CHAPTER3Two-SidedTests:GeneralApplications3.1AUnifiedFormulationInChapter2wepresentedtwo-sidedtestsforatwo-treatmentcomparisonwhennormallydistributedobservationsofcommonknownvariancearecollectedinequallysizedgroups.Thetestsweredefinedintermsofsignificancelevelsfortestingthenullhypothesisaftereachnewgroupofobservations,andpowerwasguaranteedbysettingthemaximumsamplesizeasamultipleofthatrequiredbyafixedsampletest.Weshallseethatgroupsequentialtestsdescribedinthismannercanbeappliedtoothertestingproblemswithlittleornomodification.Thereasonforthisliesinthecommonformofthejointdistributionofthesequenceofstandardizedteststatistics,{Z1,...,ZK}.SupposeagroupsequentialstudywithuptoKanalysesyieldsthesequenceofteststatistics{Z1,...,ZK}.Wesaythatthesestatisticshavethecanonicaljointdistributionwithinformationlevels{I1,...,IK}fortheparameterθif:(i)(Z1,...,ZK)ismultivariatenormal,(ii)E(Zk)=θ√Ik,k=1,...,K,and(iii)Cov(Zk1,Zk2)=√(Ik1/Ik2),1≤k1≤k2≤K.(3.1)Notethisimpliesthat{Z1,...,ZK}isaMarkovsequence,andthiswillbeimportantinsimplifyingcalculationsforgroupsequentialtests.Inthissectionweshowthatseveralcommontestingproblemsgiverisetosequencesofteststatisticswiththisjointdistribution.Itshouldbenotedthat(3.1)specifiestheconditionaldistributionof{Z1,...,ZK}given{I1,...,IK}.When,forsomereason,thereisanelementofrandomnessinthesamplesizesobservedatanalyses1toK,weconsiderpropertiesoftestsconditionalonthesequenceofIkvaluesactuallyobserved.Thus,asagroupsequentialtestprogresses,itisimportanttoensurethatthevalueofIkisnotinfluencedbythevaluesofstatisticsZ1,...,Zk−1seenpreviously,asthiscoulddestroytheproperty(3.1)onwhicherrorratecalculationsarebased;thisisasubtlepointwhichneednottroubleusnow,butweshallconsideritingreaterdepthinSection7.4andChapter17.Thefollowingexamplesillustrateageneralresultthatsequencesofstandardizedteststatisticsobtainedfrommaximumlikelihoodestimatesofaparameterinanormallinearmodelfollowthecanonicaljointdistribution(3.1).WeshalldiscussthisresultinSection3.4andproveitlater...
