Citation
Internal Pilots with the Univariate Approach to Repeated Measures

Material Information

Title:
Internal Pilots with the Univariate Approach to Repeated Measures
Creator:
Zhang, Xinrui
Publisher:
University of Florida
Publication Date:
Language:
English

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Biostatistics
Committee Chair:
CHI,YUEH-YUN
Committee Co-Chair:
GOODENOW,MAUREEN M
Committee Members:
BRUMBACK,BABETTE A
KAIRALLA,JOHN ANDREW
MULLER,KEITH E

Subjects

Subjects / Keywords:
power
sample-size
unirep

Notes

General Note:
Repeated measures of outcome are common in clinical trials and epidemiological studies. Designing studies with repeated measures requires accurate specifications of the variances and correlations in order to select an appropriate sample size. Underspecifying the variances leads to a sample size that is inadequate to detect a meaningful scientific difference, while overspecifying the variances results in an unnecessary large sample size. Both lead to waste of resources and place study participants in unwarranted risk. We extend the internal pilot design, which allows for sample size re-estimation, to repeated measures and derive approximate distributions of the final sample size and the Univariate Approach to Repeated Measures test statistic. Extensive simulations examine the impact of misspecification of the covariance matrix and demonstrate the accuracy of the approximations in controlling the Type I error rate and achieving the target power. The proposed methods are applied to a longitudinal study assessing early antiretroviral therapy for youth living with HIV. When planning a longitudinal study with Gaussian outcomes, accurate specification of the nuisance parameters in the error covariance matrix is required to have adequate power analysis. The internal pilot design preserves power and protects against covariance misspecification by using a fraction of the observed data to re-estimate covariances and adjust the sample size. Statistical testing without account for the randomness in the sample size would likely lead to the inflation of the Type I error rate. We extend the bounding approach, which modifies the critical value to ensure the maximum Type I error rate is at or below the target level. The extension applies to longitudinal internal pilot studies. The covariance structures commonly encountered in the longitudinal data, and the most general unstructured covariance matrix are considered. Extensive enumeration and simulation results highlight the advantage of the proposed approach in achieving a control of the Type I error rate while maintaining the benefits of internal pilot design in preserving power.

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UFRGP
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All applicable rights reserved by the source institution and holding location.
Embargo Date:
8/31/2018

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INTERNALPILOTSWITHTHEUNIVARIATEAPPROACHTOREPEATEDMEASURESByXINRUIZHANGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2016

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c2016XinruiZhang

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ThisdissertationisdedicatedtomyparentsFeiqingZhangandBenhuiXinfortheirunconditionalloveandsupport.

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ACKNOWLEDGMENTSIwouldrstliketothankmyadvisor,Dr.Yueh-YunChi,forhercontinuedsupportandencouragementduringthecourseofmygraduateeducationandPh.D.research.ThisworkwouldnothavebeenpossiblewithouttheinsightfuldiscussionsthatIhavehadwithher.IthasbeenanhonortobeDr.Chi'srstPh.D.student.Herguidancehasbeenindispensableinfacilitatingmydissertationresearchandpreparingmeforacareerinbiostatistics.IowemuchgratitudetomydissertationcochairDr.MaureenGoodenow.Iamverygratefultohavebeengiventheopportunitytoworkwithandlearnfromher.Iappreciateallhercontributionsoftime,insight,andnancialsupporttomakemyPh.D.experiencepossibleandproductive.TheinvaluableexperiencesIhavegainedinDr.Goodenow'slaboratoryhavebeenextremelyinstrumentalinmydevelopmentasabiostatistician.Imustalsothankmycommitteemembersfortheirguidanceandcontributioninthedevelopmentofmydissertationandinmyprofessionalgrowth.IwouldliketothankDr.KeithMullerforhisexpertiseandvaluableinsightinadaptivedesigns,whichgreatlybenetedmeinmyresearchandmygeneralunderstandingofbiostatistics.MysincerethanksgotoDr.BabetteBrumbackforherconstructivecommentstohelpmeviewmyworkfromadierentperspective,andherinterestinglecturesaboutbiostatisticalmethods,whichenhancedmyinteresttounderstandmorein-depthbiostatisticalknowledgewhenIwasaMasterstudent.IwouldliketoexpressmygratitudetoDr.JohnKairallaformakinginsightfulsuggestionsandsharinghisexperiencetohelpmegetthroughthediculttimewhenwritingthisdissertation.IwouldliketoacknowledgestaandstudentsintheDepartmentofBiostatistics,andmembersoftheGoodenowlab,pastandpresentfortheirhelpandsupport,bothscienticallyandpersonally.Mostimportantly,Imustthankmyfamilyfortheirunconditionallove,supportandencouragementthatallowedmetocomethisfar.EverythingIhaveaccomplishedis 4

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becauseofyourunwaveringsupport.Thefactthatyoualwaysbelievedinmehasenabledmetofacethechallengesandpursuemypassionswithcondence.IalsowishtothankmyboyfriendWeiXiaforalwaysbeingthereforme.Hissupportandpatiencehasmademeabetterperson,andIamfortunatetohavehiminmylife. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 12 1.1Background ................................... 12 1.2InternalPilotDesign .............................. 13 1.3UnivariateApproachtoRepeatedMeasures ................. 13 2LITERATUREREVIEW .............................. 17 2.1Stein'sTwo-StageProcedure .......................... 17 2.1.1Stage1(InternalPilotStage) ..................... 17 2.1.2Stage2 .................................. 18 2.2InternalPilotDesignforTwo-Samplet-Test ................. 19 2.2.1NaveApproach ............................. 19 2.2.2AdjustedApproach ........................... 23 2.3InternalPilotDesignforUnivariateOutcomes ................ 26 2.4InternalPilotDesignforRepeatedMeasures ................. 32 3INTERNALPILOTDESIGNFORREPEATEDMEASURES .......... 33 3.1MotivatingExample .............................. 33 3.2ExistingResultsfortheUNIREPTests .................... 34 3.2.1Notation ................................. 34 3.2.2PowerApproximationsforFixedDesign ................ 36 3.2.3TheGeneralLinearMultivariateModelforanInternalPilotDesign 37 3.3NewApproximationsforanInternalPilotDesign .............. 38 3.3.1DistributionoftheFinalSampleSize ................. 38 3.3.2ConditionalDistributionoftheFinalTestStatistic ......... 41 3.3.3MarginalDistributionoftheFinalTestStatistic ........... 43 3.4Simulations ................................... 44 3.4.1SimulationOverview .......................... 44 3.4.2Simulation1:One-GroupMultivariateDesign ............ 44 3.4.3Simulation2:Two-GroupRepeated-MeasuresDesign ........ 46 3.5ApplicationtotheMotivatingExample .................... 49 3.6Discussion .................................... 50 6

