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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
Page 1 and 2: BayesianExperimentalDesign:AReview
Page 3: theoreticframeworkforacoherentappro
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 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
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forfurtherresearchhere. 9Concluding
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Achcar,J.A.(1984).UseofBayesiananal
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Chaloner,K.andLarntz,K.(1992).Bayes
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Draper,N.R.andHunter,W.G.(1967a).Th
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Kass,R.E.andSlate,E.H.(1994).Somedi
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O'Brien,T.E.andRawlings,J.O.(1994b)
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Shannon,C.E.(1948)AMathematicaltheo
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Verdinelli,I.andWynn,H.P.(1988).Tar
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