CHAPTER19NumericalComputationsforGroupSequentialTests19.1IntroductionNumericalcalculationsforsequentialtestswithadiscreteresponsedatebacktothetwo-stageacceptancesamplingplansofDodgeandRomig(1929)andthemulti-stageplansintroducedbytheColumbiaUniversityResearchGroup(Freemanetal.,1948)whichdevelopedintotheUnitedStatesmilitarystandard,MIL-STD-105E(1989).ComputationsforbinarydatawerediscussedindetailinChapter12.Thecomputationaltaskisgreaterintheanalogouscalculationsforacontinuousresponsewheremultipleintegralsreplacemultiplesums.Armitage&Schneiderman(1958),Schneiderman(1961),Dunnett(1961)andRoseberry&Gehan(1964)computedpropertiesoftwo-stageandthree-stageproceduresfornormalobservationswithknownvarianceandArmitage,McPherson&Rowe(1969)tackledsequentialtestsinearnest,obtainingaccurateresultsforrepeatedsignificancetestswithupto200analyses.Groupsequentialdesignswereproposedinthecontextofdrug-screeningtrialswithadichotomousoutcomebySchultzetal.(1973)andforclinicaltrialswithcontinuousresponsevariablesbyMcPherson(1974)andPocock(1977).Althoughthepracticaldifferencebetweensequentialandgroupsequentialanalysisissubstantial,computationsforMcPhersonandPocock’stestsareessentiallythesameasforArmitage’s(1975)repeatedsignificancetestwithcontinuousmonitoringandasmallmaximumsamplesize.Researchinthelasttwodecadeshasproducedavarietyofgroupsequentialtestingboundaries,includingerrorspendingprocedurestohandleunpredictablegroupsizes,andmethodsformakingfrequentistinferencesintheformofP-valuesandconfidenceintervalsonterminationofagroupsequentialtest.Insomeinstances,analyticapproximationstotheerrorprobabilitiesandexpectedsamplesizesofgroupsequentialtestsareavailablebut,ingeneral,directnumericalcomputationisneededtoobtainaccurateanswersacrosstherangeofcurrentmethods.SomedetailsofnumericalmethodswhichcanbeusedtocomputepropertiesofgroupsequentialtestsareavailableintheearlypapersofArmitageetal.(1969)andMcPherson&Armitage(1971).Inthischapter,weelaborateonthesemethodsinthecaseofnormallydistributedobserv...
