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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
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 and 16: work(forexamplePilz,1991)whereBayes
Page 17 and 18: U3(X)correspondstotheBayesc-optimal
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: asanapproximationtotheexpectedutili
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
Page 69: Verdinelli,I.andWynn,H.P.(1988).Tar

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