with a probability mass function of

the form


6(y, x) =ex [y lo lg(1 x )],j 1,. .. ,(3-6)


yj = 1 or 0, and xyj is the probability that yj = 1 on the j-th trial (j = 1,..., n).

Note that this distribution belongs to the exponential family with = 1, 8j

log( ~), b(04) = log [1 + exp(04)], c(yj) = 0, and d(yj, 4) = 0, j = 1,..., n. In this

case, the mean and variance of yj are py = 'ir and of2 = xyr(1 jTy), respectively,

j= 1 ,.n.
The corresponding link function is the logit link [see McCullagh and Nelder

(1989, p. 31)] namely,

1 iT(x )iT() i :( 7
and the linear predictor is of the form


log = T(XiT X f'(x)P (3-8)

where xr(x) denotes the probability of success (that is, achieving the value 1) at x.

The mean response, xr(x), at x is then given by

exp [f' (x)P]
;'(x) = .(3-9)
1 +exp [f'(x)0]'

Model (3-9) is called the logistic regression model.

We note that the link function in (3-7) was obtained by equating rl(x) to the

canonical parameter 6. This particular link function is called the canonical link,

and its use leads to desirable statistical properties of the model, particularly in

small samples [see McCullagh and Nelder (1989, p. 32)].

3.2.1 Estimation of the Mean Response

Estimation of p in (3-3) is based on the method of maximum likelihood, which

is carried out using an iterative weighted least-squares procedure. An estimate of





24


rl(x) in (3-3) is then given by
q~x)= f'x)0,(3-10)









3.2.2 The Prediction Bias

The maximum likelihood estimate of P is, in general, a biased estimate of

p. Cordeiro and McCullagh (1991) developed an expression for the bias of P for
GLMs using the tensor methodology. Cadigan (1994) presented a method for the

computation of the bias which did not require using the tensor methodology. His

expression for the bias is approximated by

Bias(P) '-(X'WX)-1X'ZdF1,, (3-17)

where X and W are the same as in Section 3.2.1, Zd = diag(zzi,...,zon,), where zy,

is the jth diagonal element of Z = X(X'WX)-1X', F = diag(fll,..., fe,), where

for j = 1, .. ., n, fjj is given by


fy, = -ljrj Id- ,jd1 = 1 (3-18)

vy~ = nd 1, is a column victor of a ones. Inl (3-18), dennotes the~ second
derivative of p(x) with respect to rl(x) evaluated at xj (j = 1,..., n). Cordeiro

and McCullagh (1991, formula 6.3) showed that the bias of P can also be derived

approximately through a simple weighted linear regression computation of the form

Bias(0) (X'WX)-1X'W(, (3-19)

where ( -3W-1ZdF1,, and W pt1 is the role of the weight matrix.

Using (3-19), the bias of T(x) in (3-10) is approximated by

Bias [4(x)] f'(x)(X'WX)-'X'W(. (3-20)

Furthermore the prediction bias, which is the bias of [L(x) in (3-11), is

approximated by

dp-(x) 1 d21- X)
B; I- [iil(x)] Bias[4(x)] + Var[i(x)] (3-21)
dq(x) 2 dq2 X)'









wher is thle second derivative of p(x) with respect to rl(x) evaluated at
a point x in a region R, and Var [i(x)] is approximated by formula (3-15) [see
Robinson and K~huri (2003)].
3.2.3 The Mean-Squared Error of Prediction

The mean-squared error of p(x), also known as the mean-squared error of

prediction (j\!SEPl) is given by

MSE~l~x) = Ep~x)- p~)]23_22)

The right-hand side of (3-22) can be partitioned into two components, namely,


MSE[p(x)] = Var[p(x)] + {BS I- [i(x)]}2, (3-23)

since Bi I-[iI(x)] = E[p(x)] p-(x). By combining formulas (3-16) and (3-21) we
obtain


MSE [ilx)] f'(~x~r x) [X'WX]- flx) +

Bias [(x)] Var1 [i(x)] ,()~ (3-24)
dq(x) 2 dq2 X)I

where Var[4(x)] and Bias[4(x)] are approximated by formulas (3-15) and (3-20),

respectively.
3.3 Choice of Design

An important criterion for the choice of design for GLMs is the minimization

of the MSEP, given in formula (3-24), over the region R. One ill I r~~ problem in

doing so is the dependence of the MSEP on the unknown parameter vector P in
model (3-3), since rl(x), and hence p(x), depends on P. This is analogous to the

design dependence problem mentioned earlier in Section 2.1 with regard to GLMs.

3.4 Quantile Dispersion Graphs
The MSEP is defined over a region of interest, R, which is a subset of the

k-dimensional Euclidean space. Its value at a point x in R depends on the









chosen design, denoted by D, and the vector of unknown parameters, P. Let

us therefore denote such a value by -rD(x, P). We assume that P belongs to a

specified parameter space denoted by C.

In order to study the effect of the design D on MSEP, we examine the values

of -rD X, p) Within the region R for a given p in C. In particular, we consider its

values on concentric surfaces, denoted by R,,, which are obtained by reducing

the boundary of R using a shrinkage factor, v. For a given design D and a given

p in C, the values of -rD(x, p), for all x on R,,, form a data set. Let QD~p v)

denote the pth quantile of such a data set, O < p < 1. These quantiles provide a

description of the distribution of -rD X, p) foT ValueS of x on R,,. By varying v we

can cover the entire region R. Small values of -rD(x, P) throughout R are obviously

desirable .

Recall that any design criterion for GLMs suffers from the problem of the

dependence of the design on P. To address this problem, we can vary P within the

set C and then examine the corresponding values of QD p, p, V) foT a giVen design

D and given p and v. An "initial" data set that may be available on the responses

can be used to construct the parameter space C. The dependence of the quantiles

of TD(x, P) on P, for x on R,,, can be investigated by computing QD p, p, V) foT

several values of p that form a grid, C, inside C. Subsequently, the minimum and

maximum values of QD(p, p, V) over the values of p in C are obtained. This results

in the following extrema of QD(p, p, V) for each v and a given p:


Qgin(p, v) = min{QD p, p, v)} (3-25)


Qgax(p, v) =max{QD(p, P, v)}. (3-26)

Plotting these values against p results in the quantile dispersion graphs of the

mean-squared error of prediction over the region R,,. By repeating the same

process for several selected values of v, we obtain plots that portray the prediction










capability associated with the design D throughout the region R. Such plots can be

constructed for each of several candidate designs for the model. It should be noted

that for a given v, a desirable feature of a design is to have close and small values

of Qgin and Qgax over the range of p (0 < p < 1). The smallness of Q~ax indicates

small MSEP values on R,,, and the closeness of Qgin and Qgax indicates robustness

to changes in the values of P that is induced by the design D.

There are several advantages to this approach, namely,

1. the performance of a design can be evaluated throughout the region R. Other
design criteria base the evaluation of a design on a single measure, such as
D-efficiency, but do not consider the quality of prediction inside R.

2. unlike other variance-based design criteria, such as D-optimality or G-optimality,
estimation bias is taken into account in the evaluation of a given design.

3. the dependence of a design on the model's unknown parameters is clearly
depicted by the QDGs throughout the region R.

4. plotting the quantiles of the MSEP against p permits the consideration of
models with several control variables, instead of just one, as is usually the case
with other design criteria, such as D-optimality.

3.5 Examples

The type of data we consider here is of counts. Such data appear in many

applications, for example, the number of certain events within a fixed period of

time (insurance claims, accidents, deaths, births, etc.), and the frequencies in

each cell of a contingency table. There are several procedures for analyzing count

data. In many situations, a Poisson distribution is assumed. Sometimes, count

data show more variability than this distribution allows. We can then introduce

an additional overdispersion parameter to account for the extra variation. Quite

often, however, the use of an overdispersion parameter is not of much help, and

some other distributions, like the negative binomial or the gamma, may be more

appropriate [see Fahrmeir and Tutz (2001, p. 36)].









3.5.1 Poisson Response with a Logarithmic Link

Here, we consider a situation where the response follows a Poisson distribution.

Hence, the probability mass function of yj is given by

5(7, A) =exp[ {(yj log Ag Ay) log(yy!},j = 1,, ... (-7


By comparing (3-27) with (3-1) we note that the distribution of yj belongs to the

exponential family with 04 = log Ay, b(04) = Ay, c(yj) = log(yj!), and d(yj, 4) = 0.

The mean and variance of yj are E(yj) = As and Var(yj) = j = 1,...,n. The

dispersion parameter is unknown, but can be estimated or taken to be equal to

1. Nelder and Wedderburn (1972) -11---- -1. I1 taking 4 = (n p)/X2, Where X2 is

a Pearson-type statistic and p is the number of elements in the parameter vector P

[see Agresti (2002, p. 150)].
Let x be a vector of design settings in the region of interest, R. At the jth

experimental run, x = xj, the corresponding response value is yj (j = 1,..., n).

The mean response at x is denoted by A(x). The canonical link function is the

logarithmic (log) link [see McCullagh and Nelder (1989, p. 30)]


rl(x) = log A(x) = f'(x)P. (3-28)

Hence ,

^z(x) = A(x) = exp~lf'(x)0], (3-29)

where p is the maximum likelihood estimator (\!1.1-1) of P.

Using (3-16) and (3-21) it can be shown [see Appendix A.1] that the

mean-squared error of prediction (j!sl:iP) at x is approximated by

(X'WX)-]
MSE[A(x)] [A(x)] 2 fl(x) f(x) (3-30)

1 (X'WX)-]
+f' '(x) Bias(p)A(x) + f' (x) f (x)A(x)}2,
2









where

Bias(P) (X'WX)-1X'W(,

W = diag(wl,..., In~,,) with my = Ay and ( is an a x 1 vector whose jth element

is ~-, w-here zyi is the jt~h diagonal element, of Z =X(X'WX)-1X', j = 1...,n.
Here, As is given by

As = exp[f'(xj)P], j = 1,..., n.