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4AVOIDINGBIASINLONGITUDINALINTERNALPILOTSTUDIES ..... 52 4.1Motivation .................................... 52 4.2PowerConsiderationsforLongitudinalStudies ................ 53 4.2.1InputsforSampleSizeDetermination ................. 53 4.2.2LEARStructure ............................. 53 4.2.3LongitudinalInternalPilotStudy ................... 55 4.3ControllingTypeIErrorRateforLongitudinalInternalPilotStudies ... 55 4.3.1BoundingApproachforLEARStructure ............... 56 4.3.2BoundingApproachforUnstructuredCovarianceMatrix ...... 56 4.4NumericalResults ................................ 57 4.4.1Enumerations .............................. 58 4.4.2Simulations ............................... 61 4.5Discussion .................................... 63 5CONCLUSIONS ................................... 65 5.1Summary .................................... 65 5.2FutureResearch ................................. 66 APPENDIX AMATHEMATICALPROOFS ............................ 68 A.1ProofofLemma1 ................................ 68 A.2ProofofLemma2 ................................ 69 A.3ProofofLemma3 ................................ 70 A.4ProofofLemma4 ................................ 70 BSUPPLEMENTALRESULTS ............................ 72 CSIMULATIONDETAILS .............................. 75 DPARAMETERSOFTHEMOTIVATINGEXAMPLE .............. 79 REFERENCES ....................................... 82 BIOGRAPHICALSKETCH ................................ 86 7

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LISTOFTABLES Table page 3-1Notationforthegenerallinearmultivariatemodelwithinternalpilotdesign. .. 39 8

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LISTOFFIGURES Figure page 1-1Flowchartofaninternalpilotdesignwithrepeatedmeasures. .......... 14 3-1DataandparametersrequiredtoplanalongitudinalstudyinvestigatingtheeectofearlytreatmentontheconcentrationsofsolubleCD27,CD14andCD163. 35 3-2Simulation1(one-groupmultivariatedesign)results. ............... 46 3-3Simulation2(two-grouprepeated-measuresdesign)results. ............ 48 3-4Simulation2(two-grouprepeated-measuresdesign)resultsfordierentsizesofthestage1sample. .................................. 49 3-5DistributionofnalsamplesizefortheCMtestwhencovarianceismisspecied. 51 4-1PlotsofpredictedTypeIerrorrateasafunctionof=2=20(log2scale)forCMtest. ........................................ 59 4-2Plotsofpredictedpowerasafunctionof=2=20(log2scale)forCMtest. .. 60 4-3EmpiricalTypeIerrorrateforCMtest. ...................... 62 4-4EmpiricalpowerforCMtest. ............................ 63 B-1Simulation1(one-groupmultivariatedesign)results:DistributionofN+forn0=17(n1=9). ...................................... 73 B-2Simulation2(two-grouprepeated-measuresdesign)results:DistributionofN+forn0=44(n1=22). ................................ 74 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyINTERNALPILOTSWITHTHEUNIVARIATEAPPROACHTOREPEATEDMEASURESByXinruiZhangAugust2016Chair:Yueh-YunChiCochair:MaureenM.GoodenowMajor:BiostatisticsRepeatedmeasuresofoutcomearecommoninclinicaltrialsandepidemiologicalstudies.Designingstudieswithrepeatedmeasuresrequiresaccuratespecicationsofthevariancesandcorrelationsinordertoselectanappropriatesamplesize.Underspecifyingthevariancesleadstoasamplesizethatisinadequatetodetectameaningfulscienticdierence,whileoverspecifyingthevariancesresultsinanunnecessarylargesamplesize.Bothleadtowasteofresourcesandplacestudyparticipantsinunwarrantedrisk.Weextendtheinternalpilotdesign,whichallowsforsamplesizere-estimation,torepeatedmeasuresandderiveapproximatedistributionsofthenalsamplesizeandtheUnivariateApproachtoRepeatedMeasuresteststatistic.ExtensivesimulationsexaminetheimpactofmisspecicationofthecovariancematrixanddemonstratetheaccuracyoftheapproximationsincontrollingtheTypeIerrorrateandachievingthetargetpower.TheproposedmethodsareappliedtoalongitudinalstudyassessingearlyantiretroviraltherapyforyouthlivingwithHIV.WhenplanningalongitudinalstudywithGaussianoutcomes,accuratespecicationofthenuisanceparametersintheerrorcovariancematrixisrequiredtohaveadequatepoweranalysis.Aninternalpilotdesignpreservespowerandprotectsagainstcovariancemisspecicationbyusingafractionoftheobserveddatatore-estimatecovarianceandadjustthesamplesize.Statisticaltestingwithoutaccountfortherandomnessinthe 10

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samplesizewouldlikelyleadtotheinationoftheTypeIerrorrate.Weextendtheboundingapproach,whichmodiesthecriticalvaluetoensurethemaximumTypeIerrorrateisatorbelowthetargetlevel.Theextensionappliestolongitudinalinternalpilotstudies.Thecovariancestructurescommonlyencounteredinthelongitudinaldata,andthemostgeneralunstructuredcovariancematrixareconsidered.ExtensiveenumerationandsimulationresultshighlighttheadvantageoftheproposedapproachincontrollingtheTypeIerrorratewhilemaintainingthebenetsofaninternalpilotdesigninpreservingpower. 11

