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D-andE-optimality,buttheconditionsunderwhichtheaboveholdsdonotseemeasyto satisfy.Pilzdidnotgiveexplicitdesignsandexaminetheirpracticalimplicationsandhis estimation.Inqualitycontrolandinclinicaltrialspredictionoffutureobservationscanbe workissomewhatabstract. 2.4OtherUtilityFunctions ofspecialinterest.Inthesecasesthe<strong>Bayesian</strong>approachusespredictiveanalysiswhichcan 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
Page 1 and 2: BayesianExperimentalDesign:AReview
Page 3 and 4: theoreticframeworkforacoherentappro
Page 5 and 6: imenttoinvestigatebioavailability.T
Page 7 and 8: whendesigninganexperiment.Thismight
Page 9 and 10: alphabeticaldesigncriteriaareintrod
Page 11 and 12: p.10-11).Thiscriterionmaximizesthei
Page 13 and 14: constraintkck=w.Aminimaxapproachlea
Page 15: work(forexamplePilz,1991)whereBayes
Page 19 and 20: designs. lettingA=I,theintegralin(7
Page 21 and 22: eectsiareindependentandnormallydist
Page 23 and 24: section2.1.Thesecondandthirdstagest
Page 25 and 26: edividedequallybetweenthetwoandthis
Page 27 and 28: 4Nonlineardesignproblems 4.1Introdu
Page 29 and 30: asadesigncriterion.Similarlythedesi
Page 31 and 32: asanapproximationtotheexpectedutili
Page 33 and 34: and(19).Thesecriteriaareasymptotica
Page 35 and 36: localoptimalitycriterionconcaveonH.
Page 37 and 38: educestoanequivalentlinearproblem.
Page 39 and 40: exampleWu,1985).But 2.timetrendsors
Page 41 and 42: hastwodoselevelsonly,withhalfofthea
Page 43 and 44: founddesignsnumerically.AsdidChalon
Page 45 and 46: (1986)whousedapriordistribution,asa
Page 47 and 48: eviewthisworkherethereaderisreferre
Page 49 and 50: thecorrespondingvalueofxn+1.Onesolu
Page 51 and 52: Theygaveanumberofnumericalexamplesu
Page 53 and 54: obusttospecicationofthepriordistrib
Page 55 and 56: forfurtherresearchhere. 9Concluding
Page 57 and 58: Achcar,J.A.(1984).UseofBayesiananal
Page 59 and 60: Chaloner,K.andLarntz,K.(1992).Bayes
Page 61 and 62: Draper,N.R.andHunter,W.G.(1967a).Th
Page 63 and 64: Kass,R.E.andSlate,E.H.(1994).Somedi
Page 65 and 66: O'Brien,T.E.andRawlings,J.O.(1994b)
Page 67 and 68:
Shannon,C.E.(1948)AMathematicaltheo
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Verdinelli,I.andWynn,H.P.(1988).Tar
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