It should be noted that the MSEP at a point x in R depends on the unknown

parameter, p. Let us therefore denote the value of M~SE[A(x)] by D X, P>.
3.5.2 Poisson Response with a Square Root Link

Here, the relationship between the mean response and the linear predictor is


rl(x) ==f'(x)0. (3 31)

Using the MLE of P we get the estimate of the mean response to be


p(x) = A(x) = [f'(x)0]2. (3-32)

Equations (3-16) and (3-21) show [see Appendix A.2] that the mean-squared error

of prediction ( \! SE Pi) at x is approximated by

(X'WX) -
MS[~x] 4A(x)f'(x) f(x) (3-33)

(X'WX)-1
+ {2f'(x) Bias(#) +Xx f'(x) f(x)}2l

where Bias(P) is approximated by formula (3-19), W = diag(wl,..., In~,,) with

my = 4, and the jth element of ( is ,- where zy, is the jth diagonal element
of Z = X(X'WX)-1X', j = 1,..., n. It should be noted that the expression of

MSEr~lii(x)] changes with the choice of the link function. It also depends on the
unknown p.










3.5.3 A Numerical Example

In this section, we present an example that illustrates the intplenientation of

the QDGs for the log-linear model (that is, using a logarithmic link function) with

a Poisson distribution. We also assess the robustness of the QDGs to the form of

the link function and the distribution of the response variable.

The data considered in this example were taken front Piegorsch et al. (1988).

In a biomedical study of the ininuno-activating ability of two agents, TNF (tumor

necrosis factor) and IFN (interferon), to induce cell differentiation, the number of

cells that exhibited differentiation after exposure to TNF and/or IFN was recorded

using a 4 x 4 factorial design. At each of the 16 dose combinations of TNF/IFN,

200 cells were examined. The number 7 of cells differentiating in one trial, and the

corresponding dose levels of the two factors are given in Table :31.

Table :31. Experimental design and response values.


Y X1 X2 xl *

11 0 0 -1 -1
18 0 4 -1 -0.92
20 0 20 -1 -0.6
:39 0 100 -1 1
22 1 0 -0.98 -1
:38 1 4 -0.98 -0.92
52 1 20 -0.98 -0.6
69 1 100 -0.98 1
:31 10 0 -0.8 -1
68 10 4 -0.8 -0.92
69 10 20 -0.8 -0.6
128 10 100 -0.8 1
102 100 0 1 -1
171 100 4 1 -0.92
180 100 20 1 -0.6
19:3 100 100 1 1










Model (3-28) was fitted to the data set in Table 3-1 using the following model

for the linear predictor

rl(x) = So + #1x + 02 2 11 2 2 12 1 2- (3 3


Here, xl and x2 denote the coded values xi = (Xi 50)/50, with X1 and X2 being

the actual values of agents TNF and IFN, respectively. The dispersion parameter

was taken to be 1.

The experimental region, R, of the agent combinations in the coded space is

a square with -1 I xl < 1, -1 I x2 < 1. The parameter maximum likelihood

estimates and their standard errors for model (3-34) are shown in Table 3-2.

Table 3-2. Parameter estimates and model analysis using log link.


Parameter Estimate Std. error P-value

Po 6.7330 0.2316 < 0.0001
P1 0.7939 0.0386 < 0.0001
2~ 0.3668 0.0364 < 0.0001
P11 -1.9718 0.2344 < 0.0001
22a -0.5075 0.1153 < 0.0001
12a -0.1522 0.0350 < 0.0001

Scaled Deviance = 54.1883, DF= 10.


We refer to the design in Table 3-1 as D1. We also consider another design,

D2, namely, a face-centered cube with replicated center and face-center points given

in Table 3-3. Designs D1 and D2 have different design settings.

For each of the two designs, we consider the distribution of -rD(x, p) on each

of the several concentric squares, R,, which are obtained by a reduction of the

boundary of R using a shrinkage factor v, 0.5 < v < 1. Thus, R, is determined by

the inequalities


ai + (1 v) (bi ai) < xi < bi (1 v) (bi ai), i = 1, 2,









Table 3-3. Design D2*


X1 X2 1i 2

0 0 -1 -1
0 100 -1 1
100 100 1 1
100 0 1 -1
50 0 0 -1
100 50 1 0
50 100 0 1
0 50 -1 0
50 0 0 -1
100 50 1 0
50 100 0 1
0 50 -1 0
50 50 0 0
50 50 0 0
50 50 0 0
50 50 0 0


where ai and bi are the bounds on xi in R (i=1,2), that is, al = -1, a2 = -1, bl

1, b2 = 1 (SeO F1gure 3-1).

To investigate the dependence of -rD(x, P) on P, a parameter space was

established. For each parameter, a range consisting of the parameter's point

estimate plus/minus four standard errors (from Table 3-2) was considered. A

subset, C, of this parameter space was obtained by selecting three points within

each parameter range, namely, the point estimate and the two end points. Thus,

the number of points in C is 36 = 729. The same parameter space was used for the

two designs.

For each design and a selected value of P in C, quantiles of the distribution

of -rD(x, p) are obtained for x e R,,, where v is one of several values chosen from

the interval (0.5, 1]. The number of points chosen on each R,, was 1000, consisting

of 250 points on each side. The quantiles are calculated for p = 0(0.05)1. The

procedure is repeated for other values of P in the subset C. Then, QB""(p, v) and










Qg'"(p, v) were calculated using formulas (3-25) and (3-26). PROC GENMOD

in SAS [SAS Institute Inc. (2000)] was used to calculate the parameter estimates.

All other computations were done using the R software (Version 1.6). For better

representation, we take the natural logarithm (log) of the quantiles of the MSEP

when plotting the QDGs.

To compare the two designs, we examine the corresponding QDGs shown in

Figure 3-2. We note that the maximum quantiles of D1 are above those of D2 foT

most values of p, indicating that D2 has better prediction capability than D1. The

values of QB""(p, v) and Qg'"(p, v) for both designs are at some distance from each

other, indicating sensitivity to the parameter values. As v decreases (that is, as

we get closer to the center of R), there is a slight decrease in the distance between

QB""(p, v) and Qg'"(p, v) for D2, While the quantiles of D1 do not change much.

Thus, D2 becomes slightly more robust to the parameter values near the design

center, but the quantiles for D1 remain the same.

To assess the sensitivity of the QDGs to the form of the link function, we use

the same data and assume the same Poisson distribution, as before, but employ a

different link, namely, the square root.

We start with fitting model (3-31) to the data set in Table 3-1 using model

(3-34) for the linear predictor. The dispersion parameter 4 was taken to be 1. The

same two designs D1 and D2 arT used as before. Our region of interest, R, and

the several concentric squares, R,,, remain unchanged. The parameter estimates

and their standard errors in Table 3-4 using the data set based on design D1,

however, change. The change occurs due to the different link function used. The

corresponding quantiles were computed in a similar fashion.

The QDGs for both designs are compared using the two different links.

Figure 3-3 gives a comparison of the QDGs for D1 using the Poisson distribution

with a log link and a square root link, and Figure 3-4 shows the same for D2-










Table :34. Parameter estimates and model analysis using square root link.


Parameter Estimate Std. error P-value

/So 17.7924 0.92:31 < 0.0001
/31 :3.51:37 0.1722 < 0.0001
732 1.541:3 0.1807 < 0.0001
7311 -6.526:3 0.8782 < 0.0001
7322 -2.0:301 0.50:35 < 0.0001
7312 -0.2:352 0.1852 0.2042

Scaled Deviance = 58.1596; DF = 10.


The canonical link for the Poisson distribution is the log link. Misspecifying

the link as square root changes the robustness of the designs to the parameter

values. The distance between the maximum and minimum quantiles for design

D1 increases slightly as we change the link from log to square root. This indicates

a slight increase in sensitivity to the parameter values for the square root link.

The maximum and minimum quantiles for D2 for the square root link are further

apart from each other than the log link case. Thus, D2 l0Ses its robustness to the

parameter values as we alter the link.

Next we try to study the robustness of the QDGs to different distributions.

We use four distributions, namely, Poisson distribution with = 1, Poisson

distribution with 4 = 0.1824 (~ is estimated by using PROC GENMOD in

SAS [options: scale = Pearson]), negative binomial distribution, and gamma

distribution, each with log link. The model for the linear predictor remains the

same and the experimental region R also remains unchanged. The parameter

estimates and their respective standard errors, however, change.

Figure :35 gives the comparison of the QDGs with these four distributions.

The distance between the maximum and the minimum quantiles for both designs

increases a little for the Poisson distribution with an estimated 4, indicating

a slight loss in robustness to the parameter values. For the negative binomial






36;



distribution, the designs become more sensitive to the parameters. In the case of

the ganina distribution, the quantile plots for both D1 and D2 are Very clOSe to

one another. We also notice that the distance between the nmaxiniun and nxininiun

is smaller for the ganina distribution by comparison to the estimated Poisson

or the negative hinomial distributions. Both designs show similar prediction

capabilities for all four distributions. Overall, the shape of the QDGs has been

affected by the distribution, but the superiority of D2 over D1, in this example,

appears to be unaffected by the distribution.














N O











-1.0 -0.5 0.0 0.5 1.0
x1


Figure 3-1. Concentric squares within the region R.





























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CHAPTER 4
BIAS IN MULTIVARIATE GENERALIZED LINEAR MODELS

4.1 Introduction

It is well known that maximum likelihood estimates (AILEs) are biased when

the sample size, n, is small. For a likelihood function with a single parameter to

be estimated, the bias to order ( of the MLEs was given by Bartlett (1953). A

generalization of the bias formula for a likelihood function with several parameters

was established by Cox and Snell (1968).

Using Cox and Snell's (1968) results, Cordeiro and Alc~ullagh (1991)

developed a first-order approximation of the bias in the special case of generalized

linear models (GLMs). In this chapter, we provide a further extension of the bias

formula that applies to multivariate GLMs where several response variables are

under consideration in a given experimental situation.

An expression for the bias of MLEs of the parameters in a multivariate GLM

setup is very important for many reasons. One primary reason is the need to

obtain an expression for the so-called mean-squared error of prediction (j !SEP: i)

that can he utilized as a criterion for the comparison of designs for multivariate

GLMs [see blukhopadhyay and K~huri (2005)]. The MSEP incorporates both

the variance and hias associated with estimating the mean responses under

consideration. Thus the MSEP can provide useful information concerning the

quality of estimation in a multivariate GLM situation.

