赵占平-云南大学博士河南省黄淮学院数学系

BayesiansamplesizecalculationsforcomparingtwobinomialproportionsZhan-PingZhao12Nian-shengTang11.DepartmentofStatistics,YunnanUniversity,Kunming650091,China2.DepartmentofMathematics,HuanghuaiUniversity,Zhumadian463000,ChinaAbstract:Samplesizedeterminationisapopularprobleminmedicalstudies.AnewapproachtocalculatethesamplesizeisdevelopedbycombiningBayesianandfrequentistviewswhenahypothesistestbetweentwobinomialproportionsisconducted.ThesamplesizeiscalculatedaccordingtoBayesianposteriordecisionfunctionandthepowerofthemostpowerfultest.Samplesizesfortwocasesthatthetwoproportionsareequaltoafixedortoarandomvalueareinvestigated.Twonumericalexamplesareusedtoillustratetheproposedmethodologies.Keywords:Bayesdecisionfunction;Bayesfactor;Binomialproportions;Powerofthemostpowerfultest;Samplesizedetermination.1IntroductionSamplesizedeterminationiscommonlyencounteredinmodernmedicalstudiesandstatisticalanalysisfortwoindependentbinomialexperiments,andalotofmethodsincludingfrequentistandBayesianviewswereproposedtocalculatesamplesizesoftwoindependentbinomialexperiments.Forexample,Sahai&Khurshid(1996)gaveapointprocedureforthedeterminationofsamplesizesonthebasisoftestingdifferenceinproportionsfortwoindependentbinomialdesign;Cesana(2004)presentedatwo-stepprocedurecombiningpowerandtheprobabilityofobtainingapreciseestimatetocalculatesamplesizesonthebasisoftestingdifferencebetweentwounpairedproportions;Agresti(2002)proposedanapproximatesamplesizeformulaforattainingtherequiredpowerwithagiventypeIerrorrateonbasisoftestingdifferenceinproportionsfortwoindependentbinomialstudy;Fleiss(1981)showedmorepreciseformulasforthedeterminationofsamplesizesonthebasisoftestingdifferenceinproportionsfortwoindependentbinomialtrials.Alltheabovementionedmethodsrequireapointspecificationofinterestingtreatmentdifference,atypeIerrorrateandpowerandvariances,andtheydidn’tconsideruncertaintyinpointspecification.Therefore,Bayesianmethodswerepresentedtoobtaintherequiredsamplesizesintwoindependentbinomialexperimentsforthefollowingreasons:(i)theyprovidealanguageforspecifyinguncertaintyandcanproperlypropagateuncertainty;(ii)elicitationisformallyapartoftheBayesianparadigm;(iii)theBayesianparadigmcanbeusedtodeterminenotreatment,somethingthatcannotbedoneusingcurrentinterpretationofthefrequentistparadigm(Weiss,1997).Inparticular,Josephetal.(1997)andPham-Gia&Turkkan(2003)presentedaBayesiansamplesizecalculationprocedureforintervalestimatorsofthedifferencebetweentwoproportions.KatsisandToman(1999)proposedaBayesianmethodforcalculatingsamplesizewhenahypothesistestbetweentwobinomialproportionsisconducted;Stameyetal.(2005)consideredtheimpactoftestpropertiesontherequiredsamplesizefortheBayesiandesignproblemforcomparingtwoproportionswitherror-pronedata.AlthoughtheseBayesianmethodsareveryuseful,theyarecomputationallyburdensome.Inthispaper,analternativeBayesianapproachforobtainingtheoptimalsamplesizewhencomparingtwobinomialproportionsisThisworkisfullysupportedbygrantsfromNSFC(10561008),NSFYN(2004A0002M),andPh.D.SpecialScientificReasearchFoundationofChineseUniversity(20060673002),andbyprogramforNewCenturyExcellentTalentsinUniversity.Correspondenceto:Zhan-PingZhao,DepartmentofMathematics,HuanghuaiUniversity,Zhumadian463000,China.Email:[e-mail]-PearsonLemma,Bayesianposteriordecisionfunctionisthemostpowerfultestunder0-1lossfunction.Theposteriorriskcanbesufficientlysmallwhensamplesizenissufficientlylarge.Thuswecanobtaintheoptimalsamplesizebysolvingfornin011)(β-=WP(0βistypeIIerrorprobability).Themainpurposeofthispaperistodiscussthedeterminationoftheoptimalsamplesizeandtodevelopcomputingapproachforobtainingtheoptimalsamplesizeonthebasisof0-1lossfunction.Thestructureofthepaperisasfollows.InSection2,hypothesistesting21210:ppppH≠←=istransformedintoBayesposteriordecision.TheoptimalsamplesizearepresentedforthecaseswhenthetwoproportionsareequaltoafixedorarandomvalueinSection3.NumericalresultsareusedtoshowtheproposedmethodologiesinSection4.2BayesdecisionfunctionandthemostpowerfultestSupposeobservationsfromtwoindependentpopulationsarebinaryinwhichthecharacteristicofinterestappearsinthetwoindependentpopulationswithproportionsp1andp2,respectively.Forsimplicity,weassumethatsamplesizesinthetwopopulationsareequal,anddenotethecommonsamplesizeasn.Hence,undertheabovenotation,wehave),pBi(n,~X11),pBi(n,~X22and1Xand2Xareindependent,wherep)Bi(n,denotesbinomialdistributionwiththenumberoftrialsnandsuccessprobabilityp.Inthispaper,ourinterestistothedeterminationofBayesiansamplesizeforthefollowinghypothesis:.pp:Hppp:H211210≠↔==(1)ToobtaintheaboveBayesiansamplesize,weassumethatthepriorprobabilityare0πand1πof0Hand1H,Respectively,where110=+ππ.Further,weassumethatpriordistributionofp1andp2aregivenby21102110),(FIFIppfHHππ+=Where1Fisadistributionwithasupportonthelineofppp==21,eitheradeneneratedistributionatafixedvalueofporwithadensityfunction),()1()(11βαβαBpppf---=forrandomvaluep,.Further,thepriordistributionof1pisobviouslyindependentofthatof2pandpri