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andnonlinearmodels.Auniedviewofthetopicispresentedbyputtingexperimental designinadecisiontheoreticframework.Thisframeworkjustiesmanyoptimality ThispaperreviewstheliteratureonBayesianexperimentaldesign,bothforlinear KathrynChaloner1andIsabellaVerdinelli2
criteria,andopensnewpossibilities.Variousdesigncriteriabecomepartofasingle, Abstract
Key-wordsandPhrases:DecisionTheory.HierarchicalLinearModels.LogisticRegression.NonlinearDesign.NonlinearModels.OptimalDesign.OptimalityCriteria.Utility Functions. coherentapproach.
ofnon-Bayesiandesignonlyasneededforthedevelopment.DasGupta(1995)presentsa 1Introduction Hunter(1984)andintherecentbookbyPukelsheim(1993).Ford,KitsosandTitterington (1989)reviewednon-Bayesiandesignfornonlinearmodels.Thispaperconsidersthetheory complementaryreviewofBayesianandnon-Bayesianoptimaldesign. Non-BayesianexperimentaldesignforlinearmodelshasbeenreviewedbySteinbergand
treatmentstostudy,deningthetreatmentsprecisely,choosingblockingfactors,choosing specifyingallaspectsofanexperimentandchoosingthevaluesofvariablesthatcanbe 1.1ExperimentalDesign
howtorandomize,specifyingtheexperimentalunitstobeused,specifyingalengthoftime controlledbeforetheexperimentstarts.Controlvariablesmightinclude:choosingwhich Thedesignofexperimentsisanimportantpartofscienticresearch.Designinvolves
fortheexperimenttobeperformed,choosingasamplesizeandchoosingtheproportionof observationstoallocatetoeachtreatment.Theseareallrelevantaspectsindesign. lis,Minnesota UniversitadiRoma\LaSapienza"andVisitingAssociateProfessor,DepartmentofStatistics,Carnegie MellonUniversity,Pittsburgh,Pennsylvania152131
2IsabellaVerdinelliisAssociateProfessor,DipartimentodiStatistica,ProbabilitaeStatisticheApplicate, 1KathrynChalonerisAssociateProfessor,DepartmentofStatistics,UniversityofMinnesota,Minneapo

experimentoftenhelpinselectingimportantfeaturesthatdependonthespecicproblem. Indesigningaclinicaltrial,whereexperimentsaretobeperformedonpeople,thechoiceof and,ifnot,whattouseasacontroltreatment,isaproblemthatdoesnotariseindesigning thetreatmentstobecomparedisacentralethicalissue.Whetherornotaplaceboisethical Commonsense,availableresources,andknowledgeofthemotivationforcarryingoutthe
aeldexperimenttocompareyieldsfromdierentfertilizercombinations.Notallaspects forthecontrolvariableshowevercanbesimplyexpressedinamathematicalframework. Thisproblemhasbeenconsideredatlengthinthescienticliteratureandisfocusedonin thispaper. ofexperimentaldesignaresusceptibletoabstractmathematicaltreatment.Choosingvalues
precisespecicationofthereasontheexperimentisbeingconducted.Likemostareasof experimentationand,indeed,oftenmotivatesdoingtheexperiment,Bayesianmethodsare ideallysuitedtocontributetoexperimentaldesign.Bayesiandecisiontheoryalsomotivates collectionisrestrictedbylimitedresources.Becauseinformationisusuallyavailablepriorto Whendesigninganexperiment,decisionsmustbemadebeforedatacollection,anddata
stilllagbehindthetheory.ApartfromFlournoy(1993)therearenotrue\casestudies" Bayesianstatistics,Bayesianexperimentaldesignhasgainedpopularityinthepasttwo decades.ButalsolikemanyareasofBayesianstatistics,applicationstoactualexperiments thatweknowof,whereBayesianideashavebeenformallyappliedtothedesignofanactual presentedinthispaper. scienticexperimentbeforeitisdone.Thisisaveryimportantareaforfuturework.There arehoweverseveralexamplesofexamininganexperimentinaBayesiandesignframework afterithasbeendone:forexampleClyde,MullerandParmigiani(1994)andsomeexamples
shouldthereforebeselectedtoachievesmallvariabilityfortheestimatorchosen.Much dependshoweveronwhatistobeestimated,andhowitwillbeestimated.Specifyingthe interestcanbeimprovedbyappropriatelyselectingthevaluesofthecontrolvariables.In estimationproblems,estimatorswithsmallvarianceareusuallydesirable.Controlvariables Thebasicideainexperimentaldesignisthatstatisticalinferenceaboutthequantitiesof
purposeoftheexperimentgeneratesvariouscriteriaforthechoiceofadesign. WeaddressthefundamentalprinciplesofdesignbyprovidingageneralBayesiandecision 2

theoreticframeworkforacoherentapproach.MostoftheworkinBayesiandesigncanbe
Bayesiandesignsandthat,sometimes,theexperimentsusedinpracticeareapproximately includedasspecialcasesinthisgeneralstructure.Theusefulnessofthisapproachandthe
Bayesoptimal. withsomeexamples.ThreeexamplesarepresentedinSection1.2.Theywillbeexamined improvementthatcanbeobtainedoverdesigningwithinthenon-Bayesiantheoryisshown
1.2Examples againinSections3and6toillustratethetypeofimprovementthatcanbeobtainedovernonthesamenumberofobservationstoeachgroupisapossibility,butdierentchoicesofthese
eectsareofinterest.Assumealsothatthevarianceoftheobservationsisknown.Assigning measurementsontheexperimentalunitsandthegroupscorrespondtottreatmentswhose observationsmustbedividedamongtgroupsinanoptimalway.Theobservationsare Example1.Considertheone-wayanalysisofvariancemodelwhereagiventotalofn
proportionsmightbemoreappropriate,dependingonthetypeofexperiment.Thesame one-wayanalysisofvariancemodelcanbeusedeitherinatrialtostudytheeectoft dierenttreatments,orwhentheeectsoft?1similartreatmentsaretobecomparedwith astandardcontrol,or,perhaps,whenagivendrugistobetestedattdierentdoselevels. Intuitively,itmaybesensibletohavedierentdesignsforeachofthesesituations. Asecondonetakesobservationsintdierentgroupstostudytheeectofincreasinglevels z1;z2;:::ztofacertaindrug.Bothexperimentsaremodeledthroughaonewayanalysisof experiments.Oneexperimentconsistsincomparingt?1similartreatmentswithacontrol. variancebuttheyareessentiallydierent.Section3presentsawaytousepriorknowledge TogiveanideaofthedierenceaBayesianapproachcanmake,considerthefollowingtwo
intheseexamples.Apriordistributionusedintherstexperimentwillnotnecessarilybe
weredonetoassessthepotencyofindividualbatchesofdrug.Thelaboratoryperformed twoexperimentsexactlythesameandthesamedesignwouldbechosenforbothofthem. appropriateinthesecond.Anon-Bayesianapproachtodesignwouldtypicallyconsiderthe TheBayesianapproachhasinsteadmoreexibilityasisshowninSection3. Example2.AtaUniversityofMinnesotalaboratorylargenumbersofanimalexperiments 3

logisticregressionsonmanydierentdrugsandbiologicmaterial. sametypeofdesignwasusedforeachoftheexperiments.Thedesignusuallyconsistedof tenanimalswereexposedtothehighestdose.Theresponsesmeasuredwerethenumberof sixequallyspaceddoseswithtenmiceexposedtoeachdose.Sixtyanimalswererequiredfor eachexperiment.Occasionallylessthansixtyanimalswereavailableinwhichcaselessthan Foroneparticulardrugunderstudy,54similarexperimentswereperformedandthe
90%)ofthemicediedatthehighlevelsandalowerproportion(20%or10%)diedatthe onthepotencyofthebatchofdrugandbychance.Typicallyahighproportion(80%or lowlevels.Aftereachexperiment,thepotencyofthebatchwascalculatedusingmaximum likelihoodtoestimatetheLD50,thedoseatwhichtheprobabilityofamicedyingwas survivingmiceoneweekafterbeinggiventhedose.Dierentnumbersofmicedieddepending
estimatedtobe0.50.Twotypicaldatasetsaregivenintable1fromaexperimentsthat lookedatthepotencyofdierentconcentrationsofalbumen. DoseNumberNumberDoseNumberNumber 2.5 3.0 3.5ExposedDead 10 0 1 1TABLE1. 2.5 3.0 3.5ExposedDead 10 02
ThisexamplewillbediscussedfurtherinSection6.2.Thedesignused,sixequallyspaced 4.0 4.5 5.0 10 356 4.0 4.5 5.0 10 14
doseswithtenanimalsateachdose,waschosenforconvenience.Itisstraightforwardto BATCH1 BATCH2 7
dotheexperiment,andwithlargenumbersofexperiments,itissimplertousethesame 8
designeachtime.Itisnaturaltoaskwhethertheexperimentscouldhavebeendesigned dierently.Itisalsonaturaltouseresultsfromthissetofexperimentsforconstructing thesamersttwomomentsasthesample.4
apriordistributiontouseforsubsequentexperiments.The54estimatesofLD50canbe thoughtofasamplefromadistributionofpossiblevalues.Wethereforeconsideraprior distributionfortheLD50thatreasonablyreectstheobservedsampleandhasapproximately

imenttoinvestigatebioavailability.Theexperiment,describedina1979unpublishedPhD aminophyllinetoanumberofhorsesbyintra-gastricadministration.Bloodsampleswere thesisbyButtonatTexasA&MUniversity,consistedofgiving15mg/kgoftheophyllineas thendrawnatdierenttimes,t,afterinjectionandtheconcentrationofdrug,y,measured. Example3.Atkinson,Chaloner,HerzbergandJuritz(1993)examinedadesignedexper-
Thevalueofywasmodeledtoberelatedtotthroughanopenone-compartmentmodelwith Theobservationerrorsareindependentandnormallydistributedwithmeanzero.The unknownparameters(1;2;3)aresuchthat2>1.Attimet=0theexpectedresponse rst-orderabsorptioninput:y=3(e?1t?e?2t)+: iszeroand,astincreases,itincreasesuptoamaximumandthendecreasestozeroast getslarger.Severalquantitiesareofinterestincludingtheareaundertheexpectedresponse curve,thetimeatwhichthemaximumisreached,andthevalueatthemaximum.The designproblemischoosingthetimesatwhichtotakebloodsamples.Thedesignusedin
experimenterandconstructedBayesianoptimaldesignsunderseveralpriordistributions measurementateachtimeandthetimesareapproximatelyuniformonalogscale. HerzbergandJuritz(1993)lookedattheeciencyofthe18pointdesignusedbythe Button'sthesisisfairlytypicaloftheseexperimentsandisan18pointdesignwithone
suggestedbythedata.ThisisfurtherdiscussedinSection6.4. 1.3OverviewoftheBayesianApproach Asinexample2,thisisanothercaseofanonlineardesignproblem.Atkinson,Chaloner,
design.Lindley'sargumentisessentiallythefollowing. theorytoaverageoverthesamplespace.Asthesamplehasnotyetbeenobserved,the (1961),Lindley(1972,page19and20)presentedadecisiontheoryapproachtoexperimental generalprincipleofaveragingoverwhatisunknownapplies.FollowingRaiaandSchlaifer ExperimentaldesignistheonlysituationwhereitismeaningfulwithintheBayesian
observed.BasedonyadecisiondwillbechosenfromsomesetD.Thedecisionisintwo parts:rsttheselectionof,andthenthechoiceofaterminaldecisiond.Theunknown AdesignmustbechosenfromsomesetH,anddatayfromasamplespaceYwillbe 5

parametersare,andtheparameterspaceis.Ageneralutilityfunctionisoftheform U(d;;;y). Foranydesign,theexpectedutilityofthebestdecisionisgivenby
maximizing:(1)U()=max wherep()denotesaprobabilitydensityfunctionwithrespecttoanappropriatemeasure. TheBayesiansolutiontotheexperimentaldesignproblemisprovidedbythedesign U()=ZYmax d2DZU(d;;;y)p(jy;)p(yj)ddy; (1)
istospecifyautilityfunctionreectingthepurposeoftheexperiment,regardthedesign choiceasadecisionproblem,andselectadesignthatmaximizestheexpectedutility. Inotherwords,Lindley'sargumentsuggeststhatagoodwayfordesigningexperiments 2HZYmax d2DZU(d;;;y)p(jy;)p(yj)ddy: (2)
notnecessarilyalsooptimalforprediction.Evenrestrictingattentiontooptimaldesignsfor estimation,thereareavarietyofcriteriathatleadtodierentdesigns,dependingonwhat ofBayesianexperimentaldesign.Selectingautilityfunctionthatappropriatelydescribes thegoalsofagivenexperimentisveryimportant.Adesignthatisoptimalforestimationis ThepresentpaperpursuesLindley'sapproachasaunifyingformulationforthetheory
justication.Wheninferenceabouttheparametersisthemaingoaloftheanalysis,for expressesvariousreasonsforcarryingoutanexperiment. istobeestimatedandwhatutilityfunctionisused.Thechoiceofautility(orloss)function
example,autilityfunctionbasedonShannoninformationleadstoBayesianD-optimality inthenormallinearmodel(see,Bernardo,1979).Inaddition,Shannoninformationcanbe (Box,1982)suchasA-optimality,D-optimalityandotherscanbegivendecisiontheoretic Inthelinearmodel,theanalogsofwidelyknownnon-Bayesianalphabeticaldesigncriteria
usedforpredictionandinmixedutilityfunctionsthatdescribeseveralsimultaneousgoals foranexperiment.Bayesianequivalentsofsomeotherpopularoptimalitycriteriacanalso bederivedbychoosingappropriateutilityfunctions.Some,butnotallofthealphabetical optimalitycriteria,haveautility-basedBayesianversion. 6

whendesigninganexperiment.Thismightbethecase,forexample,insettingslikereliability andqualitycontrolwherethefuturelevelofoutputhastobekeptontarget,orinclinical trialswhenitisimportanttoobtaininformationonhowpatientswillrespondtosome treatment.ForthesetypesofproblemsthepredictiveBayesianapproachisappropriatefor Therearecaseswherepredictionmightbeconsideredmoreimportantthaninference
bothdesignandanalysis.Foradetailedtreatmentofthistopic,seeGeisser(1993).
Ball,SmithandVerdinelli(1993)consideredthisproblemforthelinearmodelwithinthe unnecessaryforinferenceinaBayesianexperiment:itis\merelyuseful".Randomization isanimportantpracticalaspectofdesign,especiallyinclinicaltrials.Verdinelli(1990)and morespecicissues.ForexampleasarguedbyLindleyandNovick(1981)randomizationis Otherutilityfunctionscanbedevisedfordesigningexperimentsthattakeintoaccount
totheexperimenter.ThematrixXTXisdenotedbynManditisoftenreferredtoasthe theoryofBayesianoptimalexperimentaldesign.
informationmatrix,sincetheFisherinformationmatrixisequalto?2nM.Ifniobservations TherowsofX,xTj;j=1:::nareelementsofacompactsetXofdesignpointsavailable 1.4Notation
aretakenatthepointxi2X,thentheinformationmatrixcanbewrittenasnP(ni=n)xixTi Inthelinearmodelwithnindependentobservations,Xstandsforankdesignmatrix.
withPni=n=1.FollowingFedorov(1972,page62)andmanyotherauthors,denei=
notationsnMandnM()fortheinformationmatrix.Insomesituations,itmaybeof ni=nsonM=nPixixTi.Adesigncannowbeseenasaprobabilitymeasureonthe
interesttondexactoptimaldesignswheretheprobabilitymeasureissuchthat,fora regionXofdesignpoints.Itisusuallyconvenienttorelaxtherequirementfortheni'sto
speciedn,thevaluesniareallintegers. beintegerssothatthedesignproblembecomesthatofndinganoptimaldesignmeasure fromthesetofallprobabilitymeasuresonX;thissetisdenotedH.Wewilluseboth
by(1)and(2)inSection1.3,arepossible.Fornonlinearmodels,expectedutilitiesdonot haveaclosedformrepresentation.Approximationsarethereforerequired.Itisoftenstill Insomecases,usingalinearmodel,exactcalculationsforexpectedutility,U()asgiven 7