4.2 Bias in a General Multivariate Setting

We begin by defining a situation where n q-dimensional observation vectors

yl, -, y,z have been collected. These vectors are assumed to be independent,

but the responses at the j-th run (i.e. the elements of yj, j = 1,..., n) can









be correlated. Let P be a p. x 1 vector of unknown parameters such that P =

(pl,. ., P3. )'. The likelihood and log-likelihood functions are, respectively,

L = b(y,),ad(4-1)
j= 1

1 = log6;(yy,0). (4-2)
j= 1

where by is the density function of yj. Cordeiro and McCullagh (1991, p. 634,

formula 4.1) give the following expression for the bias, correct up to order ~, of a

single element of the MLE, P, ;?i, Pa(a = 1, p.) in a general multiparameter
family of distributions:
P- p- p-
Bias(fia) -Cln I'"[I(j,,,r + Krst)], (4-3)
r=1 s=1 t=1

where I"" is the (u, v)th (u, v = 1,...,p) element of the inverse of Fisher

information matrix,

I(p) =~pp2 E

Jr, s I = p E pap a2 1 ;nd K rst, = F E p ,pa~3 r, ; t 1 p..

Stacking up these expressions, we get the bias correct up to order of 4 as follows:



Bias[P] = 1 i '-I 44



In later sections we shall develop the forms of Krst and Je,rs (r, S, t = 1, .., p.) for a

multivariate GLM. For simplicity, we shall refer to 6j (yj, P) in formulae (4-1) and

(4-2) as by (yj).
4.3 Multivariate GLM

In this section, we define the multivariate GLM setup and find an expression

for the log likelihood function. GLMs for independent q-dimensional observations









yl,...,y, and k control variables xl,..., xk are characterized by the following
structure:

1. The yj's are independent with densities from the exponential family

si (yi) =exp [ {y'.0 b(04) } + c(ys, )], j =1... n, (4 5)

where b(-), c(-) are known scalar functions and 4 is a dispersion parameter,
possibly unknown. The vector Os, j = 1,..., n, consists of q elements. The
mean vector and variance-covariance matrix of yj are

86(0j) 1 82b(04) 1 8#; .j
rw '0 = -- respectively.
80, aeae (848' ae8''


2. A linear predictor at the jth run given by

rl = Z'(xj)P, j = 1,. ., n, (4-6)

where xj (xjl ,...,xjk)', Z(xj) sy z~fi(xj), fi(x) is a known vector
function of x, p (P',,.., 0',)' is a p.-dimensional vector of unknown
parameters, and pi (Pil, ipiy)' is a vector of unknown parameters
for the i-th response, which is of order pi x 1 (i 1 .. ,q), where CL, pi = p..
For simplicity, we shall drop the use of two subscripts and instead refer to all
the elements of p as (P1,... ,pp. )', where the first pi elements come from z,,
the next p2 elements from p2, and so on.

3. The linear predictor rlj is related to the mean 4w through the function
h : RV H RV,
rwj = h(rl ), j = 1, n. (4-7)
If the inverse of h, denoted by g, exists, where g : RV H RV, then

g(p j) = rl = Z'(xj)P, j = 1, ..,n, (4-8)

and g is called the link function.

Using (4-5), the log likelihood function is


1= [ {ly'.0, -(04)} +c(yy, )].l (4 9)
j= 1

We shall use the log-likelihood function, 1, to derive the expressions for Krst and


Jt,rs (r, S, t = 1, .. ,p.) in formula (4-3).









Estimation of the Mean

The maximum likelihood estimate (j!1.1 ) of P can be obtained by using an

iterated weighted least-squares procedure. An estimate of rl (j = 1,..., n), the

linear predictor at the j-th run, is then given by


7 = Z'(xj)P, j = 1, n, (4-10)


and the corresponding estimate of the mean, 4wj, is


[sj = h[Z'(xj)P], j = 1,..., n, (4-11)

where p is the MLE of P. The variance-covariance matrix of P is approximately

given by [see Fahrmeir and Tutz (2001, p. 106)],


Var(P) '- (X'WX)- (4-12)

where X = [Z(xl),...,Z(x,~)]' and W is a block-diagonal matrix of the form,
W ~ ~ ~ ~ ~ ~~87 = iqW,.,,,Wy=( E ,...,~n. Here, s?'is the

first-order partial derivative matrix of rl taken with respect to pw' (j = 1, .. ., n)

evaluated at the jth run.

4.4 Bias in a Multivariate GLM Setting

This section is concerned with the derivation of Krst and Je,rs (r, S, t=

1,...,p.) that appeared in formula (4-3) for a multivariate GLM. Since this

derivation is quite involved and lengthy, we shall break it up using two lemmas and

several appendices. For this purpose, expressions for j and ~ (r, a ,.. .

will be determined, as both are needed to develop Krst and Je,rs.

Starting with ,- we use th~e chain rule to write it a~s

r = 1. p..(4-13)









Differentiating the log-likelihood I with respect to O', (j


1,. ., n), we get


,ei~ ~ (y '.
" j= 1


db(8 )
s'.), sinc


Substituting this derivative in (4-13) give~s the following form of ~-(r


1, p.):


j=1


(4-14)


The next step is to differe~ntiate wjcith respect to s (8 ;

~(intermnediate steps are given in Appendix B.1),


1, .., p.) to obtain


82


n = f f ((x ) a ap


\f(x )


j=1




~ i(y'.
j=1


(4-15)


We now have the necessary tools to proceed with the development of Krst and Je,rs.


Lemma 4.4.1 gives details of the derivation of Krst.


f ( xj(xj ) aa~
80 dejiirw
8p' 8 '.
d &,dl
f"(x


f ((xj ) aap

f"(xy


f((xy)dp

3a V-U3 d~w iCI, .\dpf" (xj)-uP

U~rf (xl)
p'.)

f"(xy







Lemma 4.4.1. For r, s and t = 1, .. ,p.,


f((xy)l
iap~apa ap vu


"


sode dSo, dl


Krst


anrl aC ~a~i an7:


a an
as an


E [il ( ~3)1 wei firstl take th~e partial derivative of ) i,


Proof. Since Krt


with respect to pt (t = 1, .., p.) and then find expectation of the resulting quantity.
This gives,


f' (xj>X)aiJap j
a~w "e an as a
E =


Krst


(4-16)



(4-17)


a as


(y' i
j=1


p'.)


d~wjdrlj


an~~ a ao, ap


f: (xjf (X ) p"at


l'f: (xjaap )


ff(xj) '

ap;j as
+rjdw


ap; ad as So
x[ d~dj\ d


f (xjf(xj ) piPaa,


f((xy
anjif (jf X) d~P
8rl
ffxy


j=1


I~~~f:~ (4-18)j a~p~p~p


+ E l a a





47


Now, each component in (4-16)-(4-18) is developed further as follows: The
expression (4-16) can be expanded as


j=1


apd~wdrli /


d~wdej d~wjdrlj


Expression (4-16)


j=1


ii~~wj~(~(~38, \


If((xj)\" dPd~p
f"(xy)


i~ (~)


It is shown in Appendix B.2 that expression (4-17) can be simplified further as


12

j=1


apd~wi!


ii~~w'
drlj


d~wj
drl~


Expression (4-17)


Expression (4-18) can be reduced to (see Appendix B.3)


12

j=1


f((xy)
8 p'
f"(xy) d


Expression (4-18)


By adding the right-hand sides of expressions (4-19)-(4-21) we get the desired
expression for Krst.
The derivation of Je~,r is given in Lemma 4.4.2. The expressions in (4-14) and


(4-15) are used to construct the proof of this lemma.


\f "(x >X) 4d~~


Sf (xjf (X) ddP


f((xy)

f"( p)


f((xy

iirj f:(Xfcxjja)aP


(4-19)


i ljf" (xj) dpd~p


(4-20)


(4-21)





48


Lemma 4.4.2. For r, s and t = 1, .. ,p.,


12

j=1


3P, \ i3~ug / j i377:~(~>> d~wj


~~,,


E [ ~. Hence, to obtain an expression for


Proof. Recall that, 4t,rs


Jt,r,! we first multiply ,i~ using (4 14), with ~- in (4 15), and then take thet
expectation of the product.
Carrying out the above calculations results in the following three quantities
whose sum is 4t,rs: The first term of t,rs is


i dej ii~rwj
~w>


d~wj dej d~wj
drlj d~wj drlj


which can be simplified further as (see Appendix B.4)

Expression (4-22) = 0,


(4-23)


The second term of 4t,rs is


E (y'.


-e p'.)


(y'.- /- )

(4-24)


f'(xy)dp
f,, a ,


a~ ~~~f:(x j)(jaiJap


f (Xj(xj) apiPatp

f"(xy)


f(( p)


f"(xy)


(4-22)


:[Ciy
j=1


bf: (xj>X) 4d~p









This can be reduced to (see Appendix B.5)



Expression (4-24) j= aap

\ff(xj) f"(xy
(4-25)

Finally, the third term of 4t,rs is

f:((x)a f (x,) p

ji=1 jru 1:.=1 IU

(4-26)
which can be written as (details are given in Appendix B.6)



Expression (4-26) j= apat

f"(x>)- f"(x )"
(4-27)

Adding up the terms on the right-hand sides of equations (4-23), (4-25), and

(4-27) gives the desired expression for Jr,rs
4.5 Final Expression for the Bias

Having obtained the final expressions for Krst and At,,s from Lemmas 4.4.1
and 4.4.2, respectively, we can now use these expressions in formula (4-4) to get an

approximate formula for Bias(P) in a multivariate GLM situation. As in Section
4.2, we begin with the bias of a single element of P using formula (4-3),

Bias(fia ) '- i"[ts() r + Kr)] a= ]1,...,p.1, (4-28)
r,s,t=1








recall that I"" is the (u, v)th (u, v


1,...,p.) element of the inverse of Fisher


information matrix. Using the results of Lemmas 4.4.1 and 4.4.2 in formula (4-28)
gives (details are given in Appendix B.7),


I'" ~Its
r,s,t=1


j=1


d~wdej d~wjdrlj


Bias(P,)


By ii~~~f8, 8


dBwd l


aBu. 80i 8~ Buy


" C
j= 1


8~ 80 8#,// 71(~)dw


iiBwy 80,


8~~) "By


(4-29)


To simplify the notation, let us denote by Ajit and B s(j


1,. ., n) the following


expressions:


