Tests of Hypotheses for Complex Data

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Tests of Hypotheses for Complex Data
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Holt, Nathan M
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Doctorate ( Ph.D.)
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University of Florida
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Statistics
Committee Chair:
Young, Linda
Committee Members:
Daniels, Michael J
Casella, George
Mcintyre, Lauren

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Subjects / Keywords:
gstar -- overdispersion -- relative -- total -- yield
Statistics -- Dissertations, Academic -- UF
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Statistics thesis, Ph.D.
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Three statistical inference problems arising from applications in agriculture, genetics, and epidemiology are considered. Assessing interspecific competition in replacement series experiments is addressed first. In these studies, the Relative Crowding Coefficient (RCC) is a common measure of interspecific competition. However, the interpretation of RCC depends upon the validity of an assumed functional relationship in total species yield. We develop inference procedures based upon the Relative Yield Total (RYT), a function of species yields in mixture and monoculture plantings, that may be utilized to assess the plausibility of the species yield functional form under which RCC is a reasonable metric of interspecific competition. The second problem is that of selecting a model for overdispersion of count data. Based upon the union-intersection principle we describe a two-stage procedure to test the null hypothesis of no overdispersion in a Poisson log-linear model versus the alternative hypothesis of either linear or quadratic mean-overdispersion. We suppose linear overdispersion is the result of a generalized Poisson data distribution, and quadratic overdispersion is the consequence of a negative binomial data distribution. In the first stage, a union-intersection test of the null hypothesis of no overdispersion is conducted. If the first stage null hypothesis is rejected, we propose an approach for determining which form of overdispersion is more plausible. Finally, the problem of assessing spatial and temporal association amongst observations from aerial units is addressed. Here we propose an extension the spatio-temporal Moran's I (STMI) test statistic set in the generalized spatio-temporal autoregressive model (GSTAR) model framework. Using the extended STMI, a test of the null hypothesis of no spatio-temporal association is developed. Simulation studies are conducted to evaluate the small-sample properties of each proposed procedure. The proposed methodologies of the first and final problems are applied in agricultural and epidemiological settings, respectively.
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by Nathan M Holt.
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Thesis (Ph.D.)--University of Florida, 2012.
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Adviser: Young, Linda.
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TESTSOFHYPOTHESESFORCOMPLEXDATAByNATHANMORLEYHOLTADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012NathanMorleyHolt 2

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Youcan'tsoarwiththeeagleswhenyou'reouthootin'withtheowls.Mississippiproverb 3

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ACKNOWLEDGMENTS Iwouldliketothankmyadviser,LindaJ.Young,forprovidingmewithinnumerableopportunitiesandlessons,bothinworkandinlife.Iwouldalsoliketothankmycommitteemembersfortheirparticipationandfeedbackduringtheproposal,dissertationreview,anddefenseprocess.Lastly,IwouldliketothanktheDepartmentofStatisticsfacultyandgraduatestudentsforprovidingsuchchallengingandenrichingworkenvironment.Iamforeverindebtedtomyparents,Mom,Dad,Jack,andLindsay.ThankyouforhelpingmewhenIhaveneededitmost,andforyourendlesspatienceandsupport.Mylove,Christen,youhavestoodbymysidefromthebeginningandhelpedmetoseeitthroughtotheend.Iamgratefultoyouforneverallowingmetoquit;Iwouldneverhavemadeittotheendwithoutyouinmylife.MydearGrandma,youhavealwaysputasmileonmyface.Ilovesharingourchildishmomentstogether.YourHalloweencandycarepackageshavealwaysbeenatreat,andyourplumberpostcardmademe,myfriends,andsurelynumerouspostalworkerslaugh.Mybrothersandsister,Tate,Bob,MattyandDanni:Ilookforwardtoourcomingstoriestogether,andtodominatingeachofyouinthePesOlympics.KnowthatIloveeachofyou.MyniecesNinaandArden,mayyoubothsharemyloveofstatisticsandmath!Iamlookingforwardtohelpingyouwithyourhomeworkintheyearstocome.Kenny,Mike,andTommy,youhavemadetheday-to-daymemorableandfun.Surelyonedaywewilllookbackandjokeabouthoweasyitwas.(Yeah,right.)Israel,Matt,andRyan,ourfriendshipisproofthat,asthesayinggoes,theUniversityregretfullyterminatestenure,butonenevergraduatesfromOleMiss.AndAce,youhaveshownmethatyoucanteachanolddognewtricks.Iwouldnevermadeitthroughthisodysseywithoutyourlove,support,andencouragement. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 2RELATIVEYIELDTOTALINREPLACEMENTSERIESEXPERIMENTS .... 16 2.1Background ................................... 16 2.2Large-SampleTheory ............................. 20 2.2.1DesignswithAllPlantingRatios .................... 20 2.2.2Designswith1:0,1:1,and0:1PlantingRatios ............ 23 2.3SimulationStudy ................................ 23 2.4Discussion ................................... 26 2.5CollaborationandSupport ........................... 27 3UNION-INTERSECTIONTESTSFORMODELSELECTION .......... 35 3.1Background ................................... 35 3.2TestingforOverdispersioninCountDataModels .............. 37 3.2.1DeanTestsforModelSelection .................... 37 3.2.2TheVuongTestforNon-NestedModelSelection .......... 38 3.2.3TheUnion-IntersectionTest ...................... 41 3.3Two-stageTestforOverdispersion ...................... 42 3.4ResultsandDiscussion ............................ 44 3.5CollaborationandSupport ........................... 48 4MORAN-TYPERATIOSOFQUADRATICFORMSFORASSESSINGSPATIO-TEMPORALASSOCIATION ........................ 61 4.1Background ................................... 61 4.1.1Motivation ................................ 61 4.1.2LiteratureReview ............................ 62 4.1.3ModelingFramework .......................... 70 4.1.4EconometricApproachestoSpace-TimeLatticeModels ...... 73 4.2STMIinGSTARModels ............................ 77 4.2.1HypothesisTestinginGSTARModelParameters .......... 77 4.2.2GeneralizationofIst .......................... 78 4.2.3Propertiesof^Ist ............................. 78 5

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4.2.4MaximumLikelihoodEstimationinGSTARModels ......... 84 4.3Application:DetectingChangesinPatternforStandardizedCancerMortalityRates .............................. 85 4.4CollaborationandSupport ........................... 90 APPENDIX AACCOMPANIMENTTOCHAPTER2 ........................ 97 BACCOMPANIMENTTOCHAPTER3 ........................ 98 CACCOMPANIMENTTOCHAPTER4 ........................ 125 REFERENCES ....................................... 127 BIOGRAPHICALSKETCH ................................ 132 6

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LISTOFTABLES Table page 2-1Sizeofthebootstraptestingprocedure ...................... 33 2-2Powerofthebootstraptestingprocedure ..................... 34 3-1Across-referencedlistofdataappearingineachgure. ............. 49 4-1Likelihood-ratiotestsofnullhypotheses( 4 )though( 4 ). ......... 91 B-1TypeIerrorrateforDeanandVuongtestsforlinearoverdispersion,Poissonsampling. ....................................... 98 B-2TypeIerrorrateforDeanandVuongtestsforquadraticoverdispersion,Poissonsampling. .................................. 99 B-3PowerofDeanandVuongtestsforlinearoverdispersion,generalizedPoissonsamplingwith=1.1. ........................... 100 B-4PowerofDeanandVuongtestsforlinearoverdispersion,generalizedPoissonsamplingwith=2. ............................ 101 B-5PowerofDeanandVuongtestsforlinearoverdispersion,generalizedPoissonsamplingwith=3. ............................ 102 B-6PowerofDeanandVuongtestsforlinearoverdispersion,negativebinomialsamplingwiths=10. ................................ 103 B-7PowerofDeanandVuongtestsforlinearoverdispersion,negativebinomialsamplingwiths=1.25. ............................... 104 B-8PowerofDeanandVuongtestsforlinearoverdispersion,negativebinomialsamplingwiths=1. ................................. 105 B-9PowerofDeanandVuongtestsforquadraticoverdispersion,generalizedPoissonsamplingwith=1.1. .................... 106 B-10PowerofDeanandVuongtestsforquadraticoverdispersion,generalizedPoissonsamplingwith=2. ..................... 107 B-11PowerofDeanandVuongtestsforquadraticoverdispersion,generalizedPoissonsamplingwith=3. ..................... 108 B-12PowerofDeanandVuongtestsforquadraticoverdispersion,negativebinomialsamplingwiths=10. ........................... 109 B-13PowerofDeanandVuongtestsforquadraticoverdispersion,negativebinomialsamplingwiths=1.25. .......................... 110 7

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B-14PowerofDeanandVuongtestsforquadraticoverdispersion,negativebinomialsamplingwiths=1. ............................ 111 B-15TypeIerrorratefortwo-stagetestofoverdispersion,Poissonsampling. .... 112 B-16Powerfortwo-stagetestforoverdispersion,generalizedPoissonsampling,=1.1. ........................................ 113 B-17Powerfortwo-stagetestforoverdispersion,generalizedPoissonsampling,=2. ......................................... 114 B-18Powerfortwo-stagetestforoverdispersion,generalizedPoissonsampling,=3. ......................................... 115 B-19Powerfortwo-stagetestforoverdispersion,negativebinomialsampling.s=10. ................................... 116 B-20Powerfortwo-stagetestforoverdispersion,negativebinomialsampling.s=1.25. .................................. 117 B-21Powerfortwo-stagetestforoverdispersion,negativebinomialsampling.s=1. ................................... 118 B-22Correctinferenceproportionusingtwo-stagetestforoverdispersion,generalizedPoissonsampling,=1.1. ...................... 119 B-23Correctinferenceproportionusingtwo-stagetestforoverdispersion,generalizedPoissonsampling,=2. ....................... 120 B-24Correctinferenceproportionusingtwo-stagetestforoverdispersion,generalizedPoissonsampling,=3. ....................... 121 B-25Correctinferenceproportionusingtwo-stagetestforoverdispersion,negativebinomialsampling,s=10. ........................ 122 B-26Correctinferenceproportionusingtwo-stagetestforoverdispersion,negativebinomialsampling,s=1.25. ....................... 123 B-27Correctinferenceproportionusingtwo-stagetestforoverdispersion,negativebinomialsampling,s=1. ......................... 124 8

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LISTOFFIGURES Figure page 2-1Fourmodelsofinterspecicrelativeyield. ..................... 28 2-2Q-Qplotsofbootstrapp-values,n=6 ....................... 29 2-3Q-Qplotsofbootstrapp-values,n=10 ...................... 30 2-4Q-Qplotsofbootstrapp-values,n=20 ...................... 31 2-5Q-Qplotsofbootstrapp-values,n=100 ...................... 32 3-1TypeIerrorrateforDeanandVuongasymptotic0.05-leveltests,Poissonsampling,generalizedPoissonalternative. ..................... 50 3-2TypeIerrorrateforDeanandVuongasymptotic0.05-leveltests,Poissonsampling,negativebinomialalternative. ...................... 51 3-3PowerofDeanandVuongasymptotic0.05-leveltests,generalizedPoissonsampling,generalizedPoissonalternative. ..................... 52 3-4PowerofDeanandVuongasymptotic0.05-leveltests,generalizedPoissonsampling,negativebinomialalternative. ...................... 53 3-5PowerofDeanandVuongasymptotic0.05-leveltests,negativebinomialsampling,generalizedPoissonalternative. ..................... 54 3-6PowerofDeanandVuongasymptotic0.05-leveltests,negativebinomialsampling,negativebinomialalternative. ...................... 55 3-7TypeIerrorrateforasymptotic0.05-leveltwo-stageDeantests,Poissonsampling. ....................................... 56 3-8Powerforasymptotic0.05-leveltwo-stageDeantests,generalizedPoissonsampling. ....................................... 57 3-9Powerforasymptotic0.05-leveltwo-stageDeantests,negativebinomialsampling. ....................................... 58 3-10Proportionofcorrectinferencesforasymptotic0.05-leveltwo-stagetest,generalizedPoissonsampling. ........................... 59 3-11Proportionofcorrectinferencesforasymptotic0.05-leveltwo-stagetest,negativebinomialsampling. ............................. 60 4-1Cancermortalityrateamongwhitemales,1975to1984. ............. 92 4-2Cancermortalityrateamongwhitemales,1995to2004. ............. 93 4-3Thesurfaceoftheparameterspace,. ...................... 94 9

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4-4Centeredandscaledcancermortalityrateamongwhitemales,1975to1984. 95 4-5Centeredandscaledcancermortalityrateamongwhitemales,1995to2004. 96 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTESTSOFHYPOTHESESFORCOMPLEXDATAByNathanMorleyHoltAugust2012Chair:LindaJ.YoungMajor:StatisticsThreestatisticalinferenceproblemsarisingfromapplicationsinagriculture,genetics,andepidemiologyareconsidered.Assessinginterspeciccompetitioninreplacementseriesexperimentsisaddressedrst.Inthesestudies,theRelativeCrowdingCoefcient(RCC)isacommonmeasureofinterspeciccompetition.However,theinterpretationofRCCdependsuponthevalidityofanassumedfunctionalrelationshipintotalspeciesyield.WedevelopinferenceproceduresbasedupontheRelativeYieldTotal(RYT),afunctionofspeciesyieldsinmixtureandmonocultureplantings,thatmaybeutilizedtoassesstheplausibilityofthespeciesyieldfunctionalformunderwhichRCCisareasonablemetricofinterspeciccompetition.Thesecondproblemisthatofselectingamodelforoverdispersionofcountdata.Basedupontheunion-intersectionprinciplewedescribeatwo-stageproceduretotestthenullhypothesisofnooverdispersioninaPoissonlog-linearmodelversusthealternativehypothesisofeitherlinearorquadraticmean-overdispersion.WesupposelinearoverdispersionistheresultofageneralizedPoissondatadistribution,andquadraticoverdispersionistheconsequenceofanegativebinomialdatadistribution.Intherststage,aunion-intersectiontestofthenullhypothesisofnooverdispersionisconducted.Iftherststagenullhypothesisisrejected,weproposeanapproachfordeterminingwhichformofoverdispersionismoreplausible. 11

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Finally,theproblemofassessingspatialandtemporalassociationamongobservationsfromaerialunitsisaddressed.Hereweproposeanextensionthespatio-temporalMoran'sI(STMI)teststatisticsetinthegeneralizedspatio-temporalautoregressivemodel(GSTAR)modelframework.UsingtheextendedSTMI,atestofthenullhypothesisofnospatio-temporalassociationisdeveloped.Simulationstudiesareconductedtoevaluatethesmall-samplepropertiesofeachproposedprocedure.Theproposedmethodologiesoftherstandnalproblemsareappliedinagriculturalandepidemiologicalsettings,respectively. 12

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CHAPTER1INTRODUCTIONWorkinginavarietyofappliedsettingsagriculture,geneticsandepidemiologywehaveaddressedthreedifferentresearchproblems.Hereweframeeachstatisticalproblemintermsoftheparticularappliedsettinginwhichtheworkisrooted.InChapter2,weintroducethereplacementseriesexperiment,whichmaybeemployedinanagriculturalsettingtoassessinterspeciccompetitionamongplants.Inareplacementseriesexperimenttwospecies,AandB,areexamined.Therearetwocommonreplacementseriesexperimentaldesigns.Foreachexperimentalunit,thetotalnumberofindividualsofAandB,orplantingdensity,isxed.Giventheplantingdensity,AandBaresometimesplantedineachofthreeplantingratiosofAtoB:1:0;1:1;and0:1.Alternatively,AandBareplantedineachofveratiosofAtoB:1:0;3:1;1:1;1:3;and0:1.Ateachratio,axednumberofreplicateplantingcontainersaresownand,afteraperiodofgrowth,allindividualsaresacriced.Foreachplantingcontainer,totaldryyieldisrecordedbyspecies.OnemetricagronomistsusetoquantifyinterspeciccompetitionistheRelativeCrowdingCoefcient(RCC).RCCisonlyasensiblemeasureofinterspeciccompetition,however,whenthereisaspecicfunctionalrelationshipbetweentheyieldsinmixtureplantings(e.g.,1:1intherstdesign,or3:1,1:1,and1:3intheseconddesign)andthemonocultureyields(1:0and0:1).TheRelativeYieldTotal(RYT)maybeutilizedtoassessthisfunctionalrelationshipbetweenspeciesyieldsatmixtureandinmonoculture.ProceduresexisttoconductinferenceoninterspeciccompetitionusingtheRCC.HereweproposemethodsforinferencetoassessthevalidityoftheassumedfunctionalyieldrelationshipatmixtureandmonocultureusingRYT.TheproposedRYTinferenceproceduresencompassbothoftheabove-mentionedexperimentaldesignsandmaybethoughtofasagatewaytoinferenceaboutinterspeciccompetitionusingtheRCC. 13

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WeintroducethreemodelfamiliesforcountdatainChapter3:thePoisson,thenegativebinomial,andthegeneralizedPoisson.Allthreemodelfamiliessharethepropertythatthevariancedependsuponthemean.However,thefunctionalformofthisdependencediffersforthethreemodelfamilies.ForaPoisson-distributedrandomvariable,YPoisson(),themeanE(Y)==Var(Y).Foranegativebinomial-distributedrandomvariablewithscaleparameters>0,YNB(,s)impliesthatE(Y)=andVar(Y)=+s)]TJ /F10 7.97 Tf 6.59 0 Td[(12.Lastly,forageneralizedPoissonrandomvariablewithdispersionparameter>1,YGP(,)impliesthatE(Y)=andVar(Y)=.Thus,thevariancefunctionsforthesethreemodelfamiliesaretheidentity;aquadraticfunction;andalinearfunction,respectively.Inanextgenerationdesignedexperimentinvestigatinggeneexpressionyieldingtranscriptcounts,thegeneticistmaywishtodescribegeneexpressionbehaviorusingoneofthesethreeaforementionedstochasticmodels.AlthoughthePoissonmodelfamilyisthemostparsimoniousofthethree,ifgroupsofsamplecountsvaryinexcessoftheirmean,thenthegeneticistmaywishtoconsideramodelfamilythataccountsformean-overdispersion.Tothisend,weproposeatwo-stagetestingprocedureformodelselection.Intherststage,weutilizeaunion-intersectiontestofthenullhypothesisthatthegroupvarianceisequaltothegroupmean,againstthealternativehypothesisthatthegroupvarianceisalinearorquadraticfunctionofthemean.Whenevertheunion-intersectionnullhypothesisofnooverdispersionisrejected,inthesecondstage,atestfornon-nestedmodelselectionisconductedtoascertainwhethergroupvariancesaremoreplausiblyalinearorquadraticfunctionofthegroupmean.Simulationstudiesareconductedtoevaluatethesizeandpowerofthetwo-stagetestingprocedureundervariousconditions.InChapter4weintroducespatialandspatio-temporalmodelsforcontinuousresponsesrecordedataerialunits.Althoughtestsfortheabsenceofspatialassociation(clustering)havearichhistoryinthestatistical,geographic,andeconometricliterature,procedurestoassessthepresenceofspatio-temporalstructurehaveonlyrecently 14

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surfaced.Thedearthofsuchtoolsissurprisinggiventheirinherentutility.Forexample,anepidemiologistsurveillingstate-levelcancerratesovertimemaywanttodeterminewhetherornothermodelsshouldincorporatespatialandtemporaleffects.Oneproposedasymptoticmethodtotestfortheabsenceofspatio-temporalstructureisbaseduponthespatio-temporalMoran'sI(STMI)teststatistic( Lopezetal. 2011 ),butarigorousproofoftheasymptoticpropertiesoftheteststatistichasnotbeengiven.Hereageneralizedspatio-temporalautoregressive(GSTAR)frameworkisusedtodevelopageneralizationofSTMI.Anexacttestofthenullhypothesisthatthereisnospatio-temporalstructureisdeveloped,allowingtheepidemiologisttodrawinferencesaboutGSTARmodelparameters.Theproposedmethodologyisappliedtodetectspatio-temporalstructureindecadalstandardizedcancermortalityratesamongwhitemales,andtheresultsarecomparedtothecorrespondinglikelihood-ratiotest. 15

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CHAPTER2RELATIVEYIELDTOTALINREPLACEMENTSERIESEXPERIMENTS 2.1BackgroundReplacementseriesexperiments,attributedto deWit ( 1960 ),areoftenusedinstudiesofinterspeciccompetitionforsharednutrientandwaterresources( Berendse ( 1979 ); Firbank&Watkinson ( 1985 ); Harper ( 1977 ); McGilchrist&Trenbath ( 1971 );forareview,see Weigelt&Jolliffe ( 2003 )).Althoughreplacementseriesexperimentshavebeenusedinentomologicalstudies(see,e.g., Novaketal. ( 1993 )),forconcretenesswedescribetheiruseinonlyanagriculturalsettingtoassesscompetitionamongplantspecies.Foraxedtotalnumberofindividualsperexperimentalunit,referredtoastheplantingdensity,theproportionsofeachoftwospecies,AandB,arevaried( Hall 1974 ).ExpressedasaratioofthenumberofindividualsofspeciesAtospeciesB,ataxedplantingdensitytwocommonprotocolsaretoplanttheAandBat(i)ratios1:0,1:1,and0:1;or(ii)atratios1:0,3:1,1:1,1:3,and0:1.Toobtainthespeciedplantingratios,theplantingdensitymustbeeven,whileinthelatterprotocol,theplantingdensitymustbeamultipleoffour.Considerareplacementseriesexperimenttoassessinterspeciccompetitionbetweentwospecies,AandB.Ataxedplantingdensityofptotalplantsperplantingcontainer,monoculturesofspeciesAandBareplanted(i.e.plantingratios1:0and0:1),andmixturesofspeciesAandBareplantedinratiosof3:1,1:1,and1:3.Notethatp(modulo4)0,unlessthedesignexcludesplantingratios3:1and1:3,inwhichcasep(modulo2)0.Foreachplantingdensity,therearenreplicateexperimentalunits,hereplantingcontainers.Experimentalunitsareindexedbyk,k=1,2,...,n.Fork=1,2,...n,thedrytotalbiomassofeachspeciesinmonocultureaswellasineachmixture,isrecorded.LetXi:j,kdenotethemeanbiomassperindividualofspeciesAinplantingcontainerkataplantingratioofi:j,andsimilarlyletYi:j,kdenotethemean 16

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biomassperindividualofspeciesofBinplantingcontainerkataplantingratioofi:j.Foreachplantingratio,containersarelabeledatrandomfork=1,2,...,n.Indicesofinterspeciescompetitioninreplacementseriesexperiments,suchastheRelativeCrowdingCoefcient(RCC)andtheRelativeYieldTotal(RYT),havebeendiscussedin Harper ( 1977 )and Obergetal. ( 1996 ),amongothers.Wewillintroduceconceptsbyconsideringdesignswithonlythe1:0,1:1,and0:1plantingratios,andthereafterconsiderdesignsthatalsoinclude3:1and1:3plantingratios.WebeginbylettingXi:jandYi:jdenotethesamplemeandryyieldperindividualofspeciesAandspeciesBatplantingratioi:j,respectively.Whentheexperimentaldesignincorporatesonlyplantingratios1:0,1:1,and0:1,thesampleRYTis[RYT1:1=X1:1 X1:0+Y1:1 Y0:1.SeveralvariantsoftheRCCstatisticappearintheliterature.Onecommondenition(see,e.g. Obergetal. ( 1996 )),consideredthroughoutthischapter,is[RCC1=X1:1=Y1:1 X1:0=Y0:1.Althoughnotcommonintheliterature,hereweuseahatsymboltoemphasizethattheformulaepresentedarestatistics.Foralli,j,henceforthsupposeXi:j,kareindependentandidenticallydistributed(iid)withE(Xi:j,1)=i:j<1forallk,andYi:j,kareiidwithE(Yi:j,1)=i:j<1forallk.If1:0,0:1>0,thentheparametersestimatedby[RYT1:1and[RCC1aredenedtobeRYT1:1=1:1 1:0+1:1 0:1andRCC1=1:1=1:1 1:0=0:1,respectively.Laterweintroduceanalternativedenitionoftherelativecrowdingcoefcientthatincorporatesinformationfrom3:1and1:3plantingratios.Todistinguishbetweenthetworelativecrowdingcoefcientmeasures,weuseRCC1todenotethemeasurebasedupononly1:0,1:1,and0:1plantingratios,andRCC2todenote 17

