CHAPTER16Multi-ArmedTrials16.1IntroductionItisbecomingincreasinglycommontoconducttrialswiththreeormoretreatmentarms.Thesubstantialoverheadexpenseinsettingupalargeclinicaltrialmeansthatthereareeconomiesofscaleandtimeifseveralcompetingtreatmentscanbecomparedinthesametrial.Follmann,Proschan&Geller(1994,p.331)describeanexampleofahypertensionpreventiontrialinwhichsubjectswererandomizedeithertoacontrolgrouportooneofthreeinterventions,weightloss,sodiumrestriction,orstressmanagement,makingatotaloffourtreatmentarms.Theprimaryendpointwaschangeindiastolicbloodpressurefromabaselinemeasurementtakenonentrytothestudy.Weillustratethemethodsformulti-armedtrialsinthesimplestcaseofabalancedone-waylayoutcomparingthemeansofJunivariatenormaldistributions,whereJ≥3.Extensionstoothersituationswillbereferredtolater.Independentobservationsareavailablefromeacharmj=1,...,J,andtheseareassumedtobenormallydistributedwithunknownmeanµjandcommonvarianceσ2.Observationsaretakeningroupsofgpertreatmentarm.Forj=1,...,Jandk=1,...,K,let¯Xjkdenotethesamplemeanofthenk=gkresponsesfromtreatmentarmjavailableatstagek.Also,lets2kdenotethepooledwithin-armsestimateofσ2availableatstagek.16.2GlobalTests16.2.1GroupSequentialChi-SquaredTestsInsomecircumstancesitmaybeappropriatetomonitorasingleglobalteststatistic.FortestingthehypothesisofhomogeneityofJnormalmeans,wecanconsidergroupsequentialtestsbasedonmonitoringsuccessivechi-squaredandF-statistics.Forknownσ2,thetestofH0:µ1=...=µJisbasedonvaluesofthestandardizedbetween-armssumofsquaresstatisticSk=nkσ2J�j=1(¯Xjk−¯X·k)2,k=1,...,K.(16.1)Herenk=gk,thecumulativesamplesizeoneacharm,and¯X·k=1JJ�j=1¯Xjkistheoverallmeanatanalysisk.c⃝2000byChapman&Hall/CRCWeconsideragroupsequentialchi-squaredtestwhichstopstorejectH0thefirsttimeSk≥ckandacceptsH0atthefinalstage,K,ifSK