linearandnonlinearmodelsthedesignproblemcanbethoughtofaschoosingaprobability choosevaluesofthecontrolvariablesxj;j=1;:::;nfromacompactsetX.If,justasin measureoverXfromH.WewillseeinSections4,5and6thatdesignfornonlinear possible,however,toformulatetheprobleminasimilarway.Thedesignproblemisstillto
modelspresentssomechallenges.ABayesianapproachcanprovidepracticalinsightand leadtousefulsolutions. thelinearcase,wedenoteitobetheproportionofobservationsatapointxitheninboth
ter12,forsomeroundingalgorithmsanddiscussion).Withouttherelaxationtonon-integer referredtoasapproximateorcontinuousdesigns.Anapproximatedesigncanberoundedto Designswheretheproportionsarenotconstrainedtocorrespondtointegersforsomenare anexactdesignwithoutlosingtoomucheciency(seeforexamplePukelsheim,1993,Chap- Relaxingtherequirementfornitobeintegervaluesmakestheproblemmoretractable.
withtheconstraintsofincompleteblocks.Toman(1994)derivedBayesoptimalexactdesignsfortwo-andthree-levelfactorialexperiments,withandwithoutblocking.Oneofthe designsthedesignproblemisthatofahardintegerprogrammingproblem.Majumdar(1988,
importantproblemsshesolvedisthatofchoosingafractionofthefullfactorialdesign. 1992)derivedBayesianexactdesignsforatwowayanalysisofvariancemodelconsidering aspecialsubclassofpriordistributions.Thisisaparticularlyusefulapproachwhendealing
or\OptimalBayesiandesign".Oneofthemostpowerfultoolsforndingdesignsisthe GeneralEquivalenceTheorem(Kiefer,1959,Whittle,1973).Ofcoursetheremaybeother theexpectedutility.Thisformulationhasledtoaresearchareaknownas\Optimaldesign" taken.Subjecttothisconstraint,aprobabilitymeasureonXshouldbechosentomaximize Mostapproachestodesignassumethatthereisaxednumbernofobservationstobe
familiaronewithaxedsamplesize.ThisisappliedtoBayesianlineardesignproblems p.16)whoshowedthatasimplelineartransformationcanmodifytheproblemtothemore constraintssuchasaxedtotalcost,C,andeachobservationmaycostadierentamount
inChaloner(1982).Tuchscherer(1983)ndsBayesianlinearoptimaldesignsforparticular theoremcaneasilybeadaptedtodealwiththisextension.SeeforexampleCherno(1972, ci.TheproblemthenbecomestomaximizeutilitysubjecttoaxedcostC.Theequivalence
costfunctions. 8

alphabeticaldesigncriteriaareintroducedinSection2.2andareexaminedin2.3.Other designcriteriawithintheBayesiandecisiontheoryapproacharediscussedinSection2.4. 1.5StructureofthePaper Thecaseofunknownerrorvarianceisconsideredin2.5.Section3isdevotedtothesimple butimportantcaseofanalysisofvariancemodels.Theexamplesconsideredillustratethe Sections2and3ofthispaperdealwithdesignsforlinearmodels.Bayesiananalogsof
PropertiesofoptimalnonlinearBayesiandesignarediscussedinSection5.Forexampleit approaches.Localoptimalityisconsideredin4.4.Theapproximationsarecomparedin4.5. expectedutilityareinvestigatedin4.2.Section4.3dealswithsomeofthedierentBayesian eectofincorporatingpriorinformationinthelinearmodel.
isshownthatthenumberofsupportpointsinanoptimaldesignmaydependontheprior NonlinearmodelsareexaminedinSections4and5.Variouspossibleapproximationsto
distribution.Someexactresultsaregivenandtheavailablesoftwareisreviewed.Section 6considersafewotherspecicproblemsinnonlineardesignsuchassamplesizeinclinical trialsanddesignforreliabilityandqualitycontrol. 2Bayesiandesignsforthenormallinearmodel problems,suchasdesignforvariancecomponents,foramixtureoflinearmodelsandfor modeldiscrimination,arediscussedinSection8.Section9containsconcludingremarks. NonlinearproblemsgeneratedfromalinearmodelareconsideredinSection7.Additional
2.1Introduction parameters,2isknownandIisthennidentitymatrix.Supposethatthepriorinformation datayisavectorofnobservationswhereyj;2N(X;2I),isavectorofkunknown issuchthatj2isnormallydistributedwithmean0andvariance-covariancematrix2R?1, wherethekkmatrixRisknown.Recall,fromSection1.4,thatthematrixXTXisdenoted Considertheproblemofchoosingadesignforanormallinearregressionmodel.The
bynMor,equivalently,nM().Theposteriordistributionforisalsonormalwithmean vector=(nM()+R)?1(XTy+R0)andcovariancematrix2D()=2(nM()+R)?1; D()isafunctionofthedesignandofthepriorprecisionmatrix?2R. 9

2.2BayesianAlphabeticalOptimality:Overview
expectedKullback-Leiblerdistancebetweentheposteriorandthepriordistributions: thatmaximizestheexpectedgaininShannoninformationor,equivalently,maximizesthe 1986;Bernardo,1979)consideredtheexpectedgaininShannoninformationgivenbyan experimentasautilityfunction(Shannon,1948).Theseauthorsproposedchoosingadesign FollowingLindley's(1956)suggestion,severalauthors(Stone,1959a,b;DeGroot,1962,
Thepriordistributiondoesnotdependonthedesign,sothedesignmaximizingthe expectedgaininShannoninformationistheonethatmaximizes: Zlogp(jy;) p()p(y;j)ddy: (3)
Inthenormallinearregressionmodel ThisistheexpectedShannoninformationoftheposteriordistribution.Thisexpectedutility U1()mightbeappropriatewhentheexperimentisconductedforinferenceonthevector. U1()=Zlogfp(jy;)gp(y;j)ddy; (4)
ofM().Notethesymbol()isusedtodenoteadesigncriterionfunctionandU()isused itisknownasBayesD-optimality.Non-BayesianD-optimalitymaximizesthedeterminant Thisutilitythereforereducestomaximizingthefunction1()=detfnM()+Rgand U1()=?k2log(2)?k2+12logdetf?2(nM()+R)g:
todenoteanexpectedutilityfunction. apreviousdesign.Thatis,forD-optimalitychoosingtomaximizethedeterminantof (nM+XT0X0)whereXT0X0isxed,typicallyfromadesignobtainedpreviously.Thisis clearlyalgebraicallyidenticaltoBayesianD-optimalityandisdiscussedinCovey-Crump andSilvey(1970),Dykstra(1971),Evans(1979),MayerandHendrickson(1973),Johnson Inthenon-Bayesiandesignliterature,therearepapersdiscussingtheaugmentationof
andNachtsheim(1983)andHeiberger,BhaumikandHolland(1993). Avariationofnon-BayesianD-optimalityisDS-optimality,see,forexample,Silvey(1980 10

p.10-11).Thiscriterionmaximizestheinversedeterminantofthecovariancematrixfor theleastsquaresestimatorofalinearfunction lentBayesiancriterionisobtainedconsideringtheposteriordistributionof muchattentionhasbeengiventothiscriterionintheBayesianliterature,butitsuseis straightforward. =sToftheparameters.Theequiva-
interestisininferenceforandthatp()ischosentorepresentitsprobabilitydensity BayesianD-optimalitycanbederivedfromotherutilityfunctionsaswell.Assumethat in(4).Not
andwiththefunctionp()selectedasprobabilitydensityfunctionfor: function.Thefollowingutilityfunctionisassociatedwiththetruevalueoftheparameter
Thisutilityfunctionisaproperscoringrule,rstintroducedbydeFinetti(1962)fordiscrete thecontinuouscase.Spezzaferri(1988)adopted(5)fordesigningexperimentsformodel .Buehler(1971)proposeditsuseforelicitingbeliefsabout,bothinthediscreteandin U(;p();)=2p()?Zp2(~)d~: (5)
wheninterestisinestimationof,(5)reducesto discriminationandparameterestimation.Healsoshowedthatinthenormallinearmodel,
thusobtainingtheD-optimalitycriterion.Eaton,GiovagnoliandSebastiani(1994)alsouse utilityfunctionsbasedonproperscoringrulesforpredictionandalsoderiveD-optimality asaspecialcase. 2p?kfdet[nM()+R]g1=2;
throughthefollowingtwovaluedutilityfunction: AnotherjusticationofBayesianD-optimalitywasderivedbyTiaoandAfonja(1976)
where^denotesanestimatorforandaisanarbitrarilysmallpositiveconstant. U(^;;)=8>:0j^?ja; (6)
11

oftheparameters,oroflinearcombinationsofthem,aquadraticlossfunctionmightbe appropriate.Inthiscaseadesigncanbechosentomaximizethefollowingexpectedutility: Whenthespecicreasonforconductinganexperimentistoobtainapointestimate
whereAisasymmetricnonnegativedenitematrix.TheBayesprocedureyieldsasexpectedutilityU2()=?2trfAD()gandacorrespondingcriterion2()=?trfAD()g=trfA(nM()+R)?1g.Adesignthatmaximizes2()iscalledBayesA-optimal,agener- U2()=?Z(?^)TA(?^)p(y;j)ddy; (7)
orwhenminimizingthesquarederrorofpredictionatc,wherecisnotnecessarilyxed alizationofthenon-BayesianA-optimalitycriterion,thatminimizestrfAM()?1g.Thisbeusedforthiscriterion,derivedaboundonthenumberofsupportpointsinanopti-
andadistributionisspeciedonit.SeeOwen(1970),Brooks(1972,1974,1976,1977), andDuncanandDeGroot(1976).Chaloner(1984)showedhowanequivalencetheoremcan criterionalsoariseswhenminimizingtheexpectedsquarederrorlossforestimatingcT
foranalysisofvariancemodelswithtwo-wayheterogeneity.Toman(1992a)andToman andGastwirth(1993)dealtwithA-optimalityinarobustnesscontextandToman(1994) maldesignandpresentedsomeexamples.TomanandNotz(1991)consideredthiscriterion examinedA-optimalityforfactorialexperiments. ?2cTD()c;thisvariationofA-optimalityiscalledBayesc-optimalityanditparallelsthe non-Bayesianc-optimality.Thisoptimalitycriterionisalsoobtainedwhentheexpected squaredlossisusedforestimatingagivenlinearcombinationoftheparameters: wherecisxed.ABayesianmodicationofthegeometricargumentinElfving's(1952) AspecialcaseofA-optimalityiswhenrank(A)=1,thatisA=ccTandU2()=
(1991)andDette(1993a,b). theoremforc-optimalitywasgiveninChaloner(1984)andextendedinEl-KrunzandStudden Anextensionofthenotionofthec-optimalitycriterionisE-optimality,forwhichthe =cT
estimatesisminimized.AsaheuristicargumenttomotivateE-optimality,consideran maximumposteriorvarianceofallpossiblenormalizedlinearcombinationsofparameter experimenttoestimatethelinearfunction 12
=cT,forunspeciedc,withthenormalizing

constraintkck=w.Aminimaxapproachleadstosearchingforadesignthatisgoodfor dierentchoicesofc.DenotingthemaximumeigenvalueofamatrixHbymax[H],an E-optimaldesignminimizessup Thiscriterionappearsnottocorrespondtoanyutilityfunctionandso,althoughitisreferred toasBayesE-optimality,itsBayesianjustication,inadecisiontheoreticcontext,isunclear. CloselyrelatedtoBayesianE-optimalityisBayesianG-optimality.AG-optimaldesign kck=wcTD()c=w2max[D()]: (8)
1993,sect.11.6,thatstatesthatcontinuousG-optimaldesignsarenumericallyidenticalto acorrespondingcontinuousD-optimaldesign). tomaximizingautilityfunction(althoughthereisanequivalencetheorem,seePukelsheim, ischosentominimizesupx2XxTD()x.SimilarlytoE-optimality,thisdoesnotcorrespond
totheutility(6),thequadraticutilityin(7)andthefollowingexponentialutility: ingthebestofkparametersandofrankingtheparameters.Theyalsoproposed,inadditionTiaoandAfonja(1976)presentedotherutilityfunctionsaimedattheproblemsofselect- Theyconsideredtheproblemofchoosingamongaclassofbalanceddesignstoillustratethe useoftheaboveutilitiesandtoshowthatadesignoftenhastobeselectedfromalimited rangeofavailableones. U()=1?exp?2(^?)T(^?):
designcriteria.AcharacteristicofoptimalBayesiandesignmeasuresisthedependenceonthe samplesizen,sinceD()=n?1(M()+n?1R)?1.Thisidentityshowsthatanydierences betweenaBayesiandesignanditscorrespondingnon-Bayesianoneareunimportantifnis large,since,inthiscase,(M()+n?1R)isapproximatelyequaltoM().Thisisintuitively ItisimportanttorecallbrieythemainrelationsbetweenBayesianandnon-Bayesian
reasonable:inexperimentswherethesamplesizeislargetheposteriordistributionwillbe drivenbythedataandwillnotbesensitivetothepriordistribution.Incontrast,ifnis smallthepriordistributionwillhavemoreofaneectontheposteriordistributionandon 13