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themeasurethatalsoincorporates3:1and1:3plantingratioinformation.Aswewilleventuallyconsidertherelativeyieldtotalatplantingratiosotherthan1:1,wesubscriptRYTbytheplantingratioofthestatisticsorparametersappearinginthenumeratorofeachsummand.Thatis,wetake[RYTi:j=Xi:j X1:0+Yi:j Y0:1andRYTi:j=i:j 1:0+i:j 0:1. Harper ( 1977 )describesfourdistinctrelationshipsbetweentherelativeyieldsofspeciesAandB.ExamplesoftheserelativeyieldrelationshipsareillustratedinFigure 2-1 ,arenderingofasimilargureappearingin Harper ( 1977 ,p.256,Figure8/9).Foreachspecies,per-individualyieldsarerelativetotheexpectedyieldperindividualinmonoculture(e.g.speciesAyieldisscaledby1:0andspeciesByieldisscaledby0:1).BecauseRCC1andRYT1:1arefunctionsofmeanyieldsatonlyplantingratios1:0,1:1,and0:1werestrictattentiontoquadraticyieldfunctions.Whendesignsincorporate3:1and1:3plantingratios,quarticyieldfunctionsaretheoreticallypossiblebutpracticallyimplausibleandthuswillnotbeconsidered.IntypesIandIIcompetition,RYT1:1=1andthusataplantingratioof1:1therelativeyieldofAisthereectionoftherelativeyieldofBabouttheliney=1 2.IfRYT1:1=1,thenRCC1=1onlyiftherelativeyieldislinearforbothspecies.Thismaybeinterpretedasidenticalinter-andintra-speciescompetitiveeffects.WhenRYT1:1=1andRCC1>1,however,speciesAhasconcaverelativeyieldwhilespeciesBhasconvexrelativeyield.TheresearcherinfersthatthecompetitiveeffectofspeciesAonspeciesBisstrongerthanthecompetitiveeffectofspeciesAonitself.IfRYT1:1=1andRCC1<1,speciesAhasconvexrelativeyieldwhilespeciesBhasconcaverelativeyield.Thus,thecompetitiveeffectofspeciesBonspeciesAisstrongerthanthecompetitiveeffectofspeciesAonitself.CompetitiontypesIIIandIVindicatemutualisticorantagonisticinterspecicrelationships,respectively.Insuchcasestherelativecrowdingcoefcientisnota 18

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reasonablemeasureofinterspeciccompetition.Toseewhy,considertheexamplesofsuchcompetitionillustratedinthebottomleftandbottomrightplotsofFigure 2-1 .InbothcasestherelativeyieldfunctionsproduceRCC11,indicativeofalackofinterspecicinteraction,whileinbothcasesthereisastronginterspeciceffect.Thisphenomenonmaybeobservedwheneverthereexistsc2=[(0,1 2)[(1 2,1)]suchthattherelativeyieldsat1:1mixtureX1:1 X1:0candY1:1 X0:1c.Inthiscase,[RCC1=X1:1=X1:0 Y1:1=Y0:1c=c=1.Hence,onthebasisofRCC1alonetheresearcherconcludesthatneitherspeciesisout-competingtheother.TypeIIIcompetitionischaracterizedbyamutualisticrelationshipbetweenthetwospecies;thus,theinterpretationisthatintraspeciccompetitiveeffectsdominateinterspeciccompetitiveeffects.TypeIVcompetitionischaracterizedbyanantagonisticrelationshipbetweenthetwospecieswithstronginterspeciccompetitiveeffectsforbothspecies.NoteworthyisthatRYT1:16=1forcompetitiontypesIIIandIV.KnowledgeofRYT1:1isimportantinordertoassessinterspeciccompetitionwithRCC1.ThoughmethodstoconductinferenceonRCC1havebeendescribedin Obergetal. ( 1996 )etal.,noinferenceproceduresforRYT1:1haveyetbeenidentied. Novaketal. ( 1993 )proposeanexpandedformtherelativecrowdingcoefcientfordesignsincorporating3:1and1:3plantingratios. Obergetal. ( 1996 )furtherinvestigatethisalternativedenitionoftherelativecrowdingcoefcient,writingthesummarymeasureas(hatadded)[RCC2=1 3h1 3X3:1 Y3:1+X1:1 Y1:1+3X1:3 Y1:3i X1:0=Y0:1.Wetake[RCC2tobeanestimatorofRCC2,thecorrespondingmeasureinwhichparametersaresubstitutedforthecorrespondingstatistics.UponinspectionofFigure 2-1 ,itisclearthattheuseofRCC2asameasureofinterspeciccompetitionisalso 19

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problematicincompetitiontypesIIIandIV.ForcompetitiontypesIandII,inwhichRYTi:j=1foralli,j,RCC2maybeinterpretedinthesamemannerasRCC1.Hence,fordesignsincorporating3:1and1:3plantingratios,weadvocatetheuseofRCC2onlyincaseswhereevidencesuggestsRYT3:1=1,RYT1:1=1,andRYT1:3=1.Asnotedinthepreviousparagraph,noinferenceproceduresforRYTi:jhaveyetbeenidentied.Wedeveloptwosuchinferenceprocedures,onefordesignswithonlyplantingratios1:0,1:1,and0:1,andanotherfordesignsthatalsoincorporate3:1and1:3plantingratios.Asthederivationofthetheoryforthelattermaybeusedtoderivethetheoryfortheformer,werstconsiderthecasewhereplantingratios1:0,3:1,1:1,1:3,and0:1areemployed. 2.2Large-SampleTheoryAsbiomassmeasurements,eachrandomvariableXi:j,kandYi:j,kisnon-negativeandnitewithprobability1.Hence,E[(Xi:j,k)p]andE[(Yi:j,k)p]areassumedtobeniteforallp2[1,1).Recallthatforeachspeciesandplantingratiowesupposereplicateper-plantyieldmeasurementsarei.i.d.fork=1,2,...,n,withE(Xi:j,1)=i:jandE(Yi:j,1)=i:jdenotingthemeanper-plantyieldofspeciesAandspeciesBatplantingratioi:j,respectively.LetVar(Xi:j,1)=2i:j,andVar(Yi:j,1)=2i:jdenoteper-plantyieldvarianceforspeciesAandspeciesBatplantingratioi:j,respectively.DenoteCov(Xi:j,1,Yi:j,1)=()i:j.Wesupposei:j>0fori:j2f1:0,3:1,1:1,1:3gandi:j>0fori:j2f3:1,1:1,1:3,0:1g,sothatmeantotalbiomassispositivewheneveragivenspeciesisemployedataparticularplantingratio.Lastly,tosimplifynotationweletRi:jRYTi:jand^Ri:j[RYTi:j,wherei:j2f3:1,1:1,1:3g.WefurthertakeR(R3:1,R1:1,R1:3)0and^R(^R3:1,^R1:1,^R1:3)0. 2.2.1DesignswithAllPlantingRatiosWenowletZ0k=(Zk,1,Zk,2,...,Zk,8)withZk,1=X1:0,k,Zk,2=X3:1,k,Zk,3=X1:1,k,Zk,4=X1:3,k,Zk,5=Y0:1,k,Zk,6=Y3:1,k,Zk,7=Y1:1,k,andZk,8=Y1:3,k.SupposeZkareiidfork=1,2,...,nwhereZk,ihavenitesecondmomentsforallk,i.Let 20

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E(Z01)0=(1:0,3:1,1:1,1:3,0:1,3:1,1:1,1:3).Furthermore,letVar(Z1)=,where=266666666666666666666421:00000000023:1000()3:1000021:1000()1:1000020:1000()1:3000020:10000()3:100023:10000()1:100021:10000()1:300021:33777777777777777777775.UndertheassumptionthatZkareiid,andwithZ=n)]TJ /F10 7.97 Tf 6.58 0 Td[(1Pnk=1Zk,bythemultivariatecentrallimittheorem(see,e.g., Lehmann&Casella ( 1998 )),p n(Z)]TJ /F15 11.955 Tf 12.62 0 Td[();N(0,).Inordertoconstructtheasymptoticdistributionof^Rundertheaboveassumptionsweemploythemultivariatedeltamethod( Lehmann&Casella 1998 ).Todosonecessitatesassumptionsaboutthestructureofaswellastheexistenceofafunctionf:R87!R3suchthatf(Z)=^Randf()=R.Thefunctionf()mustsatisfyseveralproperties.Wewillestablishbythemultivariatedeltamethodp n(f(Z))]TJ /F4 11.955 Tf 11.96 0 Td[(f())=p n(^R)]TJ /F8 11.955 Tf 11.96 0 Td[(R);N(0,R),whereR(rf)(rf)0.Hererff@fi @tjjt=g,i=1,2,...,3andj=1,2,...,8,the38matrixofpartialderivativesoffevaluatedat.Denef:R87!R3suchthatf(t)=(f1(t),f2(t),f3(t))0andfi(t)=ti+1 t1+ti+5 t5.Thenf(Z)=(^R3:1,^R1:1,^R1:3)0,andf()=(R3:1,R1:1,R1:3)0,sothattheelementsoff(Z)andf()arethesampleandpopulationanaloguesofRelativeYieldTotalatplantingratios3:1,1:1,and1:3,respectively.Undertheassumptionthatthereexists>0suchthatmini:j(i:j,i:j)>2,andbynotingthatallpartialderivatives@fi @tjexistfork)]TJ /F8 11.955 Tf 12.12 0 Td[(tk1
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inan-neighborhoodof.Thus,withrfgivenbyrf=266664)]TJ /F18 7.97 Tf 6.59 0 Td[(3:1 21:01 1:000)]TJ /F18 7.97 Tf 6.59 0 Td[(3:1 20:11 0:100)]TJ /F18 7.97 Tf 6.59 0 Td[(1:1 21:001 1:00)]TJ /F18 7.97 Tf 6.59 0 Td[(1:1 20:101 0:10)]TJ /F18 7.97 Tf 6.59 0 Td[(1:3 21:0001 1:0)]TJ /F18 7.97 Tf 6.59 0 Td[(1:3 20:1001 0:1377775,bythemultivariatedeltamethodwehavep n(f(Z))]TJ /F4 11.955 Tf 11.96 0 Td[(f())=p n(^R)]TJ /F8 11.955 Tf 11.96 0 Td[(R);N(0,R),whereR(rf)(rf)0.SomealgebrashowsthattheithelementalongthediagonalofRisi1:0 21:02+2i 21:0+2()i 1:00:1+2i 20:1+i0:1 20:12andthatthe(i,j)thelementofRisij21:0 41:0+ij20:1 40:1.Symbolicevaluationoftheproduct(rf)(rf)0wasperformedinMATLAB,andthescriptutilizedisprovidedinAppendix A .Considernowtheproblemofmakingfamily-wiseinferenceaboutfR3:1,R1:1,R1:3g(e.g.theelementsofR).ThisistheproblemwemustaddressinordertomakeinferencesaboutinterspeciccompetitionbetweenspeciesAandBwithRCC2.WemayuseRCC2onlyifRYTi:j=1acrossallplantingratios.Thus,inordertouseRCC2wemustfailtorejectthenullhypothesisthatR=1.Thatis,wemayuseRCC2onlyifwefailtorejectthenullhypothesisH0:R=1.Tothisend,ifjRj>0and1 2RisthesquarerootmatrixofRsuchthat1 2R1 2R=R,thenp n(^R)]TJ /F8 11.955 Tf 12.67 0 Td[(R);N(0,R)impliesthatp n)]TJ /F13 5.978 Tf 7.78 3.26 Td[(1 2R(^R)]TJ /F8 11.955 Tf 11.61 0 Td[(R);N(0,I3).Therefore,n(^R)]TJ /F8 11.955 Tf 11.61 0 Td[(R)0)]TJ /F10 7.97 Tf 6.59 0 Td[(1R(^R)]TJ /F8 11.955 Tf 11.61 0 Td[(R);23,andforanyconsistentestimator^RofR, n(^R)]TJ /F8 11.955 Tf 11.96 0 Td[(R)0^)]TJ /F10 7.97 Tf 6.59 0 Td[(1R(^R)]TJ /F8 11.955 Tf 11.96 0 Td[(R);23.(2) 22

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From( 2 ),itfollowsthatunderH0:R=(R3:1,R1:1,R1:3)0=1, X2=n(^R)]TJ /F8 11.955 Tf 11.96 0 Td[(1)0^)]TJ /F10 7.97 Tf 6.59 0 Td[(1R(^R)]TJ /F8 11.955 Tf 11.95 0 Td[(1)H0;23.(2)Anasymptoticlevel-WaldtestofH0maybeconductedbyrejectingH0:R=(R3:1,R1:1,R1:3)0=1whenevern(^R)]TJ /F8 11.955 Tf 12.84 0 Td[(1)0^)]TJ /F10 7.97 Tf 6.58 0 Td[(1R(^R)]TJ /F8 11.955 Tf 12.83 0 Td[(1)>23,1)]TJ /F18 7.97 Tf 6.59 0 Td[(,wherePr(23,1)]TJ /F18 7.97 Tf 6.59 0 Td[(>23)=1)]TJ /F7 11.955 Tf 11.96 0 Td[(. 2.2.2Designswith1:0,1:1,and0:1PlantingRatiosWhentheexperimentaldesignincorporatesonlymonoculturesanda1:1plantingratiowemayconstructthelarge-sampledistributionof^R1:1byborrowingfromthecalculationsoftheprevioussection.Inthiscaseseveraldiagonalelementsofarezero,andhencethedistributionofZ1isnowdegenerate.However,inwhatfollowsweshallseethattheborrowedcalculationsareindependentofalldegeneraterandomvariablesinZ1.WewillthusarriveatthesamesolutionhadweredenedZkwithdegeneraterandomvariablesomitted.Movingforward,supposenowf:R8!Risthefunctionf2wehavealreadydened.Thegradientoff2(t)evaluatedatt=isthenthesecondrowofrf.Hence,bytheunivariatedeltamethod(see,e.g., Lehmann&Casella ( 1998 )),p n(f2(Z))]TJ /F4 11.955 Tf 11.95 0 Td[(f2())=p n(^R1:1)]TJ /F4 11.955 Tf 11.95 0 Td[(R1:1);N(0,2R),where2R1:11:0 21:02+21:1 21:0+2()1:1 1:00:1+21:1 20:1+1:10:1 20:12.With^2Rtheestimatorobtainedbysubstitutingsamplestatisticsforpopulationparameters,wemayconductanasymptotictwo-sidedlevel-testofH0:R1:1=1byrejectingH0wheneverp n(^R1:1)]TJ /F6 11.955 Tf 11.95 0 Td[(1) ^R>tn)]TJ /F10 7.97 Tf 6.58 0 Td[(1,1)]TJ /F18 7.97 Tf 6.59 0 Td[(=2,wheretn)]TJ /F10 7.97 Tf 6.58 0 Td[(1,1)]TJ /F18 7.97 Tf 6.59 0 Td[(=2isthe1)]TJ /F18 7.97 Tf 13.15 4.71 Td[( 2quantileofatdistributionwithdf=n)]TJ /F6 11.955 Tf 11.95 0 Td[(1. 23

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2.3SimulationStudyAsimulationstudywasconductedtoassessthesizeandpoweroftheasymptoticWaldtestingprocedureofH0:R=1usingtheteststatisticgivenby( 2 ).Thestudydesignwasa342factorialarrangementofthreefactors:thenumberofreplicatesnperplantingratio,n2f6,10,20,100g;ascalingparametersforthemeanvectorsuchthatthemeansincreasewithincreasings,s2f2,5,10g;andthetypeofinterspeciescompetition(I,II,III,orIV).ForcompetitiontypesIandIIthenullhypothesisH0:R=1istrue,andforcompetitiontypesIIIandIVthenullhypothesisisfalse.Thus,therewere48settingsofthesimulationparameters,24forwhichthenullhypothesiswastrueand24forwhichthenullhypothesiswasfalse.Onethousandsamplesweregeneratedforeachcombinationofthesimulationparameters.Foreachsimulatedsample,per-plantyieldsweregeneratedfroman8-dimensionalGumbel(=2)copulawithgammadistributionmarginsusingthecopulapackageinR( Yan 2007 ).Marginalgammadistributionswereparameterizedbyaandb,withmeanabandvarianceab2.Thebparameterwasxedatonesothateachmarginalmeanisequaltotheaparameter.Weletdenotethevectorofmarginalmeans.Competitiontypedenedthevectorofmarginalgammameans,.ThefourfunctionalformsofcompetitiontypeconsideredinthesimulationexperimentareplottedinFigure 2-1 .LettcorrespondtotheproportionofspeciesYplantsinacontainersothat,e.g.,t=0forplantingratio1:0andt=0.25forplantingratio3:1.ToassessWaldtestsize,underTypeIcompetitionthefunctionsfX(t)=(1)]TJ /F4 11.955 Tf 11.73 0 Td[(t)andgY(t)=twereusedtoconstructthesimulationmeansofspeciesXandY,respectivelysothatthevectorofmarginalgammameansis=s[fX(0),fX(.25),fX(.5),fX(.75),gY(0),gY(.25),gY(.5),gY(.75)]0.Notethatthemeansareincreasedbythefactors,whichiswhywerefertoitasascalingparameter.UnderTypeIIcompetition,thefunctionsfX(t)=1)]TJ 12.1 9.67 Td[(p tandgY(t)= 24

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p twereusedtocalculatethesimulationmeansofspeciesXandY,respectively.ToassessthepoweroftheWaldtest,underTypeIIIcompetition,thefunctionsfX(t)=p 1)]TJ /F4 11.955 Tf 11.95 0 Td[(tandgY(t)=p twereusedtocalculatethesimulationmeansofspeciesXandY,respectively.Lastly,thefunctionsfX(t)=(1)]TJ /F4 11.955 Tf 12.53 0 Td[(t)2andgY(t)=t2wereusedtocalculatethesimulationmeansofspeciesXandY,respectively,underTypeIVcompetition.Thisdata-generatingschemeprovidedpositivelycorrelatedrandomsamplesofpositiverandomvariablesforwhichthemarginalmeanvectorisequalto.AlthoughthecorrelationstructureoftherandomsamplesdoesnotagreewiththetheoreticalcorrelationstructureintherandomvectorsfZigdenedinx 2.2.1 ,thecopulacorrelationstructurerepresentsaworstcasescenario.Forexample,ifanimportantmean-relatedeffectisunobservedorignored,suchasthepositioningofplantingcontainerswithinagreenhouse,thentheremayappeartobeadependentcorrelationstructureamongobservationsthataretheoreticallyuncorrelated.Furthermore,gammamarginaldistributionswithb=1areasymmetricwithaskeweduppertail.Foreachofthe1000simulatedsamples,indexedbyi,theteststatisticX2igivenby( 2 )wascalculated.Then,foreachsimulatedsample,1000bootstrapsamplesweredrawn,indexedherebyb.Foreachbootstrapsampletheteststatistic,X2i,bwascalculated.Let^Riand^R,idenoteestimatorsofRandRintheithsimulatedsample.Further,let^Ri,band^R,i,bdenoteestimatorsofRandRinthebthbootstrapreplicatedrawnfromtheithsimulatedsample.Followingthesecondguidelineof Hall&Wilson ( 1991 ),thebootstrapteststatisticX2i,bis X2i,b=n(^Ri,b)]TJ /F6 11.955 Tf 13.02 2.65 Td[(^Ri)0^)]TJ /F5 7.97 Tf 0 -10.76 Td[(R,i,b(^Ri,b)]TJ /F6 11.955 Tf 13.02 2.65 Td[(^Ri).(2)Toassessempiricalsizeandpoweroftheasymptotic0.05-levelWaldtest,foreachsimulatedsampletheupper95thpercentileofthebootstrapteststatistics,X2i,(95),wasfound.Thus,theempiricalsizeunderTypesIandIIcompetition(denotedSizeinTable 25

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2-1 )andtheempiricalpowerunderTypesIIIandIVcompetition(denotedPowerinTable 2-2 )areeachequalto10)]TJ /F10 7.97 Tf 6.59 0 Td[(3P1000i=1I(X2i>X2i,(95)).Foreachsimulatedsamplei,thebootstrapp-valueispi=10)]TJ /F10 7.97 Tf 6.58 0 Td[(3P1000b=1I(X2i,b>X2i).Themeanofthebootstrapp-values,pB=10)]TJ /F10 7.97 Tf 6.59 0 Td[(3P1000i=1pi,isreportedinTables 2-1 and 2-2 ,alongwiththemedianofthebootstrapp-values,median(pB).Figures 2-2 2-3 2-4 ,and 2-5 containpanelsofplotsoftheorderedfpigvaluesagainstprobabilitypointsofauniform(0,1)distribution.Withineachpanel,thex-labelofeachgureliststhescaleparameter,s,whilethetitleliststhestatusofthenullhypothesiswiththeRomannumeralinparenthesisdenotingcompetitiontype. 2.4DiscussionWhenalldistributionalassumptionsaremet,foranyhypothesistestthep-valuesunderH0areU(0,1)distributed.OnthebasisofFigures 2-2 2-3 2-4 ,and 2-5 ,whenH0istruethebootstrapp-valuesfpigappeartomatchtheU(0,1)probabilitypointswell,withagreementimprovingassandnincrease.Forallsandnthedisagreementismorepronouncedforpi>0.4.Fromapracticalstandpointthisisnotsomuchofanissue,becauseonewouldfailtorejectH0atanyreasonablesignicancelevelifpi>0.4.WhenH0isfalsethebootstrapp-valuesfpigdivergefromtheU(0,1)probabilitypoints,withmanymorepinearzerothanexpectedwerethepiuniformlydistributedon(0,1).Asbothsandnincrease,sodoesthenumberofpinearzero.InTable 2-1 ,theempiricalsizeofthebootstrapp-valuesislowerthanexpectedfora0.05-levelsignicancetest.However,asbothn,sincrease,theempiricalsizeapproachesthenominal0.05level.Also,asbothnandsincrease,pBandmedian(pB)approachthemeanandmedian,respectively,ofaU(0,1)randomvariable.Ultimately,however,acrossallvaluesofnands,andforbothTypesIandIIcompetition,theempiricalsizeaisbiasedlow.InTable 2-2 ,forn10theempiricalpowerofthebootstraptestdependsuponcompetitiontype.Allotherconsiderationsequal,theTypeIIIcompetitiverelationship 26

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yieldsgreaterpowerthantheTypeIVcompetitiverelationship.Asnandsincrease,however,powerapproachesoneandthemeanandmedianofthebootstrapp-valuesrapidlyapproachzero. 2.5CollaborationandSupportWethankNehaRana,DepartmentofAgronomy,UniversityofFlorida,forpresentingtheproblemandprovidingthethedatasetsthatmotivatedthiswork.Dr.RanaexploresthecontroloftwovarietiesofSmutgrassin Rana ( 2012 ). 27

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Figure2-1. Fourmodelsofinterspecicrelativeyield.Forclarity,RCCiisdenotedRCC(i),i=1,2.Furthermore,RYTiscalculatedateachplantingratio,withRYT(i:j)denotingRYTatplantingratioi:j.Adaptedfrom Harper ( 1977 ,p.256,Figure8/9). 28

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Figure2-2. Q-Qplotsofbootstrapp-values,n=6.ForeachCompetitionType(I,II,III,orIV)andforeachscalevalue,1000simulatedsamplesweregeneratedandtheteststatistic,X2,calculated.Foreachsimulatedsample,1000bootstrapsamplesweregeneratedandforeachthebootstrapteststatistic,X2b,wascalculated.Thebootstrapp-valueisequaltoPbI(X2b>X2)=1000. 29

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Figure2-3. Q-Qplotsofbootstrapp-values,n=10.ForeachCompetitionType(I,II,III,orIV)andforeachscalevalue,1000simulatedsamplesweregeneratedandtheteststatistic,X2,calculated.Foreachsimulatedsample,1000bootstrapsamplesweregenerated.Foreachbootstrapsampletheteststatistic,X2b,wascalculated.Thebootstrapp-valueisequaltoPbI(X2b>X2)=1000. 30

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Figure2-4. Q-Qplotsofbootstrapp-values,n=20.ForeachCompetitionType(I,II,III,orIV)andforeachscalevalue,1000simulatedsamplesweregeneratedandtheteststatistic,X2,calculated.Foreachsimulatedsample,1000bootstrapsamplesweregenerated.Foreachbootstrapsampletheteststatistic,X2b,wascalculated.Thebootstrapp-valueisequaltoPbI(X2b>X2)=1000. 31

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Figure2-5. Q-Qplotsofbootstrapp-values,n=100.ForeachCompetitionType(I,II,III,orIV)andforeachscalevalue,1000simulatedsamplesweregeneratedandtheteststatistic,X2,calculated.Foreachsimulatedsample,1000bootstrapsamplesweregenerated.Foreachbootstrapsampletheteststatistic,X2b,wascalculated.Thebootstrapp-valueisequaltoPbI(X2b>X2)=1000. 32

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Table2-1. Theempiricalsizeofthebootstraptestingprocedureunderacrossallsimulationstudies.Foreachnumberofreplicatesperplantingratio,n,andforeachscaleparameter,s,themeanofthebootstrapp-valueacrossall1000simulatedsamplesispBandthemedianbootstrapp-valuesismedian(pB).Theempiricalsize(Size)istheproportionoftimesthesimulatedteststatistic,X2,exceededthe95thquantileofthebootstrapteststatistics,X2b. nsCompetitionTypepBmedian(pB)Size 62I0.4540.430.02062II0.4560.4260.03265I0.4790.4560.01765II0.4910.4650.011610I0.4990.4890.020610II0.4860.4640.013102I0.4730.4470.032102II0.470.4460.032105I0.4850.4720.034105II0.4940.4720.0211010I0.4860.4780.0201010II0.5020.5010.021202I0.4890.480.034202II0.4840.4680.028205I0.4920.4810.027205II0.4950.4820.0302010I0.5070.4960.0312010II0.4990.4960.0321002I0.4950.4810.0391002II0.5040.510.0351005I0.4950.4850.0451005II0.5080.4990.03110010I0.5010.4940.04010010II0.5010.480.046 33

