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EXACTUNCONDITIONALINFERENCEFOR2x3AND2x2x2 CONTINGENCYTABLES By GERALDG.CRANS ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 1999 ©Copyright1999 by GeraldG.Crans Tomyfather ACKNOWLEDGEMENTS IwouldliketoexpressmygratitudetoProfessorJonathanShusterforhisguidance and patience throughout this dissertation. The workingenvironment was perfect for me, and his suggestions and modifications were invaluable to this work. His contributiontothisdissertationwascrucial. IwouldalsoliketothankProfessorRandolphCarterforgivingmetheopportunity toworkwithhisgroupasastatistician. Thejobhasbeenquiteinteresting,andthe amountofpracticalstatisticalknowledgethatIhaveobtained, fromworkingwith him,hasbeensubstantial. MyappreciationextendstoProfessorsMalay GhoshandAndrew Rosalskyfor readingthisdissertationandprovidinghelpfulsuggestions. Finally, Iwould liketothankmyfatherforhisconfidenceandencouragement throughoutmylife. iv TABLEOFCONTENTS ACKNOWLEDGEMENTS iv ABSTRACT vii CHAPTERS 1 LITERATUREREVIEW 1 1.1 Introduction 1 1.2 AHistoryoftheMaximizationMethod 4 1.3 MotivationforanAlternativetoConditionalMethods 13 1.4 StatisticalPowerComparisons 14 2 MAXIMIZINGTHENULLPOWERFUNCTION 20 2.1 Introduction 20 2.2 FindingtheMaximumof}{B) 21 2.3 FindingtheMaximumofg(9) 26 3 THE2x3CONTINGENCYTABLE 30 3.1 Introduction — 30 3.2 TheConditionalApproachtothe2x3ContingencyTable — 32 3.3 DeterminingSampleSizesUsingtheConditionalApproach — 35 3.4 TheAsymptoticApproachtothe2x3ContingencyTable 42 3.5 AnExactUnconditionalTestforthe2x3ContingencyTable . 43 3.6 Results 47 4 THE2x2x2CONTINGENCYTABLE 51 4.1 Introduction 51 4.2 TheConditionalApproachtothe2x2x2ContingencyTable 53 4.3 DeterminingSampleSizeRequirementsUsingtheConditional Approach 56 4.4 TheAsymptoticApproachtothe2x2x2ContingencyTable 63 4.5 AnExactUnconditionalTestforthe2x2x2ContingencyTable 65 4.6 Results 67 v APPENDICES A THESETf/3 69 B CHAPTER3TABLES 70 C CHAPTER3GRAPHS 78 D CHAPTER4TABLES 85 E CHAPTER4GRAPHS 94 F FORTRANPROGRAMS 105 REFERENCES 146 BIOGRAPHICALSKETCH 148 vi AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulfillment oftheRequirementsfortheDegreeof DoctorofPhilosophy EXACTUNCONDITIONALINFERENCEFOR2x3AND2x2x2 CONTINGENCYTABLES By GeraldG.Crans May1999 Chairman: JonathanJ.Shuster Majordepartment: Statistics Exactunconditionalhypothesistestingmethodsinthecontextofthe2x2bino- mialtrial,2x2multinomialtrial,andthecomparisonofcorrelatedproportionsex- hibitasuperiorpoweradvantagewhencomparedwithexactconditionalapproaches. However,untilnowtherehavebeennoattemptsmadetocompareunconditionaland conditionalmethodsinamoreelaboratesetting, i.e. testingscenariosthatinvolve morethanjustasinglenuisanceparameter. Thiswasduetotheunavailabilityof techniquespresentforcomputingthemaximumofnullpowerfunctionsinvolvingsev- eralnuisanceparameters. Amethodologyisdevelopedhereinwhichprovidesameanstocomputethemax- imumofnullpowerfunctionsthatareoftwopossibleforms. Thesetwoformscor- respondtomultinomialandbinomiallikelihoods,whichcancontainseveralnuisance parameters. Asspecialcases,the2x3contingencytabletocomparetwoindependent trinomialdistributions,andthe2x2x2contingencytabletocompareindependent proportionsattwodifferentstrataareconsidered. vii Fortheequalsamplesizecase,exactcriticalvaluesofthePearsonchi-squaretest forcomparingtwoindependenttrinomialdistributionsarecomputedandtabulated forn= 10(1)70, a = 0.025and a= 0.05. Tofacilitatecomparisons, atechnique fordeterminingsamplesizerequirementswhenusingtheconditionalapproachwas developed. For a = 0.025and apowerof0.80, samplesize requirements forthe conditionaltestwereoftentimessmallerthanthosefortheexactunconditionalchi squareand, infrequently, thesamplesizerequirementswerelarger. Graphsofthe nullpowerfunctionsandthecomparisonofthecriticalregionssuggestthatthecon- ditionalp-valueisabetterteststatisticthanPearson's. Forthe2x2x2tableexactcriticalvaluesoftheMantelHaenszelteststatistic