islittlepriorinformationavailable,optimalBayesiandesignsareclosetothecorresponding non-Bayesianones.Hence,whenanoninformativepriordistributionisusedforinference, asmayoftenbethecase,thereisnoadvantagetousingtheBayesianapproachfordesign. thedesign. Lettingn!1isequivalenttoR!0andasimilarlimitingresultisseen.Whenthere
beadaptedtoallowfordesignswheretheoptimalchoiceofnM()maybesingular.For optimaldesignsareagainspecialcasesofBayesiandesignbutcorrespondtoapointmass priordistributionratherthannoninformativeness.Thisisdiscussedfurtherinsection4. Thislimitingbehaviorisnotseenindesignfornonlinearmodelswhereusualnon-Bayesian
Bayesiandesigncriteria,nosuchadaptationisrequired.ThematrixRisnon-singularfor irrespectiveofwhethernM()aloneis. aproperinformativepriordistribution,sothematrixnM()+Risalwaysnon-singular Notealsothatnon-Bayesiandesigncriteria,suchasc-optimalityandDS-optimalitymust
statistics.Intheareaofexperimentaldesign,asetofpapersbyBrooks(Brooks1972, 1974,1976,1977)wereinspiredbyworkofLindley'sonthechoiceofvariablesinmultiple 2.3BayesianAlphabeticalOptimality:RelatedWork. regression(Lindley1968).BrooksfollowedLindley'sapproachtomotivatetheproblemof choosingthebestsubsetofregressorsandthedesignpointsinalinearregressionmodel. Inthe1970'sLindley'sworkhadaprofoundinuenceonmanyaspectsofBayesian
Predictingthefuturevalueofthedependentvariableisthegoaloftheexperimentand thepredictorisobtainedsubstitutingtheBayesianestimatorintheregressionfunction, ratherthanconsideringthepredictivedistributionforthefutureobservation.Aquadratic lossfunction,pluscosts,isusedtoevaluatethedierenceofthefuturevalueofyandits predictor.BayesA-optimalitywithaddedcostsisthedesigncriterionderived.Inhis1974 paper,Brooksalsolookedforoptimalsamplesizeusingthesamelossfunctionandinhis 1977paperhedealtwithdesignproblemswhencontrollingforthefuturevalueofytobe atapreassignedvaluey0.ThesettingconsideredinBrooks'earlypapersistoogeneralto allowformanyexplicitsolutionsandfewspecialcasesareexplored.Straightlineregression isexaminedinhis1976paper.Brooks'workcanbeseenasastatementofthegeneral 14

work(forexamplePilz,1991)whereBayesiandesigncriteriaareseenmostlyasextensions ofthecorrespondingnon-Bayesiancriteria,thefocusoftenbeingplacedinshowingthat principlethattheBayesianmethodhasawayfordealingwiththedesignproblem.Bayes
non-Bayesiancriteriaarelimitingcaseswhendiusepriorinformationisconsidered.See optimalitycriteriaareconsideredaselementsofaclassoflinearcriteria.Thislastfeature
alsoFedorov(1980,1981). showstheinuenceofFedorov's1972book.ItisalsofoundinPukelsheim(1980)andinPilz's
theproblemthatsubstitutesthevalueof2withitspriormeanwhereveritappearsin thenalexpressionofthecriterion.Thisapproachwasalsousedbyotherauthors,for Theydenedoptimalitycriteriawithoutadecisiontheorybasedframeworkandsohaveno exampleSinha(1970),Guttman(1971)andmorerecentlyPukelsheim(1993,chapter11). Brooksalsoexaminedthecaseof2beingunknownandusedthesimplesolutionto
clearextensiontothecasewhere2isunknown.Incontrast,withadecisiontheorybased 1979a,b,c,d,1981a,b,cNatherandPilz1980,GladitzandPilz1982a,b,Bandemer, framework,theextensiontothecasewhere2isunknownisconceptuallyeasybut,asis NatherandPilz1987).Seealsothemonograph,Pilz(1983)andtherevisedreprintofthe showninSection2.5,algebraicallyhard. monograph,Pilz(1991).Hisapproachisverygeneral,withnodistributionalassumptionsfor themodelorforthepriordistribution.PilzdenedBayesalphabeticaloptimalitycriteriaas PilzdealtwithBayesexperimentaldesignsforalinearmodelinaseriesofpapers(Pilz
admissibleandcompleteclassesofdesignstondconditionsfortheexistenceofBayesdesigns ofageneral\Linearoptimalitycriterion".D-optimalityandE-optimalitydonotfallinto thissetting,soPilzoftenderivedseparateresultsforthesecriteria.Themethodologyused throughoutPilz'workhastheavorofclassicaldecisiontheory.Forexample,heconsidered anextensionofthecorrespondingnon-Bayesiancriteriaandlookedatthemasspecialcases
inanadmissibleclass.Pilzalsoadaptedmuchoftheexistingtheoryonoptimaldesignto theBayesiancase.HeusedWhittle's(1973)generalversionoftheequivalencetheoremto alsoshowedthatundercertainconditions,Bayesalphabeticaldesignscanbeconstructed asA-optimaldesignsforatransformedmodel.Insomecases,A-optimalitycoincideswith ndrelationsamongthedierentdesigncriteriaandtondboundsforthedesigns.Pilz 15

D-andE-optimality,buttheconditionsunderwhichtheaboveholdsdonotseemeasyto satisfy.Pilzdidnotgiveexplicitdesignsandexaminetheirpracticalimplicationsandhis
estimation.Inqualitycontrolandinclinicaltrialspredictionoffutureobservationscanbe workissomewhatabstract. 2.4OtherUtilityFunctions ofspecialinterest.InthesecasestheBayesianapproachusespredictiveanalysiswhichcan alsobehelpfulindesigningtheexperiment.TheexpectedgaininShannoninformationon afutureobservationyn+1isusedratherthantheexpectedgainininformationonthevector Asnotedinsection1,incertainexperiments,predictioncanbemoreimportantthan
p(yn+1)(priorpredictive)onyn+1istheequivalentofthequantity(3)insection2.2.The p(yn+1jy;)=Rp(yn+1j)p(jy;)d(posteriorpredictive)andthemarginaldistribution ofparameters.TheexpectedKullback-Leiblerdistancebetweenthepredictivedistribution
theexpectedutility:U3()=Zlogp(yn+1jy;)p(y;yn+1j)dydyn+1: priorpredictivedistributiondoesnotdependonthedesignandthedesignthatmaximizes theexpectedgaininShannoninformationonyn+1isequivalenttothedesignthatmaximizes
experiments.Inthenormallinearmodel,maximizingU3()withrespecttocorresponds ThisutilityfunctionhasbeenusedbySanMartiniandSpezzaferri(1984)foramodel selectionproblemandbyVerdinelli,PolsonandSingpurwalla(1993)foracceleratedlifetest (9)
wherethenextobservationisgoingtobetakenatthepointxn+12X.Thisisequivalent tominimizingthepredictivevariance tomaximizing ?12nlog(2)+1+logh2xTn+1D()xn+1+2io;
Inthespecialcaseofpredictionofyn+1ataxedpointc=xn+1,thedesignmaximizing 2n+1=2[xTn+1D()xn+1+1]: 16

U3(X)correspondstotheBayesc-optimaldesignpresentedinsection2.2. sponsevariabley.Inthesecases,onemightbeinterestednotonlywithinferenceonthe parameters,butalsowithobtainingalargevalueoftheoutcome.Experimentationmight beconsideredonlyifthedesignproposedisexpectedtoproducealargevalueofoutcome aswellasalargevalueofinformation.Insuchcases,onepossibilityistolookforadesign YetanothersituationiswheretheexperimenterisconcernedwiththevalueoftherethatmaximizesacombinationoftheexpectedtotaloutputandtheexpectedShannoninfor-
expectedutility:U4()=ZhyT1+logp(jy;)ip(y;j)dyd: mationfortheposteriordistribution.VerdinelliandKadane(1992)proposedthefollowing
aectthechoiceofthedesignthroughtheratio=.AdesignmaximizingU4()isequivalent willingtoattachtothetwocomponentsofU4().Inthenormallinearmodel,theseweights Thenon-negativeweightsandexpresstherelativecontributionthattheexperimenteris (10)
toadesignmaximizingZyT1p(y)dy+2logdetfD()g: oftheexperimentisbothinferenceabouttheparametersandpredictionaboutthefuture observation.ItisgivenbyacombinationofU1()andU3(),namely: U5()=Zlogp(yn+1jy;)p(y;yn+1j)dydyn+1+!Zlogp(jy;)p(y;j)dyd:(11) Verdinelli(1992)suggestedtheuseofanotherexpectedutilityfunctionwhenthegoal
AsinU4(),theweightsand!expresstherelativecontributionofthepredictiveand thesameunits.InthelinearmodeltheexpectedutilityU5ismaximizedbyadesignthat theinferentialcomponentsoftheutility.Inthiscase,thetwocomponentsareexpressedin maximizes ?2nlog(2)+1+logh2xTn+1D()xn+1+2io?!2nklog(2)+k?logdet(?2D?1())o: 17

Thisisequivalenttominimizing2n+1detf2D()g,where2n+1isthepredictivevariance,
2.5UnknownVariance denedearlier.Itturnsoutthattheweightsand!donotaectthechoiceofthedesign. (1995).Sheexamineddesignwhenthepurposeoftheexperimentishypothesistesting. YetanotherformulationofthedesignproblemasadecisionproblemisgiveninToman
inducedbytheutilityfunctionsoftheearliersectionsmayneedtobemodied,although conceptuallythegoalofmaximizingautilityremainsthesame.Letthepriordistribution for(;2)beconjugateinthenormal-invertedgammafamily:j2N(0;2R?1)and ?2j;Ga(;),sothatp(2j;)/(2)?(+1)expf??2g.Thisimpliesthatboth Ifthevariance2inthelinearmodelofsection2.1isunknownthentheoptimalitycriteria
denotethequantity(2+n)?1n(y?X0)ThI?X(nM()+R)?1XTi(y?X0)+2oand degreesoffreedom,meanvectorandscalematrix(seeforexampleDeGroot1970,sec5.6 orBoxandTiao1973page117).Recallthat=(nM()+R)?1(XTy+R0).Leth(;y) thepriorandtheposteriormarginaldistributionsforaremultivariatetdistributions.
leta==.Thepriorandposteriormarginaldistributionsforare: Denotebyt[m;;]theprobabilitydistributionofanm-variatetrandomvariablewith
Thedistributionofyconditionalonaloneismultivariatet:yjt2[n;X;aI].In addition,themarginaldistributionofthedatayismultivariatet: t2hk;0;aR?1iand jy;t2+nhk;;h(;y)(nM()+R)?1i:
andtheposteriorpredictivedistributionforyn+1,anewobservationatxn+1,isunivariatet: yn+1jy;t2+n[1;xn+1;h(;y)fxn+1(nM()+R)?1xn+1+1g]. Evaluatingtheexpectedutilitiespresentedinsections2.2and2.4isnowamorecompli- yjt2hn;X0;a[I?X(nM()+R)?1XT]?1i;
catedtask.TheintegralsthatdeneU1;U3;U4andU5arenowintractablesincenoclosedmalapproximations(12)or(13)describedlater,insection4.2,areneededtondBayesianformexpressioncanbederived.Numericalapproachesorapproximations,suchasthenor- 18

designs. lettingA=I,theintegralin(7)reducestoRtrVar(jy)p(y)dywhereVar(jy)denotesthe posteriorcovariancematrixandp(y)isthemarginaldistributionofy.TheA-optimality criterionreducestondingadesignthatminimizes ThingsaresomewhatsimplerforA-optimalityandU2.IntheexpressionforU2(),
Theintegralintheaboveformulaisequalto[2n(2?1)?1+2]=(2+n),whichdoesnot dependony.HenceBayesA-optimalityisinsensitivetotheknowledgeof2andinthis senseitisarobustcriterionforchoosingadesign.SeealsoChaloner(1984).Thisfeature 2+n?2tr(nM()+R)?1Zh(;y)p(y)dy: 2+n
3Designforanalysisofvariancemodels distributionaldistancesareinuencedbythepriordistributionon2. 3.1Introduction ofA-optimalitymakesitappealingtouse.Itremainstobeseenhowdesigndevelopedfrom
welldenedoptimalitycriteriaforthelinearmodel.Thissectiondealswiththeimportant specialcaseofmodelsfortheanalysisofvariance.InthesecasescriteriafromSection2 sometimesallowthederivationofexplicitformsforoptimaldesigns.Twodierentwaysof buildingnormalpriordistributionsforthevectorareexamined.Bayesianoptimaldesigns Insection2,weshowedhowadecisiontheoreticsettingforexperimentaldesignleadsto
areconsideredwhenhaspriormean0andcovariancematrix2R?1,asinsection2.1. Inaddition,Bayesianoptimaldesignsunderahierarchicalpriordistribution,asinLindley andSmith(1972),arealsoderived.Thehierarchicalnormallinearmodelcanbeusedto representdierentexperimentalsettings.Agivencriterionlike,say,BayesD-optimality yieldsdierentdesignsforvariouschoicesofthehierarchicalstructurethatdescribesthe
thematrixnMissimplydiagfn1;n2;:::ntg,whereniisthenumberofobservationsinthe experiment. 3.2AnalysisofVarianceModels Intheonewayanalysisofvariancemodel,whentheeectsofttreatmentsareofinterest, 19

observationsniortheproportionsofobservationsi=ni=noneachtreatment. i-thgroup.Choosinganoptimaldesignforthismodelconsistsinchoosingthenumberof onewayanalysisofvariancemodelusingtheA-optimalitycriterion,denedinsection2.2. Inoneofthecasestheyexamined,oneofthettreatmentsisacontrolandthecontrastsof interestcomparethet?1treatmentstothecontrol. DuncanandDeGroot(1976)consideredtheproblemofBayesianoptimaldesignforthe
tothetreatmentsineachoftheblocks.Owen(1970)andGiovagnoliandVerdinelli(1983) nij,thenumberofobservationstakenonthei-thtreatmentinthej-thblock.Iftheblock sizeskjarexed,thisisthesameaschoosingtheproportionsij=nij=kjofunitstoassign treatmentsandbblocks.Thechoiceofadesignforthismodelisequivalenttothechoiceof Inthetwo-waycase,withthesecondfactorbeingablockingvariable,theremightbet
treatmentsisacontrolandtheparametersofinterestarethecontrastsofthetreatments consideredBayesiandesignsforthetwo-waymodelwithtreatmentsandblocks.Oneofthe withthecontrol.OwendealtwithA-optimalitywhileGiovagnoliandVerdinelliexamineda classofcriteriaproposed,inanon-Bayesiancontext,byKiefer(1975).Theclassisdened foraparameterp0asp=fk?1tr[D()]pg1=p.BayesianA-optimalityisaspecialcase whenp=1,BayesianD-optimalityresultswhenp!0andBayesianE-optimalitywhen p!1.Havingdenedthisclass,GiovagnoliandVerdinellithenfocusedonD-optimal designs.SimeoneandVerdinelli(1989)usednonlinearprogrammingtechniquestoderive E-optimalBayesdesignsforthesamemodel. 1995).DesignsformodelswithtwoblockingfactorswereexaminedbyTomanandNotz (1991),whomainlyconsideredA-optimalitycriterion,butalsopresentedsolutionsforDandE-optimality. BayesiandesignsforanalysisofvariancemodelswerederivedinToman(1992a,1994,
oft?1newtreatmentscomparedwithacontrolfori=2;:::;t.Assumethatthetreatment theonewayanalysisofvariancemodel.Let=(1;2;:::t)Trepresentthetreatment eectsandsupposetheexperimentisdesignedtostudythecontrastsi?1oftheeects 3.3Example1Continued FollowingDuncanandDeGroot(1976)letusnowconsidertheA-optimalitycriterionin
20