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Table2-2. Theempiricalpowerofthebootstraptestingprocedureunderacrossallsimulationstudies.Foreachnumberofreplicatesperplantingratio,n,andforeachscaleparameter,s,themeanofthebootstrapp-valuesacrossall1000simulatedsamplesispBandthemedianbootstrapp-valueismedian(pB).Theempiricalsize(Power)istheproportionoftimesthesimulatedteststatistic,X2,exceededthe95thquantileofthebootstrapteststatistics,X2b. nsTypepBmedian(pB)Power 62III0.1130.0630.43762IV0.2540.2040.07565III0.0470.0210.73265IV0.1230.0820.316610III0.0210.0070.893610IV0.0540.0360.649102III0.0560.0180.691102IV0.1510.0960.288105III0.0130.0020.931105IV0.0360.0150.8011010III0.0020.00.9941010IV0.0070.0020.975202III0.0160.0020.919202IV0.0350.0130.810205III0.0010.00.997205IV0.0020.00.9992010III0.00.01.02010IV0.00.01.01002III0.00.01.01002IV0.00.01.01005III0.00.01.01005IV0.00.01.010010III0.00.01.010010IV0.00.01.0 34

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CHAPTER3UNION-INTERSECTIONTESTSFORMODELSELECTION 3.1BackgroundNextgenerationdesignedexperimentsinvestigatinggeneexpressionoftenyieldtranscriptcounts,fYig.Insuchexperiments,givencovariatesfXig,countsaregenerallymodeledusingaknownparametricmodelfamily,typicallytakentobethePoisson(),>0family.IftheratioofthePearson-2statistictoitsdegreesoffreedomissubstantiallygreaterthanone,however,onemaydoubttheplausibilityofthesuppositionthatthemodelfamilyisPoissonandinsteadfavoramodelfamilyoverdispersedrelativetothemean.Inthegeneralizedlinearmodelframework,suchoverdispersionisusuallyassumedtobelinearorquadraticinnature(seepages199and324-238of McCullagh&Nelder ( 1989 )).WefocusourinvestigationuponassessingwhethermeanoverdispersionrelativetothePoissondistributionispresentand,ifso,whetherlinearorquadraticoverdispersionismoreplausible.GiventhemeanofarandomvariableYwithsupportZf0,1,2,...g,linearoverdispersionisattributedtoageneralizedPoissonmodelforwhichVar(Y)=,>1.QuadraticoverdispersionisattributedtoanegativebinomialmodelforwhichVar(Y)=+1 s2,s>0.Following Joe&Zhu ( 2005 ),wesubsequentlyidentifythenegativebinomialasaPoisson-gammamixturewhileframingthelinearlyoverdispersedgeneralizedPoissonasamixtureofPoissonswithanalternativePoissonmixingscheme.QuadraticoverdispersionrelativetothemeanofaPoissondistributionisincorporatedbytakingtheresponsedistributionfamilytobethenegativebinomial.Inthiscase,theprobabilitymassfunction(pmf)isf(yjp,s)=\(y+s) y!\(s)ps(1)]TJ /F4 11.955 Tf 12.76 0 Td[(p)yI(y2Z),wheretheparameterspaceisf(p,s)jp2(0,1),s>0g.ForthenegativebinomialrandomvariableYNB(p,s),E(Y)=s(1)]TJ /F5 7.97 Tf 6.59 0 Td[(p) pandVar(Y)=s(1)]TJ /F5 7.97 Tf 6.59 0 Td[(p) p2.Takings(1)]TJ /F5 7.97 Tf 6.58 0 Td[(p) p,itmaybeshownthatVar(Y)=+s)]TJ /F10 7.97 Tf 6.58 0 Td[(12. 35

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LinearoverdispersionrelativetothemeanofaPoissondeviatemaybeintroducedwiththegeneralizedPoissondistributionfamily.Beginningwiththenegativebinomialdensityfunction, Consul&Jain ( 1973 )arriveatthegeneralizedPoissondensityfunctionbyconsideringstobelargeandptobesmall,taking=s=pand'=p,andusingSterling'sapproximationforthegammafunctions.ThenYGP(,')hasdensityf(yj,')=(+'y)y)]TJ /F13 5.978 Tf 5.75 0 Td[(1exp()]TJ /F18 7.97 Tf 6.58 0 Td[()]TJ /F18 7.97 Tf 6.59 0 Td[('y) y!I(y2Z)withrespecttocountingmeasure,andtheparameterspaceisf>0,'2(0,1)g.ForYGP(,'),E(Y)= 1)]TJ /F18 7.97 Tf 6.58 0 Td[('andVar(Y)= (1)]TJ /F18 7.97 Tf 6.58 0 Td[(')3=(1)]TJ /F7 11.955 Tf 11.96 0 Td[('))]TJ /F10 7.97 Tf 6.59 0 Td[(2 1)]TJ /F18 7.97 Tf 6.58 0 Td[(',sothatoverdispersionislinearrelativetothemean.Forxedmeanandvariance2>, Joe&Zhu ( 2005 )comparethenegativebinomialtothegeneralizedPoissonandnotethat...theGPdistributionhas[a]heaviertailwhiletheNBdistributionhasmoremassatzero.However,...inmanysituations,theGPdistributionisquiteclosetotheNBdistributioninoverallshape.Thismayexplainwhybothdistributionstequallywellinmanydatasets.Thus,distinguishingbetweenlinearandquadraticoverdispersionislikelytobechallenging. Vuong ( 1989 )approachestheproblemofmodelselectionamongpairsofcandidatemodelfamiliesbyconstructingacollectionofasymptotictests.Theappropriate Vuong ( 1989 )testisprescribedonthebasisoftherelationshipbetweenthecompetingmodelfamilies,whichareclassiedaccordingtooneofthreemutually-exclusivegroups:nested,overlapping,orstrictlynon-nested. Yangetal. ( 2007 )employthe Vuong ( 1989 )testtoselectbetweenthenegativebinomialandgeneralizedPoissonmodelfamilies. Dean&Lawless ( 1989 )and Dean ( 1992 )constructscorestatisticstotestforoverdispersioncharacterizedbythefunctionalrelationshipto. Dean&Lawless ( 1989 )testforquadraticoverdispersionrelativetothemeanofaPoissondistributionwherebythevariancefunctionunderthealternativehypothesisisthatofanegativebinomialdistribution. Dean ( 1992 )derivesascoretestforlinear(e.g.generalizedPoisson)overdispersionrelativetothemeanofaPoissondistribution. Yangetal. ( 2007 )derivethe Vuong ( 1989 )testforselectingbetweentheGPandNBmodel. 36

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3.2TestingforOverdispersioninCountDataModels 3.2.1DeanTestsforModelSelectionConsiderasampleofindependentobservationsYi,i=1,2,...,n,withE(Yi)=i=i(xi,).OurgoalistoconductinferencesonVar(Yijxi).WeconsiderthreepossiblestructuralformsofVar(Yijxi):(i)Poissonvariation,withVar(Yijxi)=i;(ii)linearoverdispersionrelativetoaPoisson,withVar(Yijxi)=i(1+);or(iii),quadraticoverdispersionrelativetoaPoisson,withVar(Yijxi)=i(1+i).Correspondingtoascoretestcomparing(i)and(ii),hypothesesareH(GP,P)0:E(Yijxi)=iandVar(Yijxi)=i(1+)and=0H(GP,P)1:E(Yijxi)=iandVar(Yijxi)=i(1+)and>0,wherethesuperscripts(GP,P)denotethehypothesistestwithlinearalternative. Dean ( 1992 )showedthatthecorrectedoradjustedteststatisticP0B=Pni=1h(Yi)]TJ /F6 11.955 Tf 12.67 0 Td[(^i)2)]TJ /F4 11.955 Tf 11.95 0 Td[(Yi+^hii^ii (2Pni=1^2i)1=2isasymptoticallystandardnormallydistributedunderH0whenE(Yijxi)=iandVar(Yijxi)=i(1+).Correspondingtoatestcomparing(i)and(iii),thehypothesesareH(NB,P)0:E(Yijxi)=iandVar(Yijxi)=i(1+i)and=0H(NB,P)1:E(Yijxi)=iandVar(Yijxi)=i(1+i)and>0,wherethesuperscript(NB,P)denoteshypothesistestswithaquadraticalternative.Thequadraticoverdispersionparameterisnowsoasnottoconfusetherolesofand. Dean ( 1992 )showedthatthecorrectedoradjustedteststatisticP0C=1 p 2nnXi=1"(Yi)]TJ /F6 11.955 Tf 13.92 0 Td[(^i)2)]TJ /F4 11.955 Tf 11.96 0 Td[(Yi+^hii^i ^i# 37

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isasymptoticallystandardnormallydistributedunderH0whenE(Yijxi)=iandVar(Yijxi)=i(1+i).NotethatthemodelfamilycorrespondingtoH(GP,P)0cannotbedistinguishedfromthatofH(NB,P)0becauseunderH(GP,P)0andalsoH(NB,P)0wehaveE(Yijxi)=iandVar(Yijxi)=i. 3.2.2TheVuongTestforNon-NestedModelSelectionThe Vuong ( 1989 )modelselectionprocedureisusedfordeterminingwhichprobabilitymodel,ForG,isbetterforagivenadataset.First,FandGmustbeclassiedasnested,overlapping,orstrictlynon-nested.ModelfamiliesFff(yj)j2gandGfg(yj)j2)]TJ /F2 11.955 Tf 6.77 0 Td[(garestrictlynon-nestedifandonlyifF\G=;.Forstrictlynon-nestedmodelsthe Vuong ( 1989 )procedureemployshypothesesH0:E0"log f(Yjx,^) g(Yjx,^)!#=0Hf1:E0"log f(Yjx,^) g(Yjx,^)!#>0Hg1:E0"log f(Yjx,^) g(Yjx,^)!#<0wheretheexpectationistakenwithrespecttothetrue,unknownprobabilitymeasure.Forstrictlynon-nestedmodelssatisfyingacollectionofassumptions,Theorem5.1of Vuong ( 1989 )describestheasymptoticbehavioroftheVuongteststatistic,V,undereachoftheabovehypotheses.UnderH0,V;N(0,1);underHf1,V!1a.s.;andunderHg1,V!a.s.ToconstructtheteststatisticV,thestatisticofinterestisthelog-likelihoodratio,LR(^,^)Pni=1logf(Yijxi,^) g(Yijxi,^),forwhichtheestimatedvarianceis,inthenotationof Vuong ( 1989 ),^!2n)]TJ /F10 7.97 Tf 6.59 0 Td[(1Pni=1hlogf(Yijxi,^) g(Yijxi,^)i2)]TJ /F14 11.955 Tf 12.41 13.27 Td[(hn)]TJ /F10 7.97 Tf 6.58 0 Td[(1Pni=1logf(Yijxi,^) g(Yijxi,^)i2.Hence,the Vuong ( 1989 )teststatisticforstrictlynon-nestedmodelfamiliesisVn)]TJ /F10 7.97 Tf 6.58 0 Td[(1=2LR(^,^)=^!.Anasymptoticlevel-testofH0isperformedbyrejectingH0infavorofHf1forV>z1)]TJ /F18 7.97 Tf 6.58 0 Td[(=2,rejectingH0infavorofHg1forV<)]TJ /F4 11.955 Tf 9.3 0 Td[(z1)]TJ /F18 7.97 Tf 6.59 0 Td[(=2,andfailingtorejectH0otherwise.UnderH0themodel 38

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familiesFandGareequivalent.UnderHf1theresearcherconcludesthatFisbetterthanG,whiletheoppositeconclusionismadeunderHg1. Yangetal. ( 2007 )usedthe Vuong ( 1989 )testingproceduretoselectbetweenlinear(generalizedPoisson)andquadratic(negativebinomial)overdispersedalternativestothePoissonmodelfamily.TakeFff(yj,')j>0,'2(0,1)g,thefamilyofoverdispersed(generalized)Poissondistributions,andGfg(yjp,s)js>0,p2(0,1)gthefamilyofnegativebinomialdistributions.Expandinguponthenotationintroducedintheprevioussection,thenullhypothesisforchoosingbetweenthegeneralizedPoissonandnegativebinomialmodelsisH(GP,NB)0.UnderH(GP,NB)0,thereisnoevidencetodistinguishbetweenthegeneralizedPoissonandnegativebinomialmodelfamilies.ThelinearlyoverdispersedgeneralizedPoissonalternativewillbedenotedH(GP)1andthequadraticoverdispersed(negativebinomial)alternativewillbedenotedH(NB)1.InthiscasetheVuongteststatisticV(GP,NB)isn)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2timestheratioofthelog-likelihoodratio,LR,totheestimatedstandarddeviationofLR,wherebothareevaluatedatthemaximumlikelihoodestimators(MLEs)obtainedfromatoftherespectivegeneralizedlinearmodel(GLM).HereLR(^,^',^p,^sjy)=nXi=1logf(Yijxi,^) g(Yijxi,^)=nXi=1log^(^+^'yi)yi)]TJ /F10 7.97 Tf 6.58 0 Td[(1exp()]TJ /F6 11.955 Tf 10.03 2.65 Td[(^)]TJ /F6 11.955 Tf 13.63 0 Td[(^'yi)=yi! \(yi+^s)^p^s(1)]TJ /F6 11.955 Tf 12.1 0 Td[(^p)yi=yi!\(^s)andbysimilarsubstitutionofMLEsfortherespectivegeneralizedPoissonandnegativebinomialparametersweobtaintheestimatedvariance^!2(^,^',^p,^sjy).ThegeneralizedPoissonversusnegativebinomial Vuong ( 1989 )teststatisticisthusV(GP,NB)=[n^!2(^,^',^p,^sjy)])]TJ /F10 7.97 Tf 6.59 0 Td[(1=2LR(^,^',^p,^sjy).Wefurtherapply Vuong ( 1989 )testsofnon-nestedhypothesestoselectbetweenthePoissonandbothlinear(generalizedPoisson)andquadratic(negativebinomial)alternatives.Teststatisticsarecalculatedintheabovemanner(i.e.FisthegeneralizedPoissonfamilyandGisthePoissonfamily),with 39

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necessaryparameterspaceboundaryconditionstoensuretheclassesofgeneralizedPoissonandnegativebinomialmodelsdonotincludethePoissonmodelfamily.Inkeepingwiththenotationoftheprevioussection,hypothesesandteststatisticsforchoosingbetweenthePoissonandgeneralizedPoissonareH(GP,P)0,H(GP,P)1,andV(GP,P),respectively.HypothesesandteststatisticsforchoosingbetweenthePoissonandnegativebinomialareH(NB,P)0,H(NB,P)1,andV(NB,P),respectively.Inbothcasesweconsiderone-sidedalternativesasweviewthePoissonlog-linearGLMtobethedefaultcountdatamodel.Oneissuewiththe Vuong ( 1989 )testarisesasfollows:consideranANOVA-typedesignwithobservationsmadeonunitsineachoftwoindependentgroups.Thedatageneratingmechanismisassumedtobeoverdispersedrelativetothemean.Intheeventthattherstgroupmean,1,isequaltothesecondgroupmean,2,thetwogroupsarereallyonesupergroupwithmean1=2andvariance2.Underthenullhypothesisofnogroupeffectwemayestimatethesupergroupsamplemeanandvariancewithyands2,respectively.Becausethereisonlyonegroup,wehaveonlyoneorderedpair(y,s2)withwhichwemustestimatethegeneralizedPoissonornegativebinomialdispersionparameter.InthegeneralizedPoissoncase,whereVar(Y)=,weobtain^=s2=y.Inthenegativebinomialcase,whereVar(Y)=+(1=k)2,weobtain^k==(s2=)]TJ /F6 11.955 Tf 11.96 0 Td[(1).Withoutasecondgroup,itisdifculttoarguewhichmodelisbetter.Inthetwosamplecasewedistinguishbetweenthereducedmodelofacommongroupmean1=2andthefullmodelofpossiblydistinct1,2.Underthereducedmodel,allobservationsarefromasinglegroupandthusthe Vuong ( 1989 )testwillnotbeeffective.Inthefullmodelwehavetwoorderedpairs(y1,s21)and(y2,s22)fromwithwhichwemayestimatedispersionparameters.Assumingaconstantdispersionparameteracrossgroupswemaynowobtaintwoestimatesofthedispersionparameter.Forexample,inthegeneralizedPoissoncasewehave^1=s21=y1and^2=s22=y2.If^1^2thenwemightreasonablysupposethedatageneratingmechanismis 40

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overdispersedPoisson,whileif,say,^1>^2byalargeamountwemightsupposethedatageneratingmechanismisnotoverdispersedPoisson. 3.2.3TheUnion-IntersectionTestWeusetheunion-intersectionprincipleof Roy ( 1953 )toconstructatestofthenullhypothesisthatthereisneitherlinearnorquadraticoverdispersionrelativetothePoissonH0:H(GP,P)0\H(NB,P)0againstthealternativethatthereiseitherlinearorquadraticoverdispersionrelativetothePoisson,H1:H(GP,P)1[H(NB,P)1.Weevaluatethepowerofunion-intersectiontestsbaseduponboth Dean ( 1992 )and Vuong ( 1989 )procedures.Forbrevitywedescribeonlytheapplicationoftheunion-intersectionprincipleto Dean ( 1992 )tests.Applicationto Vuong ( 1989 )proceduresfollowsthesamedevelopment.LetZiYiforalliandletthen2matrixT=(Y,Z).WeconsidertestingH(GP,P)0andH(GP,P)1ontheYcomponentofTwhilesimultaneouslytestingH(NB,P)0andH(NB,P)1ontheZcomponentofT.Weusethisconstructiontoavoidhavingtosimultaneouslyspecifytwo,possiblydifferentprobabilitymodelsforYunderthealternativeofinterest.Tobeexplicit:H(GP,P)0:E(Yijxi)=iandVar(Yijxi)=i(1+)and=0H(NB,P)0:E(Zijxi)=iandVar(Zijxi)=i(1+i)and=0H(GP,P)1:E(Yijxi)=iandVar(Yijxi)=i(1+)and>0H(NB,P)1:E(Zijxi)=iandVar(Zijxi)=i(1+i)and>0Nowconsidertestingtheunion-intersectionnullhypothesisH0:fVar(Yijxi)=i(1+)and=0g\fVar(Zijxi)=i(1+i)and=0gversusthealternativehypothesisH1:fVar(Yijxi)=i(1+)and>0g[fVar(Zijxi)=i(1+i)and>0g. 41

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Moresuccinctly,wemayexpresstheunion-intersectionhypothesesasH0:f=0g\f=0gandH1:f>0g[f>0g.Usingtheadjustedscoretestsof Dean&Lawless ( 1989 )and Dean ( 1992 ),therejectionregionforaunion-intersectiontestofH0isfP0B(Y):P0B(Y)>w1)]TJ /F18 7.97 Tf 6.59 0 Td[(g[fP0C(Z):P0C(Z)>w1)]TJ /F18 7.97 Tf 6.59 0 Td[(g=fT=(Y,Z):maxB,CP0i(T)>w1)]TJ /F18 7.97 Tf 6.59 0 Td[(g,whereitisunderstoodthatP0B(T)=P0B(Y)andP0C(T)=P0C(Z).Regardingourchoiceof,thep-valuefortestingtheunion-intersectionnullhypothesisisp-value=Pr(rejectH0jH0istrue)=PrmaxB,CP0i>w1)]TJ /F18 7.97 Tf 6.59 0 Td[(Pr(P0B>w1)]TJ /F18 7.97 Tf 6.59 0 Td[()+Pr(P0C>w1)]TJ /F18 7.97 Tf 6.59 0 Td[().Bysetting0=1 2,thep-valueofaconservativeasymptoticlevel-testofH0isp-value=Pr(P0B(Y)>w1)]TJ /F18 7.97 Tf 6.58 0 Td[(0)+Pr(P0C(Z)>w1)]TJ /F18 7.97 Tf 6.59 0 Td[(0),whereP0BandP0Caretheadjustedscoreteststatisticsof Dean&Lawless ( 1989 )and Dean ( 1992 ). 3.3Two-stageTestforOverdispersionWeproposeatwo-stageproceduretodetectlinearorquadraticoverdispersionand,whenoverdispersionisdetected,tosecondarilydeterminewhethertheoverdispersionmoreplausiblyresultsfromageneralizedPoissonornegativebinomialmechanism.Intherststage,duringwhichweassesswhetherornotlinearorquadraticoverdispersionrelativetothemeanispresent,weproposeunion-intersectiontestsbaseduponeither Dean ( 1992 )or Vuong ( 1989 )testingprocedures.Inthesecondstageweevaluatetheeffectivenessofusingthe Vuong ( 1989 )testforoverdispersedmodelselectionproposed 42

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by Yangetal. ( 2007 )todeterminewhetherageneralizedPoissonornegativebinomialmodelistobepreferred.Weusethephrasetwo-stagetestinreferencetoproceduresemployingarststageunion-intersectionDeantestoflinearorquadraticoverdispersion.Uponrejectionoftherststageunion-intersectionnullhypothesisofneitherlinearnorquadraticoverdispersion,asecondstage Vuong ( 1989 )generalizedPoissonversusnegativebinomialmodelselectiontestofthe Yangetal. ( 2007 )typeisemployed.Thesizeandpowerofthetwo-stagetestingprocedurewereevaluatedviaasimulationstudy.Asymptoticlevel-0.05testsandi=2groupswereusedthroughout.A425factorialstudydesignwasusedtoevaluatethesizeofthetwo-stageprocedure,inwhichthefactorsweregroupsize(ni),groupmeans(fig),andthefold-changeamonggroupmeans,().Acrossallsimulationsdesignedtoassesssize,Poissonrandomsamplesweredrawnwithn1=n2.Fourlevelsofgroupsizewereconsidered,withn12f5,10,50,100g.Fourlevelsofgroupmeanswereconsidered,with12f2,24,28,210g.Themeanofthesecondgroup,21,wherethefold-change2f1.5,2,2.5,5,10g.Toassesstwo-stagetestpower,a23425factorialdesignwasused.Inthepowerassessmentweconsideredresponsedistributionandoverdispersioninadditiontotheotherfactorsmentionedinthesizeassessment.Weconsideredtworesponsedistributions,thegeneralizedPoissonfamilyandthenegativebinomialfamily.Foreachmodelfamilyweconsideredthreelevelsofoverdispersion.RecallingthatforYGP(,)wehaveVar(Y)=,forgeneralizedPoissonsamplesweconsidered2f1.1,2,3g.RecallingthatforYNB(,s)wehaveVar(Y)=+s)]TJ /F10 7.97 Tf 6.59 0 Td[(12,fornegativebinomialsamplesweconsidereds2f1,1.25,10g.Foreachoftwogroupsforeachresponsedistributionandoverdispersionparameter,randomsamplesofsizen1weregeneratedwithn12f5,10,50,100g;12f2,24,28,210g;and2f1.5,2,2.5,5,10g.Thus,atotalofof425+23425=560combinationsofsimulationstudyparameterswereconsidered. 43

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Foreachcombinationofsimulationparameters,1000simulatedsamplesweregenerated.Foreachoftwogroups,n1independentobservationsweregeneratedfromthespecieddistributionfamilywiththerstgroupmeanequalto1andthesecondgroupmeanequalto2=1.GeneralizedPoissondataweregeneratedforgroupi,i=1,2,withmeaniandvariancei.Negativebinomialdataweregeneratedforgroupi2f1,2gwithmeaniandvariancei+(1=s)2i,i=1,2,wheresdenotesthenegativebinomialsizeparameter.Priortoevaluatingthetwo-stagetestingprocedurewerstexaminedthestudydatatoassessthesizeandpowerofthe Dean ( 1992 )and Vuong ( 1989 )testsofthenullhypothesesH(GP,P)0andH(NB,P)0.Thegoalofthispreliminaryreviewofthedatawastoascertainwhetherthe Dean ( 1992 )or Vuong ( 1989 )testshadsizeclosertothenominal0.05asymptoticlevelandpowerclosertothenominal0.95level.Ifeitherthe Dean ( 1992 )or Vuong ( 1989 )testperformedsubstantiallybetterthantheother,thentherststageunion-intersectiontestofthetwo-stageprocedurewouldbebasedupononlythebetter-performingcandidate,aswededucedthatthetwo-stageprocedurebasedupontheworse-performingcandidatewouldyieldless-satisfactoryresults.Aswillbepresentedinx 3.4 ,the Dean ( 1992 )testperformedbetterinthepreliminaryexamination,andthusformedthebasisoftherststageunion-intersectiontestinthetwo-stageprocedure.RandomsamplesweregeneratedintheRenvironment,usingbuilt-inPoissonandnegativebinomialfunctionsrpoisandrnbinom,respectively.GeneralizedPoissonrandomsamplesweregeneratedusingthefunctionrgenPoisdevelopedbytheauthorsforthispurpose.Poisson,overdispersedPoisson,andnegativebinomialparameterestimateswereobtainedbyttinganappropriategeneralizedlinearmodelusingthefunctionsglm,VGAM::vglm,andMASS::glm.nb,respectively.Followingthemodelstatement,defaultargumentstoeachfunctionwereutilized. 44