Ag = (d 8i~w e By) 80 8#,

Byu 8
B- =io (


By~i~l 8p 80w f~ 8 rjdl


iBy) 80, 8 8 #,\
+


Byu5ei 8 8 8#,i
+7j3~ Iii 7:/I


/80, \


8#,-jrl


Formula (4-29) can then be written as


r,s,t 1


f(( p)
Isr [lt"f ( j(jaiJap


f((xy)dp


I. f' (xj() )piPa~p


if:(xjf (X) ddp
By j
fl{


80l,
f((xy


/ q(j JPd~p


f: (xi ) O' ~ f ((x )~p


f" (xi )OP f(xi )P


















get Bias(P) correct up to order as follows:


j=l s t~


By Il


4.6 Example: Bivariate Binary Distribution

Consider a bivariate binary response situation in which mj subjects, or

experimental units, are tested at the jth run (jth level of x), j = 1,..., n. The
measurement taken from the wth subject at the jth run for response 1 is termed

yjwl (j = 1,..., n; w = 1,..., mj; I = 1, 2). The responses yjst are 1 or 0 depending
on whether a particular outcome, labeled as "success" is observed or not. Thus we

define yjw = (yjwl, yjw2)1 aS the vector of bivariate responses from the wth subject
at the jth experimental run. If yj, for w = 1, .. m are independent then the

p.m.f. (probability mass function) of yy y ym) sgvnb




where the four cell probabilities ;rr = (xy j, Kj2 ij3 iTj4)' for the j-th level of x, can
be expressed as xyzl = P(yjwl = 1,yjw2 1,Tj2 = fwjl = w2j, = 0), K~j3


f((xy)a f: (x ) f((x y)~




(4-30)
After obtaining the biases of each element of from (4-30), we stack them up to


f"(x )




fl(x )


f" (xjf () rqpbP


f(( )


f: (xjy~ X) ~qpb









(4-31)









Phyjst = 0, yjw2 i rj4 = #jwl = 0, yjw2 = 0) (these probabilities are the same
for all w = 1,. ., mj). Note that xyir + iTj2 + Tj3 + Tj4 *) = E *)

Following a sell_t----- al by Cox (1972, p. 115) and McCullagh and Nelder (1989,

p. 226), we can write (4-32) as


i= 1

where zyr L=l my-1jwl9)w2, xj2 L, 1 jwll 9)w2 j3 Lw 1 #Wl 9)W2

and mj ti1, zys t= E l(1 y w)(1 yjw2). We note from the form of the p.m.f

in (4-33) [see Casella and Berger (2002, p. 180) and Fahrmeir and Tutz (2001, p.

70, formula 3.12)] that the responses zj = (zy I._, zy3)l 011oW the multinomial

distribution with parameters mj and Trr = (jTy j2) ij3)1 E*)n

Let us write (4-33) as


fij(zi) oc exp [zjl log ryit7._.log.Tj2zi j31g ~j3 +mj :jij lOgl j jij 4-4)


Compnlaring (434) withr (4-5) we: nlote that = 1_ = log _E (i =
1, 2, 3), b(04) = mi log(1 xS_,7i), anld c(y., i) = 0 The mean of zyi is

r (-j1, I-j2 Ij3) Where pplj = mjyst, ii._. = myj2ja Ij3 mj Tj3 and the
variance-covariance matrix is Ey, where




j~~~~~~ ;j-j 1j 2 j 1, (4-35)
-iT j1 -iT j2 iT iTj3)

Our next objective is to model the relationship between pyj and the linear

predictor rlj = (r941, rlj2, rj3)1 208t1101 (4-8) gives


8(Vj) = rlj = Z'(xj)P, j = 1,...~ ,n,









where, xj = (xjl,..., Zjk)' Z'(xj) = etlff(xj), fi(x) is a known vector function,

and p = [P', P, p'231 iS a p.-dimensional vector of unknown parameters. The

P 's, (i = 1, 2, 3) correspond to the three responses and are of order pi x 1, where


The corresponding link function used here is [see Agresti (2002, pp. 267-274)

and Fahrmeir and Tutz (2001, p. 73)]


( 1-C l=1 fjl / -,2,


(4-36)


which can be expressed as


: 1, 2, 3). An estimate of rl is given by


(4-37)


Note that 8ji = rlys (i


and the corresponding estimate of the mean, pj is


(4-38)


where p is the MLE of P.

4.6.1 Bias for a Bivariate Binary Distribution

In this section, we determine the different components of Bias(P), as given in

formula (4-31), for a bivariate binary distribution.

To fnd ,we take the inverse of ~. This is a well-known property of

Jacobian matrices. Recall that Ey =C~ .8 Hec, =E- which can be

obti:dnedl frrom (435). Thle ne:xt stecp is t.o search for: anl explresio: n for i-. From

the link function defined earlier, we know that 8j = rlj (j = 1,..., n), thus
=C" uE hence = Ey. Thl~e only unklnownl comrponlents nlow

left, in the bias formulal in (4-31) are ji ) and. -( s) (dl = ,...,p.), the


[L = h[Z'(xi)0],









first-order partial derivatives of a and "'taken with respect, to ipel, respectively.

Starting with -(a~j ) each element, of a~jis different~iatedd with respect, to

pa~ (d = 1 ,p.). Suppose that thle the (i, k)th elementn of is deinoteld by Cjik;
we can then proceed as follows:

8iCjik r ik jO
ap ~w r p (4-39)

A similar technique is used to compute -( )( 1 .,p)

To complete our ta~sk, we put1 the expressions for j, j~, -( dji), and

S( ) (dl = 1,. .) in form."ular (431). This gives anr approximately expl~ression forI
Bias(P) for the bivariate binary distribution.
4.6.2 Numerical Example

Ashford and Sowden (1970) published a study concerning two respiratory

ailments of working coal miners who were classified as smokers without radiologfical

evidence of pneumoconiosis, aged between 20 and 64 (inclusive) at the time of their

examination. Each respondent was classified according to whether or not he is

suffering from breathlessness (B) or wheeze (W). Each of the response variables has

two levels and all four combinations are possible. One aim of the investigation was

to study how the associated probabilities concerning breathlessness and wheeze are

related to age. The design and resulting data are presented in Table 4-1.

The fitted model is (4-6) with


rmj = P1 + P2 j, rj2 = 3 + 4 j, and r9j3 =5 + ,n i j = 1, 9. (4-40)

Here, x denotes the coded value x = (age 42)/5, with age representing the

midpoint of each age group. The link functions are the same as described in

equation (4-36).
Estimates of the elements of p = (P1, P2, P6)' and the corresponding

standard errors, obtained by using PROC LOGISTIC in SAS [SAS Institute Inc.










Table 4-1. Experimental design and response values.


Age-group
in years

20-24
25-29
30-34
35-39
40-44
45-49
50-54
55-59
60-64
Total


Breathlessness
Wheeze No wheeze


No breathlessness
Wheeze No wheeze


Total


1952
1791
2113
2783
2274
2393
2090
1750
1136
18282


9
23
54
121
169
26;9
404
406
372
1827


95
105
177
257
273
324
245
225
132
1833


1841
16354
1863
2357
1778
1712
1324
967
526
14022


(2000)], are shown in Table 4-2. The estimated mean response values at the

various experimental runs are obtained using the value of P (from Table 4-2) in

formula (4-38). These values, which are di;11 pli a in Table 4-3, are then used in

formula (4-31) to obtain an estimate of the bias vector, which is


Bias(P) = (-0.0008, 0.0003, -0.0023, 0.0005, -0.0005, 0.0001)'.


Table 4-2. Maximum likelihood estimates and standard errors.


Parameter

P1

P3
P4
s
P6


Estimate Std. error P-value


-2.4273
0.5498
-3.4778
0.5154
-2.0090
0.2006


0.0349
0.0141
0.0567
0.0224
0.0253
0.0112


.0001
.0001
.0001
.0001
.0001
.0001


Scaled deviance= 41.4646, DF= 21.






56


Table 4-3. Estimated mean response values.


Age-group
mn years


20-24
25-29
30-34
35-39
40-44
45-49
50-54
55-59
60-64


17. 7941
27.6945
54.9577
120.2306
160.1702
267.4753
356.9859
434.7472
386.8044


7.1418
10.7396
20.5912
43.5240
56.0216
90.3894
116.5588
137.1485
117.8979


119.9571
167.8823
259.0217
243.3589
286.6114
269.7771
231.7049
145.3899















CHAPTER 5
COMPARISON OF DESIGNS FOR MULTIVARIATE GENERALIZED LINEAR
MODELS

5.1 Introduction

In this chapter, we present a technique based on using quantile dispersion

'I,.rlph' (QDGs) for the comparison of designs for multivariate generalized linear

models (GLMs). Such models are considered whenever several response variables

can be measured for each setting of a group of control variables, and the response

variables are adequately represented by GLMs. The mean-squared error of

prediction (jl!SEP i) matrix is used to assess the quality of prediction associated

with a given design. The MSEP incorporates both the prediction variance and

the prediction bias, which results from using maximum likelihood estimates of

the parameters of the fitted linear predictor. For a given design, quantiles of a

scalar-valued function of the MSEP matrix are obtained within a certain region of

interest. The quantiles depend on the unknown parameters of the linear predictor.

The dispersion of these quantiles over the space of the unknown parameters is

determined and then depicted by the quantile dispersion graphs. An application

of the proposed methodology is presented using the special case of the bivariate

binary distribution.

5.2 Multivariate Generalized Linear Models

Section 4.3 shows that in analogy to the univariate case, multivariate

generalized linear models (GLMs) require the specification of the following three

components:

1. the data set under consideration consists of a independent q-dimensional
random variables, yl, .. ,y,. The distribution of yj (j = 1,. ., n) belongs to









the exponential family with the density function

6(yj |84, )> = exp[ {y 0; b(04)} )s ] j=1 .,n (5-1)

where b(-), c(-) are known scalar functions and 4 is a dispersion parameter,
possibly unknown. The vector Os, j = 1,..., n, consists of q elements. The
mean vector, 4w, and variance-covariance matrix, Cj, of yj are

86(0j) 1 82b(04)
rw Ey= respectively. (5-2)


2. a q-dimensional linear predictor, rl, which is a polynomial function in k control
variables, x x,. ., Zk, and is of the form

rl(x) = Z'(x)P, (5-3)

where x = (xl,..., xk)', Z(x) 0{ ifsf(x), fi(x) is a known vector function
of x, p (0',, .., 0',)' is a p.-dimensional vector of unknown parameters, and
Pi (Pil,... ipi,)' is a vector of unknown parameters for the i-th response
of order pi x 1(i 1 .. ,q), where CL pi p.. For simplicity, we shall
drop the use of two subscripts and instead refer to all the elements of P as
(P1,..., P3. )', where the first pi elements come from p,, the next p2 elements
from p2, and so on.