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CHAPTER1INTRODUCTION 1.1BackgroundSamplesizedeterminationplaysacriticalroleinplanningaclinicaltrialandepidemiologicalstudy.Asamplesizetoolowcanleadtoimpreciseestimatesandlackofpowertodetectclinicallyrelevanteects.Ontheotherhand,asamplesizeunnecessarilylargecanwastetimeandresources.ForGaussianoutcomes,oneoftherequiredinputsforsamplesizedeterminationisthenuisanceparametersinthecovariancematrix.Forexample,astudycomparingmeanresponsesoftwogroupsrequiresanaccurateguessoftheerrorvariances.Inpractice,valuesofthenuisanceparametersareuncertainintheplanningphase.Commonpracticesincludeusingvaluesobtainedfromthepublishedliterature,pilotstudypriortothecurrentstudy,andexperts'opinions.Inaccuratepowercalculationmayoccurwhenthedesignsofthepreviousstudiesaredierentfromthecurrentone.Toavoidtheuncertaintyinndingvalidvaluesforthenuisanceparameters,avarietyoftwo-stageprocedureshavebeenproposed.Stein[ 1 ]consideredusingdatacollectedatstage1tore-estimatethenuisanceparametersandadjustthenalsamplesize.WittesandBrittain[ 2 ]modiedStein'sprocedureandintroducedaninternalpilotdesign.Aninternalpilotdesignisdistinctfromageneraladaptivedesign.Aninternalpilotstudyallowsusingthestage1(internalpilotstage)datatore-estimatethesamplesize,butdoesnotpermitinterimdataanalysesoranyearlystopping.Thegroupassignmentofaninternalpilotstudyisnotcompletelyblindedatthetimeofsamplesizere-estimation.Inaninternalpilotdesign,samplesizere-estimationisbasedontherevisedestimatesofthenuisanceparametersandthesametreatmenteectsinitiallyspeciedbeforeanydatacollection. 12

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1.2InternalPilotDesignThetwo-stageinternalpilotdesignmodiedfromtheStein'sprocedure[ 1 ],wasrstintroducedin1990byWittesandBrittain[ 2 ]foratwo-samplettestcomparison.Thedesigncanbeeasilyappliedtostudiesofrepeatedmeasures.Figure 1-1 illustratestheowofthetwostagesinaninternalpilotdesignwithrepeatedmeasures.Stage1isknownasthe\internalpilotstage"whentheinitialpoweranalysis,basedonapriorguessoftheerrorcovariancematrix0,isperformed,andtheinternalpilotdataarecollected.Here,n0istheinitialsamplesize,and(0<<1)isthepre-speciedproportiontodeterminethesizeoftheinternalpilotsample(n1=n0).Attheendofthestage1,theerrorcovariancematrixcanbeestimatedandusedtoperformthesecondpoweranalysisatstage2.Thesecondpoweranalysisisbasedonexactsamespecicationsoftheinitialpoweranalysis,exceptwitharevisederrorcovariancematrix^1.Tworulesarecommonlyusedtodeterminethenalsamplesizebasedonthesecondpoweranalysis.WittesandBrittain[ 2 ]requiredthenalsamplesizetobeatleastaslargeastheinitiallyplannedsamplesize(i.e.,N+n0).BirkettandDay[ 3 ]notedthattheresulting\restrictedrule"mayleadtoanunnecessarilylargesamplesizeandhighpowerifthepriorguessoverestimatestheerrorvariance.Inturn,theyrecommendedusingthe\unrestrictedrule,"toonlyrequirethenalsamplesizetobenolessthanthesizeofthestage1sample(i.e.,N+n1).OncethenalsamplesizeN+isdetermined,N2=N+)]TJ /F3 11.955 Tf 12.38 0 Td[(n1additionalsubjectsaretobeenrolledduringstage2.Attheendofstudy,allN+observationswillbeusedtocomputethenalteststatisticandtesttheprimaryhypothesis.WenotethatbothN+andN2arerandomvariables.ClassicresultsforthenalteststatisticnolongerholdbecauseoftherandomnessinN+. 1.3UnivariateApproachtoRepeatedMeasuresWhenmodelingcorrelatedresponseswithGaussianerrors,generallinearmultivariatemodelandlinearmixedmodelaretwomostcommonlyusedtechniques.Linearmixed 13

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Figure1-1. Flowchartofaninternalpilotdesignwithrepeatedmeasures.Redhighlightsrepresentstage1processes.Bluehighlightsrepresentstage2processes. 14

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modelspermitmissingormistimeddata,whichbringsexibilitytohandleunbalanceddesigns,andallowmodelingboththeresponsemeananditscovariance.Inspiteofthemodels'popularity,theaccuracyofpowerapproximationsforlinearmixedmodelshasnotbeenfullystudied[ 4 5 ].Inaddition,linearmixedmodeltestscaninatetheTypeIerrorratebadly,especiallyinsmallsamples[ 5 ].Incontrast,generallinearmultivariatemodelsrequirebalanceddatawithnomissingobservationsandnotime-varyingcovariates.Althoughmultivariatemodelshavemorerestrictions,goodpowermethodsareavailableformultivariatemodeltests.Moreover,thetestscancontroltheTypeIerrorratewelleveninsmallsamples,andarerobustagainstthecovariancemisspecication[ 5 ].Mulleretal.[ 5 6 ]recommendedusingtestsofgenerallinearmultivariatemodeltoanalyzethecorrelatedGaussianoutcomes,wheneverpossible,inlieuoftheanalysisoflinearmixedmodel.Thehypothesistestsusedinthegenerallinealmultivariatemodelsettingfallintotwocategories:UnivariateApproachtoRepeatedMeasures(UNIREP)testsandMultivariateApproachtoRepeatedMeasures(MULTIREP)tests.Smithetal.[ 7 ]andSchatzo[ 8 ]discussedhypothesistestingoffourMULTIREPtests:(1)Roy'slargestroot(RLR),(2)Wilks'likelihoodratio(WLK),(3)Pillai-Bartletttrace(PBT),and(4)Hotelling-Lawleytrace(HLT).TheMULTIREPanalysisallowsanunstructuredcovariancematrix.Originally,theUNIREPtestsweredevelopedassumingacompoundsymmetrycovariancestructure.TherearevecommonlyusedUNIREPtests:(1)theuncorrectedtest(UN)[ 9 10 ],(2)theBoxconservativetest(Box)[ 11 ],(3)theGeisser-Greenhoustest(GG)[ 11 12 ],(4)theHuynh-Feldttest(HF)[ 13 ],and(5)theChi-Mullertest(CM)[ 14 ].Whentheassumptionofcompoundsymmetrycovariancematrixisviolated,thecorrectedUNIREPtests,suchasGG,HFandCM,allowtheUNIREPanalysistobeappliedtoanycovariancestructures.Mulleretal.[ 5 15 ]evaluatedthepowerperformancesofUNIREPandMULTIREPtests,andnoticednotestisuniformlymostpowerful.ThechoicebetweenUNIREPand 15