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TablesofsummarystatisticsassociatedwitheachgurearerelegatedtoAppendix B .Table 3-1 cross-referencestablesassociatedwitheachgure. 3.4ResultsandDiscussionInthepreliminaryexaminationofthe Dean ( 1992 )and Vuong ( 1989 )tests,the Dean ( 1992 )testshavemoredesirablesizeandpowerthanthe Vuong ( 1989 )testsovertherangeofsimulationparameters.Whenthedata-generatingmechanismisPoissontheempiricalTypeIerrorratefortheasymptotic0.05-level Dean ( 1992 )testisnolessthan0.035yetdoesnotexceed0.078.Thecorrespondingasymptotic0.05-level Vuong ( 1989 )testsizeisminimally0andmaximally0.132.ForPoissonsamples,thereisnoclearpatterninsimulationstudyparametercombinationsforwhicheitherofthe Dean ( 1992 )testnullhypothesesH(GP,P)0orH(NB,P)0iserroneouslyrejected.UnderPoissonsampling,the Vuong ( 1989 )testnullhypothesisH(GP,P)0empiricalTypeIerrorrateishigherthanthenominal0.05levelwhenn1=5and128.Whenn1>5,forall1andtheempiricalTypeIerrorrateofthe Vuong ( 1989 )testnullhypothesisH(GP,P)0decreaseswellbelowthenominal0.05levelasn1increases.Forexample,whenn1=100,theempiricalTypeIerrorrateisnomorethan1/10aslargeasthenominal0.05asymptoticlevel.Forthe Vuong ( 1989 )testofH(NB,P)0,underPoissonsamplingtheempiricalTypeIerrorrateneverexceeds0.005whenthenominallevelis0.05,andasn1increasestheempiricalTypeIerrorrateofthe Vuong ( 1989 )testofH(NB,P)0ismoreoftenthannotzero.SeeFigures 3-1 and 3-2 forillustration.Whenthedata-generatingmechanismisgeneralizedPoisson,preliminaryexaminationofempiricalpowerforasymptotic0.05-level Dean ( 1992 )and Vuong ( 1989 )testsindicatesthatpowerincreasesasgroupsize,n1,dispersionparameter,,and1increase,asillustratedinFigures 3-3 and 3-4 .Forgroupsizesn110theobservedincreaseinpowerismarkedasincreases,andlesssoas1increases.Forgroupsofsizen110, Dean ( 1992 )testpoweragainstlinearlyoverdispersedalternativesishighwhenoverdispersionismoderateorlarge(i.e.2f2,3g).Relativetothe Dean ( 1992 ) 45

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test,the Vuong ( 1989 )testforlinearlyoverdispersedalternativesperformsespeciallypoorlywhenn110,thoughresultsforlargen1aremoreconsistentwiththoseofthe Dean ( 1992 )test.Neitherthe Dean ( 1992 )northe Vuong ( 1989 )testperformswellwheneithergroupsizesaresmall(10),orwhenthereislittleoverdispersion(1).Inpreliminaryexaminationofempiricalpoweroftheasymptotic0.05-level Dean ( 1992 )and Vuong ( 1989 )testsagainstquadraticoverdispersedalternatives,appearinginFigures 3-5 and 3-6 ,bothtestsperformpoorlywhenthereislittleoverdispersion(s=10),regardlessofgroupsize.Powerimprovesasbothoverdispersionandgroupsizeincrease,withgroupsizeappearingtohaveamorepronouncedaffectonpowerthanoverdispersion.Againthe Dean ( 1992 )testappearstooutperformthe Vuong ( 1989 )test.Forgroupsizesn110the Dean ( 1992 )testperformsbetterwhenthefold-change>2.75.Asthe Dean ( 1992 )testperformsbetterthanthe Vuong ( 1989 )acrossPoisson,generalizedPoisson,andnegativebinomialsampling,weimplementthetwo-stageoverdispersiontestusingtheunion-intersection Dean ( 1992 )testintherststage.Figures 3-7 3-8 ,and 3-9 illustrateTypeIerrorandpowerforasymptotic0.05-leveltwo-stageunion-intersectiontests.Asdiscussedaboveweemploythetwo-stageprocedurebaseduponarst-stage Dean ( 1992 )test.WhenthedataarePoissonsamples,inFigure 3-7 wenotethattheempiricalTypeIerrorratefortherst-stage Dean ( 1992 )union-intersectiontestisroughlyequaltothenominal0.05levelacrossallsamplesizes.AsseeninFigure 3-8 ,forgeneralizedPoissonsampling,thedispersionparameteraswellasthegroupsizen1appeartostronglyaffectempiricalpower.Forsmalloverdispersion,where=1.1,regardlessofsamplesizetheempiricalpoweriswellbelow0.95asanticipatedforthenominal0.05leveltest.However,when2,empiricalpowerfortherst-stageunionintersectiontestimprovesasbothdispersionandgroupsizen1increase.Withnegativebinomialsampling,rst-stageunion-intersectionempiricalpowerisgreaterthangeneralizedPoissonsampling 46

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empiricalpowerforagivengroupsize,1,andcomparabledispersionparameter(seeFigure 3-9 ).Noteworthyistheapparentassociationbetweenfoldchangeandpowerfortherst-stagetest.ForbothgeneralizedPoissonandnegativebinomialsamples,foragivendispersionparameterandagivengroupsize,powerisgreaterwhenthefoldchange>2.75.WhendataarePoissondistributedtherst-stageunion-intersectiontestissomewhatconservativeforlargen1.Wedonotconsiderthistobealimitation,however,becausethepurposeoftheproposedmodelselectiontestisjustthat-toaidtheresearcherinmodelselection.OftheGLMsconsideredthelog-linearmodelisthemostwidelyknownandparsimonious.Withoutoverwhelmingevidenceinsupportofanalternativewechoosetousethemostparsimoniousmodel.Thus,aconservativeselectionmechanismeffectivelyreinforcestheprincipleofparsimony.Figures 3-10 and 3-11 illustratetheproportionofcorrectinferencesinthetwo-stagetestingprocedurewhensamplesaregeneralizedPoissonandnegativebinomial,respectively.ForgeneralizedPoissonsamples,weconcludeinferencesarecorrectifboththerst-stageunion-intersection Dean ( 1992 )testofnooverdispersionisrejectedandthesecond-stageVuongtestforgeneralizedPoissonversusnegativebinomialmodelfamiliesyieldsV>0.Fornegativebinomialsamplesweconcludeinferencesarecorrectifboththerst-stageunion-intersectiontestofnooverdispersionisrejectedandthesecond-stageVuongtestforgeneralizedPoissonversusnegativebinomialmodelfamiliesyieldsV<0.Notethatforsmalln1andlittleoverdispersionwerarelymakethecorrectinference,butassamplesizeandoverdispersionincreaseourabilitytodrawthecorrectinferenceimprovesdramatically.Inparticular,whenn150,forbothgeneralizedPoissonandnegativebinomialsamplingandeithermoderateorlargeoverdispersion,thetwo-stageprocedureyieldscorrectinferencesmorethanhalfthetime.Ifthefoldchange>2.75,thetwo-stageprocedureperformsevenbetter,yieldingcorrectinferenceswithanestimatedprobabilitygreaterthan0.8.Hence,the 47

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two-stageproceduremaybeofuseinlargestudiesyieldingcountdataforwhichaprioriknowledgeindicatesmoderatetolargeoverdispersion.Insummary,weproposethefollowingmodelselectionprocedure: 1. FirstStage:Conductaunion-intersection Dean ( 1992 )testofH0:f=0g\f=0gversusthealternativeH1:f>0g[f>0g.Foranasymptoticlevel-union-intersectiontest,rejectthenullhypothesisH0whenevermaxB,CP0i>w1)]TJ /F18 7.97 Tf 6.59 0 Td[(=2,wherePr(Ww1)]TJ /F18 7.97 Tf 6.59 0 Td[(=2thenH0isrejected,andwemustchoosewhetherthedata-generatingmechanismismoreplausiblythegeneralizedPoissonornegativebinomialdisributionfamily.Proceedto 2 2. SecondStage:Conducta Vuong ( 1989 )testformodelselectionbetweenthegeneralizedPoissonandnegativebinomialdistributionfamilies,withH(GP,NB)0:V(GP,NB)=0,H(GP)1:V(GP,NB)>0,andH(NB)1:V(GP,NB)<0. (a) IfV(GP,NB)>0thenwerejectH(GP,NB)0andinferthatthegeneralizedPoissonmodelfamilyismostplausiblythedata-generatingmechanism. (b) IfV(GP,NB)<0thenwerejectH(GP,NB)0andinferthatthenegativemodelfamilyismostplausiblythedata-generatingmechanism.Inlightofthesimulationstudyresults,werecommendtheuseoftheproposedtwo-stagemodelselectionprocedureoutlinedinSteps 1 and 2 foronefactorexperimentaldesignswithtwolevels,andn50observationsineachgroup.IfthereisapriorievidencetosuggestsampleswillbeoverdispersedrelativetothePoissonmodelfamilythentheproposedmodelselectionproceduremaybeusedinexperimentswithgroupsizesn10. 3.5CollaborationandSupportThisworkrepresentsthecollaborativeeffortsofamultidisciplinaryteam,eachmemberofwhichhasbeeninstrumentalinthecompletionoftheendeavor.WearegratefulforthesignicantcontributionsandcollaborativeeffortsofProfessorLaurenM.McIntyreandMrs.YajieYang,DepartmentofGeneticsandMicrobiology,Universityof 48

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Table3-1. Across-referencedlistofdataappearingineachgure. FigureTable(s) 3-1 B-1 3-2 B-2 3-3 B-3 B-4 B-5 3-4 B-6 B-7 B-8 3-5 B-9 B-10 B-11 3-6 B-12 B-13 B-14 3-7 B-15 3-8 B-16 B-17 B-18 3-9 B-19 B-20 B-21 3-10 B-22 B-23 B-24 3-11 B-25 B-26 B-27 Florida;andAssociateProfessorAnnL.Oberg,DepartmentofBiostatistics,MayoClinic.WearegratefultoDistinguishedProfessorGeorgeCasella,DepartmentofStatistics,UniversityofFloridaforhisinsightfulcritiqueandhelpfulcomments. 49

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Figure3-1. TypeIerrorrateforDeanandVuongasymptotic0.05-leveltests,Poissonsampling.UnderthealternativehypothesisthevariancefunctionisalinearfunctionofthemeanwithVar(Yi)=i,>1.TheDeanteststatisticisP0BandtheVuongteststatisticisV(GP,P).Blueplottingcharactersareusedforcaseswherefoldchange>2.75.Theredhorizontallineisaplotofy=0.05. 50

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Figure3-2. TypeIerrorrateforDeanandVuongasymptotic0.05-leveltests,Poissonsampling.UnderthealternativehypothesisthevariancefunctionisaquadraticfunctionofthemeanwithVar(Yi)=i+k)]TJ /F10 7.97 Tf 6.59 0 Td[(12i,k2(0,1).TheDeanteststatisticisP0CandtheVuongteststatisticisV(NB,P).Blueplottingcharactersareusedforcaseswherefoldchange>2.75.Theredhorizontallineisaplotofy=0.05. 51

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Figure3-3. PowerofDeanandVuongasymptotic0.05-leveltests,generalizedPoissonsampling.UnderthealternativehypothesisthevariancefunctionisalinearfunctionofthemeanwithVar(Yi)=i,>1.TheDeanteststatisticisP0BandtheVuongteststatisticisV(GP,P).Circular()plottingcharactersrepresentDeantestpowerwhiletriangular(4)plottingcharactersrepresentVuongtestpower.Blueplottingcharactersareusedforcaseswherefoldchange>2.75.Theredhorizontallineisaplotofy=0.95. 52

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Figure3-4. PowerofDeanandVuongasymptotic0.05-leveltests,generalizedPoissonsampling.UnderthealternativehypothesisthevariancefunctionisaquadraticfunctionofthemeanwithVar(Yi)=i+k)]TJ /F10 7.97 Tf 6.59 0 Td[(12i,k2(0,1).TheDeanteststatisticisP0CandtheVuongteststatisticisV(NB,P).Circular()plottingcharactersrepresentDeantestpowerwhiletriangular(4)plottingcharactersrepresentVuongtestpower.Blueplottingcharactersareusedforcaseswherefoldchange>2.75.Theredhorizontallineisaplotofy=0.95. 53

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Figure3-5. PowerofDeanandVuongasymptotic0.05-leveltests,negativebinomialsampling.UnderthealternativehypothesisthevariancefunctionisalinearfunctionofthemeanwithVar(Yi)=i,>1.TheDeanteststatisticisP0BandtheVuongteststatisticisV(GP,P).Circular()plottingcharactersrepresentDeantestpowerwhiletriangular(4)plottingcharactersrepresentVuongtestpower.Blueplottingcharactersareusedforcaseswherefoldchange>2.75.Theredhorizontallineisaplotofy=0.95. 54

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Figure3-6. PowerofDeanandVuongasymptotic0.05-leveltests,negativebinomialsampling.UnderthealternativehypothesisthevariancefunctionisaquadraticfunctionofthemeanwithVar(Yi)=i+k)]TJ /F10 7.97 Tf 6.59 0 Td[(12i,k2(0,1).TheDeanteststatisticisP0CandtheVuongteststatisticisV(NB,P).Circular()plottingcharactersrepresentDeantestpowerwhiletriangular(4)plottingcharactersrepresentVuongtestpower.Blueplottingcharactersareusedforcaseswherefoldchange>2.75.Theredhorizontallineisaplotofy=0.95. 55

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Figure3-7. TypeIerrorrateforasymptotic0.05-leveltwo-stageDeantests,Poissonsampling.ThenullhypothesisofnooverdispersionisrejectedifmaxB,CP0i>1.96.Blueplottingcharactersareusedforcaseswherefoldchange>2.75. 56

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Figure3-8. Powerforasymptotic0.05-leveltwo-stageDeantests,generalizedPoissonsampling.ThenullhypothesisofnooverdispersionisrejectedifmaxB,CP0i>1.96.Blueplottingcharactersareusedforcaseswherefoldchange>2.75.Theredhorizontallineisaplotofy=0.95. 57

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Figure3-9. Powerforasymptotic0.05-leveltwo-stageDeantests,negativebinomialsampling.ThenullhypothesisofnooverdispersionisrejectedifmaxB,CP0i>1.96.Blueplottingcharactersareusedforcaseswherefoldchange>2.75.Theredhorizontallineisaplotofy=0.95. 58

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Figure3-10. Proportionofcorrectinferencesforasymptotic0.05-leveltwo-stagetest,generalizedPoissonsampling.Therst-stagenullhypothesisofnooverdispersionisrejectedifmaxB,CP0i>1.96.Inthesecondstage,thedatageneratingmechanismisinferredtobethegeneralizedPoissonfamilyifV(GP,NB)>0.Blueplottingcharactersareusedforcaseswherefoldchange>2.75. 59

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Figure3-11. Proportionofcorrectinferencesforasymptotic0.05-leveltwo-stagetest,negativebinomialsampling.Therst-stagenullhypothesisofnooverdispersionisrejectedifmaxB,CP0i>1.96.Inthesecondstage,thedatageneratingmechanismisinferredtobethenegativebinomialfamilyifV(GP,NB)<0.Blueplottingcharactersareusedforcaseswherefoldchange>2.75. 60

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CHAPTER4MORAN-TYPERATIOSOFQUADRATICFORMSFORASSESSINGSPATIO-TEMPORALASSOCIATION 4.1Background 4.1.1MotivationInamapofstandardizedcancermortalityratesamongwhitemalesintheUnitedStatesfrom1975through1984,highermortalityratesappearclusteredinthenortheast(Figure 4-1 ).Incontrast,forthe1995to2004decade,theclusteringpatternofmortalityratesappearstobedifferent,withhigherratesnowoccurringinthesoutheast(Figure 4-2 ).Thefocusofthisworkisthedevelopmentofmethodstoassesswhetherthisapparentchangeinclusteringpatternisstatisticallysignicant. Moran ( 1948 )wasperhapsthersttoconsiderassessingthepresenceofspatialclusteringwithinasinglemap.Sincethenmanyscientists,acrossawiderangeofdisciplines,havefurtherexploredstatisticalproceduresforassessingspatialclusteringamongelementsofavectorofresponsesYwithrespecttoaspatialweightingorconnectivitymatrixW.ForcancermortalityratesamongwhitemalesacrossthespatialdomainD,thecollectionof48contiguousstates,letsidenotetheithelementofD.ThusD=fs1,s2,...,s48g.Letwijbethe(i,j)thelementofW.Byconventionwii=0foralli.Spatialassociationisoftendenedintermsofrst-nearestneighbors,regionsthatshareacommonborder,inwhichcasewij=1ifsiandsjarerst-nearestneighborandwij=0otherwise.Toillustrate,selectisuchthatsirepresentsthestateofFlorida.Thenarst-nearestneighborsweightingschemeyieldswij=1forjcorrespondingtoGeorgiaandAlabama,andwij=0otherwise.Throughoutouranalysisofstandardizedcancermortalityrateswetakewij=0whensiandsjmeetatacorner,sothate.g.wij=0foriandjcorrespondingtoNewMexicoandUtah.WithYiY(si),theithelementofY,thesingle-mapproblemof Moran ( 1948 )and Moran ( 1950 )istodeterminewhetherornotfYigarespatiallyindependent,givenaspatialweightingstructureW.Inamoregeneralsettinginwhichobservationsaremade 61

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overtime,wedeneYtjYt(sj)tobetheobservationcorrespondingtoregionsjattimet,fort=1,2,...,T.WithYtfYtjg,severalpropertiesofthespatio-temporalprocessfYtgTt=1areofinterest.Herethespatio-temporalassociationinducedbyaspatialweightingorconnectivitymatrixWandmethodstoevaluatethatassociationareusedtoassesschangesinpattern.Thischapterisorganizedasfollows.WerstreviewcurrentMoran-typemethodsforassessingspatialassociationamongtheelementsofY1,e.g.,whenT=1.Next,wedescribecurrentapproachestomodelingspatio-temporalprocesses.FromthisvantagepointweelucidateacontemporaryMoran-typestatisticproposedforspatio-temporalassociation,whichweextend. 4.1.2LiteratureReviewSupposeDfs1,s2,...,sngZ2isanitespatialdomainona2-dimensionallattice,whereZf...,)]TJ /F6 11.955 Tf 9.3 0 Td[(1,0,1,...,g.HenceforthwemayrefertotheelementsofDaspoints,(aerial)regions,orpossiblystatesorcounties.Fort=1,...,Tandsi2D,letYtiYt(si)denoteacontinuousrandomvariableofinterestobservedatregionsiattimet,andletYt(Yt1,Yt2,...,Ytn)0.Throughout,theprime(0)denotesmatrixtransposeandallvectorsareassumedtobecolumnvectors.AnmnmatrixAhas(i,j)thelementaij;weinterchangeAandfaijgfreely.Ifm=n,wemayalsousediag(A)(a11,a22,...,ann)0todenotethe(columnvectorof)diagonalelementsofA.TheleftKroneckerproductABofthempmatrixAandthenqmatrixBisthemnpqmatrixcomposedofmpblocks,with(i,j)blockaijB.Finally,thevectorizedformofthempmatrixAisvec(A)(a01,a02,...,a0p)0,whereaiistheithcolumnofA.GivenDfs1,s2,...,sngZ2,latticeedgescorrespondtospatialassociation.Letthespatialweightwij2Rrepresentaxedandknownconnectionbetweenpointssiandsj,withtheconventionthatwii=0.Wehaveseenthatinarst-nearestneighborspatialassociationstructurewij=1ifsiandsjshareacommonborderandwij=0otherwise.Regardlessofthespatialweightingregimewemayletthennmatrix 62

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Wfwijg. Cliff&Ord ( 1981 ,pg.16)callWageneralizedweightingmatrixinwhichwijistheeffectof[region]jon[region]i.Associationneednotbesymmetric,sothatwij6=wji.However,foreaseofcalculations,wemaysupposeWissymmetric.IfW6=W0thenwemaytakethegeneralizedweightingmatrixtobe1 2(W+W0),whichissymmetric.Inparticular,notethatforannnasymmetricgeneralizedweightingmatrixWandavectorx2Rn,x0Wx=x0W0x=x0[1 2(W+W0)]x.Thus,withoutlossofgenerality,wemaysupposeWissymmetric.Ingeneral,themechanismbywhichweightsfwijgareassigneddoesnotinduceametriconDD7!Rbecause,forsi,sj2D,wij=0doesnotimplythatsi=sj.Finally,wemayalsorequireWtoberow-standardizedsothatW1n=(1,1,...,1)0;thisisaffectedbyapplyingtoWthemapW7!diag(W1n))]TJ /F10 7.97 Tf 6.58 0 Td[(1W. Moran ( 1948 )exploredspatialassociationinbinaryprocessesoveralatticeD.Articulatinghisnotionofspatialassociation, Moran ( 1948 ,pg.243)stated: Itissometimesnecessarytoconsiderthegeographicaldistributionofsomequalityorphenomenoninthecountiesorstatesofacountry,andoneofthequestionswemaythenaskiswhetherthefactorscausingthephenomenonmayberegardedasstatisticallyindependentindifferentcounties,or,ontheotherhand,whetherthepresenceofsomequalityinacountymakesitspresenceinneighbouringcountiesmoreorlesslikely.Hence,forxedt,spatialindependencebetweenregionssiandsjimpliesCov(Yti,Ytj)=0. Moran ( 1948 )thensuggestedonepotentialapplicationofhiswork:studiesofdiseaseincidenceacrosssomecollectionofaerialregions.Twoyearslater,inexaminingspatialassociationamongcontinuousprocessesobservedoverarectangularJKlatticeD, Moran ( 1950 )proposedalarge-sampletestbaseduponthestatisticr11.Asthisstatisticisbasedononlyasinglemap,forthemomentwedropthetimeindex.LetYjkdenotetheobservationmadeatthe(j,k)thelementofD.AsinglerealizationofacontinuousspatialprocessoverDisY(Y11,Y12,...,Y1K,Y21,...,YJK)0,whereYjkareassumedtobeidenticallydistributedwithE(Yjk)and0
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lackingself-adjacency.Thatis,for(j,k),(j0,k0)2f(j,k):1jJ,1kKg,wrs=1ifeitherj=j0andjk)]TJ /F4 11.955 Tf 12.13 0 Td[(k0j=1,orifk=k0andjj)]TJ /F4 11.955 Tf 12.13 0 Td[(j0j=1,andwrs=0otherwise.ThenwithZjkYjk)]TJ /F6 11.955 Tf 13.89 2.66 Td[(Y, Moran ( 1950 )denedr11n PjPkwjkI,whereIPJj=1PK)]TJ /F10 7.97 Tf 6.59 0 Td[(1k=1ZjkZj(k+1)+PJ)]TJ /F10 7.97 Tf 6.59 0 Td[(1j=1PKk=1ZjkZ(j+1)k PjPkZ2jk.Notethefollowingaboutthecomponentsofr11.First,PjPkwjkistwicethetotalnumberofedgesontheJKlattice.InmatrixnotationwehaveZ=MY,whereMI)]TJ /F4 11.955 Tf 12.24 0 Td[(n)]TJ /F10 7.97 Tf 6.59 0 Td[(1110isaprojectionoperator.Thus,withWthegeneralizedweightingmatrixdenedinthepreviousparagraphthenumeratorofImaybewritten,inmatrixnotation,Z0WZ=Y0M0WMYandthedenominatorofYmaybewrittenZ0Z=Y0MY.HereE(Y)==1.ThusE(Z)=(I)]TJ /F4 11.955 Tf 12.79 0 Td[(n)]TJ /F10 7.97 Tf 6.59 0 Td[(1110)1=0.IfthereisnospatialassociationoverthelatticeDthenby Moran ( 1948 ),Var(Y)Y=2Iand,inthiscase,wehaveVar(Z)Z=Var(MY)=2MM0=2M.UnderthefurtherassumptionthatYismultivariatenormallydistributed,theexpectationofthenumeratorofIisE(Z0WZ)=tr(WZ)=tr[WM]=2tr[W(I)]TJ /F4 11.955 Tf 11.96 0 Td[(n)]TJ /F10 7.97 Tf 6.59 0 Td[(1110)]=2[tr(W))]TJ /F4 11.955 Tf 11.95 0 Td[(n)]TJ /F10 7.97 Tf 6.59 0 Td[(1tr(W110)]=)]TJ /F7 11.955 Tf 9.3 0 Td[(2 ntr(W110)=)]TJ /F7 11.955 Tf 9.3 0 Td[(2 nJXj=1KXk=1wjkandtheexpectationofthedenominatorisE(Z0Z)=tr(Z)=2tr(M)=(n)]TJ /F6 11.955 Tf 11.96 0 Td[(1)2.ByCorollary 1 ofx 4.2.3 ,IisindependentofZ0Z.Thus,underspatialindependence,E[I(Z0Z)]=E(I)E[Z0Z]. 64