3. a link function g : RV RV, which relates the linear predictor rl(x)=
[rll(x), rlu(x)]' to the mean response yw(x) = [p1(x), .. p(x)]' at a given
point x in a region of interest, R,

rl(x) = g [p(x)]. (5-4)

If the inverse of g, denoted by h, exists, where h : RV RV, then

rW(x) = h[rl(x)] = h[Z'(x)P]. (5-5)

5.2.1 Estimating the Mean Response

The maximum likelihood estimate (j!1.1 ) of P can be obtained by using an

iterated weighted least-squares procedure [see Fahrmeir and Tutz (2001, p. 106)].

An estimate of rl(x), is then given by


)(x) = Z'(x)P, j = 1, ..., n,


(5-6)









and the corresponding estimate of the mean, pw(x), is


[t(x) = h[Z'(x)P], j = 1,..., n, (5-7)

where p is the MLE of P.
5.2.2 Prediction Variance

The variance-covariance matrix of P is approximately given by [see Fahrmeir

and Tutz (2001, p. 106)]

Var(P) (X'WX)- (5-8)

where X = [Z(xl),...,Z(x,~)]' and W is a block-diagonal matrix of the form,
W= dly[1,..,W], y =( E ) j = 1,...l,n. Here,477 = 7(xj) (j=

1,..., n) is the linear predictor evaluated at xj = (xjl,... Z jk)l wj = p ~Xj), iS the

mean response at xy, and ~-is the first-order partial derivative matrix of q(x)
with respect to pw'(x) evaluated at xj (j = 1,..., n).

The distribution of the MLE, P, is .I-i-mptotically normal with mean P and a

variance-covariance matrix (X'WX)-1 [see Fahrmeir and Tutz (2001, p. 106)] for

large n. Based on Wald's (1943) results, we approximately have


( )[a()a )~x (5-9)

where p. is the total number of parameters and Var(P) (X'WX)-1, W is

an estimate of W using maximum likelihood estimation. Thus an approximate

100(1 a~) confidence region for P is given by

C = {7 : (4 cy)'[(X'WVX)](4 y) < ?i. .}. (5-10)

From (5-6) and (5-8) we approximately get


Var [i(x)] = Z'(x) Var(p)Z(x) Z'(x)(X'WX)- Z(x).


(5-11)









A first-order multivariable Taylor series expansion of C(x) = h[ (x)] around rl(x)

gives
Sh[rl(x)]
C(x) h[rl(x)] + [4(x) rl(x)]. (5-12)
8l' (x)
Replacing h[rl(x)] with pw(x) in the above expansion results in

8 p (x)
AL(x) p(x) + [ (x) rl(x)]. (5-13)
8l' (x)

Thus, Var [Ai(x)] is approximately given by

Var [Ai(x)] Var [4(x)]
8l' (x) 84l(x)

We refer to Var [pi(x)] as the prediction variance matrix. Using the expression of

Var [4(x)] from (5-11), Var [Ai(x)] can be approximately written as

8 p (x) 8 p' (x)
Var [C(x)]- [Z'(x) (X'WX)-1Z(x)] .(5-14)
84'l(x) 84l(x)

5.2.3 Prediction Bias

In this section, we use the expression of Bias(P) from formula (4-31) in

C'! l .ter 4 to find an expression for the bias associated with estimating pw(x).
Formuula (4-31) gave the following expression for Bias(P) correct up to order :

~(x )P ((x ) ((xy)





j=1 s,t=1 =
l \f"( )an f" (x ) a/,f"(xy) a

f: (X) ~pf:(xi"`i~l f(5-15)P









where I"" is the (u, v)th (u, v = 1,...,p.) element of the inverse of Fisher

information matrix,

I(0) = E.

Here, I is the log-likelihood function obtained using (5-1), that is,


I)l= [4{y 04 b(0))}I +(7y3, d)],
j= 1

S(i = 1, .. .1 ; dl = 1 ,p.) is th~e vector of first-order pa~rtia~l derivatives of P,
with respect to Pd. By Ajt (j = 1,..., n; t = 1,..., p.) and Bjs (s = 1,..., p.) we

denote the q x q matrices:


Aj, = + +


B -+

In te abve epresions denotes the matrix of first-order partial derivative

of pw(x) with respect to rl'(x) evaluated at xj (j = 1,..n) denotes the

matrix of first-order partial derivative of 8j wit;h repect~ to r ad-



respectively, taken with respect to Pd.

Using (5-6) and (5-15) jointly, the bias of ~(x) is given by


Bias[ (x)] = Z'(x) Bias[P]. (5-16)


From the expansion of pL(x) = h[ (x)] in a neighborhood of rl(x) by a first-order

approximation of Taylor series as in (5-13), we have

8pw(x)
11(x) pw(x) [4(x) rl(x)].
8l' (x)









Taking expectation on both sides of the above equation, we get an approximate

expression for the prediction hias, that is,

8pw(x) 8pw(x)
Bias [Ci(x)] '- ias [il(x)] =Z'(x) Bias [P]. (5-17)
8l' (x) 84l'(x)

5.2.4 Mean-squared Error of Prediction

The mean-squared error of pi(x), also known as the mean-squared error of

prediction (jl!SEPl), is given by


MSE[ i(x)] = E[( i(x) pw(x))( i(x) pw(x))']

= Var[ i(x)] + {Bias[Ci(x)]} {Bias[ i(x)]}'. (5-18)


Using formulae (5-14) and (5-17) we get an approximate expression for the MSEP
in the multivariate case.

5.3 Criterion for Comparing Designs

In the univariate case, when comparing two designs, my- D1 and D2, design

D1 is deemed better than D2, in terms of prediction capability, if the MSEP of D1

is smaller than the MSEP of D2. In the multivariate case with several response

variables, the MSEP is a matrix. A design comparison can then he based on a

scalar-valued function of the MSEP. One possible choice is the largest eigenvalue of

the MSEP matrix. Obviously, small values of such a function are desirable. Other

possible choices include the trace and the determinant of the MSEP. Here, we

consider the largest eigfenvalue of the MSEP as a criterion for comparing designs for

multivariate GLMs. Such a criterion function is denoted hv EMSEP.

Design Dependence Problem

One 1!! I r~~ problem with the use of the MSEP is that it depends on P, the

vector of unknown parameters of the fitted linear predictor model. We are then

faced with a dilemma, since the purpose of a design is to estimate the unknown

parameters using the data generated by the design in the experiment. To actually










construct the best design, however, one needs to know the true values of the

parameters. Other design optimality criteria such as A-, D-, E- optimality, which

are variance-based, also suffer from the same problem. Common approaches to

solving this design dependence problem were summarized in Section 2.1. In this

chapter, we extend the use of the quantile dispersion (lrl', approach, mentioned

earlier in Section 5.1, to multivariate GLMs.

5.4 Quantile Dispersion Graphs

In this graphical technique, the distribution of a scalar-valued function of

the MSEP, such as EMSEP, is examined throughout the experimental region,

R. This function is defined over a region of interest, R, which is a subset of the

k-dimensional Euclidean space. Its value at a point x in R depends on the chosen

design, denoted by D, and the vector of unknown parameters, P. Let us therefore

denote such a value by -rD(x, p). The distribution of -rD~ X,) OVer R is determined

in terms of its quantiles. These quantiles are obtained on several concentric

surfaces inside R as follows: for a given design D and a given P, we denote the

p-th quantile of TrD(x, p) on R,, by QD ~p,0, v). Here, R,, is the surface of a region

obtained by reducing, or shrinking, the boundary of R by a shrinkage factor v, and

p is the vector of unknown parameters in model (5-3). By varying the value of v

we can cover the entire region R.

Recall that any design comparison criterion for a GLM suffers from the

problem of the dependence of the design on P. To address this problem, we can

vary p within a specified parameter space, C, and then examine the corresponding

values of QD p, p, V) foT a giVen design D and given p and v. The set C is chosen

to be the (1 a~)100; confidence region on P given by formula (5-10). An "initial"

data set that may be available on the responses can be used to construct such a

confidence region. Thus the dependence of the quantiles of TD X, P) On P, foT X On

R,,, can be investigated by computing QD p, p, V) foT SeV6Tra ValueS of p that form










a grid, C, of points randomly chosen from C. The minimum and maximum values

of QD p, p, V) are Subsequently obtained over the values of P in C, for fixed p and

v, and for a given design D. We can therefore compute the following functions:


Qgin (p, v) =min { QD (p p, v) }, (5-19)

Q~ax(p, v) =max {QD(p, p, v)}. (5-20)


For a fixed v, plotting these values against p results in the quantile disper-

sion yI'rll' (QDGs) of the EMSEP over the region R,,. By repeating the same

process for several selected values of v we obtain plots that portray the prediction

capability associated with the design D throughout the region R.

Small and close values of the minimum and maximum quantiles in (5-19) and

(5-20) of a design D over the range of p (0 < p < 1) are highly desirable. The

smallness of Q~ax indicates small EMSEP values on R,,, and the closeness of Qgin

and Qgax indicates robustness to changes in the values of the parameter vector, P.

5.5 Bivariate Binary Distribution

In many experimental situations, several responses may be observed for

the same subject. For example, in a drug testing experiment, in addition to the

standard binary response of success or failure of the drug, some measure of the

side effects of the drug may be of interest. This results in two responses, efficacy

and toxicity of the drug. The efficacy response is 1 if the drug used has the desired

therapeutic effect; toxicity response is 1 if the drug causes unwanted side effects.

These responses are often modelled separately, as in Peruca and Pisani (1989), with

the assumption that they are uncorrelated. However, since the two responses come

from the same subject, it is prudent to allow for correlation in the responses, that

is, to consider them as a bivariate response.









5.5.1 Bivariate Binary Distribution with a Logit Link

Here, we consider a bivariate binary situation similar to the one described
in Section 4.6. Formula (4-33) in CHI Ilpter 4 shows that the response zj

(zyi, zy2, Xj3)l fo OWS the multinomial distribution with parameters mj and

;T~ = (iTy1,wy ij, Tj3)', j = 1,..., n. Refer back to Section 4.6 for the definitions

of zj, xyj and m The mean of zj is 4w = (pylii._, py3) Where pyrj = mjyst,
il._. = myj2j, Ij3 = mj Tj3, and the variance-covariance matrix of zj is Ey, where



xy1(1 Tj ) -iTT Ty2 jT 1 -Tj3



Let x be any point in the region of interest, R. The mean response at x will be

denoted by ;r(x), where xr(x) = [rlx),~ra(K2X),3 rsX~l With xzl(x), x2a(x) and K3 ~X)

being probabilities similar to the elements of syj, but are evaluated at x. Hence,

pw(x) = ;r(x). Note, however, that at the j-th run, pw(xj) = rwj, since xr(xj) = ;rr
(j = 1,..., n). Our objective here is to find a form for g, the link function, such
that

rl(x) = g [p(x)].