eectsiareindependentandnormallydistributedwithpriormeansiandvariance2i. Theuseofutilityfunction(7)forBayesianA-optimality,leadstoanoptimalproportionof
andontheithnewtreatment observationsonthecontrol1=max(0;1+Ptj=11=n2j i=max8

independentandidenticallydistributed.Theoptimalproportionofobservationsonthe control,1,dependsagainontheratios1=2=n21and2=2=n2.When1and2are
aboutthenewtreatmentsispreciseandnoobservationsneedtobetakenonthem. onthecontrol,justasinBayesianA-optimality.Similarlyif2islargethepriorinformation ispreciseknowledgeoftheeectofthecontrol,anditmaybeoptimaltotakenoobservations bothsmall,thenon-BayesianD-optimaldesignisobtained,thatplacesthesameproportions ofobservations1=tonthenewtreatmentsandonthecontrol.Incontrast,if1islarge,there
onthattreatmentintheposteriordistributionwillbefromthepriorinformation.Some experimentersmightwellndthisfeatureunappealing:somemightarguethatthisisnot evenanexperiment.Inimplementingsuchadesigntheassumptionisclearlycriticalthat thepriorinformationreallydoesrepresentaccurateinformationabouttheexperimental Whentheoptimaldesigntakesnoobservationsonatreatment,thentheonlyinformation
unitsinthisparticularexperiment.Thisisalwaysanimportantassumptiontoexamine, non-Bayesiandesign.Butofcourseitisinexactlythesecasesofprecisepriorinformation especiallywhentheoptimalBayesiandesignissodierentfromthecorrespondingoptimal
oninferenceandonyieldingalargevalueofthetotaloutput.Inthiscase,theoptimal or,equivalently,ofsmallplannedsamplesize,thatBayesianoptimaldesignscanimprove overnon-Bayesiandesignsifthecriticalassumptionholds.
suchthatif2?1Fthen1=1andif2?1Gthen1=0.Hencetheoptimal proportion1onthecontroldependsbothonthepriormeansandonthepriorvariances. Itcanbeshownthattherearetwothresholdvalues,FandG,functionsof1and2only, Similarresultsareobtainedwhentheutilityfunctionchosenis(10)andconcernisboth
3.4Hierarchicalformforthepriordistribution designdoesnottakeobservationsonthenewtreatmentsifthepriormean2ofthenew observationsaretakenonthecontrolif2islargecomparedto1. treatmenteectissmallcomparedwiththepriormeanofthecontroleect1.Similarlyno
Thebasicmodelconsistsofthreestages.Therststageisthesamplingdistributionand itisjusttheusualnormallinearmodelwithavectorofparameters,say,asdescribedin TheuseofahierarchicalnormallinearmodelismotivatedbyLindleyandSmith(1972). 22

section2.1.Thesecondandthirdstagestogetherareusedtomodelthepriordistributionfor throughonestageonly.Wenowconsiderpriordistributionsspeciedintwostages.The distributionofattherststageisexpressedthroughavectorofhyperparametersanda secondstageisaddedtospecifythedistributionofthehyperparameters. .Thelinearmodelsofearliersectionsareobtainedwhenthepriordistributionisexpressed
suchthattheyijareindependentandyijjiN(i;2); Forexampleintheonewayanalysisofvariancemodel,letthesamplingdistributionbe
therststageofthepriordistributionisthat,conditionalonsomevalue,theiare with2known.Torepresenttheinformationthatallthegroupeectsiaresimilar,then fromthesamedistributionijN(;2).Thesecondstageofthepriordistribution oftheparametersiissuchthattheiareexchangeable,butnotindependent.Thei'sare representstheuncertaintyin:forexampleN(0;!2).Thenthemarginaldistribution independentwithmean,andwiththesameknownvariance2.Thatistheiareasample positivelycorrelated,representingthattheyarebelievedtobesimilar.Evenif!2!1, suggeststhespecicationsfortherststageoftheprior,thatdescribetherelationship representingvaguepriorknowledge,thedistributionstillretainsacorrelationstructure. existingamongtheelementsof.Atthesecondstage,knowledgeislikelytobeweak, soitisnaturaltoexpressthisbyassumingadistributionforthehyperparametersthatis dispersed.Underthistypeofpriordistribution,themarginaldistributionofthedata,yis AsLindleyandSmith(1972,page7)remarked,itisthetypeofexperimentthatoften
formallythatofarandomeectsmodelratherthanaxedeectsmodel. optimality(4)andA-optimality(7)aredierentthanunderapriordistributionsetonly inonestage.Relativelylittleresearchhasbeendoneondesignwithahierarchicalprior distributionandmoreworkisneededinthisarea. Underthishierarchicalstructure,theBayesianoptimaldesigncriteriaderivedfromD-
23

3.5Example1Continued thepriordistributionissuchthat,conditionalon1and2,thecontroleectisnormally distributedwithmean1andvariance21,andthet?1treatmenteects(2;3;:::;t)are normalwithameanvector12,where1isa(t?1)vectorofone's,andavariancematrix 2I.Thevariances21and2areassumedtobeknown.Letthepriordistributionof1and Assumeagainthereisacontrolgroupandt?1treatmentgroups.Therststageof
apartfromthefactthat1and2arethoughttobedierent.Collapsingthetwo-stages givesasingularpriorprecisionmatrix?2R: 2beatandimpropertorepresentthatnotmuchisknownabouttreatmentsandcontrol,
whereJ=11T.ThematrixRissuchthatthemeanofthecontroleectisindependentof R= (t?1)n22640: 2 0:(t?1)I?J375; 0T
same,2say,with02(t?1)?1,andtheproportionofobservationsonthecontrolis themeaneectsofthenewtreatments,thatthenewtreatmentsmeansareexchangeable, butnotindependentofeachother,andthatthepriordistributionisnon-informativewith respecttothecontrol.Thesymmetrybuiltintothismodelissuchthatforanyoftheutility functionsofsection2,theoptimalproportionsofobservationsoneachnewtreatmentisthe chosen,forBayesianD-optimality,andinterestisoninferenceoneitherthevector= (1;2;:::;t)oronthecontrastsi?1;fori=2;:::;t,theoptimalproportionof observationsonthecontrol1,canbeexpressedasafunctionoftheratio=2=n2. 1=1?(t?1)2.
When!1thepriorvarianceforthenewtreatmentsissmallcomparedwiththeerror Inthiscasetoo,dierentdesignsaregeneratedfromdierentutilities.When(4)is
treatments.Intuitively,tocomparetwoindependenttreatments,theobservationswould halfoftheobservationsareonthecontrol,andtherestareequallydividedamongthenew variance2=n.Thisimpliesthatthenewtreatmentsarebelievedtobeverysimilarto eachother(whichmightoftenbethecaseinpractice)andlim!11()=1=2.Hence 24

edividedequallybetweenthetwoandthisis,essentially,thesituationwhen!1.In
on.When!1andthenewtreatmentmeansarebelievedtobeverysimilarthen, independent,andalltreatments,controlandnew,getanequalallocationofobservations. 0,thanlim!01()=1=t.Exchangeabilityinthiscasereducestothenewtreatmentsbeing contrast,ifthepriorvarianceforthenewtreatmentsislargecomparedwith2=n,thatis!
forthislimitingcase,allalphabeticallyoptimalcriteriacoincide.Incontrast,whenthenew treatmentmeansareindependent,representedby!0,wegetagainthesquarerootrule lim!11()=1=2.ThissolutionisthesamefoundforD-optimalitywhen!1.Infact, Thedesignderivedfrom(7),issuchthattheA-optimalproportion1alsodependsonly
giveninSection3.2.
aresponsecurvedescribedbyalowdegreepolynomial.Hence Verdinelli,1980)wherethepriormeansofeachgroupareformedtostudytheeectsof thepriordistributionintwostages.Attherststagethepriormeansofeachgroupareon increasinglevelsz1;z2;:::zkofagivendrug.Thisexperimenttoocanbemodeledspecifying AsmentionedinSection1.2,letusconsidernowadierentexperiment(Smithand
wherei=1;2;:::;t;r

thepolynomialattherststage,asrepresentedbytheratiobetweensampleandprior
0.0 0.08
0.0 0.10 0.20
0.0 0.15
0.0 0.2
delta=0.00
delta=0.25
delta=0.50
0.0 0.06
0.0 0.10
0.0 0.15
delta=0.00
delta=0.25
delta=0.50
delta=1.00
delta=1.00
variances.Theoptimalproportionsofobservationsonthetgroupsvaryfromthenon- Figure1
delta=3.00
delta=3.00
BayesianD-optimaldesignfortheonewaymodeli=1=t,whenissmall,tothenon- LeftD-optimalproportionsfork=7groupsandastraightlineattherststageoftheprior
BayesianD-optimaldesignsforthepolynomialchosenwhenislarge.Thislastcasecor- RightD-optimalproportionsfork=9groupsandaparabolaattherststageoftheprior
respondstoassumingastrongpriorknowledgeaboutthepolynomialrelationshipforthet
delta=6.00
delta=6.00
groups,whilenotconsideringtheone-waystructureparticularlyrelevant. 26
0.0 0.2 0.4
0.0 0.2 0.4
0.0 0.15 0.30
0.0 0.15
0.0 0.2

4Nonlineardesignproblems 4.1Introduction expectedutilitybutapproximationsmusttypicallybeusedastheexactexpectedutilityis oftenacomplicatedintegral.Designscanstillbedenotedbyaprobabilitymeasureover thecoecientsofalinearmodelisofinterest.Suchproblemsarereferredtoas\nonlinear designproblems".Itwillbeshownthatthedesignproblemcanbeformulatedasmaximizing Designismoredicultwhenthemodelisnotlinearorwhenanonlinearfunctionof
thedesignspaceXandthesetofallsuchmeasuresbedenotedH.Themeasuresmaybe arbitraryprobabilitymeasuresrepresentingapproximate,orcontinuous,designs,ormeasures 4.2Approximationstoexpectedutility correspondingtoexactdesignswhichhavemass1=nonn,notnecessarilydistinct,points.
(1985,p.224),andinvolveeithertheexpectedFisherinformationmatrixorthematrixof posteriordistribution.Severalnormalapproximationsarepossible,seeforexampleBerger secondderivativesofthelogarithmofeitherthelikelihoodortheposteriordensity.The expectedFisherinformationmatrixforamodelwithunknownparameters,adesign Mostapproximationstoexpectedutilityinvolveusinganormalapproximationtothe
intheprevioussectionsonlineardesign,isaveryspecialcaseofI(;),whereI(;) doesnotdependon.Forconsistencywiththeliterature,andtoemphasizethatI(;) isnotnecessarilyamomentmatrix,thisseparatenotationisusedforlinearandnonlinear problems. andasamplesizeofnisdenotedbynI(;).Notethatthematrixofmoments,M,used
be: Let^denotethemaximumlikelihoodestimateof.Onenormalapproximationmight
nativeapproximationis:In(12)theposteriornormalapproximationonlydependsonthedatathrough^.Analter- jy;N(^;[R+nI(^;)]?1) jy;N(^;[nI(^;)]?1): (12)
where^nowdenotesthemodeofthejointposteriordistributionof(alsocalledthegener- 27
(13)

thepriordistribution. secondderivativesofthelogarithmofthepriordensityfunction,ortheprecisionmatrixof alizedmaximumlikelihoodestimateofasinBerger,1985,p.133),andRisthematrixof
theexpectedinformationmatrix,almostalwaysgivesabetternormalapproximationtothe theremaybereasonstopreferoneapproximationtoanother,andtheobserved,ratherthan andvarianceasthemeanandvarianceoftheapproximatingnormaldistribution,orusing theobservedratherthanexpectedFisherinformationmatrix.Althoughinspecicproblems Severalotherapproximationsarepossible,forexampleusingtheexactposteriormean
theexpectedFisherinformationmatrixisusuallyalgebraicallymuchmoretractable.Using approximationsotherthan(12)and(13)isanareaforfutureresearch. posteriordistribution,ingeneralthereisnoobviouslybestonetouse.Fordesignpurposes U1()isgivenbyequation(4),asinthelinearmodel.U1()istheexactexpectedutility, model whichinvolvesp(yj),themarginaldistributionofthedataforadesign.Asinthelinear If,forillustration,Shannoninformationisthechoiceofutilitythentheexpectedutility
utilityonlydependsonythroughsomeconsistentestimate^afurtherapproximation,of thesameorderas(12)and(13),istotakethepredictivedistributionof^tobetheprior Inmostcasesthismarginaldistributionofymustalsobeapproximated.Whentheposterior p(yj)=Zp(yj;)p()d:
distribution.Usingthisapproximationtogetherwith(12)givesanapproximatevalueof U1(): AsinearliersectionsU()willbeusedtodenoteexactexpectedutilityand()adesign criterion.Theconstanttermsandmultiplierin(14)canbedroppedtogive ?k2log(2)?k2+12ZlogdetfnI(;)gp()d: 1()=ZlogdetfnI(;)gp()d (14) (15)
28

asadesigncriterion.Similarlythedesigncriterionderivedusing(13)gives:
squarederrorlossisappropriate,sothattheutilityisU2()in(7).Denethekvectorc() tobethegradientvectorofg().Thatis,theithentryofc()is: Supposenowthattheonlyquantitytobeestimatedisafunctionofthecoecientsg()and 1R()=ZlogdetfnI(;)+Rgp()d: (16)
Then,using(12),theapproximateexpectedutilityis 2()=?Zc()TfnI(;)g?1c()p()d: ci()=@g() @i: (18) (17)
AslightlydierentapproximationinvolvingRisgivenwhen(13)isused: Shouldmorethanonefunctionofbeofinterest,thetotalexpectedlossisthesumof theexpectedlossesforallthenonlinearfunctions.Thissumcouldbeaweightedsumto representsomefunctionsbeingofmoreinterestthanothers.IfthematrixA()isthesum, 2R()=?Zc()TfR+nI(;)g?1c()p()d: (19)
orcorrespondingweightedsum,oftheindividualmatricesc()c()Tthentheapproximate expectedutilityis withasimilarexpressioninvolvingthematrixRif(13)isused.Criteria(15),(18)and (20)willbereferredtoasBayesianD-optimality,Bayesianc-optimalityandBayesianAoptimalityrespectively. 2()=?ZtrfA()[nI(;)]?1gp()d (20)
normalityitisappropriatetodesigntoensurethat,withhighprobability,theposterior mizingthecriteriadiscussedabovesubjecttosomeconstraintsthathelpensurenormality.distributionis,approximately,normal.Shesuggestedseveralapproaches,includingmaxi- Clyde(1993a)suggestedthatastheseBayesiandesigncriteriaarebasedonapproximate 29