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Therefore,theexpectationofr11underspatialindependenceisE(r11)=n PjPkwjkE(I)=n PjPkwjkE(Z0WZ) E(Z0Z)=n PjPkwjk)]TJ /F18 7.97 Tf 6.59 0 Td[(2 nPjPkwjk (n)]TJ /F6 11.955 Tf 11.96 0 Td[(1)2=)]TJ /F6 11.955 Tf 21.04 8.09 Td[(1 n)]TJ /F6 11.955 Tf 11.95 0 Td[(1. Moran ( 1950 )concludedhispaperbynotingthatr11isasymptoticallynormallydistributedasJ,K!1.Henceforthweusethephraseincreasingdomaintorefertoasymptoticsinwhichthenumberoflatticerowsandcolumnstendtoinnity.Soonafter Moran ( 1950 )appearedinprint, Whittle ( 1954 )introducedaclassofspatialmodelstodescribesimultaneouslythedistributionofamean-zero,continuousprocessfYigni=1overDwitharst-nearest-neighborspatialweightingmatrix.Forzero-mean,uncorrelatederrorsfig, Whittle 'smodelisY=WY+,wherethespatialassociationparameterjj<1andwherediag(W)=0.ForGaussianfig, Besag ( 1974 )allowsforpossiblynon-zeromeanfYiginthemodelof Whittle ( 1954 ).WithE(Yi)i,andwith(1,2,...,n)0, Besag denedthesimultaneousautoregressive(SAR)modeltobe Y)]TJ /F15 11.955 Tf 11.96 0 Td[(=W(Y)]TJ /F15 11.955 Tf 11.96 0 Td[()+.(4)Rearrangingterms,theSARmodelwithiiderrorsmaybewritten(I)]TJ /F7 11.955 Tf 12.12 0 Td[(W)(Y)]TJ /F15 11.955 Tf 12.13 0 Td[()=,whereN(0,2I).Hence, Besag 'smodelfortheresponsevectorisYN(,2(I)]TJ /F7 11.955 Tf 11.96 0 Td[(W))]TJ /F10 7.97 Tf 6.59 0 Td[(1(I)]TJ /F7 11.955 Tf 11.95 0 Td[(W0))]TJ /F10 7.97 Tf 6.59 0 Td[(1).Introducingaformaltestforspatialassociationrootedintheframeworkof Moran ( 1950 )andtheserialcorrelationtestingproceduresof Durbin&Watson ( 1951 ), Cliff&Ord ( 1972 )employedtheSARmodelpriorto Besag ( 1974 ).Given( 4 )inwhich=XforaknowndesignmatrixXandap)]TJ /F1 11.955 Tf 9.3 0 Td[(vectorofunknownregressioncoefcients,considertestingthenullhypothesisofnospatialassociationH0:=0againstH1:6=0.UnderH0,( 4 )reducestothelinearregressionofYonX. Cliff&Ord ( 1972 )denedtheparameterI/0W=0,andconsideredtestingH0usingthe 65

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statistice0We=e0e,whereeY)]TJ /F8 11.955 Tf 12.2 0 Td[(X^isthevectorofordinaryleastsquaresregressionresiduals. Cliff&Ord ( 1972 )furtherallowedforWtobeageneralizedweightmatrixsothatweightsmaybearbitrary,andshowedthatunderincreasing-domainasymptoticsthedistributionof^Ie0We=e0eisGaussian.Subsequentto Cliff&Ord ( 1972 ),thestatistic^I=ke0We=e0e,k2R,hasbeencalledMoran'sI,thoughincontrasttotheliteratureweuseahattoemphasizethat^Iisanestimatorofthecorrespondingpopulationquantity,I=0W=0.Noticethattakingk=n=PjPkwjkyieldsr11of Moran ( 1950 ).Furthermore,applicationoftherow-standardizationmapf:W7!diag(W1))]TJ /F10 7.97 Tf 6.58 0 Td[(1WtothegeneralizedweightingmatrixyieldsWf(W),withelementsfwjkg.Therow-standardizedgeneralizedweightmatrixWhasthedesirablepropertythatPjPkwjk=Pj(Pkwjk))]TJ /F10 7.97 Tf 6.58 0 Td[(1Pkwjk=n.ThenI=n PjPkwjk0W=0=0W=0.Thus,forarow-standardizedgeneralizedweightmatrixW,I=0W=0and^I=e0We=e0earescaledinthesamemannerthat Moran ( 1950 )employed. Besag&Moran ( 1975 )describedmaximumlikelihoodestimationandlarge-samplelikelihood-ratiotestingofmodel( 4 )parameters. Cliff&Ord ( 1981 )seeminglybroughtMoran'sIintothestatistical,geographical,andeconometriclexiconbyreviewingrelevantliteratureinparticular,their 1972 results.InsupportofinferentialproceduresincorporatingMoran'sI, Cliff&Ord ( 1981 )notedthatwhileonemayuse,forexample,maximumlikelihoodtoconductinferenceontheSARspatialassociationparameter,,thestatistic^Ianditsrsttwomomentsaresimpletocalculate.OnecaveattoSARmodelspatialinferencebaseduponMoran'sIisthat,whereastheparameter2()]TJ /F6 11.955 Tf 9.3 0 Td[(1,1)yieldstheinterpretationofincreasingspatialassociationasjj!1,thestatistic^Iisalmostsurelyabsolutelyboundedbyaconstantthatmaybelessthanunity.ForaSARmodelwith=[1], Cliff&Ord wrotetheboundingconstantintermsofaratioofthevarianceofPnk=1wjkYktothevarianceofYj.Givenaconstant-meanSARmodel deJongetal. ( 1984 )improvedupontheabsoluteboundingconstantof Cliff&Ord 66

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( 1981 )byidentifyingconstantscL,cU2RthesmallestandlargesteigenvaluesofMWM,respectivelysuchthat^I2[cL,cU]withprobabilityone.AfundamentalshiftintheMoran'sIliteratureisevidentfollowingthepublicationof Tiefelsdorf&Boots ( 1995 ),whoderivedundergeneralconditionstheexactdistributionof^Iaswellaslowerandupperboundsforthestatistic.Theauthorsemployedthefollowingargument.ConsiderthegeneralSARmodel(I)]TJ /F7 11.955 Tf 12.79 0 Td[(W)(Y)]TJ /F8 11.955 Tf 12.8 0 Td[(X)=,N(0,2I),whereWisasymmetric,row-standardizedgeneralizedweightmatrix.Here,Moran'sIstatistic ^I=(Y)]TJ /F8 11.955 Tf 11.96 0 Td[(X^)0W(Y)]TJ /F8 11.955 Tf 11.96 0 Td[(X^) (Y)]TJ /F8 11.955 Tf 11.96 0 Td[(X^)0(Y)]TJ /F8 11.955 Tf 11.96 0 Td[(X^)(4)isanestimatorofthespatialassociationparameterI=(Y)]TJ /F8 11.955 Tf 11.95 0 Td[(X)0W(Y)]TJ /F8 11.955 Tf 11.95 0 Td[(X) (Y)]TJ /F8 11.955 Tf 11.96 -.01 Td[(X)0(Y)]TJ /F8 11.955 Tf 11.95 .01 Td[(X)=0W 0ConsidertestingH0:=0versusH1:6=0usingMoran'sI,andsupposeH0istrue.ThentheSARmodelreducestothesimplelinearregressionmodelY=X+,N(0,2I).WithPX(X0X))]TJ /F10 7.97 Tf 6.59 0 Td[(1X0theprojectionontothecolumnspaceofXandMI)]TJ /F8 11.955 Tf 11.95 0 Td[(P,( 4 )is^I=(Y)]TJ /F8 11.955 Tf 11.96 0 Td[(X^)0W(Y)]TJ /F8 11.955 Tf 11.96 0 Td[(X^) (Y)]TJ /F8 11.955 Tf 11.95 0 Td[(X^)0(Y)]TJ /F8 11.955 Tf 11.96 0 Td[(X^)=Y0MWMY Y0MY=e0We e0e. (4)Notingthate=Y)]TJ /F8 11.955 Tf 11.95 0 Td[(X^=MY=M(X+)=MX+M=M,wemaywriteexpression( 4 )intermsoftheunobservableregressionerrors,,as^I=0MWM=0M.BecauseMWMandMcommute,by,e.g.Corollary21.13.2of Harville ( 1997 ),thereexistsanorthogonalmatrix)]TJ /F1 11.955 Tf 10.26 0 Td[(thatsimultaneouslydiagonalizes 67

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MWMandM.Thus,MWM=DW)]TJ /F9 7.97 Tf 6.94 5.15 Td[(0,andM=DM)]TJ /F9 7.97 Tf 6.94 5.14 Td[(0,Notingthat)]TJ /F15 11.955 Tf 6.94 0 Td[(N(0,I)providedN(0,I),underH0:=0wehave^I=e0We e0e=0MWM 0M=()]TJ /F10 7.97 Tf 6.58 0 Td[(1)0MWM()]TJ /F10 7.97 Tf 6.59 0 Td[(1) ()]TJ /F10 7.97 Tf 6.59 0 Td[(1)0M()]TJ /F10 7.97 Tf 6.58 0 Td[(1)d=0MWM 0M=0DW)]TJ /F9 7.97 Tf 6.94 5.15 Td[(0 0DM)]TJ /F9 7.97 Tf 6.94 5.15 Td[(0d=0DW 0DM=Pni=1i2i Pn)]TJ /F5 7.97 Tf 6.58 0 Td[(pi=12i,whered=denotesequalityinprobabilitymeasure;iiidN(0,1);andDW=diag(1,2,...,n)aretheneigenvaluesofMWMofwhichparenecessarilyzeroandtheremainingn)]TJ /F4 11.955 Tf 12.98 0 Td[(parenotnecessarilynon-zero.BecauseMisaprojection,itseigenvaluesalongthediagonalofDMarezerosandones.Furthermore,asrank(M)=n)]TJ /F4 11.955 Tf 10.94 0 Td[(ptherearen)]TJ /F4 11.955 Tf 10.94 0 Td[(ponesalongthediagonalofDM,andwemaypermutethecolumnsof)]TJ /F1 11.955 Tf 10.26 0 Td[(sothattherstn)]TJ /F4 11.955 Tf 9.92 0 Td[(pelementsalongdiag(DM)equaloneandtheremainingpelementsarezero.ToevaluateprobabilitystatementsunderH0, Tiefelsdorf&Boots ( 1995 )appliedtheresultsof Imhof ( 1961 )toapproximatetheintegralPr(^ItjH0),t2R.Alternatively,MonteCarlomethodsmaybereadilyemployedtoapproximatePr(^ItjH0),t2R. Oden ( 1995 ), Waldhor ( 1996 )and Assuncao&Reis ( 1999 )consideredMoran'sIstatisticsadjustedforpopulationsize. Tiefelsdorf ( 1998 )evaluatedtherstandsecondmomentsof^IunderH1,aswellasthecorrelationbetween^IQand^IR,where^IQand^IRareMoran'sIstatisticsbaseduponSARswithidenticalerror,design,andconnectivitystructurebutwithpossiblydifferentweights.Inthepresenceofnuisanceparameters Pinkse ( 2004 )developedasymptotictheoryinordertoconductinferenceonratiosofquadraticforms,suchasMoran'sI. Kelejian&Prucha ( 2001 )developedlarge-sample 68

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theoryforthedistributionofratiosofquadraticformsinestimatedandobservedmodelerrorsthatisapplicabletoalargeclassofMoran'sIstatistics. Tiefelsdorf ( 2002 )describedthesaddlepointandothermethodsforobtainingapproximationsofPr(^It),underboththenullandalternativehypotheses.Consideringamean-zeroSAR, Lietal. ( 2007 )proposedanalternativespatialindependencetestofH0:=0basedupontheapproximateprolelikelihoodestimator(APLE)of. Jacqmin-Gaddaetal. ( 1997 ), Lin&Zhang ( 2007 ),and Zhang&Lin ( 2008 )proposedgeneralizationsofMoran'sIforlog-linearandothercountdatamodels. Zhang&Lin ( 2007 )decomposedtheclassicalMoran'sIstatistic,writing^Iasalinearcombinationofcross-products,eachsummandrepresentingadistinctpartitionoftheparameterspacef:2()]TJ /F6 11.955 Tf 9.3 0 Td[(1,1)g. Wartenberg ( 1985 )proposedamultivariateversionofMoran'sIstatisticrootedina(spatial)principalcomponentsanalysisframework. Wartenberg 'smatrix-valuedstatisticcontainsclassicalMoran'sIstatisticsalongthediagonal,oneforeachvariableconsidered,whiletheoff-diagonalelementsarecross-productsthatrepresentspatialassociationbetweeneachpairofdistinctvariables. Lee ( 2001 )developedabivariatespatialassociationmeasure,thecomponentsofwhicharefunctionsofMoran'sIandPearson'scorrelationcoefcient.GivenamultivariatetimeseriesfYig, Lopez&Chasco ( 2007 )describedanensembleof(univariate)Moran'sI-typestatisticstoassessspatialassociationamongtheelementsofYtandYs,t6=s. Lopezetal. ( 2011 )introducedaspatio-temporalMoran'sI(STMI)statistictoquantifyspatio-temporalassociationinananalogousmannertothequanticationofspatialassociationthrough^I.Givenasequenceofn1responsevectorsYt,letY0(Y01,Y02,...,Y0T).ForthennspatialweightingmatrixWthatcodesspatialconnectivityamongstfs1,s2,...,sng,assumedtobestaticfort=1,2,...,T,andwiththebandedTTmatrixAwithonesinthe(i+1,i)thpositionandzeroselsewhere,takingWITW+AIn, STMI=nt PjPkwjk(Y)]TJ /F6 11.955 Tf 13.37 2.66 Td[(Y)0W(Y)]TJ /F6 11.955 Tf 13.36 2.66 Td[(Y) (Y)]TJ /F6 11.955 Tf 13.36 2.66 Td[(Y)0(Y)]TJ /F6 11.955 Tf 13.37 2.66 Td[(Y).(4) 69

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HenceforthweuseIstand^Isttodenotethepopulationandsampleanalogsof( 4 ),respectively.Weproceedtodescribeaclassofspatio-temporalmodelsinwhichageneralizationofIstarisesinamannerconsistentwithMoran'sIandsuchthat,forthespecialcasewhereT=1,Moran'sIisexactlyIst.Wethencalculatemomentsof^Ist,andproposemethodsbywhich^Istmaybeusedtoconductinferenceinaspatio-temporalcontext. 4.1.3ModelingFrameworkTothisendwenowintroducethespatio-temporalautoregressivemovingaverage(STARMA)model( Ali ( 1979 ); Pfeifer&Deutsch ( 1980b ))byfollowingthenotationof Cressie&Wikle ( 2011 ,pg.344).ForknowngeneralizedweightmatricesfFkmgandfGlmgandunknownparametersfkmgandflmgtheSTARMAmodelis Yt=rXk=0 MkXm=1kmFkm!Yt)]TJ /F5 7.97 Tf 6.58 0 Td[(k+sXl=0 QlXm=1lmGlm!t)]TJ /F5 7.97 Tf 6.58 0 Td[(l,t=1,2,...,T(4)whereftgareiidmean-zerovectorswithVar(1)=.In( 4 )thefkmgareautoregressivespatialparameters,andtheflmgaremoving-averagespatialparameters.ThexedmatricesfFkmgandfGlmgaccountforthespatialweightingofcurrentandpriorresponseanderrorvectors,respectively,inYt.Theerrorcovariancematrixaccountsforcorrelationamongtheelementsof1.Intheabsenceofamoving-averagestructure,s=0,Ql=1,G01Iand011,andthesecondtermontheright-handsideof( 4 )reducestosXl=0 QlXm=1lmGlm!t)]TJ /F5 7.97 Tf 6.59 0 Td[(l=01G01t=t.Thusintheabsenceofamoving-averagestructure,wehave Yt=rXk=0 MkXm=1kmFkm!Yt)]TJ /F5 7.97 Tf 6.58 0 Td[(k+t,t=1,2,...,T.(4)Henceforthwesuppose( 4 )hasnomoving-averagestructure.Therindexin( 4 )maybethoughtofasthespatialautoregressiveorderoftheprocess,andtheMkindex 70

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isusedtoaccountforthepossibilitythattheautoregressivespatialstructurevariesintime.Forexample,ifr=1,M0=3andM1=2,( 4 )containstertiaryspatialstructureinthecontemporaryresponseYtandsecondaryautoregressivespatialstructureintheone-laggedresponse,Yt)]TJ /F10 7.97 Tf 6.58 0 Td[(1.ForeaseofnotationwemaytakeMk=1forallkandwedropthem,sothatthetheautoregressivespatialstructuresubscriptyieldingthemodel Yt=rXk=0kFkYt)]TJ /F5 7.97 Tf 6.58 0 Td[(k+t,t=1,2,...,T,(4)thespatio-temporalautoregressive(STAR)processof Pfeifer&Deutsch ( 1980a )withacontemporaneoustimeterm(e.g. Pfeifer&Deutsch ( 1980a )'sSTARprocesswithanadditionalterm,0F0,accountingforspatialassociationamongsttheresponse,Yt).Morerecently, Terzi ( 1995 )exploredtheSTARprocessmodelwithcontemporaneousspatialassociation,which Terzi calledthegeneralizedSTAR(GSTAR)process.InthefollowingdiscussionwemaintaintheassumptionthatMk=1forallk,andthensubsequentlyrelaxthisassumptiontoemploythefullmodelframeworkof Terzi ( 1995 )givenby( 4 ).ForbrevitywehenceforthwritekFkBk,k=0,1,...,r,andforidentiabilitywefurthersupposethediagonalelementsofB0areidentically0and(I)]TJ /F8 11.955 Tf 12.39 0 Td[(B0))]TJ /F10 7.97 Tf 6.58 0 Td[(1exists( Cressie&Wikle 2011 ).TodeveloptheSARasaspecialcaseof( 4 ),supposeT=1sothatobservationsaremadeatonlyonetimepoint.Thennecessarilyr=0andthemodelis Y1=B0Y1+1,(I)]TJ /F8 11.955 Tf 11.95 0 Td[(B0)Y1=1.(4)IfisGaussian,thenN(0,)andthus( 4 )isamean-zeroSARprocess.AllowingforthemeanE(Y)todependlinearlyuponan(np)matrixofcovariatesXthroughanunknownvectorofpregressionparametersallowsfortheextensionof( 4 ),toY1)]TJ /F8 11.955 Tf 11.96 0 Td[(X=B0(Y1)]TJ /F8 11.955 Tf 11.95 0 Td[(X)+1,(I)]TJ /F8 11.955 Tf 11.96 0 Td[(B0)(Y1)]TJ /F8 11.955 Tf 11.95 0 Td[(X)=1,aSARmodelwith=X2Rn. 71

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Returningtostructure( 4 ),forT>1,wehaveYt=rXk=0BkYk+tt=1,2,...,T.ToillustratethemannerinwhichspatialandtemporalassociationarisesintheGSTARprocess,supposeT=3andr=1.TheprocessisthusY1=B0Y1+1,Y1=(I)]TJ /F8 11.955 Tf 11.96 0 Td[(B0))]TJ /F10 7.97 Tf 6.58 0 Td[(11Y2=B0Y2+B1Y1+2,Y2=(I)]TJ /F8 11.955 Tf 11.96 0 Td[(B0))]TJ /F10 7.97 Tf 6.58 0 Td[(1B1Y1+(I)]TJ /F8 11.955 Tf 11.95 0 Td[(B0))]TJ /F10 7.97 Tf 6.59 0 Td[(12Y3=B0Y3+B1Y2+3,Y3=(I)]TJ /F8 11.955 Tf 11.96 0 Td[(B0))]TJ /F10 7.97 Tf 6.58 0 Td[(1B1Y2+(I)]TJ /F8 11.955 Tf 11.95 0 Td[(B0))]TJ /F10 7.97 Tf 6.59 0 Td[(13.TakingA(I)]TJ /F8 11.955 Tf 11.96 0 Td[(B0))]TJ /F10 7.97 Tf 6.59 0 Td[(1,wehaveY1=A1Y2=AB1Y1+A2Y3=AB1Y2+A3.HereE(Yt)=0forallt2f1,2,3g.With=I,assumingerrorvectorsaremutuallyindependentandmutuallyindependentoffYtg,thevarianceofeachresponsevectoris,by C.1 ,Var(Y1)=AA0,Var(Y2)=1Xk=0(AB1)kAA0(B01A0)k,andVar(Y3)=2Xk=0(AB1)kAA0(B01A0)k. 72

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Covariancematricesarethen,by C.2 ,Cov(Y2,Y1)=AB1Var(Y1),Cov(Y3,Y2)=AB1Var(Y2),andCov(Y3,Y1)=(AB1)2Var(Y1).InregardstothestructureofB0andB1,recallthatwehaveassumedA=(I)]TJ /F8 11.955 Tf 12.48 0 Td[(B0))]TJ /F10 7.97 Tf 6.58 0 Td[(1exists.Given( 4 ),wenowrelaxtherestrictionsplaceduponfMkgandconsiderthecasewhereM0=1andM1=2.Thatis,wesupposethereisaprimaryspatialstructureinthecontemporaryresponsevectorandapossiblysecondaryspatialautoregressivestructureintheone-laggedresponse.AsB0=01F01wherediag(F01)=0,bytaking012()]TJ /F6 11.955 Tf 9.29 0 Td[(1,1)andF01tobearow-standardizedweightmatrixsuchthatF011=1,thedifferencematrixI)]TJ /F8 11.955 Tf 12.11 0 Td[(B0isdiagonallydominantandhenceinvertibleforall012()]TJ /F6 11.955 Tf 9.3 0 Td[(1,1). Cressie&Wikle ( 2011 )donotdescribethestructureofB1.Wechoosetoconsiderthestructureimposedby Terzi ( 1995 ),wherebyF11=IandF12=F01.Thus,B1=11I+12F01,yieldingarst-orderautoregressivestructurewithparameter11inadditiontoarst-orderspatialautoregressivestructurewithparameter12.Forthisexample, Terzi ( 1995 )providessufcientconditionsforthestationarityofthespatio-temporalprocessfYtg.Theseconditionsare: 1. alleigenvaluesofB0=01F01arelessthanoneinabsolutevalue,and 2. alleigenvaluesof(I)]TJ /F8 11.955 Tf 12.11 0 Td[(B0))]TJ /F10 7.97 Tf 6.58 0 Td[(1B1=(I)]TJ /F7 11.955 Tf 12.1 0 Td[(01F01))]TJ /F10 7.97 Tf 6.58 0 Td[(1[11F11+12F12]arelessthanoneinabsolutevalue.Asconditions 1 and 2 dependuponthespecicationoftheweightmatricesfFijg,inpracticetheparameterspacef(01,11,12):fYtgisstationarygisidentiedbymeansofagridsearch.Wemayinterpretf01,11,12ginthefollowingmanner.IfthereisneitherspatialnortemporalassociationamongtheelementsoffYtg,then01=11=12=0.IfforeachtthereisspatialassociationamongtheelementsofYt,butthereisnotemporal 73

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associationamongtheelementsoffYtg,then016=0and11=12=0.Lastly,ifthereisbothspatialandtemporalassociationamongtheelementsoffYtgthen016=0and116=0or126=0.Here11allowsforsame-sitetemporalassociation,while12controlsspatio-temporalassociation. 4.1.4EconometricApproachestoSpace-TimeLatticeModelsAbovewefollowaspatio-temporalmodelframeworkcommonlyfoundinthestatisticsliterature.Intheeconometricsliteraturespatio-temporalregressionmodelshavebeenwidelyexplored.Timeseriesofcrosssectionswereinvestigatedin Balestra&Nerlove ( 1966 ), Wallace&Hussain ( 1969 ), Maddala ( 1971 ),and Mundlak ( 1978 ),tonameafew. Balestra&Nerlove ( 1966 )and Nerlove ( 1971 )employedavariancecomponentsmodelwithlaggedintra-regionobservationstoallowforspatio-temporalassociation,thoughtheydidnotconsiderinter-regionlagging.Notethattheseauthorsindexedprimarilybyregionandsecondarilybytime(e.g.timeisthefastest-movingindex). Wallace&Hussain ( 1969 )usedonlyvariancecomponentsanddidnotallowforlaggeddependentvariables. Maddala ( 1971 )and Mundlak ( 1978 )followed Balestra&Nerlove ( 1966 ),usingrandomeffectstoaccountforspatialandtemporalassociation. Baltagi ( 1981 )allowedfortime-laggeddependentvariablesinasimultaneousequationswitherrorcomponentsmodel,butapparentlyconsideredonlytemporalassociationbetweentheelementsoffYtgTt=1andnotspatialassociationwithintheelementsofYtforeacht. Bhargava&Sargan ( 1983 )exploreddynamicpanelmodelsbutdonotallowinter-regionrelationships. Anderson&Hsiao ( 1981 1982 )examinedacollectionofautoregressivemodelsforcross-sectionalunitsintime.Theydiscussedtheeffectofinitialconditionassumptionsonmodelinterpretationandaccountedforcross-sectionalassociationusingarandomeffectcoupledwithanautoregressiveterm.AlongwithGaussianMLEstheauthorsproposedcovarianceestimatorsformodelparameters.Inneitherpaperdidtheauthors 74

