CHAPTER7FlexibleMonitoring:TheErrorSpendingApproach7.1UnpredictableInformationSequencesThetestsdescribedinChapters2to6arealloriginallydesignedforafixednumber,K,ofequallysizedgroupsofobservationswhichgiverisetoequallyspacedinformationlevels{I1,...,IK}.Wehaveexplainedhowthe“significancelevelapproach”canbeusedtoadapttheseteststounequalinformationsequencesinawaywhichmaintainstheTypeIerrorrateclosetoitsnominalvalue.However,thismethodbreaksdownasinformationincrementsbecomemoreandmoreuneven.Also,thesignificancelevelapproachrequiresthemaximumpossiblenumberofanalyses,K,tobefixedatthedesignstage.InthischapterweshalldescribeamoreflexiblewayofdealingwithunpredictableinformationsequenceswhichguaranteesTypeIerrorexactlyequaltoαforanyobservedinformationsequence.Thismethodhasthefurtheradvantageofnotrequiringthemaximumnumberofanalysestobefixedinadvance.Therearesoundpracticalreasonswhyobservedinformationlevelsmaybeunpredictable.Administratively,itisconvenienttoscheduleinterimanalysesatfixedcalendartimes,butifpatientsarerecruitedatanunevenrate,thenumberofnewobservationsbetweenanalyseswillvary.Whentestingaparameterinalinearmodel,asdiscussedinSection3.4,incrementsininformationdependoncovariatevaluesofthenewsubjects.Inthegroupsequentiallog-ranktestofSection3.7,theinformationIkisapproximatelyonequarterofthetotalnumberofdeathsrecordedbyanalysisk.Clearly,methodsforhandlingunpredictableinformationsequencesareanimportantpartofthestatisticalmethodologyforsuchtrials.Inthefollowing,weshallsupposeagroupsequentialtestconcernsaparameterθ,andestimatesˆθ(k)andinformationlevelsIk={Var(ˆθ(k))}−1,k=1,2,...,ariseatsuccessiveanalyses.Themaximumpossiblenumberofanalysesmaynotbecompletelydeterminedatthestartofthestudy;forexample,samplingmaycontinueuntilIkreachessomepre-specifiedvalueifthestoppingruledoesnotimplyearliertermination.Nevertheless,itisconvenienttorefertothemaximumnumberofanalysesthatwouldariseinaparticularrealizationiftherewerenoearlystopping,andwedenotethisnumberbyK.Weassumethesequen...