aretypicallysatisedbythedesignmaximizingtheBayesiancriterion.Shealsolookedat TheconstraintssheusedaredevelopedfromtheideasofSlate(1991)andKassandSlate
approximatenormality.HamiltonandWatts(1985)andPazmanandPronzato(1992a,b), otherwaysofcombiningthetwoobjectivesofmaximizingapproximateutilityandattaining areactivebutforlargesamplesizesposteriornormalityismorelikelyandsotheconstraints (1994)whogavediagnosticsforposteriornormality.Forsmallsamplesizestheconstraints
tookarelatednon-Bayesianapproach.
4.3Bayesiancriteria MarkovChainMonteCarlomethodsbuttheeectivenessofthissuggestioninspecicproblemshasyettobedemonstrated. MullerandParmigiani(1995)suggestedestimatingtheexactexpectedutilityusing
of(19),thatis: cientandunknownLD50,denotedby.Thecriterionhemaximizedwastheunivariatecasecriteria.Tsutakawaconsideredaoneparameterlogisticregressionwithknownslopecoe- areTsutakawa(1972)andZacks(1977).TheybothusedthematrixI(;)intheirdesign SomeoftheearliestpapersputtingdesignfornonlinearmodelsinaBayesianperspective
wheretheintegrandisascalar.Tsutakawanumericallyfounddesignsmaximizing(21), restrictingthedesignstoequallyspaceddesignpointswithequalnumbersofBernoulli observationsateachdesignpoint.HegavetheargumentsofSection4.2tojustify(19).In ()=?ZfR+nI(;)g?1p()d: (21)
alaterpaper(Tsutakawa1980),heextendedsimilarideastodesignfortheestimationof otherpercentileresponses. tialfamilywithknownscaleparameterandwheresomefunctionofthemeanislinearin formforthesemodelsandZacksconsidereddesignsthatmaximizetheexpectedvalueofthe anexplanatoryvariable.Thisclassofgeneralizedlinearmodelsincludesquantalresponse modelsandmodelsforexponentiallifetimes.TheFisherinformationmatrixhasacommon Zacks(1977)consideredproblemswherethedataaretobesampledfromanexponen-
determinantofI(;),thatis: 30

asanapproximationtotheexpectedutility(4). Zacksexaminedseveralexamplesandalsofoundoptimalmultistagedesignsforquantal responseexperiments.Notethatthecriterionin(22),unlike(15),isnotreadilyinterpretable ()=ZdetfnI(;)gp()d: (22)
\robustexperimentaldesign",wasdevelopedintheeldofpharmacokineticsandbiological proceduresaredescribedinWalterandPronzato(1985)andPronzatoandWalter(1985, modeling.ThiswasinitiallydevelopedwithoutmentionofanyBayesianmotivation.These 1987,1988),seealsoLandaw(1982,1984).Thisworkalsorelatestoworkondynamic AsimilarapproachtothatofTsutakawaandZackstodesignfornonlinearmodels,called
systemsasinMehra(1974)andGoodwinandPayne(1977). suchas: Inadditiontousingthecriterion(22)PronzatoandWalteralsousedseveralothercriteria
Theyalsodiscussedandderiveddesignsbasedonminimaxcriteria. ()=?Z[detI(;)]?1p()d: ()=detIZp()d;: (23)
macokineticandbiologicalmodels.AfeaturewhichmakesthesemethodsdierentisthatThereisarichrelatedliterature,mostlynon-Bayesian,ondesign,forcomplexphar- (24)
LaunayandIliadis(1988),MalletandMentre(1988),D'ArgenioandVanGuilder(1988), andD'Argenio(1983),thisimportantworkisnotinthemainstreamstatisticsliteraturebut ThomasethandCobelli(1988)andD'Argenio(1990).Withafewexceptions,suchasKatz oftenallowancesaremadeforinter-andintra-subjectvariability.Anotherfeatureofsuch
inthescienticliteratureofpharmacokineticsandmathematicalbiology. modelsthatisoftenusedisnon-constanterrorvariance.Furtherreferencescanbefoundin
butionof^byaonepointdistribution.Theonepointwouldrepresenta\bestguess".This 4.4Localoptimality Acrudeapproximationtoexpectedutilitywouldbetoapproximatethemarginaldistri- 31

distributionontheparametervalues.DraperandHunter(1967a)extendtheworkofBox approach,knownaslocaloptimality,hasbeenusedextensivelyinnonlineardesignandis
andLucas.White(1973,1975)showedhowresultsfromlineardesigntheorycanbeadapted usedlocaloptimality,BoxandLucassuggestedextendingthisbytakingintoaccountaprior duetoCherno(1953,1962).ItisalsousedinthepioneeringpaperofBoxandLucas(1959)
toapplytolocaloptimalityinnonlinearmodelsandshealsoderivedlocallyoptimaldesigns wheretheimportantissuesindesignfornonlinearregressionwereidentied.Althoughthey
asbeingapproximatelyBayesianalthoughitistypicallynotjustiedinthiswayandis forbinaryregressionexperiments.
LocalD-optimalityinvolveschoosingthedesignmaximizing usuallyusedinanon-Bayesianframework. Aslocaloptimalityisaverycrudeapproximationtoexpectedutility,itcanbeconsidered Theexperimenterisrequiredtospecifyabestguess,0fortheunknownparameters.
foraxedvalue0.Similarly,localc-optimalityistochoosetomaximize: 20()=?cT(0)I(0;)?1c(0)=?trA(0)I(0;)?1 10()=detfI(0;)g: (25)
whichcanclearlybegeneralizedtolocalA-optimality.Asin(18)and(19)thevectorc(0) isthegradientvectorofthefunctionofinterest,evaluatedat0.Typicallyc(0)dependson 0asdoesthematrixA(0)=c(0)c(0)T.Ifmorethanonefunctionoftheparametersisof (26)
4.5Comparisonoftheapproximations Toourknowledgeversionsof(25)and(26)involvingthematrixRhavenotbeenused. interestthenthematrixA(0)isthe,possiblyweighted,sumofmatricescorrespondingto theindividualfunctions.Theweightsaretherelativeimportanceofeachnonlinearfunction.
criteria(15)and(16)and,forBayesianc-optimalityandminimizingsquarederrorloss,(18) becompared.ForBayesianD-optimalityandmaximizingShannoninformationwecompare Thevariouswaystoapproximate(1)presentedearlierandtheirimplicationswillnow 32

and(19).Thesecriteriaareasymptoticapproximationsofthesameorder.Severalaspects dodistinguishthem.Thecriteria(16)and(19) giveoptimaldesignswhichdependonthesamplesize requirethespecicationofR.
Thecriteria(15)and(18)alternatively, avoidtechnicalproblemsusingpriordistributionswithunboundedsupportwhere,for
canbeinterpretedasaprocedurewhereadierentpriordistributionwillbeusedfor discussedinTsutakawa,1972). adesignwithboundedsupport,I(;)maybearbitrarilyclosetobeingsingular(as
theanalysisthanwasusedinthedesignstage.Anoninformativepriordistribution willbeusedintheanalysis,hencegivingRidenticallyzero,butallavailableprior informationwillbeusedinthedesignprocessandaninformativep()willbeusedto
forsimilarreasonsthesecriteriaareappealinginanon-Bayesianframeworkwhereit averageoverintheintegral.ThisechoestheideagiveninTsutakawa(1972)ofusing dierentpriordistributionsfordesignandforanalysis.(SeealsoEtzioneandKadane, 1993).
sizes,orforcaseswherethematrixRcorrespondstoimpreciseinformation,therewillbe Forthesereasonsweprefer(15)and(18)over(16)and(19).Butnotethatforlargesample isacceptedthatpriorinformationmustbeusedindesignbutshouldnotbeusedin
verylittledierencebetweenthetwosetsofcriteria. theanalysis.IndeedthisisthemotivationofPronzatoandWalter(1985,1987).
orthesecondderivativeofthelogarithmoftheposteriordonotappeartohavebeeninformempirically.Foraspecialcaseofexample3,Atkinson,Chaloner,HerzbergandJuritzvestigatedandmightgivebetterdesigns,especiallyforsmallsamples.Similarlynotmuchisknowningeneralabouthowwellthesecriteria,whichapproximateexpectedutility,per- Versionsofthesecriteriausingtheobservedratherthanexpectedinformationmatrix,
33

(1993)showedbysimulationthattheBayesiancriteriadowellempirically.Clyde(1993) alsopresentedsomesimulations.ArecentpaperbySun,TsutakawaandLu(1995)showed bysimulationthatthenumericalapproximationofTsutakawa(1972)fordesignintheone parameterlogisticregressionexampleisremarkablyaccurate.
notalwaysbeenfullyunderstood.ForexampleAtkinsonandDonev(1992)present\Five nosingleapproachcancomfortablybelabeledasthedenitive\Bayesiannonlineardesign criterion".Thecriteriaderivedinthissectionareallapproximationstotheideal.Thishas 4.6Discussion
versionsofBayesianD-optimality"inTable19.1.Theyexplainthatthe(15)corresponds Apartfromtheidealapproachofmaximizingexactexpectedutilitypreciselyasinsay(1),
to\pre-posteriorexpectedloss"butdonotexplainthatitisShannoninformationasutility ratherthanloss,anditisanapproximation.
suchasthosegivenby(15)whicharetheexpectation,overapriordistributionofalocal 5.1Introduction 5OptimalnonlinearBayesiandesign
optimalitycriterion.Werefertosuchcriteriaas\Bayesiandesigncriteria".Thesedesign criteriaareconcaveonH,thespaceofallprobabilitymeasuresonX.Subjecttosome regularityconditions,anequivalencetheoremcanbederived.Theequivalencetheoremwas Chaloner(1987)andChalonerandLarntz(1986,1988,1989)developedtheuseofcriteria
statedbyWhittle(1973)inthecontextoflineardesignproblems,butitsapplicationto nonlinearproblemswasnotthenapparentandtheregularityconditionsrequiredforitsuse
designcanbefoundusingnumericaloptimizationandthetheoremmakesiteasytocheck inthenonlinearcasenotstated.SeealsoLauter(1974,1976)andDubov(1977).The
whetherthecandidatedesignisindeedgloballyoptimaloverH. theoremstatesthat,inordertoverifythatadesignmeasureisoptimal,itisnecessaryonly tocheckthattheappropriatedirectionalderivativeatthatdesignmeasure,inthedirectionof allonepointdesignmeasuresiseverywherenonpositive.Acandidateoptimalapproximate Thetheoremappliestoanycriterionthatisanaverage,overapriordistribution,ofa 34

localoptimalitycriterionconcaveonH.Mostofthecriteriaincommonusage,including thosegivenby(15),(16),(18),(19)and(20)satisfythiscondition. measureis,whenthelimitexists: Foracriterion()thederivativeatadesignmeasureinthedirectionofanother
d(;x). denotedx.Thedirectionalderivativeof()inthedirectionxisD(;x)andisdenoted TheextremepointsofHarethemeasuresputtingpointmassatasinglexinXandare D(;)=lim#01[f(1?)?g?()]:
wherekisthedimensionof. Forexample()denedby(15),BayesianD-optimality,thederivativeis:
thereisatleastonedesignsuchthat()isnite,that()iscontinuousonHinsome Regularityconditionsthataresucientfortheequivalencetheoremtoholdarethat d(;x)=ZtrI(;x)I(;)?1p()d?k;
topologysuchasweakconvergence,andthatthederivativesd(;x)of()existandare continuousinx.
easilyderived. denition.ABayesianDs-criterionanditscorrespondingequivalencetheoremcanalsobe forwardunderthegeneralapproachofmaximizingexpectedutility.ForBayesianc-optimalityandA-optimalitynoextensionisrequiredasnuisanceparametersareinherentintheirTheextensionofBayesiancriteriatosituationsinvolvingnuisanceparametersisstraight- fortheminimumnumberofpointsinanoptimaldesign.Thisisdiscussedinthefollowing section. 5.2Numberofsupportpoints overanitedimensionalspaceandsotheequivalencetheoremdoesnotprovideanybound Unlikeinlinearproblemsthecriterionfunction()isnotnecessarilyaconcavefunction
Inmostnon-Bayesianlinearproblemsanupperboundonthenumberofsupportpoints 35

theboundreliesonthefactthatthematrixMdependsonlyontherstfewmomentsof numberofunknownparametersandthedesigntakesanequalnumberofobservationsat inanoptimaldesignisavailable,seePukelsheim(1993,p.188-9).Forlinearmodelsderiving thedesignmeasureandCaratheodory'stheoremisused.TheD-optimalitycriterionin eachpoint(Silvey,1980,p.42,andPukelsheim,1993,section9.5forpolynomialmodels). linearmodelstypicallyleadstoanoptimalnumberofsupportpointsthatisthesameasthe ertiesarereadilyexamined.Theyarenotveryappealinginpractice,however,astheydo notallowforcheckingofthemodelaftertheexperimentisperformed. Designsonasmallnumberofsupportpointsareeasytondandtheirtheoreticalprop-
modelsthereisnosuchboundavailableonthenumberofsupportpoints.Althoughthe onanitedimensionalmomentspaceandsoCaratheodory'stheoremcannotbeinvoked. criteriaareconcaveonH,thespaceofprobabilitymeasures,theyarenotconcavefunctions models(see,forexampleCherno,1972,p.27andChaloner,1984).Incontrastfornonlinear TheboundalsoappliestomostlocaloptimalitycriteriaandBayesiancriteriaforlinear
pointsinanoptimalBayesiandesignincreasesasthepriordistributionbecomesmoredispersed.Theyfoundthatforpriordistributionsthathavesupportoveraverysmallregion theBayesianoptimaldesignsarealmostthesameasthelocallyoptimaldesignandthey havethesamenumberofsupportpointsasthenumberofunknownparameters.Formore ChalonerandLarntz(1986,1989)gavetherstexamplesofhowthenumberofsupport
canbecheckedwithdatafromtheexperiment.ThisisdiscussedfurtherinSection8.5. dispersedpriordistributionstherearemoresupportpoints.Thisisausefulfeatureforade-
Ridout(1994),Chaloner(1993)andAtkinson,Chaloner,JuritzandHerzberg(1993). signas,iftherearemoresupportpointsthanunknownparameters,themodelassumptions
5.3ExactResults xedcanbefoundinAtkinsonandDonev(1992),O'BrienandRawlings(1994a,b,c), OtherexamplesofBayesiannonlineardesignswherethenumberofsupportpointsisnot
(1992)andWu(1988).Foraparticularvalueoftheunknownparameterstheproblemoften forexampleWhite(1975),Kitsos,TitteringtonandTorsney(1988),Ford,TorsneyandWu Forlocaloptimalitythereareseveralpapersderivingclosedformexpressionsfordesigns: 36