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allowforinter-regionalassociation,andonlyintra-regiontemporalassociationwasconsidered. Anselin&Kelejian ( 1997 )describedaMoran'sIstatisticinthecontextofaspatialautoregressivemodelwithboundaryeffects.Ignoringboundaryeffects,theirmodelmaybewrittenasY=X+WY+Y+.AllvectorsandmatricesareoflengthN.TheYmatrixisNG,sothatthereareGendogenousregressor(s)foreachelementoftheresponsevectorY.TheSTARMAmodelconsideredhereinisthusdistinguishedfromthemodelof Anselin&Kelejian ( 1997 )bytheallowanceforspatiallyandtemporallyweightedYterms.Furtherdistinguishingthepresentpaperfrom Anselin&Kelejian ( 1997 )isthemethodofestimation;hereweproposetheuseofgeneralizedleastsquareswhereas Anselin&Kelejian ( 1997 )employedthemethodofinstrumentalvariables.Morerecently, Kelejian&Prucha ( 2004 )exploredsimultaneousequationsmodelsinwhichregionsmaybetemporallyassociatedinageneralmanner.Though Kelejian&Prucha ( 2004 )didnotidentifytheconnection,asubcollectionofmodelsdenedintheirpaperisequivalenttothesubcollectionofSTARMAmodelsdenedby( 4 ).(Thisconnectionwasnotedin Sain&Cressie ( 2007 ),whoexploreddynamicconditionalautoregressive(CAR)models.)Themannerinwhichcovariatesenterthemodelaswellasthemannerofintroducingspatio-temporalassociationdistinguishthesimultaneousequationsmodelof Kelejian&Prucha ( 2004 )fromtheSTARMAmodeldenedby( 4 ).Foreachaerialunitthesimultaneousequationsmodelof Kelejian&Prucha ( 2004 )allowsfortheuseofthesamecovariatevaluesacrosstime(withpossiblydifferentregressioncoefcients).Forqualitativefeaturessucharestrictionmightstillbeuseful,butdynamicfeaturessuchasaerialunitpopulationmayvaryintime.Asintheprevioussection,letYtdenotethen1vectorofregionalobservationsattimet,andYtWYtdenotethevectoroflinearcombinationsoftheelementsofYt,withWannnknownspatialweightmatrix.WitherrorvectorsfUjg, Kelejian&Prucha 75

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( 2004 )denethesimultaneousequationsmodel Y=YB+XC+Y)]TJ /F6 11.955 Tf 18.69 0 Td[(+U(4)withY[Y1,Y2,...,Ym],X[x1,x2,...,xk],YY1,Y2,...,Ym,U[U1,U2,...,Um].InthisconstructionthemmmatrixB,thekmmatrixC,andthemmmatrix)]TJ /F1 11.955 Tf 10.27 0 Td[(areallunknownparametermatrices(alongwiththecovariancestructureoffUjg).NotingthatY=[WY1,WY2,...,WYm])]TJ /F6 11.955 Tf 10.82 0 Td[(=WY)]TJ /F1 11.955 Tf 28.45 0 Td[(,simultaneousequationsmodel( 4 )maybewrittenY=YB+XC+WY)]TJ /F6 11.955 Tf 31.12 0 Td[(+U.Vectorizingtheleft-andright-handsidesof( 4 )yieldsvec(Y)=(B0In)vec(Y)+(C0In)vec(X)+()]TJ /F9 7.97 Tf 6.94 5.15 Td[(0W)vec(Y)+vec(U);groupingliketerms,wehave vec(Y)=(B0In)+()]TJ /F9 7.97 Tf 6.94 5.15 Td[(0W)vec(Y)+(C0In)vec(X)+vec(U).(4)Hence,fortimet2f1,2,...mgthemodelis Yt=mXi=1DitYi+kXr=1crtxr+Ut,(4)whereDijbijIn+ijW.Though Kelejian&Prucha ( 2004 )donotidentifytheconnection,whenk=0asubcollectionofmodelsdenedbyexpression( 4 )isequivalenttothesubcollectionofSTARMAmodelsdenedby( 4 ).(Thisconnectionisnotedin Sain&Cressie ( 2007 ),whoexploredynamicconditionalautoregressive(CAR)models.)Themannerinwhichcovariatesenterthemodelaswellasthemannerofintroducingspatio-temporalassociationdistinguishthesimultaneousequationsmodelof Kelejian 76

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&Prucha ( 2004 )fromthegeneralizedSTARmodeldenedby( 4 ).From( 4 )covariatesfxijgenterthemodelforY11,therstaerialunitattimet=1,byPkr=1cr1x1r,wherexpqisthepthelementofthen1vectorxq.Similarly,covariatesenterthemodelforY12,therstaerialunitattimet=2,byPkr=1cr2x1r.Hence,foreachaerialunitthesimultaneousequationsmodelof Kelejian&Prucha ( 2004 )onlyallowsfortheuseofthesamecovariatevaluesacrosstime(withpossiblydifferentregressioncoefcients).Forqualitativefeaturessucharestrictionmightstillbeuseful,butdynamicfeaturessuchasaerialunitpopulationmayvaryintime. Laietal. ( 2008 )usedinstrumentalvariablesandgeneralizedmethodofmomentstoextendtheresultsof Anderson&Hsiao ( 1981 )regardingparameterestimationindynamicpanelmodels. Wang&Kockelman ( 2007 )and Baltagi&Bresson ( 2011 )exploredtheseeminglyunrelatedregressions(SUR)modelframeworkof Baltagi ( 1980 ),withthe Baltagi&Bresson ( 2011 )paperaddressingmaximumlikelihoodparameterestimationandaclassofscoretestsfortheSURmodel. Arraizetal. ( 2010 )examinedtheuseofinstrumentalvariablesinSARmodels. Kelejian&Prucha ( 2010 )consideredtheeffectofmissingdatainthesimultaneousequationsframeworkof Kelejian&Prucha ( 2004 ),butgeneralizedtoincorporatespatiallyandtemporallylaggedcovariates.BothpapersmayofferamoresuitableframeworkinwhichSTMImaybeextendedtoaccountfornon-zeromeanprocesses,aswellasindependentvariables.Suchexplorationisafutureendeavorofthisauthor. 4.2STMIinGSTARModels 4.2.1HypothesisTestinginGSTARModelParametersWenowdescribeanextensionofSTMIthatmaybeemployedtodetectspatialorspatio-temporalassociationinGSTARmodels.RecalltheGSTARregressionmodel,( 4 ),withr=1,M0=1andM1=2:Yt=01F01Yt+(11F11+12F12)Yt)]TJ /F10 7.97 Tf 6.58 0 Td[(1+t,t=1,2,...,T, 77

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wheretiidN(0,I);F11=I;andF01=F12=Wforasymmetric,row-standardizedweightmatrixW.Giventheinterpretationofthespatialandtemporalparametersf01,11,12gdiscussedintheconcludingparagraphofx 4.1.3 ,wedenethefollowinghypothesesofinterest: 1. H0:f01=0,11=0,12=0g:thereisneitherspatialnortemporalassociationamongfYtg. 2. HS:f016=0,11=0,12=0g:thereisspatialbutnotemporalassociationamongfYtg. 3. HT:f01=0g\[f116=0g[f126=0g]:thereistemporalbutnospatialassociation.Here( 4 )yieldsTindependentautoregressivetimeseriesif116=0and12=0.Alternatively,( 4 )yieldsaone-laggedspatialautoregressiveprocessif11=0and126=0. 4. HST:f016=0g\[f116=0g[f126=0g]:thereisbothspatialandtemporalassociation.InthiscaseYtdependsuponbothcontemporaryobservationsinYtaswellaspastobservationsYt)]TJ /F10 7.97 Tf 6.58 0 Td[(1,Yt)]TJ /F10 7.97 Tf 6.58 0 Td[(2,....InthefollowingsubsectionweintroduceatestofH0baseduponinferenceforaMoran'sI-typeparameter.AstheparameterspaceintheGSTARmodeldependsuponfFijg,andhencesubsequentestimationproceduresareconditionaluponthestructureoffFijg,thetestsubsequentlyproposedisusefultoascertainwhetherornotimplementingsuchprocedures(e.g.estimationoffijg)maybefruitful. 4.2.2GeneralizationofIstCorrespondingtoGSTARmodel( 4 )wedenethelowerblock-triangularnTnTspatio-temporalweightingmatrixCfCijgasfollows.TakeCii=PM0m=1F0m;Ci,i)]TJ /F10 7.97 Tf 6.59 0 Td[(1=PM1m=1F1m;Ci,i)]TJ /F10 7.97 Tf 6.59 0 Td[(2=PM2m=1F2m;andsoforth,withCi,i)]TJ /F5 7.97 Tf 6.59 0 Td[(v0forv>minfr,i)]TJ /F6 11.955 Tf 12.89 0 Td[(1g.ThoughallweightmatricesfFijgarerow-standardizedwhenconstructingthelikelihood,inkeepingwithe.g. Cliff&Ord ( 1981 )atthispointwedenefCijgtobebinarycodedweightsthatmaybescaledtoeliminateamultiplicativeconstantin^Ist.ApplytoCthesymmetrizationmap,yieldingC1 2[C+(C)0]with(r,s)thelementcrs.LetY0=(Y01,Y02,...,Y0T).Parallelingthe Cliff&Ord ( 1972 )generalizationofMoran'sI, 78

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considerageneralizationof( 4 )of Lopezetal. ( 2011 )denedtobe IstnT PrPscrsY0CY Y0Y.(4)Forarealizationofthespatio-temporalprocess,thesampleanalogis ^IstnT PrPscrs(Y)]TJ /F6 11.955 Tf 13.37 2.66 Td[(Y)0C(Y)]TJ /F6 11.955 Tf 13.36 2.66 Td[(Y) (Y)]TJ /F6 11.955 Tf 13.37 2.65 Td[(Y)0(Y)]TJ /F6 11.955 Tf 13.36 2.65 Td[(Y).(4)Whenthereisonlyonetimeperiodandoneweightmatrix,i.e.whenT=1andM0=1,thenmodel( 4 )isaSAR.Inaddition,iftheelementsofY1arespatiallyindependent(i.e.if01=0)then^IstisequaltothegeneralizedMoran'sIstatisticof Cliff&Ord ( 1972 )foraconstant-meanprocessgivenby( 4 ).Furthermore,ifT>1;r=1;M0=1;M1=2;F01=W=F12forabinary-codedspatialweightingmatrixW;andF11=I;then^IstistheSTMIstatisticof Lopezetal. ( 2011 )givenby( 4 ).NotethatthemodelframeworkyieldingtheSTMIstatisticof Lopezetal. ( 2011 )correspondstothestructureexploredindetailinthelatterpartofx 4.1.3 ,wherethereiscontemporaryspatialstructureinadditiontobothrst-orderautoregressivestructureandrst-orderspatialautoregressivestructure. 4.2.3Propertiesof^IstWeemploythefollowinglemmatoidentifytherstmomentsofIstand^Ist. Lemma1( Conniffe&Spencer ( 2001 )). Forann1randomvectorXN(0,2I),asymmetricnnmatrixA,andasymmetricandidempotentnnmatrixBwith1rB=rank(B)
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whereZ1N(0,2Ir)andisasymmetricmatrix; 2. showthatZ1=kZ1kisindependentofkZ1k;and 3. showthatthereexistcontinuousfunctionsf:R7!Randg:Rr7!Rsuchthat f(kZ1k)=Z01Z1andgZ1 kZ1k=Z01Z1 Z01Z1;(4)whichprovestheresultbecausefunctionsofindependentrandomvectorsareindpendent. Proof. Step1.BecauseBisidempotentandA=BGB,itfollowsthatAB=BA.Thus,byCorollary21.13.2of Harville ( 1997 )thereexistsanorthogonalmatrixPsuchthatA=PDAP0andB=PDBP0,whereDAandDBarediagonalmatriceswiththeeigenvaluesofAandBalongthediagonal,repsectively.Borrowingfromtheargumentsof Smith ( 1989 ),permutethecolumnsofPsothattherstrelementsalongthediagonalofDBareonesandtheremainingn)]TJ /F4 11.955 Tf 12.34 0 Td[(relementsarezero.LetP=[P1P2],wherethecolumnsofthenrmatrixP1aretheeigenvectorscorrespondingtotherstr(non-zero)eigenvaluesofBalongthediagonalofDB,andwherethecolumnsofthen(n)]TJ /F4 11.955 Tf 12.32 0 Td[(r)matrixP2aretheeigenvectorscorrespondingtoremainingn)]TJ /F4 11.955 Tf 11.97 0 Td[(r(zero)eigenvaluesalongthediagonalofDB.NotethatP1P01+P2P02=In,P01P1=Ir,P02P2=In)]TJ /F5 7.97 Tf 6.58 0 Td[(r,andP01P2=0.LetC(F)andN(F)denotethecolumnspaceandnullspace,respectively,ofamatrixF.NotethatbecauseC(A)=C(BGB)C(B),itfollowsthatN(B)N(A)andrank(A)r=rank(B).Foreachcolumn,pofP2,Bp=0andsop2N(B))p2N(A).Thus,AP2=P2A=0.Furthermore,becauseA=PDAP0,inwritingP=[P1P2]itfollowsthattheeigenvaluesofAassociatedwiththeeigenvectorsofP2arenecessarilyzero.Thus,therstrelementsalongthediagonalofDAarethe(notnecessarily)non-zeroeigenvaluesofAandtheremainingr+1,r+2,...,nelementsalongthediagonalofDAarethenecessarilyzeroeigenvaluesofAassociatedwiththeeigenvectorsofP2.LetDA,1beanrrdiagonalmatrixofthenotnecessarilynon-zero 80

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eigenvaluesofAcorrespondingtotheeigenvectorsthatarethecolumnsofP1,andletDA,2=0denotethe(n)]TJ /F4 11.955 Tf 12.04 0 Td[(r)(n)]TJ /F4 11.955 Tf 12.05 0 Td[(r)zeromatrix.ThusDA=diag(DA,1,DA,2).Similarly,wemaywriteDB=diag(DB,1,DB,2),whereDB,1=IrbecauseBisarank-rprojection,andDB,2=0(n)]TJ /F5 7.97 Tf 6.58 0 Td[(r)(n)]TJ /F5 7.97 Tf 6.59 0 Td[(r).WithZ1=P01XN(0,2Ir),wehaveX0AX X0BX=X0(P1P01+P2P02)A(P1P01+P2P02)X X0BX=X0P1P01AP1P01X X0BX=Z01P01AP1Z1 X0PDBP0X=Z01P01AP1Z1 X0P1DB,1P01X=Z01P01AP1Z1 Z01Z1=Z01Z1 Z01Z1, (4)whereP01AP1issymmetric.ButA=P1DA,1P01andso=P01AP1=P01P1DA,1P01P1=DA,1.Thus,from( 4 )wehave X0AX X0BX=Z01DA,1Z1 Z01Z1,(4)andsotheratioofquadraticformsX0AX=X0BXisequaltoaratioofquadraticformsinther1randomvectorZ1N(0,2Ir).Step2.TherandomvectorZ1issphericallysymmetricbecauseforanyorthogonalmatrix)]TJ /F1 11.955 Tf 6.94 0 Td[(,wehaveZ1d=Z1N(0,2Ir).ByTheorem2.5(iii)of Fangetal. ( 1990 ),thereexistsarandomvariablerandarandomvectorUsuchthatZ1d=rU,whereUisuniformlydistributedonthesurfaceoftheunitsphereinRr.AsPr(Z1=0)=0,byTheorem2.3of Fangetal. ( 1990 )wehavekZ1kd=randZ1=kZ1kd=U,andfurthermorekZ1kandZ1=kZ1kareindependent.Step3.Fort2Randt2Rr,denefunctionsf:R7!Randg:Rr7!Rbyf(t)=t2andg(t)=t0DA,1t t0t. 81

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Notethatfiscontinuous,andgiscontinuouswithprobabilityonewithrespecttotheprobabilitymeasurecorrespondingtoZ1.Furthermore,f(kZ1k)=kZ1k2andgZ1 kZ1k=Z01DA,1Z1 Z01Z1,sothatg(Z1=kZ1k)isequaltoexpression( 4 ).Butfiscontinuousandgisalmost-everywherecontinuouswithrespecttotheprobabilitymeasurecorrespondingtoZ1,andbecausekZ1kandZ1=kZ1kareindependent,itfollowsthatf(kZ1k)andg(Z1=kZ1k)areindependent.Asf(kZ1k)=Z01Z1istheexpressionappearinginthedenominatorof( 4 ),itfollowsthattheratioofquadraticforms( 4 )isindependentofZ01Z1.NotingthatX0BX=Z01Z1andX0AX X0BX=Z01DA,1Z1 Z01Z,wehaveestablishedthatX0BXisindependentofX0AX=X0BX. Corollary1. UndertheconditionsofLemma 1 ,EX0AX X0BX=E(X0AX) E(X0BX). Proof. BecauseX0AX0=X0BXisindependentofX0BX,theexpectationoftheirproductisequaltotheproductoftheirexpectations.ButX0AX0 X0BXX0BX=X0AX,andthusbyarearrangementoftermswemaywritetheexpectationoftheratioofquadraticformsastheratiooftheexpectationsofthequadraticforms. InordertondE(Ist)webeginbyrow-standardizingthespatio-temporalweightmatrixC.ThenPrPscrs=nTandwemaythusdroptheconstantinIstand^Ist.Under 82

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H0:f01=0,11=0,12=0g,YtiidN(0,I).ThusIst=Y0CY Y0Y=Y0CY Y0IY,andtheconditionsofLemma 1 aretriviallysatisedbecauseC=ICI.Thus,byCorollary 1 ,E(Ist)=EY0CY Y0Y=E(Y0CY) E(Y0Y).UnderH0,YN(0,I)andhenceY0Y2nT.Thus,E(Y0Y)H0=nT.Furthermore,thexedandknownweightmatrixCissymmetric,andthusforanorthogonalmatrix)]TJ /F1 11.955 Tf 6.94 0 Td[(,C=DC)]TJ /F9 7.97 Tf 6.94 5.15 Td[(0.Notingthatforanorthogonalmatrix)]TJ /F1 11.955 Tf 6.94 0 Td[(,XN(0,I)=)XN(0,I),Y0CY=Y0DC)]TJ /F9 7.97 Tf 6.94 5.15 Td[(0Yd=Y0DCY=nTXi=1iY2i,wherefigaretheeigenvaluesofCthatarefoundalongthediagonalofthe(diagonal)matrixDC.Hence,E(Y0CY)H0=E(nTXi=1iY2i)=nTXi=1iE(Y2i)=nTXi=1i.Therefore, E(Ist)H0=E(Y0CY) E(Y0Y)=(nT))]TJ /F10 7.97 Tf 6.59 0 Td[(1nTXi=1i.(4)Nowconsider^Istgivenby( 4 ).WithMI)]TJ /F6 11.955 Tf 12.34 0 Td[((nT))]TJ /F10 7.97 Tf 6.58 0 Td[(1110,underH0,Y)]TJ /F6 11.955 Tf 13.74 2.65 Td[(Y=MYN(0,M),wherewehavemadeuseofthefactthatMisaprojectionoperator.UndertheassumptionthatChasbeenscaledbytheconstantin( 4 ),^Ist=(Y)]TJ /F6 11.955 Tf 13.37 2.65 Td[(Y)0C(Y)]TJ /F6 11.955 Tf 13.37 2.65 Td[(Y) (Y)]TJ /F6 11.955 Tf 13.37 2.65 Td[(Y)0(Y)]TJ /F6 11.955 Tf 13.37 2.65 Td[(Y)=Y0MCMY Y0MY.ButMCM=M[MCM]M,andbecauseYH0N(0,I),underH0theconditionsofLemma 1 aresatised.Thus,byCorollary 1 E(^Ist)=EY0MCMY Y0MY=E(Y0MCMY) E(Y0MY).(4) 83

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Furthermore,becauseMCMandMcommute,thetwomatricesmaybesimultaneouslydiagonalizedbyanorthogonalmatrix,)]TJ /F6 11.955 Tf 6.94 0 Td[(.Thus,MCM=DC)]TJ /F9 7.97 Tf 6.94 5.14 Td[(0andM=DM)]TJ /F9 7.97 Tf 6.94 5.14 Td[(0.BecauseMisaranknT)]TJ /F6 11.955 Tf 12 0 Td[(1projection,theelementsalongthediagonalofDMareoneswithasinglezero.Wemaythuspermutetherowsof)]TJ /F1 11.955 Tf 10.27 0 Td[(sothattherstnT)]TJ /F6 11.955 Tf 12.12 0 Td[(1elementsalongthediagonalofDMareonesandthenTthelementiszero.UnderH0,YN(0,I)andtheorthogonalrotation)]TJ /F9 7.97 Tf 6.94 5.14 Td[(0Yd=YN(0,I),itfollowsthatE(Y0MCMY)=E(Y0DC)]TJ /F9 7.97 Tf 6.94 5.15 Td[(0Y)=E(Y0DCY)=nTXi=1iE(Y2i),wherefigaretheeigenvaluesofMCM.BecauseYH0N(0,I),underH0Y2iareiid21randomvariableswithE(Y2i)=1foralli.Therefore,E(Y0MCMY)=nTXi=1i.Furthermore,E(Y0MY)=PnT)]TJ /F10 7.97 Tf 6.59 0 Td[(1i=1E(Y2i)=nT)]TJ /F6 11.955 Tf 11.96 0 Td[(1.Thus,E(^Ist)=(nT)]TJ /F6 11.955 Tf 11.95 0 Td[(1))]TJ /F10 7.97 Tf 6.59 0 Td[(1nTXi=1i.UnderH0,weseethatingeneral^IstisabiasedestimatorofE(Ist).Fortunately,wemaycapitalizeuponelementsoftheabovecalculationstoproposeaprocedureforconductinginferenceonIstunderH0.Recallthat^Ist=Y0MCMY Y0MY=Y0DC)]TJ /F9 7.97 Tf 6.94 5.14 Td[(0Y Y0DM)]TJ /F9 7.97 Tf 6.94 5.15 Td[(0Yd=Y0DCY Y0DMY=PnTi=1iY2i PnT)]TJ /F10 7.97 Tf 6.58 0 Td[(1i=1Y2i.ToconductinferenceunderH0onIst,theexactnulldistributionof^Istmaybereadilyconstructedviasimulation.Tothisend,theeigenvaluesfigofthexedandknownmatrixMCMareobtained.ForalargenumberS,wegeneratesamplesZ(s)N(0,I), 84

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s=1,2,...,Sand,foreachsample,obtainarandomdeviate RsPnTi=1i(Z(s)i)2 PnT)]TJ /F10 7.97 Tf 6.59 0 Td[(1i=1(Z(s)i)2.(4)fromthenulldistributionofY.Foragivenlevel2(0,1),alevel-testofH0:f01=0,11=0,12=0gisconductedbyrejectingH0iftheobserved^Istislessthanthe 2thorderstatisticoffRsgorgreaterthanthe(1)]TJ /F18 7.97 Tf 13.15 4.71 Td[( 2)thorderstatisticoffRsg.Anothermorestraight-forwardmannerbywhichonemayconductalevel-testofH0istogenerateZ(s)N(0,I),s=1,2,...,Sandcalculate^I(s)stforeachsample.Foreachs,thenulldistributionofRsisequaltothenulldistributionof^I(s)st.Theformerprocedureislesscomputational-resourcedemanding,however,asforeachsnomatrixmultiplicationisrequired. 4.2.4MaximumLikelihoodEstimationinGSTARModelsConsidertheGSTARmodelgivenby( 4 )withR=1;M0=1;andM1=2(i.e.withcontemporaneousspatialassociation,rst-orderautoregressiveassociationandrst-orderspatialautoregressiveassociation).Followingtheexampleinx 4.1.3 ,E(Yt)=0becauseeachYtmaybewrittenasalinearcombinationofmean-zeroerrorvectorsftg.LetB001F01;B111F11+12F12andAI)]TJ /F8 11.955 Tf 12.25 0 Td[(B0.DeneijCov(Yi,Yt),i,j2f1,2,...,Tg.Byinductiononi,itmaybeshownthati+1,i+1=AB1i,iB01A0+AA0,i=1,2,3,...,T)]TJ /F6 11.955 Tf 12.72 0 Td[(1.Furthermore,byinductiononiitmaybeshownthatfori>j,i+1,j=AB1ij.Withfijg;(01,11,12)0;andy0=(y01,y02,...,y0T),thelikelihoodofgivenfytgis L(jy)=(2)nT=2jj)]TJ /F13 5.978 Tf 7.78 3.26 Td[(1 2exp)]TJ /F6 11.955 Tf 10.49 8.09 Td[(1 2y0)]TJ /F10 7.97 Tf 6.59 0 Td[(1y.(4)Recallnowstationarityconditions 1 and 2 : 1. alleigenvaluesofB0=01F01arelessthanoneinabsolutevalue,and 2. alleigenvaluesof(I)]TJ /F8 11.955 Tf 12.11 0 Td[(B0))]TJ /F10 7.97 Tf 6.58 0 Td[(1B1=(I)]TJ /F7 11.955 Tf 12.1 0 Td[(01F01))]TJ /F10 7.97 Tf 6.58 0 Td[(1[11F11+12F12]arelessthanoneinabsolutevalue. 85