From Section 5.2, x = (xl, .. ., xk)' is VeCtoT Of COntrOl Variables, Z'(x) = etiff (x),

fi(x) is a known vector function, and p = [P'z,0 P p31 iS a p.-dimensional vector
of unknown parameters. The Pi's (i = 1, 2, 3) correspond to the three responses

(outcomes in the multinomial distribution) and are of order pi x 1, where Ci pi =


The corresponding link function used here is [see Fahrmeir and Tutz (2001,

p. 73) and Agresti (2002, pp. 267-274)]


fl~x =lo 3= ) =1,2,3 (5-22)
~i~() k*( (x)=









Note that rli(xy)


rqj = Opi (i


1, 2, 3; j= 1, .. ., n). Hence,


exp [ff(x)04]
;ii(x) = 2
1 + CI elxplff(x)00l


1, 2, 3.


where pi is the MLE of Pi for i = 1, 2, 3.

Using formulas (5-14) and (5-17), it can be shown (see Appendix C) that the
mean-squared error of prediction (jl\!SEPi) at x is approximately given by


8pw(x)
'- [Z'(x) (X'WX)
8l' (x)


8p'l(x)
'Z(x)]
84l(x)


MSE[jiL(x)]


(5-23)


where X, Z,and Bias(P) are as defined in Section 5.2.2, and the first-order partial

derivative of pw with respect to rl' evaluated at x is


-xT1(x)iTT(x)


-xT1(x)7


8pw(x)
8l' (x)


(5-24)


1(x)[1 iT (x)]

2~ X> iT1 (X)

T3 (X)iT1 (X)


X2 X> 1 iT2 X)

-iT3 (X)T2 (X)


iT3 X) 1


( Ey )


and W = (1; I[W1,...,W,] with Wy


1,..., n, and


(ixjl+i44) 1 1
iXj17j4 Xj4 4
1 (ixj2+iXj4 1
xy4 Xj27j4 iXj4
1 1 (ixjs+iXj4)
\ Wy4 j4 iXj3i7j4


89l 1
dBy my


Note that


1E=exp (rqi)


exp[ff (xj)04]
3 i
1+ cg=,exp[ff(x,)p0t


1,2,3; j= 1,...,n.


It should also be noted here that the MSEP is a 3 x 3 matrix and depends on P.


+ Zdr'(x)\~ ~1\/ Biasl(X)U Z(x Bias(#) I


r3 X)

T3(x) ,

iT3 X)I










5.5.2 Numerical Example

The data set considered here is taken from Gennings et al. (1994, pp. 429-451).

In a combination drug therapy study on male mice, the pain relieving (analgesic)

ability of two drugs, namely, Ag-tetrahydro-cannabinol (AS-THC) and morphine

sulfate are studied. Though both drugs are analgesic (i.e. provide pain relief), they

are also associated with adverse side-effects. The two responses are UI~, pain relief,

and y2, side-effect. The response yl takes the value 1 if a mouse takes more than

8 seconds to flick its tail when placed under a heat lanip, and yl = 0, otherwise.

Response y1 = 1 is considered good because the mouse does not feel pain when

placed under a heat lanip for at least 8 seconds due to the pain relieving (analgesic)

ability of the drugs. The side effect response, y2, WRs (letermlille( by recording the

rectal temperature of the mouse after 60 minutes following drug administration.

This response is equal to 1 when the rectal temperature of the mouse drops below

:35 C (resulting in hypothermia) after the drug administration and is equal to

zero, otherwise. Hypothermia (y2 = 1) is a tOXic or harmful side-effect. Thus each

of the response variables has two levels and all four combinations are possible. One

aim of the investigation was to study how the associated probabilities concerning

pain relief and hypothermia are related to dose levels of the two drugs.

For the pain relief and side-effect responses, :35 groups of mice (six animals per

group) front a 5 x 7 factorial experiment were randomly assigned to receive the

treatments, where a treatment consists of a single injection of one of the possible

combinations of morphine sulfate (0, 2, 4, 6, 8 nig/kg) and A"-THC (0, 0.5, 1.0,

2.5, 5.0, 10.0, 15.0 nig/kg). Thus we have :35 runs with 6 experimental units (mice)

in each run. The design and the resulting data are presented in Table 5-1.









Model (5-3) was fitted to the data in Table 5-1 using the following first-degree

models for the linear predictors


rli(x) = P1+ x+ 02 3 ,

l2 (X) = P4 5 1 + 6 2)

r/3 (X) = P7 + 8 1 9 3. (5-25)


Here, xl and x2 represent the dose levels of the drugs, morphine sulfate and

a9-THC, respectively. The link functions are the same as those described in

formula (5-22).

The experimental region, R, is rectangular in shape with R : {0 < xl < 8, O <

x2 < 15}. The parameter estimates and their standard errors for the above models,

obtained by using PROC LOGISTIC in SAS [SAS Institute Inc. (2000)], are shown

in Table 5-2.

Let us now refer to the design in Table 5-1 as D1. We compare this design

with another design, D2. The additional design differs in design settings from

D1, but has the same number of experimental runs (= 35) and the same number

of mice per run (= 6). Design D2 1S a 32 faCtorial With the center point (4, 7.5)

replicated three times and all other points replicated four times. The listing of the

two designs is given in Table 5-3. Figure 5-1 shows the design points of D1 and



For each design in Table 5-3 we study the distribution of the EMSEP on

each of several concentric rectangles, R,, which are obtained by a reduction of the

boundary of R using a shrinkage factor v, 0.5 < v < 1. Thus, R,, is determined by

the inequalities


ai + (1 v) (bi ai) < xi < bi (1 v) (bi ai), i = 1, 2,










where ai and bi are the bounds on xi in R (i=1,2), that is, al = 0, a2 = 0, bl =

8, b2 = 15. Four or five values of v are typically used in the calculation of the

quantiles.

To investigate the dependence of EMSEP on P we consider C to be the

95' confidence region on P [see formula (5-10)] and C to be a set of 500 points

randomly chosen from C. For each design and a selected value of P in C, quantiles

of the distribution of the EMSEP are obtained for x e R,,. The number of points

chosen on each R,, was 500, consisting of 125 points on each side. The quantiles

are calculated for p = 0(0.05)1 (p from 0 to 1 in steps of 0.05). The procedure

is repeated for all the values of p in C. Then, Q"in(p, v) and Q"a"p ) r

calculated using formulas (5-19) and (5-20). The R software (Version 2.0.1) was

used to carry out the numerical investigations and draw the QDGs.

To compare the two designs, we examine the corresponding QDGs shown in

Figure 5-2. For v = 1, we note that the maximum quantiles of D1 are above those

of D2 for most values of p, indicating that D2 has better prediction capability than

D1. The values of QB""(p, v) and Qg'"(p, v) for both designs are close to each

other for most of the values of p indicating that they are robust to changes in the

parameter values. As v decreases (that is, as we get closer to the center of R),

there is a decrease in the distance between the maximum quantiles of D1 and D2

indicating that they have similar prediction capabilities near the center of R. So,

overall, D2 RppearS to be the better design, in terms of prediction capability, for