educestoanequivalentlinearproblem. tionstothisaresimplespecialcases:thesecasesarenotveryusefulinpractice,buttheyBayesiandesigncriteriarequiresthatdesignsbefoundbynumericaloptimization.Excep- giveinsightintopropertiesoftheoptimaldesignsformorerealisticandpracticalsituations. Exact,algebraicresultsarequitediculttoderiveasnoneofthetoolsfromlocaloptimality FindingoptimalBayesiandesignsalgebraicallyismuchharderandthusimplementing
areveryhelpful. withonlytwosupportpoints,itispossibletoexamineexactlyhowthetransitionfrom aonepointoptimaldesigntoatwopointoptimaldesignoccursasthepriordistribution ischanged.MukhopadhyayandHaines(1995),DetteandNeugebauer(1995a,b),Dette andSperlich(1994b)andHaines(1995)allconsideredsomenonlinearregressionproblems InChaloner(1993)forexample,inaoneparameterproblem,withpriordistributions
isofaparticularform.Looselyspeakingtheseresultscanbegeneralizedtosaythatifthe priordistributionisnottoodispersedanddoesnothaveheavytailsthenanoptimalBayesian designhasthesamenumberofsupportpointsasthereareunknownparameters.Haines involvinganexponentialmeanfunction,andgaveconditionsunderwhichtheoptimaldesign
distributionwithnitesupport,theproblemreducestoaparticularconvexprogramming problem. 5.4DesignSoftware (1995)gaveaninsightfulgeometricinterpretationofthisanddemonstratedhow,foraprior
softwaremustbereadilyavailable.ChalonerandLarntz(1988)describesuchsoftwarefor designsystemistheobject-orientedenvironmentofClyde(1993b),developedwithinXLISP- logisticregression.ThesearemenudrivenFORTRANprogramsthatareeasytouseand compileandareavailablefromtheauthorsbyemail.AmorepowerfulandexibleBayesian ItisclearthatifBayesiandesignsfornonlinearproblemsaretobeusedinpracticethen
STAT(Tierney,1990).Thissystemenablesbothexactdesignsandapproximatedesign measurestobeeasilyfoundforbothlinearandnonlinearproblems.Locallyoptimaldesigns andnon-BayesianlineardesignscanalsobefoundasaspecialcaseofBayesiandesigns.The systemalsoallowsforconstraintsintheoptimizationprocessassuggestedinClyde(1993a). 37

eality.Theavailabilityofthissoftwaremakesitstraightforwardtoderivedesignsfora Thispowerfulsoftwareenvironmentisalittlediculttouseinitiallybutcanbeeasily varietyofpriordistributions,modelassumptionsandcriteriaandsoexaminerobustness. Thesoftwareisavailablefromtheauthor(byemail)andrequirestheNPSOLFORTRAN adaptedtosolveamultitudeofdesignproblemsandmakesBayesiandesignaverypractical
isdescribedinClyde(1993b). libraryofGilletal(1986)tobeloaded.Documentation,installationandavailabilitybyftp Gibbssampler. forproceduresroundingcontinuousdesignmeasurestoexactdesigns. quiredthenPukelsheimandRieder(1992)andPukelsheim(1993,p.424)canbeconsulted Warner(1993)describessomeothersoftware,whichwehavenotexamined,usingthe
5.5Sequentialdesign Whensoftwareprovidesacontinuous(approximate)designandanexactdesignisre-
isnothingtobegainedbydesigningsequentially.Thisiseasilyseenwhentheerrorvariance asaxeddesignprocedure.Inmostlineardesignproblems,however,bothBayesianand non-Bayesian,theoptimalsequentialprocedureisthexed,non-sequentialprocedure.There 2isknown:theposteriorutilitydependsonthedesign,butdoesnotdependonthedata Inanydesignproblemanoptimalsequentialdesignproceduremustbeatleastasgood
y.Forthecasewhen2isunknownitisnotsoclear.ForA-optimalityand2unknown thedatay,orafunctionofysuchas^,andthereshouldbeagainfromchoosingdesign sequentialdesignisbetter.Fornonlinearproblemstheposteriorutilityclearlydependson withaconjugatepriordistribution,theanalysisofsection2.5showsthatthereisnothingto begainedbysequentialdesigninthiscase.Forotherlinearproblemsitisunclearwhether
doseforeachoneofthe60animalscouldbedecidedupononeatatimeandtheextensive pointssequentially. statisticalliteratureonsequentialdesignofbinaryresponseexperimentsconsulted(seefor experimentsofexample2doneintheUniversityofMinnesotalaboratory.Theoreticallythe Sequentialdesign,however,maybeunrealisticinpractice.Considerforillustrationthe
38

exampleWu,1985).But 2.timetrendsorseasonaleectsmaybeintroducediftheexperimentalconditionschange 1.asdeathoverthe7daysfollowinginjectionofthedrugistheresponse,theexperiment overtime.Similaranimalsmightnotalwaysbeavailableandthedrugsdeteriorate overtime. wouldbeprolongedfromatotalof7daystomanymonths.
3.theprobabilityoferrorindosesandcalculationsisincreasedwhen60calculationsare donetodeterminethenextdose.Anon-sequentialprocedureiseasilyimplemented
solvedtheBayesiansequentialdesignproblemexactlyforaverysmallandsimplebinary Kuo(1983)whodevelopsproceduresfornonparametricbinaryregression.Freeman(1970) exampleBerryandFristedt(1985)whoreviewedtheextensiveworkinbanditproblemsand SeveralpowerfulsequentialBayesiandesignprocedureshavebeendeveloped:see,for andrequireslesstrainingoflaboratorysta.
5.6Discussion regressionexperiment.Wedonotattempttoreviewthisworkhere. thattheexperimentalconditionsfromonebatchtothenextmightbedierent. andRidout,1995)mightprovetobemorepractical.Thereisapracticalconcern,however, Batchsequentialproceduresratherthanfullysequentialprocedures(asinZacks,1977,
eredfornonlineardesignas,unlikeinalinearmodel,theposteriorutilityofadesigndepends onthedata.Anexperimentermaybewillingtospecifyaninformativepriordistribution indesigningtheexperimentbutmayprefertouseanoninformativepriordistributionfor inference. Whatevercriterionisused,Bayesianornon-Bayesian,priorinformationmustbeconsid-
6.1Binaryresponsemodels 6Specicnonlineardesignproblems Flournoy(1993),andClyde,MullerandParmigiani(1994)alluseBayesiandesignideas Tsutakawa(1972,1980),Owen(1975),Zacks(1977),ChalonerandLarntz(1989), 39

inbinaryregressionmodels.Thesemodelsareimportantandhavemanyapplicationsin toxicologyandreliabilitystudies.Theyarealsointerestingfromadesignperspectivebe-
extreme.Incontrastconsiderabinaryregressionwithabinaryresponsevariablewhich causetheyaresoverydierentfromlinearregressionmodels.
is,\success"or\failure".Supposethattheprobabilityofsuccessnearoneextremeofthe istotakehalftheobservationsatoneextremeoftheintervalandtheotherhalfattheother isstraightforwardtoshowthatthelinearD-optimaldesign,underavaguepriordistribution, Consider,forexample,asimplelinearregressionmodelandacloseddesignintervalX.It
designintervalXiscloseto0andattheotherextremeitiscloseto1.Adesignthat putsallobservationsatthetwoextremesofXwouldbeveryinecient.Therewouldbea goodchancethattheexperimentwillyieldnousefulinformation:alltheresponsesatthe highvalueofxmightbesuccessesandalltheresponsesattheothervalueofxmightbe informative. failures.Inthiscasethelikelihoodhasnowelldenedmodeandtheexperimentisnotvery regressionarespreadthroughoutovertheintervalXand,asthesupportofthepriordistributiongetswider,thenumberofsupportpointsoftheoptimaldesignincrease.Good designsforbinaryregressionproblemshave,therefore,quitedierentpropertiesthangood designsforlinearregressionproblems. ItcanbeshownthatdesignpointsoftheBayesianD-optimalitycriterionforabinary
regressionexperimentsonseveraldierentdrugsandbiologicmaterial. wasatoneofseveralconcentrations:120,121,122or124mg/mlandasimilardesignwas 6.2Example2continued
usedforeachofthe54experiments.Thedesignwasadesignof6equallyspaceddosesof Foroneparticulardrugunderstudy,54similarexperimentswereperformed.Thedrug Recallexample2wheretheUniversityofMinnesotalaboratoryperformedmanylogistic
2.5,3.0,3.5,4.0,4.5and5.0,with10miceexposedtoeachdose.60animalswereused ineachexperiment.Sometimeslessthan60animalswereavailableinwhichcaselessthan 10animalswereexposedtothehighestdose.Theresponsesmeasuredwerethenumber ofsurvivingmice,usually7daysafterbeinggiventhedose.EstimatesoftheLD50were 40

hastwodoselevelsonly,withhalfoftheanimalsateachdose. calculatedforeachexperimentandtheseestimatesrangefrom3.2to4.2andtheslopes ofdrug. rangefromabout-4.0to-1.5.TheLD50'swereusedtoestimatethepotencyofeachbatch
thesetof54estimatescanbeusedtoconstructapriordistributiontodesignfutureexperiments.The54estimatescanbethoughtofasamplefromadistributionofpossiblevalues CouldBayesiandesignideashavebeenusefulinthisexample?Toexaminethisquestion Thedesigndoesnotcorrespondtoanylocallyoptimaldesign,asalocallyoptimaldesign
thatmightbeencounteredinfutureexperiments.ApriordistributionwasthereforeconstructedwheretheLD50andtheslopebothhaveindependentBetadistributions,Beta(4,4),andtheLD50liesbetween3.2and4.2andtheslopebetween-4.0and1.5.Thispriordistributionreasonablyreectsthesampleandhas,approximately,thesamersttwomoments asthesample.Itreectstheactualvaluesobtainedintheexperimentsperformed.Before
ingtheposteriorvarianceoftheLD50iseasilyfoundusingthesoftwareofChalonerand sameinterval. moreuncertaintyandsoasecondpriordistributionwasconstructedwhichisuniformonthe theexperimentswereperformedamorerealisticpriordistributionmightbeonerepresenting
Larntz(1989)orClyde(1993b).Itisa4pointdesign,symmetricaroundthepriormean fortheLD50of3.6,andittakesobservationsat3.07,3.47,3.73,4.13withweights0.30, 0.20,0.20and0.30respectively.Thedesignpointsarenotequallyspaced.Underthisprior FortheindependentBeta(4,4)distributionstheBayesian2-optimaldesignforminimiz-
distribution,thedesignactuallyusedinthelab,withanequalnumberofanimalsateach the2-optimaldesign.A5pointdesignobtainedbyomittingthehighestdoseof5.0from of6equallyspaceddosesbetween2.5and5.0,hasa2-criterionvalue1.52timesofthatof the6pointdesignanddividingthe60animalsequallybetweentheremaining5doseshas acriterionvalueof1.29timesthatoftheoptimalvalue.Ifthelowestdoseinthe5point designisalsoomittedandtheanimalsequallydividedbetweentheremaining4dosesof 3.0,3.5,4.0,and4.5thecriterionvalueis1.13timesthatoftheoptimalvalue.Thus,if theBeta(4,4)distributionsreectedtheexperimenters'beliefswell,iftheywerewillingto usetheoptimalBayesiandesigntheycouldhavereducedthevariabilityoftheirestimates 41

designusinga4pointdesignbyomittingthetwoextremedesignpointsoftheirdesign. considerably.Iftheywantedtouseequallyspaceddosesatconvenientvaluesspaced0.5 unitsapartandincludeintegervaluestheycouldhavegotveryclosetoanoptimalBayesian
weredone.Inthiscasetheoptimaldesignisa5pointdesign,againcenteredat3.6units, takingobservationsat2.78,3.21,3.6,3.99,4.42withweights0.28,0.15,0.14,0.15and expectandso,forfurtherillustration,considerthepriordistributionthatisuniformoverthe sameinterval.Thispriordistributionmighthaverepresentedbeliefsbeforetheexperiments TheBeta(4,4)priordistributionscorrespondtoquiteaccurateknowledgeofvaluesto
0.28respectively.Althoughthepointsarealmostequallyspacedthereismoremassatthe extremesthanatthecenterpoints.Theequallyspaced,equalweight,designsconsidered criterionvalue1.02timesthatoftheoptimaloneandthe4pointdesignwithequalweight earlierareamazinglyecientforthispriordistribution.The6pointdesignusedbythe experimenterswithequalweightat2.5,3.0,3.5,4.0,4.5,5.0hasacriterionvalue1.13times
distributionthenthedesigntheyusedisclosetotheBayesianoptimaldesign.Thisexample at3.0,3.5,4.0,4.5hasacriterionvalue1.08oftheoptimalvalue. thatoftheoptimalvalue,the5pointdesignwithequalweightat2.5,3.0,3.5,4.0,4.5hasa
has,therefore,notillustratedthatBayesiandesigncouldhavegreatlyimprovedeciency ofestimationinthislaboratory,butratherillustratedthatwhattheyweredoingmaywell havebeenclosetobeingoptimalinaBayesiansense. Iftheexpectationsoftheexperimenterscouldbereasonablyrepresentedbytheuniform
zeroandvariance2.TheexpectedFisherinformationmatrixforthesemodelsdependson wehaveyi=f(xi;)+ei.Theerrorseiareindependentandnormallydistributedwithmean isrelatedtoexplanatoryvariablesxbyanonlinearfunctionf(x;).Thatisfori=1;:::;n, 6.3Nonlinearregressionmodels
thegradientvectorg(x;)andis Inanonlinearregressionmodel,themeanofanormallydistributedresponsevariabley
Designfornonlinearregressionmodelshasrecentlyreceivedconsiderableattentionand nI(;)=nXi=1g(xi;)gT(xi;): 42

founddesignsnumerically.AsdidChalonerandLarntz(1986,1989),theseauthorsall notedthatasthepriordistributionbecomesmoredispersedthenumberofsupportpoints Bayesiancriteria,suchas1-optimalityand2-optimality,havebeeninuential.Pronzato,Huang,Walter,LeRouxandFrydman(1989),Huang,WalterandPronzato(1991) Sun(1995),DetteandNeugebauer(1995a),DetteandNeugebauer(1995b)andDetteand typicallyincreases.Chaloner(1993),MukhopadhyayandHaines(1994),He,Studdenand andAtkinson,Chaloner,HerzbergandJuritz(1993)focusedoncompartmentalmodelsand
Sperlich(1994b)allexaminesimplespecialcasesandproveoptimalityanalytically. ometricinterpretationsofBayesianoptimaldesignsandalsoidentiesseveralparallelsbetweenoptimalBayesiandesignandotherareas.ThepaperbyDetteandSperlich(1994a) isalsonoteworthyasitusesanexpansionoftheStieltjestransformofthedesignmeasure. Theresultprovidesadierentperspectiveonthenumericaloptimizationproblemandgives TheimportantpaperbyHaines(1995)isquitedierentandintroducessomenovelge-
valuableexamples. 6.4Example3continued an18pointdesignwiththeobservationsapproximatelyequallyspacedinthelogarithmof time.The18pointdesigntakesoneobservationattimes(inhours)0.166,0.333,.5,.666, timesatwhichtotakebloodsamplestomeasurethelevelofadrug.Theexperimenterused 1,1.5,2,2.5,3,4,5,6,8,10,12,24,30,and48.Atkinson,Chaloner,HerzbergandJuritz Example3isacaseofdesignfornonlinearregression.Thedesignproblemistochoose
constructedBayesianoptimaldesignsundertwopriordistributionssuggestedbythedata. Theyalsoconstructedlocallyoptimaldesigns.Undereachpriordistributionseparate2- andthemaximumconcentrationcmax).Oneofthetwopriordistributionsissuchthat1has optimaldesignswereconstructedforestimatingeachofthethreefunctionsofinterest(the areaundertheexpectedresponsecurveorAUC,thetimetomaximumconcentrationortmax
threequantities. 4:2984:0:theparameter3istakentobepointmassat21.80.Forthispriordistribution the18pointdesignusedbytheexperimenterisactuallyfairlyecientforestimatingthese auniformdistributionon:05884:04and,independently,2hasauniformdistributionon
43