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MULTIREPtestsdependsonthedegreetowhichthesphericityassumption(i.e.,alleigenvaluesofthecovariancematrixareequal)hasbeenviolated.TheUNIREPtestshavemorepowerwhenthecovariancestructureisclosetosphericity,whiletheMULTIREPtestshavebetterpowerperformancewhenthecovariancestructureisfarawayfromsphericity[ 5 ].Whenconsideringinternalpilotdesignwithrepeatedmeasures,severalspecialcaseshavebeendiscussedusingeitherUNIREPorMULTIREPanalysis.Park[ 16 ]highlightedaspecialcase,min(a;b)=1,forMULTIREPanalysis.Here,aandbistherankofthebetween-subjectandwithin-subjectcontrastmatrix,respectively.Thisspecialcaseappliestoonegrouptestortwogroupcomparisons.Whenmin(a;b)=1,Park[ 16 ]provedthegenerallinearmultivariatemodelwithMULTIREPtestscanalwaysbetransformedtoahypothesisequivalentunivariatemodel.Gurkaetal.[ 17 ]consideredarestrictedclassoflinearmixedmodelsthatcanbestatedasgenerallinearmultivariatemodelswithsphericalcovariancematrix.Gurkaetal.[ 17 ]demonstratedsuchrestrictedmodelswithUNIREPtestscanbetransformedtoasetofunivariatemodels.Subsequently,theinternalpilottheorydevelopedforunivariateoutcomes[ 18 { 20 ]canbeappliedtobothspecialcases.Currently,thereisnointernalpilottheorygenerallyapplicabletorepeatedmeasures.Theaimofthisdissertationistodevelopanalyticalmethodsforinternalpilotdesignwiththeunivariateapproachtorepeatedmeasures. 16

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CHAPTER2LITERATUREREVIEW 2.1Stein'sTwo-StageProcedureStein'stwo-stageprocedurewasrstdevelopedforthesinglegroupsettingin1945[ 1 ],butcanbeeasilyextendedtotwo-samplecomparisons.Considerastudycomparingthemeansoftwoindependentgroups,Xii:i:dN(x;2)andYji:i:dN(y;2).Forthesakeofsimplicity,commongroupvarianceandequalsamplesizeareassumed.Wedene=x)]TJ /F3 11.955 Tf 12.61 0 Td[(y,theprimaryresearchquestionistotestthenullhypothesisH0:=0againstthealternativehypothesisHa:6=0.ThestudyhastargetTypeIerrorratet,andtargetpowerPt.ThetargetpowerisrelatedtothetargetTypeIIerrorratet(whichequals1)]TJ /F3 11.955 Tf 11.95 0 Td[(Pt).Theinitialtotalsamplesizen0isdeterminedby n0=420(Zt=2+Zt)2 2t:(2{1)Here,tistheclinicallyrelevantdierencesbetweenthetwogroups,20istheinitialguessoftheerrorvariance,andZqrepresentstheupperqquantileofthestandardnormaldistribution. 2.1.1Stage1(InternalPilotStage)Aftertheinitialtotalsamplesizeisdetermined,insteadofrecruitingalln0subjects,weenrolln1=n0subjects(n1=2pergroup)toformthestage1(internalpilot)sample.Here,(0<<1)isapre-speciedproportiontodeterminethesizeofthestage1sample.Attheendofstage1,thepooledvariances21=(s2X1+s2Y1)=2canbeobtained,wheres2X1ands2Y1denotethewithingroupsamplevariancesinstage1.Subsequently,anadjustedsamplesizeNadjcanbecalculatedusingaresizingformula Nadj=4s21(tt=2;1+tt;1)2 2t:(2{2)Theresizingformulareplacesthenormalquantileswiththequantilesofthetdistribution,andtheinitialguessoferrorvariance20withthestage1varianceestimator 17

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s21[ 21 22 ].Here,tq;1representstheupperqquantileofthetdistributionwith1degreesoffreedom,and1=n1)]TJ /F1 11.955 Tf 11.75 0 Td[(2isthedegreesoffreedomofthestage1(internalpilot)sample. 2.1.2Stage2Atthebeginningofstage2,thenaltotalsamplesizeN+isdeterminedbasedontheruleofnalsamplesize.Theruleofselectingnalsamplesizemustbedenedbeforestartingastudywiththeinternalpilotdesign.WittesandBrittain[ 2 ]suggestedusingN+=max(n0;Nadj),whichisreferredtoasthe\restricted"rulebyWittesetal.[ 23 ].Therestrictedruledoesnotallowthenalsamplesizetodecreaseandbesmallerthantheinitialsamplesizen0.BirkettandDay[ 3 ]arguedtherestrictedrulecouldcauseunnecessarilyhighpowerif20overestimates2.Instead,theyrecommendedusingN+=max(n1;Nadj)toallowdecreaseinsamplesize,whichistermedas\unrestricted"rule[ 23 ].AlesscommonlyusedruleofselectingnalsamplesizewasproposedbyGouldandShih[ 24 ].Theysuggestedincreasingthesamplesizeonlywhenw1n0
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Stein'steststatistictsusesallN+observationstocalculatethedierenceofthesamplemeansinthenumerator,whileonlythestage1sample(n1observations)isusedtoobtainthevarianceestimators21inthedenominator.Theteststatistictsfollowsatdistributionunderthenullwith1=n1)]TJ /F1 11.955 Tf 12.3 0 Td[(2degreesoffreedom[ 1 ].For(n1)]TJ /F1 11.955 Tf 12.31 0 Td[(2)s21=22(n1)]TJ /F1 11.955 Tf 12.13 0 Td[(2),andtheconditionaldistributionof(X)]TJ /F1 11.955 Tf 13.9 3.03 Td[(Y)=p (42=n+),givens21,isstandardnormalunderthenull.Notethatconditioningons21isthesameasxingn+inadvance,wheren+=N+(s21).Butwhenn+isxed,(X)]TJ /F1 11.955 Tf 14.39 3.02 Td[(Y)=p (42=n+)isindependentofs21.Consequently,Stein'steststatistictshasanulltdistribution.ThedistributionoftsunderthenullcanalsobeobtainedasaspecialcaseofthegenerallinearunivariatemodeldiscussedbyCoeyandMuller[ 18 ].ThisdistributionalpropertyensuresStein'sprocedurehastheexactcontrolofthelevel,andcanguaranteepowertobeatleastthetargetedlevelofPt[ 25 ].However,Stein'sprocedureissubjectedtocriticismfornotusingallthedatatoestimatethevariance. 2.2InternalPilotDesignforTwo-Samplet-TestMostresearchfortheinternalpilotdesignhasbeenfocusedonthetwo-samplettest.WittesandBrittain[ 2 ]useda\nave"ttestwiththerestrictedrule.Itwascalled\nave"becausethedistributionoft-statisticisnotmodiedtoaccountfortherandomnessinN+.BirkettandDay[ 3 ]examinedthesametestwiththeunrestrictedrule.Bothreportedthatthenavet-testinatestheTypeIerrorrate,especiallywhenthesizeoftheinternalpilotsampleissmall(n1<20).TheinationoftheTypeIerrorrateiscausedbythesystematicunderestimationofthevariance[ 23 ].VariousapproacheshavebeenproposedtocontroltheTypeIerrorrateusingalternativevarianceestimators[ 25 { 27 ],adownweighted-level[ 23 ],oranadjustedcriticalvalue[ 28 ]. 2.2.1NaveApproachDuetothelossofinformationfornotusingalltherelevantdatatoestimatevariance,Stein'sprocedureisnotwellacceptedinepidemiologicalstudiesorclinicaltrials.WittesandBrittain[ 2 ]modiedStein'sprocedureandintroducedtheinternalpilotdesignin 19