withcontinuitycorrectionarecomputedandtabulatedforn=10(1)50,a=0.025 anda=0.05,wherethesamplesizenisthesameforeachbinomialtrial. Again, amethodwasdevelopedtodeterminesamplesizerequirementsfortheconditional approach,socomparisonscouldbemade. Fora—0.025andapowerof0.80,sample sizerequirementsforeachmethodwereestablished. Theresultsprovedinconclusive, asneithermethodoutperformedtheother. However,graphsofthenullpowerfunc- tionsindicated that theconditional p-valueismuch morestablethanthe Mantel Haenszelstatistic,andcomparisonsoftherespectivecriticalregionsindicatethatthe conditionalp-valueisthebetterstatistic. viii CHAPTER 1 LITERATUREREVIEW 1.1 Introduction Inhypothesistestingprocedureswhereboththenullandalternativehypotheses aresimple,theoptimalsolutionforagivenlevelaiscompletelyspecifiedbytheNey- manPearsonlemma. However,whenthedistributionoftheteststatistic,underH0, dependsonadditionalparameterswearenotinterestedin,formallycallednuisance parameters,thenthesolutionisanythingbutclear. Suppose,forexample,thatwe areinterestedinconductingahypothesistestabouttheparameter9,butthedistri- butionofthestatisticusedforthetestingprocedurealsodependsonthenuisance parameter7. Theproblemwearefacedwithishowtoconductourhypothesistest inthepresenceofthisunknownnuisanceparameter. Therearenumeroussolutionstothenuisanceparameterproblem,andBasu(1977) summarizesthese. Ofthemethodshereviews,onlythreewillbeconsideredinwhat follows. Italsoshouldbenotedthatofthesethree,onlytwoareregularlyusedin practice. Thefirst,andperhapsmostcommonmethod,isusingasymptotictheory. Here, allparametersexceptthosebeingtestedarereplacedbyestimates. Inmost casestheestimatesarethemaximumlikelihoodestimates,whichundercertainreg- ularityconditionsareconsistent. Sincetheyareconsistent,wecanuselimittheory resultssuchasSlutsky'stheorem(inconjunctionwiththecentrallimittheorem)to derivetheasymptoticnulldistributionoftheteststatistic. Itfrequentlyturnsout thattheresultingdistributionissimpleandwelltabulated. Asanexample,thePear- sonchi-squareteststatisticforcomparingtwoindependentbinomialproportionshas 1 2 anasymptoticchi-squarenulldistributionwithonedegreeoffreedom. Theadvantage ofusingasymptoticmethodsissimplicity. However,asthisisanasymptoticresult, itmaynotworkwellforsmallsamples. Thesecondmethodistheconditionalapproach. Sinceweoftenworkwithlikeli- hoodsthataremembersofexponentialfamilies,wecanusethemethodsofLehmann (1982)toeliminatenuisanceparameters. Forexample,supposeweareinterestedin testingahypothesisconcerningtheparameter9inthepresenceofanunknownnui- sanceparameter7. Also,supposethatXandYarerandomvariablesthatarejointly sufficientfor(#,7),andthejointprobabilitydistributionof(X,Y)isamemberofan exponentialfamily,i.e. fxY{x,y;0,i)=C{6,-y)exp(tx(8)x+t2(y)y+g(x,y)). Since7isunknown,wecanrestrictourclassofteststoa-similartests. Usingthe conditionaldistributionofXgivenY,wecandevelopauniformlymostpowerfulun- biased(UMPU)testforthehypothesisabout9. Themajorcriticismofthismethod isthatifXandYarediscrete, theconditionaldistributionofX given Ywill be discreteaswell, sothatarandomizationprocedureisneededtoobtainatestsize ofexactlya. However,sincerandomizedtestsarenotusedinpractice,theresult- ingnon-randomizedconditionaltestwillbeconservative. Anotherdisadvantageof theconditionalmethodisthatwearerestrictedtoaconditionalsamplespace. All statisticalinferenceprocedureswillbedonewithrespecttothisconditionalspace. Thiswillposeaproblemtothenon-statisticians, inthattheinterpretationofthe attainedsignificancelevel (p-value) isnolongerappealing, orforthatmattereas- ilyunderstandable. Whenconditional methodsareusedintheapplied literature, suchasmedicineoragriculture,theconditionalnatureofthep-valueisalmostnever mentionedinthepublication. The third technique is the maximization method. Here, the size ofa test is determinedbymaximizingthenullpowerfunctionoverthedomainofthenuisance