CHAPTER12BinaryData12.1ASingleBernoulliProbability12.1.1UsingtheNormalApproximationInthischapterweconsidergroupsequentialexperimentswheretheprimaryoutcomeisbinary.WestartbyrevisitingtheonesampleproblemfirstconsideredinSection3.6.1andillustratedinanapplicationtoindividualequivalencestudiesinSection6.4.Here,individualresponsesX1,X2,...areindependentBernoullirandomvariablestakingthevalueone,or“success”,withprobabilitypandzero,or“failure”,withprobability1−p.WeshallassumeresponsesaretobeobservedinuptoKstages,withmkobservationsinthekthgroup,k=1,...,K.Wedenotethecumulativesamplesizeatthekthstagebynk=m1+...+mkanddefineSk=X1+...+Xnktobethetotalnumberofsuccessesinthefirstkgroups.Themaximumlikelihoodestimateofpbasedondataavailableatstagekisthusˆp(k)=Sk/nk.SupposewewishtotestH0:p=p0againstatwo-sidedalternative.InSection3.6.1weexplainedhowthestandardizedteststatisticsZk=ˆp(k)−p0√{p0(1−p0)/nk},k=1,...,K,(12.1)maybeusedtoconstructsuchatwo-sidedtest.Thereare,however,twoproblemswiththisapproximation.First,thenormalapproximationisinaccurateifsamplesizesaresmallorpisclosetozeroorone.Second,thevarianceofˆp(k)istakentobep0(1−p0)/nkinbothTypeIerrorandpowercalculations,whereasavarianceofpA(1−pA)/nkshouldbeusedtocalculatethepowerunderthealternativep=pA.IntheexampleofSection3.6.1,atestofH0:p=p0=0.6withTypeIerrorprobabilityα=0.05andpower1−β=0.9atp=p0±0.2hasuptofourgroupsof19observationseach,andH0isrejectedatanalysiskif|Zk|≥CP(4,0.05)=2.361,k=1,...,4.UsingthenormalapproximationwithVar(Xi)=pA(1−pA)=0.16,thepoweratp=pA=0.8isfoundtobe0.939,incontrasttothevalue0.906obtainedundertheassumptionVar(Xi)=p0(1−p0)=0.24whichwasusedtocalculatethegroupsizeof19.Thesamedifficultiesarisewhenusingnormalapproximationstoapplytheone-sidedtestsofChapter4tobinarydata.Onecommonmethodofhandlingdatawhosevarianceisafunctionofthemeanistouseavariancestabilizingtransformation.Forbinarydata,onewouldworkwitharcsin√(ˆp(k))inplaceofˆp(k),buteventhenthismethodisstillapproximate.Fortunately,itispossibl...
