CHAPTER17AdaptiveTreatmentAssignment17.1AMulti-StageAdaptiveDesignInthischapterweconcentrateonthetwo-treatmentparalleldesignsofChapters2and3,inparticular,theunpairedtwo-treatmentcomparisonofSection3.1.1.Itisnotunusualforunequalrandomizationordrop-outofsomesubjectstoproduceunbalancednumbersofobservationsonthetwotreatments.Wenowconsiderintentionalimbalancebetweentreatmentsachievedbyallowingtreatmentallocationineachstagetodependonpreviouslyobservedoutcomes.Suchdesignsaresaidtousestage-wisedata-dependentorstage-wiseadaptivesamplingrules.SupposewehaveresponsesXAi∼N(µA,σ2A),i=1,2,...,forsubjectsallocatedtotreatmentAandXBi∼N(µB,σ2B),i=1,2,...,forthoseontreatmentB,wherethevariancesσ2Aandσ2Bareknownbutnotnecessarilyequal.Ateachstagek=1,...,KwecollectmAkobservationsontreatmentarmAandmBkontreatmentarmB.ThecumulativenumbersofobservationscollecteduptostagekonarmsAandBarenAk=k�j=1mAjandnBk=k�j=1mBjandwedenotethecumulativesamplemeansby¯X(k)Aand¯X(k)B,respectively.Letθ=µA−µBandsupposewewishtotestH0:θ=0.ThestandardizedstatisticatanalysiskfortestingH0isZk=(¯X(k)A−¯X(k)B)√Ik,whereIk=(σ2A/nAk+σ2B/nBk)−1.(17.1)WecantestH0usingagroupsequentialtwo-sidedtest,withorwithoutaninnerwedge,oragroupsequentialone-sidedtest.InChapters2to6,wesawhowtospecifyaboundaryforZ1,...,ZKtogetherwithaparticularinformationsequence,I1,...,IK,tosatisfyerrorprobabilityrequirements.Ifweoptforequalincrementsofinformation,weneedIk=(k/K)IK,k=1,...,K,wherethefinalinformationlevelIKisdeterminedusingconstantstabulatedinChapters2and5fortwo-sidedtestsandinChapter4forone-sidedtests.Indefiningadaptivesamplingrules,weshallassumeI1,...,IKarepre-specifiedanddesignthesamplingrulestoproducetheseinformationlevels.Thus,whenplanningthekthsetofobservations,groupsizesmAkandmBkmustbechosentosatisfy(17.1)withthepre-specifiedvalueofIk.Thereisstillc⃝2000byChapman&Hall/CRCflexibilityinthechoiceofmAkandmBkandweshallallowthesetodependontheaccumulateddatathroughthecurrentestimateofthetreatmenteffect,ˆθ(k−1)=¯X(...
