CHAPTER9RepeatedConfidenceIntervals9.1IntroductionInSection8.5wediscussedhowaconfidenceintervalforaparameter,θ,couldbeconstructeduponterminationofagroupsequentialprocedure.Inthischapterweareconcernedwiththeconstructionofasequenceofintervalestimatesforθthatwecanformatanyinterimlookatthedata.Themultiple-looksproblemaffectstheconstructionofconfidenceintervalsjustasitaffectssignificancelevelsofhypothesistests.Suppose,forexample,independentnormalobservationsX1,...,XmKareavailablesequentially,eachwithmeanθandknownvarianceσ2.Supposealsothataftereachgroupkofmobservations,k=1,...,K,weformtheusual(or“naive”)95%confidenceinterval(¯Xmk±1.96σ/√(mk))forθbasedontheobservationsaccumulatedsofar.Then,theprobabilitythatallKintervalssoformedcontainthetruevalueofθiscertainlylessthan95%.TheseprobabilitiesarelistedinTable9.1forselectedvaluesofKandcorrespondtotheprobabilitiesinTable2ofArmitage,McPherson&Rowe(1969).Repeatedconfidenceintervals,denotedRCIs,foraparameterθaredefinedasasequenceofintervalsIk,k=1,...,K,forwhichasimultaneouscoverageprobabilityismaintainedatsomelevel,1−αsay.Thedefiningpropertyofa(1−α)-levelsequenceofRCIsforθisPrθ{θ∈Ikforallk=1,...,K}=1−αforallθ.(9.1)Here,eachIk,k=1,...,K,isanintervalcomputedfromtheinformationavailableatanalysisk.WeshallusethenotationIk=(θk,θk)fortheseintervals.TheintervalIkprovidesastatisticalsummaryoftheinformationabouttheparameterθatthekthanalysis,automaticallyadjustedtocompensateforrepeatedlooksattheaccumulatingdata.Assuch,itcanbepresentedtoaDataandSafetyMonitoringBoard(DSMB)tobeconsideredwithallotherrelevantinformationwhendiscussingearlyterminationofastudy.Ifτisanyrandomstoppingtimetakingvaluesin{1,...,K},theguaranteeofsimultaneouscoverage(9.1)impliesthattheprobabilityIτcontainsθmustbeatleast1−α,i.e.,Prθ{θ∈Iτ}≥1−αforallθ.(9.2)ThispropertyshowsthatanRCIcanbeusedtosummarizeinformationaboutθonterminationandtheconfidencelevel1−αwillbemaintained,regardlessofhowthedecisiontostopthestudywasreached.Incontras...