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UnderH0:f01=0,11=0,12=0g,L(jy)isconstantasafunctionof.UnderHS,HT,andHST,however,maximizationofL(jy)isanontrivialproblem.Ineachcase,theparameterspaceisafunctionoffF01,F11,F12g.Furthermore,isanon-rectangularsubspaceofR3.Hence,numericaloptimizationroutinessubjecttoboxconstraints,suchastheL-BFGS-Balgorithmof Byrdetal. ( 1995 ),whichmaybeemployedinRusingtheoptim()function,maynotbeusedtooptimize( 4 ).WithF01thebinary-codedandrow-standardizedweightmatrixrepresentingarst-nearest-neighborhoodstructureonthecontiguous48UnitedStates,andwithF11=IandF12=F01,inFigure 4-3 weillustratethesurfaceoverwhichfYtgarestationary.Therefore,maximizationof( 4 )isachallengetoperforminanautomatedmanner.WeobtainMLestimatesbyperformingagridsearchoveracollectionfgthatsatisfystationaryconstraints 1 and 2 .InlightofthechallengesofMLestimation,aclearadvantageoftheassessmentofH0using^Istemerges.AlthoughMLestimationandinferenceforrequiresagreatdealofcare,evaluationofH0via^Istasdiscussedinx 4.2.3 isstraight-forwardandrequireslittlemorethanobtainingtheeigenvaluesofaxedandknownmatrix,C. 4.3Application:DetectingChangesinPatternforStandardizedCancerMortalityRatesConsidertheproblemofdetectingchangesinthepatternofdecadalstandardizedcancermortalityratesmentionedinx 4.1.1 .Decadalstandardizedcancermortalityratesareprovidedper100,000whitemales,bystate,forthe48contiguousUnitedStates.ThespatialdomainofinterestisDfs1,s2,...,s48g,whereeachsidenotesastate.LetYti=Yt(si)denotethestandardizedcancermortalityrateper100,000whitemalesinstatesiattimet,wheret=1forthe1975-1984decadeandwheret=2forthe1995-2004decade.Fort2f1,2gletY0t=(Yt1,Yt2,...,Yt48)denotethevectorofstandardizedcancermortalityratesattimet.LetW=fwijgdenotethebinary-codedrst-nearest-neighborweightmatrixsuchthatwij=1ifsiandsjshareacommonborder,andwij=0otherwise.Figures 4-1 86

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and 4-2 arechloroplethsofthestandardizedcancermortalityratesfort=1andt=2,respectively.WeconsideraGSTARmodelwithr=1,M0=1andM1=2i.e.withcontemporaneousspatialassociation,rst-orderautoregressiveassociationandrst-orderspatialautoregressiveassociation.Webeginbypre-processingfYtiginamannersimilartothatof Lietal. ( 2007 ).Theypre-processedagriculturallatticedatabyrstaccountingforaneast-westtrendbycenteringnorth-southcolumnsaboutthecolumnmedianandthenaccountedforanoverallmeanprocessbysubtractingfromeachcolumn-adjustedresponsethegrandmean.Herewepre-processthedatabyrstttingtheone-wayANOVAmodel,Yti=t+ti.With^2equaltotheresidualmeansquarederror,westandardizedfYtigbyapplyingtoeachobservationthetransformationYti7!(Yti)]TJ /F6 11.955 Tf 12.89 0 Td[(^t)=^.HenceforthwedenotethecenteredandscaledresponsesbyfYtig.Figures 4-4 and 4-5 arechoroplethsofthecenteredandscaledfYtig.EmbeddingfYtigintheGSTARmodelstructure( 4 ),withF01=W=F12andF11=I,wehaveY1=01WY1+1Y2=01WY2+(11I+12W)Y1+2.Theobjectiveistomakeinferencesabout0=(01,11,12).Inparticular,weconsiderthehypothesesH0:f01=0,11=0,12=0gversusH1:9ij6=0.IfH0isrejectedthenHS,HT,orHSTistrue.UnderH0,thereisneitherspatialnortemporalassociationamongfYtig.UnderHS,foreacht,spatialassociationexistsamongobservationstakenattimet,i.e.amongthefYtig,butneitherautoregressivenorspatialautoregressiveassociationispresentamongobservationsattwodifferenttimepoints.UnderHT,thereisneitherspatialnorspatialautoregressiveassociationamongobservationsatdifferenttimepoints,butthereisconstantautoregressiveassociationbetweenfY1jgandfY2jgforallj.Lastly,underHST,foreacht,thereisaspatialassociationamongtheobservations,fYtig,attimet.Furthermore,aspatio-temporalassociationispresentamongfY1igand 87

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fY2jgforsomei,j.Inthecontextofstandardizedcancermortalityrates,016=0impliesstandardizedratesareassociated,orclustered,inspace.If016=0and116=0but12=0,weinferthat,ateachtimepoint,ratesareassociatedinspaceandalsowiththerateatthesamelocationataprevioustime,butnotwithratesatotherlocationsforaprevioustime.If016=0and11=0but126=0,weinferthereisspatialandspatialautoregressiveassociationamongrates,butnotemporalassociationamongratesatthesamelocation.Lastly,if016=0,116=0and126=0,weinferageneralspatio-temporalassociationamongstfYtig.Now,considerthetestofspatio-temporalassociationbaseduponthespatio-temporalMoran'sIteststatistic.Withthebinary-codedspatio-temporalweightmatrixCfCijg,whereC11=C22=W,C21=W+I,andC12=0,webeginbytakingC1 2(C+(C)0).ThenthesumoftheelementsofCis684,whilenT=48(2)=96.Thusthescalingconstantin( 4 )is96=984.Here^Ist=0.5555.With=.05,aMonteCarloapproximationofthenulldistributionFH0^Iof^IstbaseduponS=1,000,000N(0,I)samplesand( 4 ),the=2orderstatisticofFH0^Iis)]TJ /F6 11.955 Tf 9.3 0 Td[(0.0928andtheupper(1)]TJ /F7 11.955 Tf 12.38 0 Td[(=2)orderstatisticofFH0^Iis0.0872.Because^Ist>0.0872and)]TJ /F6 11.955 Tf 8.54 2.65 Td[(^Ist<)]TJ /F6 11.955 Tf 9.29 0 Td[(0.0928,werejectH0:f01=0,11=0,12=0g.Infact,theMonteCarlodistributionofFH0^Iisminimally)]TJ /F6 11.955 Tf 9.3 0 Td[(0.210andmaximally0.279,yieldingaMonteCarlop-valueof0andthusstrongevidencetorejectH0.Theconclusionisthatspatialand/ortemporalassociationexistsamongstandardizedcancermortalityrates.Toquantifythisassociation,weproceedwithlikelihood-basedestimationoftheelementsof.Tothisendwemaximize`(jy)=logL(jy)usingagridsearchovertheparameterspace,whereistheparameterspaceoverwhichtheGSTARmodelisstationary(Figure 4-3 ).Weuseatwo-stageapproach,inwhichtherststageistomaximize`overacoarsegridon.Wethenconstructanegridcenteredonthecoarse-gridargmax`(jy),trimmedtogridpointssatisfyingstationaryconditions 1 and 2 .The 88

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two-stageprocedureyieldsargmax`(jy)=^,where^0=(0.53,0.415,0.05),and`(^jy)=)]TJ /F6 11.955 Tf 9.29 0 Td[(113.000.Wenowconductalikelihood-ratiotestofH0.Thelog-likelihoodofthefullmodelwithunrestrictedfijg2isdenoted`1=`1()whilethelog-likelihoodofthenullmodelinwhichij=0foralli,jisdenoted`0().Let^1denotetheargumentmaximizing`1overtheunrestrictedparameterspace1,andlet^0denotetheargumentmaximizing`0over0,theparameterspaceresultingfromthenull-hypothesizedconstraints.UnderH0,2[`1(^1))]TJ /F7 11.955 Tf 11.97 0 Td[(`0(^0)];23,wherethedegreesoffreedomisequaltothedimensionof( Lehmann&Romano 2005 ).Trivially^0=(0,0,0)0,and`0(^0)=)]TJ /F6 11.955 Tf 9.3 0 Td[(135.2181.Thus,thelikelihood-ratioteststatisticfortestingH0:f01=0,11=0,12=0gis 2[`1(^1))]TJ /F7 11.955 Tf 11.96 0 Td[(`0(^0)]=2()]TJ /F6 11.955 Tf 9.3 0 Td[(113)]TJ /F6 11.955 Tf 11.95 0 Td[(()]TJ /F6 11.955 Tf 9.3 0 Td[(135.2181))=44.4362.(4)Becausethep-valueisPr(23>44.4362)<110)]TJ /F10 7.97 Tf 6.58 0 Td[(9,werejectH0andconcludeeithercontemporaryspatialstructure,rst-orderautoregressivestructure,orrst-orderspatialautoregressivestructureispresent.Thus,asymptoticlikelihood-ratiotestinferenceagreeswithexactspatio-temporalMoran'sIinference.Asthereisstrongevidenceofeitherspatialortemporalassociation,weconductlikelihood-ratiotestsofthefollowingnullhypothesesversusthealternativehypothesis,H1,thatfijg2areunrestricted:H0:f016=0,11=0,12=0g, (4)H0f01=0g\[f116=0g[f126=0g], (4)H0:f016=0,11=0,126=0g,and (4)H0:f016=0,116=0,12=0g, (4)Here,( 4 )and( 4 )areexactlyHSandHTmentionedinx 4.2.1 .TheGSTARprocessesdenedby( 4 )yieldsaprocesswithcurrentspatialassociationandone-laggedspatialautoregressiveassociation.Lastly,theGSTARprocessdenedby 89

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( 4 )yieldsaprocesswithbothcurrentspatialassociationandalsoone-laggedspatialautoregressiveassociation.Wesummarizetheresultsofalllikelihood-ratiotestsinTable 4-1 .ThereisstrongevidencetorejectH0infavoroftheunrestrictedmodelin( 4 ),( 4 ),and( 4 ).However,thereisnotsufcientevidencetorejectH0denedby( 4 ).TomaintainparsimonyweabandontheunrestrictedGSTARprocessmodelinfavorofthatdenedby( 4 ).With`0(jy)thelog-likelihoodbasedupontheGSTARprocessgivenby( 4 ),andwith^0argmax`0(jy),wehave^0=(0.537,0.421,0)0.Notethatthersttwoelementsof^0arewithin1 100ofthersttwoelementsof^1.Regardingthespatialandtemporalassociationamongstandardizedcancermortalityrates,recall^0=(0.537,0.421,0)0.Hence,theestimatedGSTARmodelforstandardizedcancermortalityratesamongwhitemalesisY1=0.537WY1Y2=0.537WY2+0.421Y1.With^01=0.537,thereisrst-nearest-neighborspatialassociationamongstandardizedcancermortalityratesateachtimepoint.UponinspectionofFigures 4-1 and 4-2 ,rst-nearest-neighborspatialassociationappearsevident.InFigures 4-1 and 4-2 ,wenotethatattimet=2,ratesforalmostallwesternstatesdifferbynomorethanonedecilefromtheirdecilegroupingattimet=1(ateachtimepoint,mapcolorscorrespondtodecilegrouping).Thispatternmayexplaintheappearanceoftheautoregressiveterm,^11=0.421,thatweightseachrateattimet=2bythecorrespondingrateatt=1. 4.4CollaborationandSupportThisworkhasbeencompletedincollaborationwithProfessorLindaJ.Young,DepartmentofStatistics,UniversityofFlorida;andDr.CarolGotway-Crawford,Division 90

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ofBehavioralSurveillance,PublicHealthSurveillanceandInformaticsProgramOfce,CentersforDiseaseControlandPrevention. 91

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Table4-1. Likelihood-ratiotestsofnullhypotheses( 4 )though( 4 ).TheH0columnidentiesthenull-hypothesizedparameterspace.Thelog-likelihoodevaluatedattheMLEsunderH0andandH1are`0and`1,respectively.Degreesoffreedomforeachasymptoticlikelihood-ratio2testarelistedinthedfcolumn.Thelikelihood-ratioteststatisticisX2,andthep-value=Pr(2df>X2). H0`0`1dfX2Pr(2df>X2) f016=0,11=0,12=0g)]TJ /F1 11.955 Tf 48.51 0 Td[(117.438)]TJ /F1 11.955 Tf 9.3 0 Td[(113.00028.8750.012f01=0g\[f116=0g[f126=0g])]TJ /F1 11.955 Tf 9.29 0 Td[(125.170)]TJ /F1 11.955 Tf 9.3 0 Td[(113.000124.341<0.001f016=0,11=0,126=0g)]TJ /F1 11.955 Tf 48.51 0 Td[(116.303)]TJ /F1 11.955 Tf 9.3 0 Td[(113.00016.6070.010f016=0,116=0,12=0g)]TJ /F1 11.955 Tf 48.51 0 Td[(113.150)]TJ /F1 11.955 Tf 9.3 0 Td[(113.00010.3000.584 92

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Figure4-1. Cancermortalityrateamongwhitemales,1975to1984.Ratesarestandardizedper100,000person-years.Dataavailableathttp://ratecalc.cancer.gov/ratecalc/. 93

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Figure4-2. Cancermortalityrateamongwhitemales,1995to2004.Ratesarestandardizedper100,000person-years.Dataavailableathttp://ratecalc.cancer.gov/ratecalc/. 94

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Figure4-3. Thesurfaceoftheparameterspace,,whenF01thebinary-codedandrow-standardizedweightmatrixrepresentingarst-nearest-neighborhoodstructureonthecontiguous48UnitedStates;F11=I;andF12=F01. 95

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Figure4-4. Centeredandscaledcancermortalityrateamongwhitemales,1975to1984. 96

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Figure4-5. Centeredandscaledcancermortalityrateamongwhitemales,1995to2004. 97

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APPENDIXAACCOMPANIMENTTOCHAPTER2BelowisthetextofasimpleMATLABprogramusedtocomputeR. symss11s22s33s44s55s66s77s88s26s37s48;symsd11d12d13d21d51d52d53d61;sigma=[s110000000;0s22000s2600;00s33000s370;000s44000s48;0000s55000;0s26000s6600;00s37000s770;000s48000s88];D=[d11d12d13;d2100;0d210;00d21;d51d52d53;d6100;0d610;00d61];delta_var=(D.')*sigma*D; 98

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APPENDIXBACCOMPANIMENTTOCHAPTER3 TableB-1. TypeIerrorrateforDeanandVuongtestsforlinearoverdispersion,Poissonsampling.ThenullhypothesisisofnooverdispersionandthealternativehypothesisisofgeneralizedPoisson-typelinearoverdispersion. Deantest:Pr(P0B>1.645)Vuongtest:Pr(VGP,P>1.645)n1n115105010051050100 21.50.0530.0380.0690.0550.1230.0450.010.00122.00.0420.060.0570.060.1190.050.0110.00422.50.0460.0560.0620.0540.1320.0310.0060.00425.00.0550.0650.0640.0490.0960.0380.0090.003210.00.0520.0720.0590.0470.1080.0440.0060.004161.50.0440.0560.0520.0450.1020.040.0050.008162.00.0610.060.0510.050.1030.0420.0130.001162.50.0540.0580.060.0590.0990.0440.0010.003165.00.0550.0630.0580.0630.0870.0390.0040.0031610.00.0510.0690.0440.0610.1040.0450.0030.0012561.50.0560.0570.0580.0640.0820.0470.0060.0042562.00.0520.0630.0440.0580.0780.0380.0040.0022562.50.0570.0530.0550.0460.0830.0510.0030.0022565.00.060.0620.0560.0580.0820.0420.0050.00525610.00.0580.0650.0420.0590.0920.0410.0030.00510241.50.0440.0460.0450.0470.0850.0420.0040.00210242.00.0510.0620.0530.0420.0740.0380.0040.00210242.50.0430.0640.0570.0470.0860.0380.0090.00410245.00.0510.0730.060.0570.0620.0390.0050.004102410.00.0560.0520.0480.0670.0260.0250.0080.004 99

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TableB-2. TypeIerrorrateforDeanandVuongtestsforquadraticoverdispersion,Poissonsampling.Thenullhypothesisisofnooverdispersionandthealternativehypothesisisofnegativebinomial-typequadraticoverdispersion. Deantest:Pr(P0C>1.645)Vuongtest:Pr(VNB,P>1.645)n1n115105010051050100 21.50.0490.0350.0630.0510.00.00.0010.022.00.0460.060.0510.0590.00.00.00.022.50.0410.0440.0580.0520.0030.00.00.00125.00.0440.060.0690.0520.0010.00.00.0210.00.0440.0780.0660.0380.0010.00.00.0161.50.0410.0680.0490.0510.0010.00.00.0162.00.0520.0610.0430.0510.0020.00.00.0162.50.0480.0580.0640.0650.00.00.00.0165.00.0520.0520.0510.0560.00.00.00.01610.00.0420.0540.0580.060.0010.00.00.02561.50.0550.0560.0630.0590.00.00.00.0012562.00.0510.0530.060.0540.00.00.00.02562.50.0510.0630.0630.0450.00.00.00.02565.00.0580.0560.0520.0560.00.00.00.025610.00.0520.0510.0510.0580.00.00.00.00110241.50.0440.0560.0460.0440.00.00.00.010242.00.0490.0560.050.0390.0020.00.00.010242.50.0420.0580.0620.0410.00.00.0010.010245.00.0480.0610.0670.0550.00.00.00.0102410.00.0510.0620.0560.0580.00.00.00.0 100

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TableB-3. PowerofDeanandVuongtestsforlinearoverdispersion,generalizedPoissonsamplingwith=1.1.ThenullhypothesisisofnooverdispersionandthealternativehypothesisisofgeneralizedPoisson-typelinearoverdispersion. Deantest:Pr(P0B>1.645)Vuongtest:Pr(VGP,P>1.645)n1n115105010051050100 21.50.0730.0930.1960.2640.0790.0290.0010.00522.00.0810.0910.1780.2390.0820.0330.0010.00622.50.0810.0980.1740.2280.0840.0180.0040.00325.00.0730.090.1770.2160.0910.0350.0020.005210.00.0710.0850.1490.2080.0870.0220.0010.002161.50.0710.1120.1780.2490.0830.0230.0040.002162.00.0920.1090.1940.2680.0830.0250.0020.004162.50.080.0970.1730.2460.070.0210.0020.001165.00.070.0860.1560.2440.0750.0350.0010.0031610.00.0760.0920.1460.2080.0730.0310.0020.0062561.50.0870.1050.2020.270.0690.0230.0030.0042562.00.0790.1080.150.2290.0640.0210.0010.0022562.50.0810.1050.1470.2540.0660.0220.0030.0042565.00.090.0910.1830.2180.0550.0280.0040.00425610.00.0890.0950.1640.2080.0540.0340.0020.00910241.50.0730.0870.1750.2750.0830.0270.00.00310242.00.0750.1030.1830.2740.0620.0260.0030.00210242.50.0750.0990.190.2560.0660.0240.00.0110245.00.060.1040.1350.2080.0490.030.0020.005102410.00.0750.1020.1580.1790.0160.0180.0010.005 101

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TableB-4. PowerofDeanandVuongtestsforlinearoverdispersion,generalizedPoissonsamplingwith=2.ThenullhypothesisisofnooverdispersionandthealternativehypothesisisofgeneralizedPoisson-typelinearoverdispersion. Deantest:Pr(P0B>1.645)Vuongtest:Pr(VGP,P>1.645)n1n115105010051050100 21.50.3820.630.9931.00.0440.0730.7110.98422.00.3840.6690.9961.00.0450.0940.7590.99722.50.3790.6160.9941.00.0410.0670.7390.9925.00.3450.5990.991.00.0440.0780.7560.995210.00.3640.5710.981.00.0550.0630.7810.995161.50.4480.6950.9991.00.0360.0720.8370.999162.00.4230.6680.9981.00.0290.080.8521.0162.50.4050.6370.9971.00.0390.0590.840.997165.00.3680.6160.9911.00.0450.0660.8580.9991610.00.3470.5830.9841.00.0490.0740.8540.9992561.50.4340.690.9991.00.0420.0660.8521.02562.00.4040.7050.9981.00.0420.0780.8690.9982562.50.4020.6610.9931.00.0380.0650.8431.02565.00.3860.5940.9931.00.0480.0670.8480.99825610.00.3590.5440.9811.00.0350.0570.8670.99910241.50.4620.7251.01.00.0480.0770.860.99910242.00.4120.680.9991.00.0390.0720.8471.010242.50.4190.6490.9941.00.0410.0710.8670.99910245.00.3740.5940.991.00.0440.0730.8650.999102410.00.3730.5590.9821.00.0320.0690.8481.0 102

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TableB-5. PowerofDeanandVuongtestsforlinearoverdispersion,generalizedPoissonsamplingwith=3.ThenullhypothesisisofnooverdispersionandthealternativehypothesisisofgeneralizedPoisson-typelinearoverdispersion. Deantest:Pr(P0B>1.645)Vuongtest:Pr(VGP,P>1.645)n1n115105010051050100 21.50.6360.8581.01.00.1470.2470.981.022.00.5940.8571.01.00.1020.2560.9861.022.50.6120.8741.01.00.140.2890.9891.025.00.5840.851.01.00.110.2910.9911.0210.00.5430.8311.01.00.1070.2810.9941.0161.50.7020.9341.01.00.1210.3060.9971.0162.00.7250.9241.01.00.1140.2951.01.0162.50.6650.9351.01.00.1020.3040.9991.0165.00.6570.8871.01.00.0980.3221.01.01610.00.5750.8441.01.00.10.2891.01.02561.50.7230.9431.01.00.1160.3391.01.02562.00.6880.941.01.00.1020.2870.9991.02562.50.7170.9121.01.00.1190.2850.9991.02565.00.6070.861.01.00.1070.3081.01.025610.00.580.8251.01.00.1180.3191.01.010241.50.7140.9471.01.00.1140.3260.9991.010242.00.6890.9451.01.00.1110.3150.9981.010242.50.6850.9281.01.00.1030.3331.01.010245.00.6290.8781.01.00.1140.3291.01.0102410.00.6050.8531.01.00.0990.3091.01.0 103

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TableB-6. PowerofDeanandVuongtestsforlinearoverdispersion,negativebinomialsamplingwiths=10.ThenullhypothesisisofnooverdispersionandthealternativehypothesisisofgeneralizedPoisson-typelinearoverdispersion. Deantest:Pr(P0B>1.645)Vuongtest:Pr(VGP,P>1.645)n1n115105010051050100 21.50.1350.1960.50.7280.060.0090.0210.09122.00.1390.2250.6030.8540.0610.0180.0440.13322.50.1780.2670.7270.9330.050.0080.0630.26725.00.3090.5140.9660.9990.0370.0150.2940.787210.00.5350.7851.01.00.0430.060.7990.999161.50.7380.9331.01.00.1220.3081.01.0162.00.8090.9671.01.00.1240.4071.01.0162.50.8190.9851.01.00.1760.4611.01.0165.00.9330.9981.01.00.2580.6471.01.01610.00.9740.9991.01.00.330.8061.01.02561.51.01.01.01.00.9410.9991.01.02562.01.01.01.01.00.9230.9961.01.02562.51.01.01.01.00.9360.9951.01.02565.01.01.01.01.00.8930.9941.01.025610.01.01.01.01.00.870.9841.01.010241.51.01.01.01.00.9750.9991.01.010242.01.01.01.01.00.9760.9991.01.010242.51.01.01.01.00.9760.9991.01.010245.01.01.01.01.00.9450.9951.01.0102410.01.01.01.01.00.9380.991.01.0 104

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TableB-7. PowerofDeanandVuongtestsforlinearoverdispersion,negativebinomialsamplingwiths=1.25.ThenullhypothesisisofnooverdispersionandthealternativehypothesisisofgeneralizedPoisson-typelinearoverdispersion. Deantest:Pr(P0B>1.645)Vuongtest:Pr(VGP,P>1.645)n1n115105010051050100 21.50.6550.9091.01.00.1390.3090.9941.022.00.7210.931.01.00.160.3741.01.022.50.760.9621.01.00.1720.430.9991.025.00.9140.9961.01.00.2850.6251.01.0210.00.9660.9981.01.00.3820.8021.01.0161.50.9971.01.01.00.8380.9891.01.0162.00.9991.01.01.00.8590.9891.01.0162.50.9991.01.01.00.8720.9921.01.0165.01.01.01.01.00.8890.991.01.01610.01.01.01.01.00.8930.971.01.02561.51.01.01.01.00.9960.9981.01.02562.01.01.01.01.00.9920.9991.01.02562.51.01.01.01.00.9941.01.01.02565.01.01.01.01.00.9830.9951.01.025610.01.01.01.01.00.9690.9911.01.010241.51.01.01.01.00.9961.01.01.010242.01.01.01.01.00.9951.01.01.010242.51.01.01.01.00.9960.9991.01.010245.01.01.01.01.00.9860.9961.01.0102410.01.01.01.01.00.9750.9861.01.0 105