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1990.Theyproposedtousethe\nave"ttestforhypothesistesting.TheteststatistictnaiveusesallN+observationstocalculatethesamplemeanandestimatethevariance tnaive=^ p 4s2=N+:(2{4)Thepooledvariances2ofthenalsample(N+observations)isestimatedbytheaverageofthewithingroupsamplevariances,s2=(s2X+s2Y)=2.Thenullisrejectedwhentnaive>tt;+,where+=N+)]TJ /F1 11.955 Tf 11 0 Td[(2isthedegreesoffreedomofthenalsample.Thereasonitwascalled\nave"becausethestandardteststatisticandtdistributionforthexedsamplesizedesignisadopted,ignoringthefactthatthenalsamplesizeN+isarandomvariable.WittesandBrittaincomparedtheperformanceofxeddesignandinternalpilotdesignunderacircumstancethattheinitialtotalsamplesizeis86,halfoftheinitialsampleisusedtoformtheinternalpilotsample(=0:5),andN+isdeterminedbytherestrictedrule(i.e.,N+n0).TheirresultsshowedtheinternalpilotdesigngainssubstantialpowerwhileonlyslightlyinatingtheTypeIerrorratewhenthepriorguessoferrorvariance20underestimatesthetruevariance2.Whenanoverestimated20ispresented,becausethenalsamplesizecannotbesmallerthann0,theinternalpilotdesignyieldsexpectednalsamplesizeE(N+)n0.Inaddition,when20>2,theinternalpilotdesignwithrestrictedrulehaspowerveryclosetothatofthexeddesign,andleadstooverpoweredstudies.BirkettandDay[ 3 ]examinedtheimpactofstage1samplesizeontheperformanceoftheinternalpilotdesign.Theyconsideredanextremecaseofnoinitialsamplesizecalculation,whichisequivalenttoausualinternalpilotdesignwiththeunrestrictedruleofselectingnalsamplesize.Theyconcludedthesizeofstage1samplehasnoimportanteectontheTypeIerrorrateandpoweraslongasstage1samplehas20degreesoffreedomtoestimatethevariance(11subjectspergroupforatwo-samplestudy).However,incontrasttotheexactprotectionachievedbyStein'sprocedure,internalpilotdesignusingnaveapproachwouldresultinaninatedTypeIerrorrateduetoE(s2)<2 20

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[ 23 25 27 ].Bothrestrictedandunrestrictedruleofselectingnalsamplesizewouldleadtothesystematicalunderestimationof2.TheunderestimationofthetruevariancecanbeprovedusingtheHelmerttransformation[ 25 29 ].LetX1;X2;:::;Xni:i:dN(;2),theHelmerttransformationisappliedtoobtainan1vectorZ=[Z1Z2:::Zn]0,withZk=p k=(k+1)(Xk)]TJ /F3 11.955 Tf 12.14 0 Td[(Xk+1);k=1;:::;n)]TJ /F1 11.955 Tf 12.14 0 Td[(1,andZn=p nXn.Here,Xk=(1=k)Pki=1Xi.Inmatrixnotation,ZcanbeexpressedasZ=HX,where H=26666666666666641 p 2)]TJ /F4 7.97 Tf 14.02 4.7 Td[(1 p 200:::::::::01 p 61 p 6)]TJ /F9 11.955 Tf 9.3 13.68 Td[(q 2 30:::::::::0........................1 p k(k+1)1 p k(k+1)1 p k(k+1):::1 p k(k+1))]TJ /F9 11.955 Tf 9.3 13.47 Td[(q k k+1:::0........................1 p n1 p n1 p n1 p n1 p n::::::1 p n3777777777777775:(2{5)TheHelmertmatrixHisorthonormal.SinceXN(1n;2In),thetransformedZN(H1n;2HH0).Specically,ZN[(00:::p n)0;2In]withZi(i=1;:::;n)independentnormalrandomvariables.Furthermore,theEuclideanlengthjjZjj2=(HX)0HX=X0H0HX=X0X=jjXjj2.Inturn,Pn)]TJ /F4 7.97 Tf 6.59 0 Td[(1i=1Z2i=jjZjj2)]TJ /F3 11.955 Tf 10.12 0 Td[(Z2n=Pni=1X2i)]TJ /F3 11.955 Tf 10.13 0 Td[(nX2n=(n)]TJ /F1 11.955 Tf 12.42 0 Td[(1)s2n,wheres2nisthesamplevariance.Infact,fork=2;:::;n,wecanalwayswritePk)]TJ /F4 7.97 Tf 6.59 0 Td[(1i=1Z2i=(k)]TJ /F1 11.955 Tf 11.96 0 Td[(1)s2k,wheres2kisthesamplevarianceoftherstkvariablesofZ.Inthesettingoftheinternalpilotdesign,thestage1andnalwithingroupsamplevariancescanbeexpressedass2X1=(n1=2)]TJ /F1 11.955 Tf 13.09 0 Td[(1))]TJ /F4 7.97 Tf 6.59 0 Td[(1Pn1=2)]TJ /F4 7.97 Tf 6.59 0 Td[(1i=1Z2i,ands2X=(n+=2)]TJ /F1 11.955 Tf -418.84 -23.91 Td[(1))]TJ /F4 7.97 Tf 6.58 0 Td[(1Pn+=2)]TJ /F4 7.97 Tf 6.59 0 Td[(1i=1Z2i.Thewithingroupvariancess2Y1ands2Yhavethesimilarexpression,andthepooledvarianceestimatorss21=(s2X1+s2Y1)=2,ands2=(s2X+s2Y)=2.Theconditional 21