withmass.29,.29,.15,.22,.06attimes.25,.94,2.8,8.8and24.7.Theratio2(18)=2(t)is estimatingtmaxunderthispriordistribution.The2-optimaldesigntisavepointdesign expectedposteriorvarianceoftmaxofonly1.3timesthebestpossiblevalue.Forestimating only1.3whichmeansthatthe18pointdesignisfairlyecientforestimatingtmax,withan Specicallydenote18tobethe18pointdesignandttobethe2-optimaldesignfor
alsoavepointdesignwithmass.10,.36,.32,.16and.06attimes.37,1.1,2.4,6.1and cmaxtheoptimaldesignminimizesanappropriate2-criterionandisdenotedc.Thisis
HerzbergandJuritzshowedthatunderthispriordistributionitispossibletoimproveon 24.1.Theratio2(18)=2(c)is1.4.Againthe18pointdesignisfairlyecient.ForAUC
the18pointdesign,butnotbymuch.TheAUCwasfoundtobenotwellestimatedunder designisnotasecientforestimatingtheAUCasitistmaxandcmax.Atkinson,Chaloner, times.29,1.7,13.1and39.6andthecorrespondingratioofcriteriais3.2,andsothe18point thecorresponding2-optimaldesignisa4pointdesignputtingmass.01,.03,.26and.70at
anydesignexceptonespecicallydesignedtoestimateit.DesignsecientfortheAUC areveryinecientforestimatingtmaxandcmax.Ifthereisveryprecisepriorinformation,
6.5Samplesizeforclinicaltrials improveduponconsiderablyusingBayesiandesign. however,oriftheareaunderthecurveisofprimaryimportance,the18pointdesigncanbe theexperimentersweredoinginpracticewasclosetoaBayesianoptimaldesign. Samplesizecalculationsareespeciallyimportantinthedesignandplanningofclinical So,interestingly,thisisasimilarsituationasexample2,inthatitmaywellbethatwhat
trialstocomparetwoormoredierenttreatments.Theprimarynon-Bayesianapproachis tospecifythemagnitudeoftheeectthatthetrialshouldbeabletodetectandchoosethe
andDeMets(1980).Severalcomputerprogramsareavailabletoimplementvariationson samplesizetogivearequiredpowerforahypothesistestatthatalternative.Itisusually
thesemethods. recommendedtomakeallowancesforpatientswhodonottaketheirassignedmedication ThisisdescribedforexampleinLakatos(1988),DupontandPlummer(1990)andWu,Fisher (noncompliance)andpatientswhotakeadierentmedicationthanassigned(switchover).
44

(1986)whousedapriordistribution,asanapproximationtoapredictivedistribution,to averageoverthepower.Berry(1991)describedaBayesianapproachforaverysimple situationusingdynamicprogrammingandsequentialupdating.Achcar(1984)lookedat BayesiancalculationsofsamplesizewhensamplingfromasingleWeibulldistribution.A ApartiallyBayesianapproachtothisproblemisgiveninSpiegelhalterandFreedman
completelyBayesianapproachisadvocatedinBrooks(1987)whoconsideredtheexpected gainininformationfromatwogroupexperimentwithWeibulllifetimes.Hedealtwith thesamplesize,theproportionofobservationsineachgroup,thelengthoftimetoaccrue patientsandhowlongtofollowthem.Heobtainedsomeclosedformexpressionsforthe
theorywhentheresponsesareBernoulli. Sylvester(1988)examinesthesamplesizeforaPhaseIIclinicaltrialusingBayesiandecision andalsomadesomeapproximations.SomeofthesecalculationsaresimilartoBrooks(1982) gaininShannoninformationundernormalpriordistributionsfortheunknownparameters wherehediscussedtheinformationlost,forexponentiallifetimes,whencensoringispresent.
Forthenon-Bayesiansolutions,Shih(1995)describedaSASmacrocomputerprogramthat implementsthemethodofLakatos(1988). isbecausetherearenofreelyavailablecomputerprogramstomakethesemethodsaccessible. isbecausetheyfailtoaccountexplicitlyfornoncomplianceandswitchover.Or,perhapsthis TheseBayesianapproachesappearnottohavebeenusedmuchinpractice.Perhapsthis
6.6Othersamplesizeproblems Mukhopadhyay(1994)takeaBayesianapproachchoiceofsamplesizeforasamplefroma singlenormaldistributionwithaconjugatenormalpriordistribution.Theydenecriteria whichmakethesamplesizerobusttothefuturedata.DasGuptaandVidakovic(1994) takeaBayesianapproachtosamplesizechoiceforhypothesistestinginaonewayanalysis Decidingonthesamplesizeninanexperimentisalwayspartofdesign.DasGuptaand
ofvariancemodelofexample1wherehypothesistestingisthepurposeoftheexperiment. TheyalsogiveMathematicacodefortheirmethod. models.SeeDasGupta(1995). Thereareopportunitiesforfurtherresearchinthisareaformoregeneral,non-normal 45

6.7Designproblemsinreliabilityandqualitycontrol ponentiallifetimeswhereexperimentalunitsmay,ormaynot,besubjecttoanincreased stressandwhereunitsmaybesubjecttoahighstress,iftheydonotfailinaspecied periodoftimeunderalowstress.DeGrootandGoelcallthis\tampering".Theyderived exactBayesianoptimaldesignsunderparticularlossfunctionsandcosts.DeGrootandGoel DeGrootandGoel(1979)consideredaBayesianapproachtodesigningstudiesofex(1988)isareviewofthisworkandappearedinavolume,ClarottiandLindley(1988),deoptimaldesignsforacceleratedlifetestingwherethelifetimeshaveeitherWeibullorlog-
diagramsandtheiruseinoptimaldesign. papers:forexamplethechapterbyBarlow,MensingandSmiriga(1988)discussesinuence votedtoBayesiananalysisanddesigninreliability.Thisvolumealsocontainsotherrelevantnormaldistributionsandthelengthoftimeavailablefortheexperimentisxed.TheirmethodsareextendedinNaylor(1994).Verdinelli,PolsonandSingpurwalla(1993)discussedBayesiandesignforacceleratedlifetestingexperimentswherepredictionisthegoal.
ChalonerandLarntz(1992)tooktheapproachdescribedinSection4.2toderiveBayesian
informationinagrouptestingexperiment:theyprovidedfreesoftwarefortheirmethod. havealognormaldistribution.MitchellandScott(1987)alsodesignedtomaximizeShannon TheyusedShannoninformationin(9)asutilityandconsideredthecasewherethelifetimes
valueandtominimizethepredictivevariance. targetwhichisanimportantproblemintheTaguchiapproachtodesign.Theyproposed, 6.8Largecomputerexperiments asaBayesianalternativetonon-Bayesianmethods,tosetthepredictivemeanatthetarget VerdinelliandWynn(1988)examinedsomeaspectsofkeepinganexpectedresponseon
Thesituationcanbethoughtofashavingaresponsesurfacewhichisknowntobesmooth butitsgeneralformisunknownandthevaluesoftheresponsesurfacecanbedetermined totheproblemofchoosingthevaluesatwhichtorunalargedeterministiccomputermodel. withouterror.Sacks,Welch,MitchellandWynn(1989)reviewthiswork.Recentadvances Someexcitingrecentdevelopmentshaveoccurredinapplyingideasfromoptimaldesign
aredescribedinWelchetal(1992),Morris,MitchellandYlvisaker(1993)andBates,Buck, 46

eviewthisworkherethereaderisreferredtotheabovereferences.Thisimportantproblem RiccomagnoandWynn(1995).MuchofthisworkinvolvessequentialdesignbutnonsequentialdesignhasalsobeenfoundhelpfulasinCurrin,Mitchell,MorrisandYlvisaker hasuniqueaspects. 6.8Othernonlineardesignproblems (1991).ABayesianformulationoftheproblemhasprovedfruitful.Ratherthanattemptto
methodsinthesetypesofdesignsanddecisionmaking. decisiontheorytostudythedesignproblemofwhentoscreenfordiseaseandappliedthisto breastandcervicalcancerscreening.TheypresentedapowerfulcasefortheuseofBayesian dilutionassayproblem.Parmigiani(1993)andParmigianiandKamlet(1993)usedBayesian Ridout(1994)appliedBayesiandesignideastoaseedtestingexperimentsimilartothe
ascalculatedby(1),forclinicaldesignproblems.Theymainlyconsiderexponentialor calculationsforasimpledecisionproblemandelicitpriorprobabilitiesandutilitiesdirectly. binomialresponseswithconjugatepriordistributions.LadandDeely(1995)alsodoexact ParmigianiandBerry(1994)examinedseveralproblemsusingtheexactexpectedutility,
Bayesiandesignformultivariateresponsemodels,eitherlinearornonlinear.Draperand Hunterdevelopedandusedacriterionsimilarto(16)whereapriorprecisionisincorporated intothecriterion.intheirexamplestheyeitherusedapriorestimateforthenonlinear parameters,similartolocaloptimality,ortheyusedsequentialdesign.Itisapotentialarea ApartfromDraperandHunter(1966,1967b)littleresearchhasbeendoneinusing
responsecase. 7Nonlinearestimationwithinalinearmodel ofresearchtousecriteriawhichmorecloselyapproximateexpectedutilityinthemultivariate
tothoseinsection4.2canbeusedtogivedesigncriteria. 7.1Generalproblem interestthentheexpectedutilitycannotbecalculatedexactlyandtheproblemhasmorein commonwithnonlineardesignthanwithlineardesign.Asymptoticapproximationssimilar Whenanonlinearfunctionoftheregressioncoecientsinalinearmodelisofprimary Assumethatthemodelisasinsection2andthatanonlinearfunctionoftheparameters 47

g()isofinterest.Denethekvectorc()tobethegradientvectorofg()asin(17). Approximationssimilartothoseinsection4.2giveasquarederrorlossofeither
distribution.Asinsection4.2,thecriteria whereRiseitherthepriorprecisionmatrixorthematrixofsecondderivativesoftheprior 2c(^)T(nM)?1c(^)or;2c(^)T(nM+R)?1c(^)
and canbeexpressedasaformofA-optimality.Thatisthedesign,,shouldbechosento 2R=Z2c()T(R+nM)?1c()p(;)dd 2=Z2c()T(nM)?1c()p(;)dd
minimizeeithertrAM?1ortrA(R+M)?1withA=E[2c()c()T],theexpectationbeing overthepriordistributionof.Ifmorethanonenonlinearfunctionofisofinterest, A-optimality,itshouldbepossibletogetabetterdesignbychoosingthedesignpoints saygi()fori=1;:::;m,thenthematrixAisthesum,orpossiblytheweightedsum,of individualmatricesE[2ci()ci()T].Notehoweverthat,unlikethecasefortheusuallinear sequentially. ThisproblemisdiscussedinMandal(1978),BuonaccorsiandIyer(1984,1985),Buonaccorsi (1985),andChaloner(1989).BuonaccorsiandIyer(1986)alsoexaminedseveralother problemsinvolvingdesignfortheratioofthecoecientsinlinearmodel.Aspecialcaseof estimatingsucharatioisthecalibrationproblemwherenindependentobservationsyiare Onesuchdesignproblemisthatofestimatingtheturningpointinaquadraticregression.
whereei;i=1;:::;narenormallydistributedwithmeanzeroandvariance2.Thereare takenfromasimplelinearregressionmodel.Thatis
nobservationsyandan(n+1)stobservationyn+1forwhichitisrequiredtoestimate yi=0+1xi+ei
48

thecorrespondingvalueofxn+1.Onesolutionistoestimatethenonlinearfunctiong()= (yn+1?0)=1.BuonaccorsiandIyer(1986)discusseddesignforthisproblemusingboth takeninBarlow,MensingandSmiriga(1991)whoputapriordistributiononxn+1. localoptimalityandBayesianA-optimality.AdierentbutrelatedBayesianapproachwas
istousethelocaloptimalityapproachofCherno(1953).Theproblemofestimatingthe turningpointinaquadraticregressionwillbeusedtoillustrateanimportantlimitationof localoptimality.Thisexampleisusedtoillustratenon-BayesiannonlineardesigninFord 7.2Turningpointexample
andSilvey(1980)andFord,TitteringtonandWu(1985)andBayesiannonlineardesignin Asinothernonlineardesignproblemstheusualnon-Bayesianapproachtotheseproblems
Chaloner(1989). pointisg()=?1=(22).Denec()tobethegradientvector(0;1=(22);1=(222))T.The asymptoticvarianceofthemaximumlikelihoodestimatorofg()isthen: Supposethattheexpectationoftheresponseyatxis0+1x+2x2.Thentheturning
withnM=nPki=1ixixTidenedinSection1.Localoptimalityrequiresabestguessfor (27).Supposenowthatobservationsxicanbetakenanywhereintheinterval[?1;1]and ,0say.Thevalueof0issubstitutedinto(27)andthedesignischosentominimize 2c()TM?1c() (27)
toshowthatthelocallyoptimaldesigntakeshalftheobservationsatx=1andhalfat thatg(0)=12isthebestguessvaluetobeusedforlocaloptimality.Itisstraightforward x=0,givingnMasasingular33matrixofrank2.Thetwodesignpointsx=1and x=0aretwopointswheretheexpectedvalueofyisequalandtheturningpointx=g() ishalfwaybetweenthesetwopoints.Ifthisexperimentweretobecarriedout,usingno
earestimationwithinalinearmodel,canleadtoamatrixnMwhichminimizes(27)butis priorinformationintheestimationprocess,thenitisclearlyimpossibletotaquadratic thereforeuselessforpracticalpurposes. regressiontotwodatapointsandestimatetheturningpoint.Thelocallyoptimaldesignis Theaboveillustratesthegeneralpointthatthelocallyoptimaldesignforcasesofnonlin- 49

fromthelinearc-optimalitycasewhere,althoughtheoptimaldesignmaygiveasingular 8Otherdesignproblems matrixnM,thecontrastofinterest,cT,isalwaysestimable. singularandinthiscasethequantityofinterestg()maynotbeestimable.Thisisdierent
8.1Variancecomponentsmodels derson(1975),forexample,reviewedthistopic.MorerecentlyMukerjeeandHuda(1988) examinedoptimalityandGiovagnoliandSebastiani(1989)consideredthedesignproblem whenboththevariancecomponentsandthexedeectsareofinterest.Theapproachhas alwaysbeentouselocaloptimalityuntiltherecentpaperofLohr(1995),wholookedat Designsfortheestimationofvariancesareimportantinqualitycontrolresearch.An-
aBayesianapproachtodesign.SheusedBayesianD-optimalityandA-optimalityforthe
Bayesianmethodsfordesign. largeclassofpriordistributions.Inthecontextofhierarchicalmodelswithunknownvariancecomponentsformulti-centerclinicaltrialsStanglandMukhopadhyay(1993)alsoused whichabalanceddesignisoptimalandshowedtheoptimalityofabalanceddesignundera estimationofthevariancecomponentsortheirsumortheirratio.Shegaveconditionsunder
forexample,i()istheD-optimalitycriterionundertheithofmcandidatemodels.The averagebeingoveranumbermofmodels.Sheusedacriterion()=Pmi=1wii(),where, 8.2Mixturesoflinearmodels weight,wiontheithmodelisthepriorprobabilityonthatmodel.CookandNachtsheim (1982)appliedsuchacriteriontodesignforpolynomialregressionwhenthedegreeofthe Lauter(1974,1976)proposedadesigncriterionthatisanaverageofdesigncriteria,the
ofthepredictedmeanresponseoverthedesignintervalisproportionaltotrAiM?1where polynomialisunknown.Thecriterion,(),theyusedwasbasedonA-optimalityforpredictingtheresponseoverthedesigninterval.Fortheithmodelandadesignthevariance minimizingthevarianceofpredictionoverthedesignregion,andletMibetheinformation ratherthanaveragetheA-optimalitycriteriadirectlyCookandNachtsheimaveragedeciencycriteria.SpecicallyletibetheA-optimaldesignfortheithmodel,i=1;:::;m,for Aiisaspeciedmatrix.ThiscriterionissometimesreferredtoasQ-orL-optimality.But 50