almost all values of p and v.









































~~~~~00~~~~~00~~0000O











00000~000000000000~0O


0000~~~0000~~~000~~~~


CI~ 13 d"oi Ln o d~oi Ln o d~oi Ln
0~ ~ ~ 0~ ~ ~ 0~
H



































































00000~0000000+


CI~ 13 d"oi Ln o d~oi Ln
Ln Ln OLn Ln Ln OLn
H


ed



Oc




































D1 D2
X1 2~ 1 2~ 1 2~ 1 2~
0 0 4 10 0 0 8 15
0 0.5 4 15 0 0 4 15
0 1 6 0 0 0 4 15
0 2.5 6 0.5 0 0 4 15
0 5 6 1 4 0 4 15
0 10 6 2.5 4 0 0 15
0 15 6 5 4 0 0 15
2 0 6 10 4 0 0 15
2 0.5 6 15 8 0 0 15
2 1 8 0 8 0 0 7.5
2 2.5 8 0.5 8 0 0 7.5
2 5 8 1 8 0 0 7.5
2 10 8 2.5 8 7.5 0 7.5
2 15 8 5 8 7.5 4 7.5
4 0 8 10 8 7.5 4 7.5
4 0.5 8 15 8 7.5 4 7.5
4 1 8 15
4 2.5 8 15
4 5 8 15


Table 5-2. Maximum likelihood estimates and standard errors.


Parameter Estimate Std. error P-value
P1 -5.2899 0.7767 < 0.0001
2~ 0.7346 0.1651 < 0.0001
P3 0.8355 0.1593 < 0.0001
P4 -2.3166 0.5232 < 0.0001
s5 0.8349 0.1457 < 0.0001
P6 0.5652 0.1523 0.0002
P7 -5.6437 1.5662 0.0003
Ps 0.4188 0.3512 0.2331
9s 0.6619 0.2054 0.0013

Scaled Deviance= 78.0672, DF= 96.





Table 5-3. Designs D1 (5 x 7 factorial) and D2 (2 faCtOrial).






































* *


0 0 0 0 0


0 0 0 0 0

0 0 0


O


Figure 5-1. Designs D1 (5 x 7 factorial) and D2 (2 faCtOrial)


































o













co
u,


I I II0' 0' 0 '


a ~x (
1 1 1 I


O













co
u,


a (





T. o a

a n
1111


OO


I II I


d3SVU3 ~o salRueng


00















CHAPTER 6
OPTIMIZATION IN MULTIVARIATE GENERALIZED LINEAR MODELS

6.1 Introduction

This chapter is concerned with the simultaneous optimization of several

estimated mean responses in a multivariate GLM setup. The optimization

algorithm is based on the generalized distance approach of K~huri and Conlon

(1981). In this algorithm, we first compute the estimated mean responses using

the corresponding multiresponse data. The next step is to obtain optima for the

individual estimated mean responses. If all the individual optima are attained

at the same set of conditions on the input (control) variables or factors, then we

ii- that an "ideal optimum" has been reached. Unfortunately, this is rarely the

case. Optimal conditions for one mean response may be far from optimal or be

even physically impractical for the others. Thus we resort to finding compromise

conditions on the input variables that are 1 l.-0 .1 II.10" to all the mean responses.

The deviation from the ideal optimum is measured by a distance function expressed

in terms of the estimated mean responses along with their variance-covariance

matrix. By minimizing such a distance function we arrive at a set of conditions for

a "compromise optimum".

6.2 Multivariate Generalized Linear Models

From Section 5.2 we know that the q-dimensional response vector yj (j=

1, .. ., n) is assumed to belong to an exponential family with a probability

distribution 6(yj |84, )> [see formula (5-1)]. The linear predictor, rl(x) = Z'(x)ff,

is related to the mean response pw(x) by a link function rl(x) = g [p(x)] [refer

to equations (5-2) and (5-3)], where x = (xl,..., k)' is the vector of control









variables, p = (0',, ., 0',)' is a p.-dimensional vector of unknown parameters, and

g : RV IRV.

Using the maximum likelihood estimate (jl 1.1 ), P, of P, we can obtain an
estimate of the mean response pw(x) given by

C(x) = h[Z'(x)P]. (6-1)

The variance-covariance matrix of P is approximately given by [see formula (5-8)]


Var(0) (X'WX)-1, (6-2)

where X and W are the same as in (5-8). An approximate 100(1 c0) confidence

region for p is [see formula (5-10)]

C { : (P 7)'(X'WX) (P y) < X p~. 8

where W is obtained by estimating each of the elements of W using maximum

likelihood estimation. An approximate expression for Var [ i(x)] is given by [see

formula (5-14)]

8pw(x) 8p'l(x)
Var [ i(x)] [Z'(x) Var(p)Z(x)] .(6-4)
84'l(x) 84l(x)

An estimator of Var [C(x)] is

8pw(x) 8p'l(x)
Var[ L(x)] [Z'(x)Var(p)Z (x)] (6-5)
8l' (x) 84l(x)

where x is ob-tainled by, estimratinlg each elemrenlt of using max~imrum
likelihood estimation, and Var(P) (X'WX)-1

6.3 Simultaneous Optimization of a IVultiresponse Function

In this section, we discuss a procedure for the simultaneous optimization of

several estimated mean responses in a multivariate GL1\ setup. For this purpose

we use the generalized distance approach of K~huri and Conlon (1981).









6.3.1 Generalized Distance Approach in Multivariate GLM

Multiresponse optimization using the generalized distance approach amounts

to finding conditions on x that minimize the distance between the estimated mean

responses and their corresponding individual optimum values. Let ki (i = 1, .. ,q)

be the optimum value of 4i(x) optimized individually over the experimental region

R, where As(x), i = 1, .. ,q is the i-th element of C(x). Let it = (ky, .., Ag)' be

the vector of all the q optima. Also, let p[ i(x), k] be a distance function which

measures the distance of Ci(x) from Ct. Several choices are possible for p. We

choose here a distance function given by


p[ i(x), k] = [(Ai(x) A)'{Var[Ai(x)]}-l (Ai(x) &)]1/2, (6-6)


where Var [Ai(x)] is the estimate of Var [Ai(x)] in formula (6-5).

But, before minimizing p, we have to take into account the variability

associated with At. To do so, we apply the following procedure, given by K~huri

and Conlon (1981): Let S(p() be the true optimum value of the i-th (i = 1,..., q)

mean response optimized individually over R, and let e(() = [S(p(),...,S(P()]'.

Our objective is to find an x E R such that p[ i(x), e(()] is minimized over R. As

((0)> is unknown, trying to minimize p[Ai(x), ((0)], which is a function of e((), is

impossible. Instead, we minimize an upper bound on p[ i(x), ((0)] as follows: Let

DC be a confidence region for e((). Then, whenever e(() E DC,


p[ L(x), ((0)]l < max p[ i(x), (]. (6-7)
EEDC

The right-hand side of (6-7) serves as an overestimate of p[ i(x), e(()]. We

therefore minimize the right-hand side of (6-7) over the region R, thus adopting a

conservative distance approach to our minimization problem.









6.3.2 Construction of confidence interval on the true mean response

Rao (1973, p. 240) shows that for any continuous function 1(P), defined on a

subset of the p.-dimensional Euclidean space that contains C,


P~mmnl(y) 1- a, (6-8)
yEC yEC

where C is the 100(1 a~)~ confidence region on P given in (6-3).

The interval [minec 1(y), maxec 1(y)] defines a conservative confidence

interval on 1(P). Since the individual optimum, S(p() (i = 1, .. ) of the ith true

mean response, ps(x), can be expressed as a continuous function of the parameter

vector p, the above result in (6-8) can be used to obtain a conservative confidence

interval on S(p() using the confidence region in (6-3). Moreover, since whenever



9i(P) E [mm(4(7r), maxg(r(7), i= 1,...,q,
yEC yEC

then

P [mmn 9(r> (7 < (4 < max 9(4 () | i = 1, q] > P [P E C] a 1 a~.
yEC y6C

It follows that [minec g(ii), maxec g(ii)], i = 1, .. ,q, form conservative

simultaneous confidence intervals on the S(p() with a joint coverage probability

approximately greater than or equal to 1 a~. Now, let us denote the interval

minec gi(7), maxec Si(7)] by De(C), i = 1, .. ,q. Since g(p(0 E Di(C) for

i .. ,q if and only if e(() e xV (Di(C), where xV (Di(C) denotes the

Cartesian product of the Di(C), then


P[e(() e x( ,Di(C)]= P[S(P(0 E De(C)|i = 1,...,q] > P[0 E C] a 1-a c. (6-9)


Consequently, xy zDi(C) forms a rectangular conservative confidence region on

e((). We therefore choose this region to be the one described in Section 6.3.1, that

is, DC.









6.4 Example: Bivariate Binary Distribution

In many experimental situations, several responses may be observed for the

same subject. A typical example is from a drug testing experiment, where in

addition to the standard binary response of success or failure of the drug, some

measure of the side effects of the drug may be of interest. This results in two

responses, efficacy and toxicity of the drug. The efficacy response is 1 if the drug

used has the desired therapeutic effect; toxicity response is 1 if the drug causes

some unwanted side effects. The two responses are assumed to be correlated as

they come from the same subject. In this example, the determination of the dose

level that simultaneously maximize the efficacy and minimize the toxicity effects of

the drug is of particular interest.

Once again, we consider the bivariate binary situation similar to the one

described in Sections 4.6 and 5.5.1. The probability mass function of the response

zj is [see formula (4-33)]


by (z ) oc jTZj Zj Z3 1 -: 32 (6-10)
i= 1

where zyl L= E yjwlyjw2, xj2 1 yjwll )w2) Xj31 jl)2
and zy4 MS =1i Ci 1"" ijw 9)w2) Wel note from the fom~

of the pmf in (6-10) that the responses zj = (z l, ~._, zy3)l foll0W the multinomial

distribution with parameters mj and xyj = (xy j, iTj2) j3)', j = 1, n. The

notations yjst, yjw2, zj, xyj and mj are the same as defined in Section 4.6. The

mean of zj is pj = (pyli ._.,y) Where pyr = mj'irj, py2 Mj mj2i, I-j3 j mjj3,

and the variance-covariance matrix of zj is E where




Ey~~~~~~ = y -j 1 j j 2j 1, (6-11)
-3 1 ij3iTj j2 T Tj3)









The corresponding link function used here is


fli ~ ~ iTx)X = og3 @X) i, i = 2, 3, (6-12)

where fi(x) is a known vector function of x defined in formula (5-3). Hence we can

write,
exp[ff (x)04]
;ii(x) = ,2= 1, 2, 3.
1+ 13"=1exp,[fi'(x)00l

where pi is the MLE of Pi for i = 1, 2, 3. The first-order partial derivative of pw

with respect to rl' evaluated at x is [see formula (5-24)]

~1(x)[1 -x iT(x) --xyT(x)iT2X) --KT1 X)iT3 X)

84() -~ 2X)iT1 X) iT2 (X) 1 iT2 X)1 --K2 ~X)i3 (X)
T3 (X)iT1 X) --KT3 (X)iT2 X) iT3 (X) 1 iT3 X)1

where pw(x) = [rlx),r(x), X2 3 X) 1.

6.4.1 A Numerical Example

In this example, we study the effects of a vasodilator (drug for reducing

hypertension) and age on a group of 450 patients suffering from hypertension

diastolicc blood pressure > 99 mm Hg). Two responses were measured, namely,

yl = efficacy and y2 = tOX1City Of the vasodilator. The efficacy response yl takes

the value 1 if the blood pressure of a patient was reduced by at least 15 mm Hg

and y, = 0, otherwise. The toxicity response y2 is 1 if the patient experienced any

unwanted side effects and y2 = 0, otherwise. Three age groups and five levels of

the drug were used. The data for this example was simulated, using the R software

(Version 2.0.1).

A 3 x 5 factorial design was used to randomly assign a single dose of the

vasodilator (0, 5, 10, 15, 20 mg) to the patients in each of the three age groups.

The design points and the corresponding response values are given in Table 6-1.

Note that the center of each age group was used as the value of the age variable.









Table 6-1. Experimental design and response values.

(a) (b) (b)
Age group Age Dose xl x2 1l 2~ 3 x4 m
44-54 49 0 -1 -1 1 3 1 26 30
44-54 49 5 -1 -0.5 0 12 1 17 30
44-54 49 10 -1 0 1 25 1 3 30
44-54 49 15 -1 0.5 2 20 2 6 30
44-54 49 20 -1 1 19 6 3 2 30
55-65 60 0 0 -1 2 3 1 24 30
55-65 60 5 0 -0.5 2 7 2 19 30
55-65 60 10 0 0 5 17 3 5 30
55-65 60 15 0 0.5 12 13 2 3 30
55-65 60 20 0 1 22 3 4 1 30
66-76 71 0 1 -1 1 2 2 25 30
66-76 71 5 1 -0.5 1 4 2 23 30
66-76 71 10 1 0 15 8 4 3 30
66-76 71 15 1 0.5 20 5 5 0 30
66-76 71 20 1 1 25 0 1 4 30

(a) The center of each age group is used as the value of the
age variable.
(b) xl and x2 are the coded levels of age and dose respectively.


As in Section 6.4, we denote by zyz (j = 1,...,15) the number of patients in

the the j-th run whose diastolic blood pressure was reduced by at least 15 mm