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TableB-8. PowerofDeanandVuongtestsforlinearoverdispersion,negativebinomialsamplingwiths=1.ThenullhypothesisisofnooverdispersionandthealternativehypothesisisofgeneralizedPoisson-typelinearoverdispersion. Deantest:Pr(P0B>1.645)Vuongtest:Pr(VGP,P>1.645)n1n115105010051050100 21.50.7010.9271.01.00.1750.3970.9941.022.00.7880.9741.01.00.2350.4940.9981.022.50.8080.9771.01.00.2170.5360.9981.025.00.9430.9961.01.00.3790.7331.01.0210.00.9751.01.01.00.4730.8511.01.0161.50.9961.01.01.00.8890.9931.01.0162.00.9991.01.01.00.8910.9951.01.0162.50.9991.01.01.00.9020.9951.01.0165.01.01.01.01.00.9280.9891.01.01610.01.01.01.01.00.9180.9791.01.02561.51.01.01.01.00.9981.01.01.02562.01.01.01.01.00.9981.01.01.02562.51.01.01.01.00.9951.01.01.02565.01.01.01.01.00.9840.9991.01.025610.01.01.01.01.00.980.991.01.010241.51.01.01.01.00.9970.9991.01.010242.01.01.01.01.00.9951.01.01.010242.51.01.01.01.00.9980.9991.01.010245.01.01.01.01.00.9870.9991.01.0102410.01.01.01.01.00.9810.9941.01.0 106

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TableB-9. PowerofDeanandVuongtestsforquadraticoverdispersion,generalizedPoissonsamplingwith=1.1.Thenullhypothesisisofnooverdispersionandthealternativehypothesisisofnegativebinomial-typequadraticoverdispersion. Deantest:Pr(P0C>1.645)Vuongtest:Pr(VNB,P>1.645)n1n115105010051050100 21.50.0710.0840.1860.2670.0020.00.0010.00622.00.080.0890.1850.2550.0030.00.0010.00522.50.0750.1010.1880.2560.0030.0030.0010.00225.00.0630.0840.1870.2660.00.00.0020.004210.00.0640.0850.1760.2650.0020.0010.0010.002161.50.070.1150.1820.2640.00.00.0010.005162.00.090.1230.210.2660.0010.00.0020.002162.50.0840.0940.1860.2640.0010.00.0030.001165.00.0770.1110.2050.2630.00.0010.00.01610.00.0780.1040.1710.2740.00.00.00.0032561.50.0890.1040.1950.2770.00.0010.0010.0042562.00.0790.120.1730.2480.00.00.0010.0022562.50.080.1130.1650.2670.0010.0020.0020.0022565.00.0920.0850.1950.2480.0020.0010.0010.00525610.00.080.1090.190.2720.00.00.0010.00210241.50.0820.1010.1740.2990.0010.0010.00.00310242.00.0860.1070.1870.2710.00.00.0020.00310242.50.0820.0970.1980.2880.0010.00.0010.00510245.00.0810.1020.1580.2720.00.00.00.003102410.00.0830.1180.2060.250.00.00.00.005 107

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TableB-10. PowerofDeanandVuongtestsforquadraticoverdispersion,generalizedPoissonsamplingwith=2.Thenullhypothesisisofnooverdispersionandthealternativehypothesisisofnegativebinomial-typequadraticoverdispersion. Deantest:Pr(P0C>1.645)Vuongtest:Pr(VNB,P>1.645)n1n115105010051050100 21.50.3810.6290.9931.00.0250.0620.6970.98222.00.3750.6670.9971.00.0210.0840.7220.99122.50.3660.6430.9961.00.0190.0590.6720.98325.00.3870.670.9951.00.0260.0460.6150.968210.00.4310.6840.9981.00.0230.0380.5280.93161.50.4440.6970.9991.00.0230.0670.8230.999162.00.4430.6940.9981.00.0260.0710.8180.998162.50.4470.6740.9991.00.0260.0510.7690.991165.00.4590.7040.9991.00.0260.0480.6580.9781610.00.4620.7230.9991.00.0220.0350.580.952561.50.4490.6981.01.00.030.0670.8430.9982562.00.430.7431.01.00.0230.0760.8240.9972562.50.470.7190.9971.00.0220.0660.7880.9982565.00.4730.7011.01.00.0230.0430.6620.98525610.00.4680.7130.9991.00.0240.0320.5830.95410241.50.4740.7341.01.00.0310.080.8490.99910242.00.4570.7121.01.00.0260.0670.8060.99710242.50.4630.7080.9991.00.0280.0610.790.99310245.00.4530.7130.9991.00.0190.0370.690.983102410.00.4660.7131.01.00.0180.0410.5430.948 108

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TableB-11. PowerofDeanandVuongtestsforquadraticoverdispersion,generalizedPoissonsamplingwith=3.Thenullhypothesisisofnooverdispersionandthealternativehypothesisisofnegativebinomial-typequadraticoverdispersion. Deantest:Pr(P0C>1.645)Vuongtest:Pr(VNB,P>1.645)n1n115105010051050100 21.50.6240.8511.01.00.1130.2360.981.022.00.6090.8591.01.00.0840.2240.9831.022.50.6110.8861.01.00.1020.2410.9761.025.00.6360.891.01.00.0770.20.9731.0210.00.6290.8981.01.00.0520.1590.9611.0161.50.720.9391.01.00.1120.3050.9971.0162.00.7420.9371.01.00.1170.2650.9981.0162.50.7120.951.01.00.0960.2670.9951.0165.00.7460.9451.01.00.0810.2240.9911.01610.00.730.9421.01.00.0680.1490.9751.02561.50.7290.9491.01.00.1010.3320.9991.02562.00.7210.9571.01.00.0950.2730.9961.02562.50.7360.9431.01.00.1050.2430.9981.02565.00.7120.9191.01.00.080.2160.9941.025610.00.7360.9421.01.00.080.1640.9811.010241.50.7230.9551.01.00.110.3120.9991.010242.00.7360.9581.01.00.1050.2920.9971.010242.50.7370.9491.01.00.0940.2871.01.010245.00.7540.951.01.00.10.210.9911.0102410.00.7320.9431.01.00.0720.160.9741.0 109

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TableB-12. PowerofDeanandVuongtestsforquadraticoverdispersion,negativebinomialsamplingwiths=10.Thenullhypothesisisofnooverdispersionandthealternativehypothesisisofnegativebinomial-typequadraticoverdispersion. Deantest:Pr(P0C>1.645)Vuongtest:Pr(VNB,P>1.645)n1n115105010051050100 21.50.1210.1750.4970.7190.0010.0030.020.09122.00.1350.2030.5780.8440.0040.0020.0420.16822.50.1420.2450.6880.9140.0030.0030.0770.32625.00.2660.4410.9530.9970.0210.0180.4180.872210.00.470.7130.9991.00.040.1170.8991.0161.50.7430.9311.01.00.120.3091.01.0162.00.8090.9731.01.00.1320.4211.01.0162.50.850.9881.01.00.1860.4871.01.0165.00.9460.9991.01.00.3290.71.01.01610.00.9690.9991.01.00.3880.8531.01.02561.51.01.01.01.00.9440.9991.01.02562.01.01.01.01.00.9320.9961.01.02562.51.01.01.01.00.9420.9951.01.02565.01.01.01.01.00.8950.9951.01.025610.01.01.01.01.00.8770.9841.01.010241.51.01.01.01.00.9760.9991.01.010242.01.01.01.01.00.9760.9991.01.010242.51.01.01.01.00.9750.9991.01.010245.01.01.01.01.00.9470.9951.01.0102410.01.01.01.01.00.9380.991.01.0 110

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TableB-13. PowerofDeanandVuongtestsforquadraticoverdispersion,negativebinomialsamplingwiths=1.25.Thenullhypothesisisofnooverdispersionandthealternativehypothesisisofnegativebinomial-typequadraticoverdispersion. Deantest:Pr(P0C>1.645)Vuongtest:Pr(VNB,P>1.645)n1n115105010051050100 21.50.6410.9031.01.00.130.2970.9941.022.00.6980.9251.01.00.1370.3681.01.022.50.7450.9621.01.00.1660.4320.9991.025.00.9060.9921.01.00.320.6591.01.0210.00.9670.9981.01.00.4430.8341.01.0161.50.9991.01.01.00.8460.9891.01.0162.00.9991.01.01.00.8630.991.01.0162.51.01.01.01.00.8780.9931.01.0165.01.01.01.01.00.8950.9911.01.01610.01.01.01.01.00.9070.9721.01.02561.51.01.01.01.00.9960.9981.01.02562.01.01.01.01.00.9920.9991.01.02562.51.01.01.01.00.9941.01.01.02565.01.01.01.01.00.9830.9951.01.025610.01.01.01.01.00.970.9911.01.010241.51.01.01.01.00.9971.01.01.010242.01.01.01.01.00.9981.01.01.010242.51.01.01.01.00.9970.9991.01.010245.01.01.01.01.00.9860.9961.01.0102410.01.01.01.01.00.9760.9861.01.0 111

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TableB-14. PowerofDeanandVuongtestsforquadraticoverdispersion,negativebinomialsamplingwiths=1.Thenullhypothesisisofnooverdispersionandthealternativehypothesisisofnegativebinomial-typequadraticoverdispersion. Deantest:Pr(P0C>1.645)Vuongtest:Pr(VNB,P>1.645)n1n115105010051050100 21.50.6870.9261.01.00.1550.3860.9951.022.00.7760.971.01.00.2320.4930.9981.022.50.8030.9781.01.00.2070.5380.9991.025.00.9370.9951.01.00.3980.7681.01.0210.00.9751.01.01.00.5390.8621.01.0161.50.9971.01.01.00.8950.9931.01.0162.01.01.01.01.00.8950.9951.01.0162.51.01.01.01.00.9130.9951.01.0165.01.01.01.01.00.9380.9891.01.01610.01.01.01.01.00.9310.9791.01.02561.51.01.01.01.00.9981.01.01.02562.01.01.01.01.00.9991.01.01.02562.51.01.01.01.00.9931.01.01.02565.01.01.01.01.00.9850.9991.01.025610.01.01.01.01.00.9790.991.01.010241.51.01.01.01.00.9970.9991.01.010242.01.01.01.01.00.9941.01.01.010242.51.01.01.01.00.9980.9991.01.010245.01.01.01.01.00.9870.9991.01.0102410.01.01.01.01.00.9810.9941.01.0 112

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TableB-15. TypeIerrorratefortwo-stagetestofoverdispersion,Poissonsampling.Therststageistheunion-intersectionDeantestforeitherlinearorquadraticoverdisperison.Theunion-intersectionnullhypothesisisrejectedwhenevermaxB,CP0i>1.96. n1151050100 21.50.0380.0320.0430.03822.00.0320.0520.0350.04722.50.0360.0410.0530.03325.00.0480.0640.0570.046210.00.0480.0730.060.037161.50.0330.0460.0370.03162.00.0510.050.0320.037162.50.0430.0520.0460.047165.00.0520.0450.0380.0481610.00.0470.0510.0480.052561.50.0440.0390.0430.042562.00.0370.0450.0420.0422562.50.0450.050.0490.0382565.00.0590.0530.0460.04725610.00.0540.0640.0330.04310241.50.0320.0380.0290.02710242.00.0420.0460.0380.02310242.50.0390.0580.0540.03610245.00.0490.0660.0540.045102410.00.0590.0470.0440.06 113

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TableB-16. Powerfortwo-stagetestforoverdispersion,generalizedPoissonsampling,=1.1.Therststageistheunion-intersectionDeantestforeitherlinearorquadraticoverdisperison.Therst-stagetestisrejectedwhenevermaxB,CP0i>1.96.Asincreases,sodoesoverdispersionrelativeto. n1151050100 21.50.0570.0690.140.20222.00.0720.0770.1510.21822.50.0710.0810.1590.2125.00.0570.0870.1640.233210.00.0680.0850.1690.233161.50.0540.0970.1270.189162.00.0830.0930.1590.224162.50.0760.0840.1570.22165.00.0710.0910.1720.2381610.00.0690.10.150.2422561.50.0650.0770.1540.2172562.00.070.1040.1380.1982562.50.0710.1010.1330.2222565.00.0850.0820.1720.21525610.00.080.0940.1680.24610241.50.060.0690.1310.22610242.00.0720.0770.1540.2210242.50.0660.0850.1520.23510245.00.0690.0970.1290.219102410.00.0890.1150.1790.214 114

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TableB-17. Powerfortwo-stagetestforoverdispersion,generalizedPoissonsampling,=2.Therst-stagetestisrejectedwhenevermaxB,CP0i>1.96.Asincreases,sodoesoverdispersionrelativeto. n1151050100 21.50.3420.5920.991.022.00.3440.6420.9941.022.50.3480.6150.9931.025.00.3850.6510.9951.0210.00.4090.6740.9991.0161.50.4020.6530.9981.0162.00.4150.6660.9971.0162.50.4220.640.9991.0165.00.4430.690.9981.01610.00.4320.711.01.02561.50.3990.6510.9991.02562.00.40.7020.9981.02562.50.4330.6910.9961.02565.00.4540.6661.01.025610.00.4680.6670.9971.010241.50.4320.6961.01.010242.00.4180.6821.01.010242.50.4280.6720.9981.010245.00.4440.6770.9981.0102410.00.4380.6770.9981.0 115

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TableB-18. Powerfortwo-stagetestforoverdispersion,generalizedPoissonsampling,=3.Therststageistheunion-intersectionDeantestforeitherlinearorquadraticoverdisperison.Therst-stagetestisrejectedwhenevermaxB,CP0i>1.96.Asincreases,sodoesoverdispersionrelativeto. n1151050100 21.50.6090.8321.01.022.00.5990.8471.01.022.50.5950.871.01.025.00.6160.8851.01.0210.00.620.91.01.0161.50.6820.9291.01.0162.00.7290.9311.01.0162.50.6840.9411.01.0165.00.7250.9321.01.01610.00.7090.9351.01.02561.50.6970.9361.01.02562.00.6910.9421.01.02562.50.7220.9311.01.02565.00.6960.9181.01.025610.00.7180.9281.01.010241.50.6890.9411.01.010242.00.7030.9441.01.010242.50.720.9441.01.010245.00.7340.9371.01.0102410.00.7260.9371.01.0 116

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TableB-19. Powerfortwo-stagetestforoverdispersion,negativebinomialsampling.s=10.Therststageistheunion-intersectionDeantestforeitherlinearorquadraticoverdisperison.Therst-stagetestisrejectedwhenevermaxB,CP0i>1.96.Assdecreases,overdispersionrelativetoincreases. n1151050100 21.50.1020.1440.4330.67322.00.1130.1850.5480.81722.50.1380.2420.6840.91425.00.280.4860.9610.998210.00.5150.7471.01.0161.50.710.9191.01.0162.00.7910.9641.01.0162.50.8340.9791.01.0165.00.9530.9991.01.01610.00.9770.9991.01.02561.51.01.01.01.02562.01.01.01.01.02562.51.01.01.01.02565.01.01.01.01.025610.01.01.01.01.010241.51.01.01.01.010242.01.01.01.01.010242.51.01.01.01.010245.01.01.01.01.0102410.01.01.01.01.0 117

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TableB-20. Powerfortwo-stagetestforoverdispersion,negativebinomialsampling.s=1.25.Therststageistheunion-intersectionDeantestforeitherlinearorquadraticoverdisperison.Therst-stagetestisrejectedwhenevermaxB,CP0i>1.96.Assdecreases,overdispersionrelativetoincreases. n1151050100 21.50.6170.8871.01.022.00.690.9171.01.022.50.7320.9571.01.025.00.9130.9951.01.0210.00.9660.9991.01.0161.50.9981.01.01.0162.00.9991.01.01.0162.51.01.01.01.0165.01.01.01.01.01610.01.01.01.01.02561.51.01.01.01.02562.01.01.01.01.02562.51.01.01.01.02565.01.01.01.01.025610.01.01.01.01.010241.51.01.01.01.010242.01.01.01.01.010242.51.01.01.01.010245.01.01.01.01.0102410.01.01.01.01.0 118

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TableB-21. Powerfortwo-stagetestforoverdispersion,negativebinomialsampling.s=1.Therststageistheunion-intersectionDeantestforeitherlinearorquadraticoverdisperison.Therst-stagetestisrejectedwhenevermaxB,CP0i>1.96.Empiricalrst-stagepowerispresentedinthistable.Assdecreases,overdispersionrelativetoincreases. n1151050100 21.50.6750.911.01.022.00.7650.9651.01.022.50.7980.9781.01.025.00.9410.9961.01.0210.00.9741.01.01.0161.50.9961.01.01.0162.01.01.01.01.0162.51.01.01.01.0165.01.01.01.01.01610.01.01.01.01.02561.51.01.01.01.02562.01.01.01.01.02562.51.01.01.01.02565.01.01.01.01.025610.01.01.01.01.010241.51.01.01.01.010242.01.01.01.01.010242.51.01.01.01.010245.01.01.01.01.0102410.01.01.01.01.0 119

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TableB-22. Correctinferenceproportionusingtwo-stagetestforoverdispersion,generalizedPoissonsampling,=1.1.Therststageistheunion-intersectionDeantestforeitherlinearorquadraticoverdisperison.ThesecondstageistheVuongtestforcomparingthegeneralizedPoissonandnegativebinomialdistributionfamilies.InferenceiscorrectwhenevermaxB,CP0i>1.96andV>0.Asincreases,sodoesoverdispersionrelativeto. n1151050100 21.50.0180.0340.0650.09922.00.0270.030.0720.11722.50.0220.0310.0860.11425.00.0170.0330.0870.153210.00.0310.0440.0890.145161.50.0260.0460.0650.09162.00.0410.0450.0870.112162.50.0330.0410.0830.134165.00.0390.0450.0990.1391610.00.0270.0510.0830.1492561.50.0280.040.0710.1062562.00.0360.0450.0720.1152562.50.0290.0420.0730.1182565.00.0370.0380.1030.12325610.00.0290.040.0850.15510241.50.0290.0320.0720.12210242.00.0310.0410.0790.10310242.50.0240.0370.0750.12110245.00.0330.0410.0680.138102410.00.0440.0630.110.148 120

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TableB-23. Correctinferenceproportionusingtwo-stagetestforoverdispersion,generalizedPoissonsampling,=2.Therststageistheunion-intersectionDeantestforeitherlinearorquadraticoverdisperison.ThesecondstageistheVuongtestforcomparingthegeneralizedPoissonandnegativebinomialdistributionfamilies.InferenceiscorrectwhenevermaxB,CP0i>1.96andV>0.Asincreases,sodoesoverdispersionrelativeto. n1151050100 21.50.1290.2830.5910.6422.00.1480.3250.6770.75122.50.1530.3450.7190.82525.00.2080.3940.8410.931210.00.2270.430.9150.971161.50.20.3770.6370.683162.00.2040.3540.7230.769162.50.2240.3780.750.847165.00.2460.4560.8750.9491610.00.2760.5090.9150.9792561.50.1980.340.6320.662562.00.2070.4010.7110.8062562.50.2370.4120.7670.852565.00.250.440.8680.95625610.00.2860.4450.9420.97810241.50.2140.360.6590.70810242.00.230.3740.6970.78610242.50.2480.3960.7970.85310245.00.2560.4490.8650.946102410.00.2540.4850.9220.974 121

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TableB-24. Correctinferenceproportionusingtwo-stagetestforoverdispersion,generalizedPoissonsampling,=3.Therststageistheunion-intersectionDeantestforeitherlinearorquadraticoverdisperison.ThesecondstageistheVuongtestforcomparingthegeneralizedPoissonandnegativebinomialdistributionfamilies.InferenceiscorrectwhenevermaxB,CP0i>1.96andV>0.Asincreases,sodoesoverdispersionrelativeto. n1151050100 21.50.2320.3550.5930.69622.00.2570.440.7390.83422.50.2730.4910.8120.88625.00.3490.5790.8880.974210.00.3680.6420.9520.994161.50.3330.5030.670.744162.00.3980.5630.7860.871162.50.3810.6060.8440.932165.00.4060.680.9560.9891610.00.470.730.9881.02561.50.3740.5430.6740.7522562.00.4080.5810.7750.8752562.50.4110.5870.8410.922565.00.4410.6920.9490.99525610.00.490.7510.9840.99910241.50.3730.5450.6820.73510242.00.4190.5810.7950.85910242.50.4010.6240.850.92810245.00.4770.6930.9510.995102410.00.4720.7510.9931.0 122

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TableB-25. Correctinferenceproportionusingtwo-stagetestforoverdispersion,negativebinomialsampling,s=10.Therststageistheunion-intersectionDeantestforeitherlinearorquadraticoverdisperison.ThesecondstageistheVuongtestforcomparingthegeneralizedPoissonandnegativebinomialdistributionfamilies.InferenceiscorrectwhenevermaxB,CP0i>1.96andV<0.Assdecreases,overdispersionrelativetoincreases. n1151050100 21.50.0570.0870.2450.40822.00.0750.1180.3580.56422.50.10.150.4760.66725.00.2220.3660.7810.896210.00.4280.630.9410.989161.50.4180.5480.7050.748162.00.4950.6180.8050.89162.50.5530.6780.8690.947165.00.7290.8250.9780.9991610.00.8030.9271.01.02561.50.5270.6030.7710.8722562.00.640.6730.8890.9372562.50.6540.760.9590.982565.00.7430.8540.9941.025610.00.8330.9280.9991.010241.50.5760.6140.7750.88110242.00.6050.7060.8790.95810242.50.6350.7390.9470.98410245.00.7740.8360.9961.0 123

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TableB-26. Correctinferenceproportionusingtwo-stagetestforoverdispersion,negativebinomialsampling,s=1.25.Therststageistheunion-intersectionDeantestforeitherlinearorquadraticoverdisperison.ThesecondstageistheVuongtestforcomparingthegeneralizedPoissonandnegativebinomialdistributionfamilies.InferenceiscorrectwhenevermaxB,CP0i>1.96andV<0.Assdecreases,overdispersionrelativetoincreases. n1151050100 21.50.4050.5760.6990.7622.00.480.6190.770.84922.50.50.690.8420.91725.00.7210.8110.9630.992210.00.7830.8780.9910.999161.50.580.6520.8570.936162.00.60.6940.9120.966162.50.6280.7280.9320.984165.00.7070.8240.981.01610.00.7650.8780.9971.02561.50.5380.7080.9670.9982562.00.5870.7210.9730.9982562.50.5990.7670.9811.02565.00.6590.8090.9911.025610.00.740.8460.9971.010241.50.5840.7220.9830.99910242.00.5770.7140.980.99710242.50.570.7570.9881.010245.00.6550.8120.9961.0102410.00.7290.8480.9981.0 124

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TableB-27. Correctinferenceproportionusingtwo-stagetestforoverdispersion,negativebinomialsampling,s=1.Therststageistheunion-intersectionDeantestforeitherlinearorquadraticoverdisperison.ThesecondstageistheVuongtestforcomparingthegeneralizedPoissonandnegativebinomialdistributionfamilies.InferenceiscorrectwhenevermaxB,CP0i>1.96andV<0.Assdecreases,overdispersionrelativetoincreases. n1151050100 21.50.4480.6220.6870.77822.00.5210.6740.8010.85222.50.570.7140.8450.92625.00.7330.8180.9540.983210.00.7920.870.9910.999161.50.6150.6910.8810.964162.00.5790.6960.9130.977162.50.6250.7240.9460.984165.00.6590.8150.981.01610.00.7370.850.9971.02561.50.5680.7440.9780.9992562.00.5960.7440.9930.9992562.50.5930.7660.9881.02565.00.670.8130.9961.025610.00.70.8550.9951.010241.50.5640.7490.9911.010242.00.5780.7640.9911.010242.50.6380.7760.990.99910245.00.6320.8140.9981.0102410.00.6950.8691.01.0 125

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APPENDIXCACCOMPANIMENTTOCHAPTER4 PropositionC.1. ConsidertheGSTARmodel( 4 )withT>1;r=1;M0=1;andM1=2.LetA(I)]TJ /F7 11.955 Tf 11.96 0 Td[(01F01))]TJ /F10 7.97 Tf 6.58 0 Td[(1andB11F11+12F12.Thenfort=1,2,...,T)]TJ /F6 11.955 Tf 11.96 0 Td[(1, Var(Yt+1)=tXk=0(AB)kAA0(B0A0)k,(C)where(AB)0I. Proof. NotingthatVar(Y1)=Var(A1)=AA0.,weproceedbyinductionont.BecauseY2=ABY1+A2,itfollowsthatVar(Y2)=Var(ABY1+A2)=Var(ABY1)+Var(A2)=ABVar(Y1)B0A0+AA0=ABAA0B0A0+AA0=1Xk=0(AB)kAA0(B0A0)k.Thus,theinitialconditionholdsfort=1.Nowsupposetherelationholdsfort=q1;r=1;M0=1;andM1=2.LetA(I)]TJ /F7 11.955 Tf 11.96 0 Td[(01F01))]TJ /F10 7.97 Tf 6.58 0 Td[(1andB11F11+12F12.Thenforr,s2f1,2,...,Tgwith 126

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1s
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BIOGRAPHICALSKETCH NathanHoltwasbornandraisedinMarinCounty,California.Natehastwoolderbrothers,oneyoungerbrother,andoneyoungersister.HegraduatedSummacumLaudewithhisB.S.inmathematicsfromtheUniversityofMississippiin2004,andcompletedhisM.S.inmathematicsattheUniversityofMississippiin2006witha4.0GPA.NateearnedhisMasterofStatisticsdegreeattheUniversityofFloridain2009.HereceivedhisPh.D.fromtheDepartmentofStatisticsattheUniversityofFloridainthesummerof2012. 133