Theygaveanumberofnumericalexamplesusingthiscriteriontopredicttheuraniumcontent matrixfortheithmodel.Thentheymaximized
ofalog. TheseideasaresimilartothoseofBayesiannon-lineardesignalthoughthemotivation ()=?mXi=1witrAiMi()?1 trAiMi(i)?1:
ofCookandNachtsheimisnotBayesian.Thisisapparentthroughtheiruseofaverage eciency:itisunclearhowthiscorrespondstomaximizingexpectedutility.ABayesian directlyratherthantheireciencies.InotherwordsaBayesianapproachwouldusethe approach,withsquarederrorloss,wouldargueforanaveragingoftheA-optimalitycriteria criterion pdet(Mi),orperhapsaneciencymeasure,likethatofCookandNachtsheim,suchas utilityisconsidered,whethertoaveragei()=logdet(Mi),ori()=det(Mi),ori()= SimilarlyinusingD-optimalityaveragedoveracollectionofmodels,itisunclear,unless ()=?mXi=1witrAiMi()?1:
utilityshouldbemaximized,notexpectedeciency. Bayesianperspectiveofmaximizingexpectedutility,however,theanswerisclear:expected ()=det[Mi()]=det[Mi(i)],whereiistheD-optimaldesignfortheithmodel.Froma
rivedaversionofElfving's(1952)theoremforthiscase.DetteandStudden(1994)providedmixturesofBayesianlinearmodelcriteriainvolvingthepriorprecisionmatrix.HealsodelencetheoremcanbeappliedisinPukelsheim(1993,p.286-296).Dette(1990)gavesome generalresultsforD-optimalityandpolynomialregression.Dette(1991,1993a,1993b)used Anexcellentsummaryofthemathematicsofsuchcriteriaandhowthegeneralequiva-
(1995)gavefurthergeometricinsightintosuchcriteria. 8.3Designformodeldiscrimination furtherresultscharacterizingtheoptimaldesignintermsofitscanonicalmoments.Haines anumberofnon-Bayesianapproachestodesignhavebeensuggested.Usuallyamethod Inanexperimentwhereseveralmodelsarecompared,inordertoselectoneofthem, 51

whoalsoprovidevaluableinsightintothemixturecriteriaoftheprevioussectionandsuggest anumberofwaysofdesigningforanumberofsimultaneousobjectives.SeealsoPonceDe modelarecombined.TheseproceduresarereviewedinPukelsheimandRosenberger(1993) LeonandAtkinson(1991). fordiscriminatingbetweenmodelsandamethodforestimatingtheparameterswithineach
linearmodelsandtodesignwiththedualgoalofmodelselectionandparameterestimation. Heusedtheutilityfunctionin(5)ofsection2.2forbothproblems.Fordiscriminating betweentwomodels,thedesigncriterionhederivedleadstominimizingtheexpectationof theposteriorprobabilityofonemodel,whentheotherisassumedtobetrue.Inthecase Spezzaferri(1988)presentedaBayesianapproachtodesignforchoosingbetweentwo
ofmultivariatenormalnestedmodels,whenusingdiusepriorinformation,thiscriterionis thesameasnon-BayesianD-optimalityfortestingthehypothesis0=0,where0isthe subvectorofextraparametersinthelargermodel(see,forexample,Atkinson1972). smallermodel.Theotherfactoristheexpectationoftheposteriorprobabilityofthesmaller model,whenitisassumedtobetrue.Theoptimaldesignfordiscriminationandestimation normallinearmodels,Spezzaferrishowedthattheoptimalitycriterionusingutility(5)is givenbytheproductoftwofactors.Oneisthedeterminantoftheinformationmatrixofthe Forthedualpurposeofmodeldiscriminationandparameterestimationfortwonested
models.Theyfounddesignsthatmaximizeexpectedutilityforaxedpriordistribution maximizestheproductofthesefactors.
subjecttobeingrobustforaclassofpriordistributions.DasGupta,Mukhopadhyayand Studden(1991)constructedaframeworkforrobustBayesianexperimentaldesignforlinear 8.4Robustness
Studden(1992)gaveadetailedapproachtodesigninalinearmodelwhenthevariance Itisimportanttocheckthesensitivityofthedesigntopriordistribution.DasGuptaand
oftheresponseisproportionaltoanexponentialorpowerfunctionofthemeanresponse. designthatishighlyecientforseveraldesignproblems.TheyconsideredbothBayesian andnon-Bayesianformulationsofthedesigncriteria. Theydevelopedexamplesof\compromisedesigns"wheretheexperimenterwantstonda 52

obusttospecicationofthepriordistribution.Theyusedthedesignproblemofestimating theturningpointinaquadraticregressionastheirmotivatingexample.Theysuggesteda certaineciencyoveraclassofcloselyrelatedpriordistributions. criterionofdesigningfora\major"priordistributionsubjecttoaconstraintofattaininga SeoandLarntz(1992)suggestedsomecriteriafornonlineardesignthatmakethedesign
Bayesiandesigninthenormallinearmodelwithrespecttothepriordistribution.These distributionswherethevariancestakevaluesinspeciedintervals.Thecriteriashesuggested papersdealtmainlywiththeonewayanalysisofvariancemodel.Toallowforpossible misspecicationofpriorvariances,Toman(1992a,b)proposedusingaclassofnormalprior Toman(1992a,b)andTomanandGastwirth(1993,1994)alsoconsideredrobustnessof
eitherthedeterminantorthetraceoftheposteriorprecisionmatrix.Averagesaretaken withrespecttoadistributiononthepriorprecisionparameters. forchoosingdesignsaremaximizingtheaverage,overtheclassofposteriordistributions,of analysisofvariancemodelswhenthepriordistributionisinaclassofnitemixturesof normals.Theyusedasquaredlossfunctionandconsideredanaverageoftheposteriorrisk overtheclassofcorrespondingposteriordistributions. TomanandGastwirth(1993)examinedbothrobustestimationandrobustdesignfor
meansusingresultsfromapilotstudy.Theyassumedthattheerrorvariancesofthepilot andofthefollowupstudiestobeunknown,butthattheintervalsinwhichtheyvarycan theestimator,aminimaxcriterion,overtheclassofposteriordistributions. bespecied.Theyadoptedasquaredlossfunctionandproposedtouse,forthedesignand TomanandGastwirth(1994)suggestedspecifyingthepriordistributionontreatment
supportpointsinanoptimaldesignisoftenthesameasthenumberofparameters{inwhich casenomodelcheckingcanbedone.Inaddition,undertheassumptionthatthemodelis 8.5Modelunknown known,thedesignpointsareusuallyattheboundaryofthedesignregion{butifthelinear responsesurfaceis,asisquiteusual,alinearapproximationtosomesmoothbutunknown Amajorcriticismoftraditionaloptimaldesignforlinearmodelsisthatthenumberof
surface,thenitisattheboundaryofthisregionthattheapproximationismostinaccurate. 53

Thesecriticismsarenotnew(seeforexampleBoxandDraper,1959,SacksandYlvisaker, 1984,1985)andapplytobothBayesianandnon-Bayesianoptimaldesignforlinearmodels. maldesignsfornonlinearproblems.Inthesecasesthereisnoboundonthenumberof supportpointsinanoptimaldesignandthesupportpointsmaybespreadthroughoutthe experimentalregion.Itisunclear,however,underwhatcircumstancesthisisso. AsdiscussedinSection5.2thesecriticismssometimesdonotapplytoBayesianopti-
introducedamodiedBayesianD-optimalapproachforthespecialcaseoffactorialmodels. Theyconstructedapriordistributionwithastructurerecognizing\primary"and\potential" terms.TheresultingBayesianD-optimaldesignshaveverydesirableproperties.Indeedthey tureapproachasdescribedinsection8.2.MorerecentworkbyDuMouchelandJones(1994)Amongattemptsatincorporatingmodeluncertaintyintothedesignproblemisthemix- providedaBayesianjusticationforresolutionIVdesigns.DuMouchelandJonesshowed
eectsareexactlyzerobutratherthattheyaresmallcomparedtoothereects.DuMouchel assumedtobezerotoderiveafractionaldesigntheexperimenterdoesnotbelievethatsuch belief,thathaslongbeenheldbypractitioners,thatwhencertaininteractionsoreectsare severalcompellingexamplesoftheuseoftheirmethods.Thisworkrecognizesmodeluncer-
andJoneshavesucceededinformalizingtheotherwiseheuristicjusticationforresolution tainty,whichisalmostalwayspresentinapracticalsetting.Itspecicallyaccountsfortheoftheproposedmodel.Hederivedamethodforchoosingthescaleofthetwofactorex-
problemandalsousedaBayesianformulationtointroduceuncertaintyabouttheadequacy IVdesignsoverotherdesignswhichhavethesamevalueoftheD-optimalitycriterion. periment:thatishechosethe\high"andthe\low"levelsforeachfactorconditionalon aparticularfractionalfactorialdesignbeingused.Inthiswaythetradeoisrecognized Steinberg(1985)consideredtwo-levelfactorialexperimentstorepresentaresponsesurface
betweenchoosingdesignpointsontheboundaryofthedesignregionstomaximizeinfor- thecurvetobetisasmoothfunctionanddesignpointsarechosenbasedonthepredictive mationandchoosingthemtowardsthecenteroftheregionwherethemodelisbelieved distribution. toholdtobetterapproximation.Steinberg'sapproachisreminiscentofearlierworkby O'Hagan(1978).O'HaganconsideredaBayesianapproachtodesignforcurvettingwhere 54

forfurtherresearchhere. 9Concludingremarks problems.Inrealproblemsthemodelisalmostneverknownexactly.Thereisclearlyaneed TheseapproachesalluseBayesianideastosolvetheverypracticalaspectofrealdesign
forthechoiceofexperiment,explanatoryfactors,samplesize,andmodel.ABayesianapproachtodesigngivesamechanismforformallyincorporatingsuchinformationintothe designprocess.Thedecisiontheoreticformulationpresentedinthispapershowsthatutility ologyhasmuchtooerinexperimentaldesign,wherepriorinformationhasalwaysbeenusedBayesiandesignisanexcitingandfastdevelopingareaofresearch.TheBayesianmethod- functionscanclarifytheapproachtodesign. thatsomeexperimentersmayalreadybeactuallyusingdesignswhichcanbejustiedas approximatelyoptimalunderaBayesianformulation.AformalBayesianapproachtoexperimentaldesignmaywellleadtosubstantialimprovements.Itdoesremainregrettable, however,thatsofewrealcasestudiesappearinthestatisticalliteratureofBayesianoptimal Theexamplespresented,especiallyexamples2and3ofnonlinearproblems,illustrate
design.Thesamecanbesaidofnon-Bayesiannonlineardesignwherethereisconsiderable theoreticalresearchbutfewrealcasestudies. components.Thesimpleexamplespresentedinsection3illustratethatmoresensibledesigns canbeobtainedwhenthepriordistributionisspeciedwithinthehierarchicallinearmodel. approach.Inparticular,withinthelinearmodelcontext,thereisaneedformethodsincorporatinghierarchicallinearmodelsandhierarchicalpriordistributionsandunknownvariance TherearemanyspecicdesignproblemsthatremaintobeinvestigatedbyaBayesian
But,asremarkedbyGoldstein(1992),thereisalsotheneedfortheseideastobeappliedto actualexperiments. realitythatthemodelforanalysisisalmostneverknownwithcertaintybeforetheexperiment isdone.Theexperimentaldesignprocessshouldincorporatemodeluncertaintyintothe designprocess. Inbothlinearandnonlinearproblemsthereistheneedformethodswhichreectthe Thereisalsoaparallelneedformethodstobedevelopedforthespecicationandquanti- 55

ormaybebasedonpreviousexperimentsandpastdata.Whateverthesourceofpriorinformationverylittleguidanceisavailableonhowtocollectandquantifysuchinformation. cationofpriorbeliefs.Priorbeliefsmaybeentirelysubjective,basedonpersonalexperience, AnotableexceptiontothisistheimportantworkofGarthwaiteandDickey,forexample GarthwaiteandDickey(1988),whohavedevelopedusefulmethodsforelicitationforthe linearmodel.Itremainsachallengetodevelopmethodsforpriorelicitationfordistributions tobeusedindesignfornonlinearmodels.AwelcomebeginningisthestudyofFlournoy (1993)whogivesaniceexampleoftheentiredesignprocess,includingexpertelicitation. beenwidelyusedinthestatisticalliterature,itwouldalsobeinterestingtoseealternatives inthedesignprocess,tocarefullyconsiderthereasontheexperimentisbeingdoneandto constructedandexploredinfutureresearch. considerwhatutilityshouldbeused.AlthoughShannoninformationandsquarederrorhave Bayesiandesignalsorequiresaspecicationofautilityfunction.Itisclearlyhelpful,
thereisaneedforsoftwaretondsuchdesigns,bothexactandcontinuous.Without availableanduserfriendlysoftwarethesemethodswillnotbeusedinrealproblems.The softwareofClyde(1993b)hasthepotentialtomakeBayesiandesignsaccessibletothe scientist. AsmostBayesianmethodsfordesignrequirenumericaloptimizationandintegration
Acknowledgements manuscriptandforprovidingmanyhelpfulsuggestions. WearegratefultoRobKass,therefereesandLarryWassermanforcarefullyreadingthe
56

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