Bayesian Experimental Design - Mathematical Sciences Home Pages
Bayesian Experimental Design - Mathematical Sciences Home Pages
Bayesian Experimental Design - Mathematical Sciences Home Pages
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observationsniortheproportionsofobservationsi=ni=noneachtreatment. i-thgroup.Choosinganoptimaldesignforthismodelconsistsinchoosingthenumberof onewayanalysisofvariancemodelusingtheA-optimalitycriterion,denedinsection2.2. Inoneofthecasestheyexamined,oneofthettreatmentsisacontrolandthecontrastsof interestcomparethet?1treatmentstothecontrol. DuncanandDeGroot(1976)consideredtheproblemof<strong>Bayesian</strong>optimaldesignforthe<br />
tothetreatmentsineachoftheblocks.Owen(1970)andGiovagnoliandVerdinelli(1983) nij,thenumberofobservationstakenonthei-thtreatmentinthej-thblock.Iftheblock sizeskjarexed,thisisthesameaschoosingtheproportionsij=nij=kjofunitstoassign treatmentsandbblocks.Thechoiceofadesignforthismodelisequivalenttothechoiceof Inthetwo-waycase,withthesecondfactorbeingablockingvariable,theremightbet<br />
treatmentsisacontrolandtheparametersofinterestarethecontrastsofthetreatments considered<strong>Bayesian</strong>designsforthetwo-waymodelwithtreatmentsandblocks.Oneofthe withthecontrol.OwendealtwithA-optimalitywhileGiovagnoliandVerdinelliexamineda classofcriteriaproposed,inanon-<strong>Bayesian</strong>context,byKiefer(1975).Theclassisdened foraparameterp0asp=fk?1tr[D()]pg1=p.<strong>Bayesian</strong>A-optimalityisaspecialcase whenp=1,<strong>Bayesian</strong>D-optimalityresultswhenp!0and<strong>Bayesian</strong>E-optimalitywhen p!1.Havingdenedthisclass,GiovagnoliandVerdinellithenfocusedonD-optimal designs.SimeoneandVerdinelli(1989)usednonlinearprogrammingtechniquestoderive E-optimalBayesdesignsforthesamemodel. 1995).<strong>Design</strong>sformodelswithtwoblockingfactorswereexaminedbyTomanandNotz (1991),whomainlyconsideredA-optimalitycriterion,butalsopresentedsolutionsforDandE-optimality. <strong>Bayesian</strong>designsforanalysisofvariancemodelswerederivedinToman(1992a,1994,<br />
oft?1newtreatmentscomparedwithacontrolfori=2;:::;t.Assumethatthetreatment theonewayanalysisofvariancemodel.Let=(1;2;:::t)Trepresentthetreatment eectsandsupposetheexperimentisdesignedtostudythecontrastsi?1oftheeects 3.3Example1Continued FollowingDuncanandDeGroot(1976)letusnowconsidertheA-optimalitycriterionin<br />
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