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expectationE(s2Xjs2X1;s2Y1)=E(n+=2)]TJ /F1 11.955 Tf 11.96 0 Td[(1))]TJ /F4 7.97 Tf 6.59 0 Td[(1(n1=2)]TJ /F1 11.955 Tf 11.95 0 Td[(1)s2X1+n+=2)]TJ /F4 7.97 Tf 6.59 0 Td[(1Xi=n1=2Z2is2X1;s2Y1=(n1)]TJ /F1 11.955 Tf 11.95 0 Td[(2)s2X1+n22 n+)]TJ /F1 11.955 Tf 11.95 0 Td[(2=2+(n1)]TJ /F1 11.955 Tf 11.95 0 Td[(2)(s2X1)]TJ /F3 11.955 Tf 11.96 0 Td[(2) n+)]TJ /F1 11.955 Tf 11.95 0 Td[(2: (2{6)Here,weconditiononboths2X1ands2Y1toxthesamplesizen+,buts2Y1itselfhasnoimpactons2X.TheexpectationofwithingroupvarianceE(s2X)=EE(s2Xjs2X1;s2Y1)=2+(n1)]TJ /F1 11.955 Tf 11.96 0 Td[(2)Ehs2X1)]TJ /F3 11.955 Tf 11.95 0 Td[(2 n+)]TJ /F1 11.955 Tf 11.95 0 Td[(2i=2+(n1)]TJ /F1 11.955 Tf 11.96 0 Td[(2)Covh(s2X1)]TJ /F3 11.955 Tf 11.96 0 Td[(2);1 n+)]TJ /F1 11.955 Tf 11.96 0 Td[(2i=2+(n1)]TJ /F1 11.955 Tf 11.96 0 Td[(2)Covs2X1;1 n+)]TJ /F1 11.955 Tf 11.95 0 Td[(2: (2{7)SinceE(s2Y)hasthesimilarform,wecanobtainE(s2)=E(s2X+s2Y)=2=2+1 2(n1)]TJ /F1 11.955 Tf 11.96 0 Td[(2)hCovs2X1;1 n+)]TJ /F1 11.955 Tf 11.96 0 Td[(2+Covs2Y1;1 n+)]TJ /F1 11.955 Tf 11.95 0 Td[(2i=2+(n1)]TJ /F1 11.955 Tf 11.95 0 Td[(2)Covs21;1 n+)]TJ /F1 11.955 Tf 11.96 0 Td[(2: (2{8)Alargevalueofs21willleadtoalargesamplesizen+,soCov(s21;1 n+)]TJ /F4 7.97 Tf 6.59 0 Td[(2)<0,andE(s2)<2.Becauseofthesystematicalunderestimationoftheunderlyingvariance,thenaveapproachinatestheTypeIerrorrate.However,akeyfocusintheareaofclinicaltrialistoprovidegoodprotectionoftheTypeIerrorrate.ICHguidelineE9StatisticalPrinciplesforClinicalTrials[ 30 ]particularlyemphasizedtheimportanceofexplainingthepotentialinationoftheTypeIerrorrateforthesamplesizeadjustmenttechniques.InordertoacquirebettercontroloftheTypeIerrorrate,severaladjustedapproacheshavebeenproposed. 22

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2.2.2AdjustedApproachWittesetal.[ 23 ]adoptedthenaveteststatisticandtdistribution.TheyusedtripleintegrationtonumericallycalculatetheTypeIerrorrateandpower.Theauthorscomparedanalyticalresultstosimulationsandreportedinaccuracyoftheanalyticalresultswhenn2issmall.Extensivenumericalresultswerepresentedtoexaminetheeectofrestrictedandunrestrictedruleusingtripleintegration(for<0:9)andsimulations(for0:9).TheirresultsshowedtherestrictedruleprovidesadequatepowerwithlittleinationoftheTypeIerrorrate,whiletheunrestrictedrulecouldleadtoconsiderableTypeIerrorrateinationespeciallywhenthesizeofstage1sampleissmall(=0:25).TocontroltheTypeIerrorrate,theauthorssuggesteddecreasingthenominal-level.Basedontheirsimulations,restrictedrulewith=0:047or0.048,andunrestrictedrulewith=0:038areconsideredappropriateforsmallstage1sample(=0:25)whenthetargett=0:05.Theauthorsalsoassesseddierentsizesofthestage1sample,i.e.,n1=n0with=0:1;0:25;0:5;0:75;0:9;1,andclaimedapracticalchoiceofisbetween0.25and0.75.DenneandJennison[ 28 ]usedthenaveteststatisticbutamodieddegreesoffreedom,DJ=n1+(N+)]TJ /F3 11.955 Tf 12.63 0 Td[(n1))]TJ /F1 11.955 Tf 12.63 0 Td[(2,forthetdistribution.Thenullisrejectedwhentnaive>tt;DJ.Here,(01)isusedtoaccountfortherandomnessinthenalsamplesize.When=0or1,theproposedmethodhasthesamedegreesoffreedomasStein'sprocedureorthenaveapproach.TheauthorsconductedallsimulationsusingtheunrestrictedruleofselectingN+.Theirresultssuggestedchoosing=1=8tominimizetheabsolutemeandierencebetweentheactualandtargetedTypeIerrorrate.Theauthorsdemonstratedthebetterperformanceoftheproposedmethodascomparedwiththenaveapproachinawiderangeofsimulationparametersettings.Inaddition,theproposedmethodachievesgoodcontroloftheTypeI,IIerrorratesevenforstage1samplesizeassmallas5,andcanbegeneralizedtopairedt-testsetting.However,thechoiceof=1=8isonlybasedonsimulationresultswithoutanalyticaljustication. 23