Hg but experienced some unwanted side-effects, I. the number of patients in the

the j-th run whose diastolic blood pressure was reduced by at least 15 mm Hg

but had no unwanted side-effects after being given the drug, zy3 1S the number

patients in the j-th run who experienced only side-effects but no reduction in their

diastolic blood pressure, and zy4 is the number of patients in the the j-th run who

experience neither side-effects nor any reduction in their diastolic blood pressure.

We investigate the effect of the two factors, age and dose levels of the

vasodilator, on the means of the responses, namely, xzl(x), x2~(x) and Kr3 X>,

x = (xl, x2)' and xl and x2 are the coded levels of age and dose, respectively,

given by xl = (age 60)/11 and x2 = (dose 10)/10. It is of interest to

simultaneously minimize wil(x) and K3s(x), and maximize the mean response x2 Xx).










Here x4i(x), i = 1, 2, 3 denotes the mean of the i-th response at a given point x in

the coded experimental region, R : -1 I xl I 1, -1 I x2 i

Model (5-2) was fitted to the data in Table 6-1 using the following models for

the linear predictors


rll(x)

l2 X)

r/3 X)


P1 + 2 1 + 3 2

P4 + 5 1 6X 2 P7 22, and

gs 9 1sx 10 2 -


The link functions are the same as those described in formula (6-12). The

parameter estimates and their standard errors for the above models, obtained

by using the R software (Version 2.0.1), are shown in Table 6-2. These parameter

estimates and the estimated variance covariance matrix of P in Table 6-3 are used

jointly to find C, the approximate 95' confidence region on P.

Table 6-2. Maximum likelihood estimates and standard errors.


Parameter Estimate Std. error
P1 0.2141 0.1872
2~ 0.7629 0.1990
P3 2.5480 0.2858
P4 1.0998 0.2117
s5 -0.6744 0.1765
P6 1.4743 0.3062
P7 -2.1864 0.3766
Ps -0.9094 0.2335
9s 0.3797 0.2507
Pio 1.6782 0.3343

Scaled Deviance = 113.3755, DF =


P-value
0.2527
0.0001
< 0.0001
< 0.0001
0.0001
< 0.0001
< 0.0001
< 0.0001
0.1299
< 0.0001







83














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00000000


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CI0091~090900L~


S000~0000009
S000 000090






S090 0009000






0900r0~0900

mammoooooo

C ~00 0L ~ 00 0 300 00
0000000000


cr 0100n 0


9~00~00?0Cr 00

Oc 040000 0
0000000090










Using formula (6-9), the confidence region DC is the Cartesian product of the

individual confidence intervals, Di(C), on 9(P(), i = 1, 2, 3. To find Di(C), the

following steps are taken: For each of 1000 values of y randomly selected from

C (selection process is discussed in detail below), we compute the optimum of

x4i(x), i = 1, 2, 3, over the experimental region R. We denote the optimum of iT4(x)

by g(y(). Next, we find the minimum and maximum of these (4, i = 1, 2, 3, over

the values of y E C to get the lower and upper bounds of Di(C), respectively. The

intervals, Di(C), for i = 1, 2, 3 are given in Table 6-4. This table also includes the

individual optima (Ri) of the three estimated mean responses, #4i(x), i = 1, 2, 3,

and the locations of these optima. The next step is to use the confidence intervals

Di(C) to get the confidence region DC.

Table 6-4. The individual optima and the region DC.


Location of
individual optima De(C)*

Estimated mean Individual estimated xl x2 Lower U~pper
response optima (A) bound bound
iii 0.0362 -1 -1 0.0091 0.1113
#20.7638 -1 0.0750 0I ".el1 0.8748
ji3 0.0350 -1 -0.1070 0.0101 0.0826;

c The joint coverage probability is > 0.95 approximately.


For each of 1380 values of x selected from the region R by a grid search

(selection process is discussed in detail below), we compute the maximum of the

distance function, p[ L(x),S (]with respect to ( E DC. Let us denote the maximum

of p[AL(x),(] by pmax(x). Minimizing pmax(x) over the values of x in R, we arrive at

the minimax distance. The minimax distance and the corresponding simultaneous

optima of #1i(x), #2i(x), and ji3(x) and their locations are given in Table 6-5. Note

that increasing the number of selected values of y and x above 1000 and 1380,

respectively, does not cause any appreciable change in the results.









Table 6-5. Simultaneous optima.


Minmax p 9.6826

Simultaneous iii 0.0723
optima #20.7056
iT3 0.0387

Location of xl -0.8823
simultaneous optima x2 -0.1849


From Table 6-5 we see that the location of x corresponding to the minimax

distance is (-0.8823, -0.1849). Using the original uncodedd) settings of the

variables, age and dose, the coordinates of the minimax point are age = 50 and

dose = 8.1508 mg. This result tells us that efficacy of the vasodilator is maximized

and its toxicity effects are minimized for patients near the age group [44 54]

receiving a moderate dose level of 8.1508 mg of the drug.

6.4.2 Computer Programs Used for the Selection Process

RANDOM_DATA is a set of MATLAB programs used to generate data

points randomly from a given region. We used RANDOM_DATA to generate

*y's randomly from an ellipsoidal region, C. The required MATLAB files can be

downloaded from http://www.csit. fsu.edu/ burkardt /m_sre/random_data/

random_data.html.

A set of MATLAB routines called HEX_GRID was used to select x's from the

experimental region R. HEX_GRID computes points on a hexagonal grid defined

on a rectangular region. The required MATLAB files can be downloaded from

http://www.csit. fsu.edu/ burkardt /m_src/hex_grid/hex_grid. html.

A computer program called MCS [Huyer and Neumaier (1999)], was used in

all the above optimizations. MCS is a MATLAB program for constrained global

optimization using function values only. It is based on a multilevel coordinate

search that balances global and local searches. The local search for the optimum is









done via a sequential quadratic programming procedure. The required MATLAB

files can be downloaded from http://www~mat .univie.ac.at/ neum/software/mes/.

To run the programs, the two MATLAB 5 programs MINQ (bound constrained

quadratic program solver) and GLS (global line search) are also required. Both

MINQ and GLS are found in the same website.















CHAPTER 7
SITAINARY AND FITTIRE RESEARCH TOPICS

7.1 Summary

This dissertation is primarily concerned with development of techniques that

can he used to

1. compare designs for both univariate and multivariate generalized linear models
(GLMs) ,

2. derive an approximate expression for the bias of the maximum likelihood
estimates (AILEs) for a multivariate GLM setup, and

:3. determine the settings of the input variables or factors which simultaneously
optimize several responses in a multivariate GLM situation.

C'!s Ilter :3 demonstrates that the graphical approach of quantile dispersion

graphs (QDGs) is a convenient technique for evaluating and comparing designs for

GLMs. These graphs also provide information on the prediction capability of a

design throughout the experimental region, and on its dependence on the model's

parameters. The proposed methodology is illustrated with an example based on

designs for Poisson regression models. The example also shows that the choice of

the link function can influence the shape of the QDGs for a given distribution. For

a given link function, the example shows that the QDGs may also be influenced by

the distribution of the response variable.

The main contribution of ChI Ilpter 4 is the derivation of formula (4-31)

representing the bias of MLEs of the parameters in a multivariate GLM setting.

An application of the use of this formula in the special case of a hivariate binary

distribution was discussed in Section 4.6 of this chapter.










The first portion of ('!s Ilter 5 deals with the development of an expression for

the mean-squared error of prediction (jl!SEP i) concerning designs for multivariate

GLMs. The remaining portion compares these designs for multivariate GLMs based

on their MSEP. An example of a bivariate binary situation is used to illustrate the

comparison of designs in a multivariate GLM setup.

In C'!s Ilter 6, the generalized distance approach of K~huri and Conlon (1981)

was used to determine optimum operating conditions on the input variables that

result in a simultaneous optimization of several predicted mean responses in a

multivariate GLM situation. An example is presented to illustrate optimization in

a multivariate GLM setup.

7.2 Future Research

This section is concerned with research objectives that will be the focus of

my future investigation. The future topics include some possible extensions of the

results found in ('! .pter 3 through 6, in addition to certain problems in the context

of linear mixed effects models. A listing of these topics follows:

1. The graphical approach of QDGs allows for the comparison of several
candidate designs but is not currently used to construct "optimal" designs. It
would be desirable to develop this approach so that it can generate a "b. I I. I
design sequentially starting with an initial design.

2. Comparison of designs for GLMs in C'! Ilpter 3 and 5 is based on a distributional
form for the data under consideration. There may arise situations where we
lack a well-defined likelihood function for the multiresponse data. Thus,
a possible extension will be to compare designs for both univariate and
multivariate GLMs in these situations based on a quasi-likelihood function.

3. In C'!s Ilter 6 we base our optimization algorithm in a multivariate GLM
setup on a specific distributional form. Optimization techniques that can
be developed without assuming a distributional form are therefore needed.
For example, there may be a need to determine the settings of the input
variables which simultaneously optimize two responses, both following a
Poisson distribution. In this case, to apply the same algorithm as in ('!! Ilpter
6, we shall first need to come up with a multivariate Poisson distribution to










estimate the mean responses. Unfortunately, a closed-expression form for the
multivariate Poisson distribution is not available in the statistical literature.

4. There is a need for research on design issues for generalized linear mixed
models (GLM~ls) and non-linear mixed-effects models (NLMEs). It should
be possible to use QDGs for the comparison and evaluation of designs for
GLM~ls and NLMEs.

5. My future plan also extends to linear mixed effects models with heterogeneous
variances. The experimental error variance, in a response surface model
with a block effect, has traditionally been assumed to be constant. In some
experimental situations, however, this variance may be different for the
different blocks that make up the associated design. A study on the choice of
designs for these linear mixed effects models with heterogeneous variances is
a possible extension. Some of the procedures for testing the random effects in
these models are likelihood ratio and ANOVA-hased tests. A comparison of
the powers of these two tests for the random effects is also a future target.

6. Generalized least squares (GLS) estimates of fixed effects in mixed linear
models depend on the values of the variance components. However, these
values are typically unknown, and it is a common practice to replace them
with estimated values. The procedure of first estimating the variance
components and then using those estimates in GLS computation is often
referred to as a two-stage mixed model analysis, or estimated generalized
least squares (EGLS). With few exceptions, properties of EGLS fixed effect
estimates are unknown. The variance of the EGLS estimate depends on the
design. It is desirable to perform a diagnosis to check if it is possible to reduce
the difference between the variance of the GLS estimate and that of the EGLS
estimate by choosing an appropriate design.