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ProschanandWittes[ 25 ]usedthet-testwithanalternativevarianceestimators2PWbasedonallN+observationss2PW=~2X+~2Y 2~2X=s2X1+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()^2X~2Y=s2Y1+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()^2Y^2X=(N+)]TJ /F1 11.955 Tf 11.95 0 Td[(2)s2X)]TJ /F1 11.955 Tf 11.96 0 Td[((n1)]TJ /F1 11.955 Tf 11.95 0 Td[(2)s2X1=(N+)]TJ /F3 11.955 Tf 11.96 0 Td[(n1)^2Y=(N+)]TJ /F1 11.955 Tf 11.95 0 Td[(2)s2Y)]TJ /F1 11.955 Tf 11.95 0 Td[((n1)]TJ /F1 11.955 Tf 11.96 0 Td[(2)s2Y1=(N+)]TJ /F3 11.955 Tf 11.95 0 Td[(n1)=(n1)]TJ /F1 11.955 Tf 11.95 0 Td[(2)=(n0)]TJ /F1 11.955 Tf 11.96 0 Td[(2); (2{9)wheres2X1ands2Y1arethewithingroupsamplevariancesofthestage1sample(n1=2observationspergroup),s2Xands2Yarethewithingroupsamplevariancesofthenalsample(N+=2observationspergroup).Theauthorsproveds2PWisunbiased,anditsvarianceissmallerthanthevarianceofs21.Notethats21isthevarianceestimatorusedinStein'sprocedure.ProschanandWittes[ 25 ]proposedtousetheteststatistic tPW=^ p 4s2PW=N+:(2{10)ThenullhypothesisisrejectedwhentPW>tt;n0)]TJ /F4 7.97 Tf 6.59 0 Td[(2.TheauthorsprovedtheTypeIerrorratewouldnotexceedthetargett.Simulationresultsshowedtheproposedmethodisrobustagainstsmallstage1sample(6subjectspergroup),andchangeinthetruevarianceduringthestudy.Ingeneral,theproposedmethodhasbetterperformancethanStein'sprocedureandthenaveapproachintermsofcontrollingtheTypeIerrorrateandmaintainingpower.However,themethodisapplicableonlywhenthenalsamplesizeisselectedbasedontherestrictedrule,i.e.,N+n0. 24

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KieserandFriede[ 26 ]proposedtouseaslightlymodiedvarianceestimators2KFtoenablethecalculationofthedensityoftheteststatistictKFs2KF=s n1)]TJ /F1 11.955 Tf 11.95 0 Td[(2 N+)]TJ /F1 11.955 Tf 11.95 0 Td[(4s21+N2)]TJ /F1 11.955 Tf 11.96 0 Td[(2 N+)]TJ /F1 11.955 Tf 11.95 0 Td[(4s22tKF=^ p 4s2KF=N+: (2{11)Here,s21ands22arepooledsamplevariancesofstage1sample(n1observations)andstage2sample(N2observations),respectively.ThenullisrejectedwhentKF>tt;N+)]TJ /F4 7.97 Tf 6.59 0 Td[(4.SincethedensityoftKFcanbederived,theauthorsobtainedexplicitexpressionoftheactualTypeIerrorrate.Allsimulationsinthispaperwerebasedontheunrestrictedruleofselectingnalsamplesize.Simulationresultsshowedforaparticularn1,thereexistsauniquenalsamplesizeN+thatmaximizestheactualTypeIerrorrate.Theuniquemaximumallowstheadjustmentof-leveltocontroltheactualTypeIerrorratenotexceedthetarget.Theauthorsfurthersuggestedthatafterstage1(internalpilotstage),theadjustedsamplesizeNadjshouldbecalculatedusing100(1)]TJ /F3 11.955 Tf 12.53 0 Td[()%uppercondencelimitofs21.Bydoingthis,theprobabilityofpowernolessthanthetargetedlevelisguaranteedtobeatleast1)]TJ /F3 11.955 Tf 12.31 0 Td[(.However,Proschan[ 29 ]arguedthereisnoneedtouseamodiedtKFwiththeonlypurposeofobtainingthedistributionoftheteststatistic,sincetheexactdistributionoftnaivecanbeeasilyattainedbyusingtheHelmerttransformation.Miller[ 27 ]extendedtheworkofWittesetal.[ 23 ]andderivedasharptheoreticalboundofE(s2))]TJ /F3 11.955 Tf 11.98 0 Td[(2.Inthepaper,Millerincorporatedapracticalconsiderationthatsomesubjectsmayalreadyenrolledintothestudywhentheresultsobtainedfromthestage1sampleareavailable.SothenalsamplesizeisdeterminedbyN+=max(Nadj;n1+n2min),wheren2minisarbitraryandneedstobepre-specied.Whenn2min=0,theruleisexactlytheunrestrictedruleofselectingthenalsamplesize,whiletherulebecomestherestrictedrulewhenn2min=n0)]TJ /F3 11.955 Tf 11.96 0 Td[(n1.Miller[ 27 ]provedthebiasofs2hastightbounds )]TJ /F3 11.955 Tf 13.15 8.08 Td[(n1)]TJ /F1 11.955 Tf 11.95 0 Td[(2 n1)]TJ /F1 11.955 Tf 11.95 0 Td[(4s21 NadjE(s2))]TJ /F3 11.955 Tf 11.96 0 Td[(20;(2{12) 25

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andproposedan\almost"unbiasedvarianceestimators2MwithanadditivecorrectiontoconstructtheteststatistictMs2M=8><>:s2+(n1)]TJ /F1 11.955 Tf 11.96 0 Td[(2)s21[(n1)]TJ /F1 11.955 Tf 11.95 0 Td[(4)Nadj];ifN+=Nadjs2;ifN+=n1+n2mintM=^ p 4s2M=N+: (2{13)ThenullisrejectedwhentM>tt;N+)]TJ /F4 7.97 Tf 6.59 0 Td[(2.Simulationresultsshowedthet-testbasedons2MyieldstheTypeIerrorrateclosetothetargetedlevel.Moreover,theauthorcomparedstandarddeviationsofs2,s21,s22,s2PW,ands2M.Amongthesevarianceestimators,s2alwayshasthesmalleststandarddeviation,butthemodieds2Mhasalmostthesamestandarddeviationass2.Standarddeviationofs2PWisslightlyhigherthanthatofs2ands2Mwhentheunderlying2isrelativelysmall(<10),butincreaseddramaticallywhen2islarge. 2.3InternalPilotDesignforUnivariateOutcomesHithertowehavereviewedpublishedarticlesoninternalpilotdesignfortwosamplet-testsetting.Inthissection,wediscussresearchworkhavebeendonetoapplyinternalpilotdesigntomulti-armstudies,suchasone-wayANOVA,withunivariateoutcomes.CoeyandMuller[ 18 19 31 ]appliedtheinternalpilotdesigntothegenerallinearunivariatemodel,ofwhichthetwo-samplet-testisaspecialcase.Theyproposedusingaboundingapproach,whichadjuststhecriticalvaluetocontroltheTypeIerrorratetobenogreaterthanthetargetedlevel.Theyprovidedextensiveenumerationresultstoexaminetheeectsoftherestrictedandunrestrictedrule,thesizeoftheinitialsampleandinternalpilotsample.TheirresultssuggestedtheboundingapproachprotectspowerandcontrolstheTypeIerrorrate,especiallywhenthesamplesizeissmallormoderate.TheapproachproposedbyCoeyandMuller[ 18 ]requiresxedpredictors,i.e.,predictorscontrolledbythedesignofthestudy.Thegenerallinearunivariatemodelwithnormalerror,eN(0;2I),ispartitionedasbelowtorepresentthetwo-stagenatureof 26

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theinternalpilotdesign266666664y1(n11)y2(N21)377777775=266666664X1(n1q)X2(N2q)377777775++266666664e1(n11)e2(N21)377777775y+(N+1)=X++(N+q1)+e+(N+1): (2{14)Thedesignmatricesindierentstagesoftheinternalpilotdesignareassumedtohavethesamerank,i.e.,rank(X1)=rank(X2)=rank(X+)=r.Therankofthedesignmatricesneedstobesmallerthann1inordertoacquireanestimablevarianceestimateafterstage1^21=y01[In1)]TJ /F5 11.955 Tf 11.96 0 Td[(X1(X01X1))]TJ /F5 11.955 Tf 7.08 -4.93 Td[(X01]y1=11=n1)]TJ /F3 11.955 Tf 11.96 0 Td[(r: (2{15)Afterstage1(internalpilotstage),thenalsamplesizeN+isdeterminedbasedon^21andtheruleofnalsamplesize.Inthestage2,N2=N+)]TJ /F3 11.955 Tf 12.32 0 Td[(n1subjectsareenrolled.Attheendofstudy,thenalvarianceestimate^2obtainedfromallN+observationsisusedtoformtheteststatistic.ThegenerallinearhypothesisassociatedwiththegenerallinearunivariatemodelisH0:=0.Withoutlossofgenerality,asimpliednullhypothesisH0:=0canbeassumed.Theparameter(a1)isdenedas=C,whereCisthefullrankbetween-subjectcontrastmatrixwithdimensionsaqandranka.CoeyandMuller[ 18 ]proposedusingtheclassicteststatisticFCM(i.e.,forthexedsamplesize)to 27

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testthehypothesisFCM=SSH+=a SSE+=+SSH+=^0M)]TJ /F4 7.97 Tf 6.58 0 Td[(1+^SSE+=+^2=y0+[IN+)]TJ /F5 11.955 Tf 11.96 0 Td[(X+(X0+X+))]TJ /F5 11.955 Tf 7.08 -4.93 Td[(X0+]y+M+=C(X0+X+))]TJ /F5 11.955 Tf 7.08 -4.93 Td[(C0+=n+)]TJ /F3 11.955 Tf 11.96 0 Td[(r: (2{16)ForaparticularrealizationofN+=n+,thenullisrejectedwhenFCM>Ft;a;+.Althoughtheteststatisticandcriticalvalueforthexeddesignareadopted,thedependenceofFCMontherealizationofN+makesitsdistributionmorecomplicated.Thiscomplicationisduetotherandomnessinthedenominatoroftheteststatistic.Thenumerator,justasforthexeddesign,followsanoncentralchi-squaredistributionSSH+ 22(a;w)w=0M)]TJ /F4 7.97 Tf 6.59 0 Td[(1 2=d 2d=0M)]TJ /F4 7.97 Tf 6.58 0 Td[(1: (2{17)Thenoncentralitywisassociatedwithknownmeanandvariance,whilew(n+)representsthenoncentralityforaparticularrealizationofN+=n+toensurethetargetTypeIerrorrateandpower.Sincethenon-centralFdistributionisstochasticallyincreasingwiththenoncentrality,thecumulativedistributionfunction(CDF)ismonotonicallydecreasingwithw.Thismonotonicityensuresthelargestvalueof^21doesnotexceedd(n+)=w(n+)whenN+n+.Becausethereisnorandomnessinthestage1sample,accordingtothelinearmodeltheory,1^21=22(1).Theauthorsexpressedtheprobabilityofaparticularrealization 28

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ofthenalsamplesizebyboundingtherangeofstage1varianceestimate^21Pr(N+=n+)=Prhd(n+)]TJ /F1 11.955 Tf 11.96 0 Td[(1) w(n+)]TJ /F1 11.955 Tf 11.96 0 Td[(1)<^21d(n+) w(n+)i=Prhd(n+)]TJ /F1 11.955 Tf 11.96 0 Td[(1) w(n+)]TJ /F1 11.955 Tf 11.96 0 Td[(1)1 2<1^21 2d(n+) w(n+)1 2i=Pr[qL<1^21 2qU]: (2{18)Afterobtainingthedistributionofthenalsamplesize,CoeyandMuller[ 18 ]furtherderivedthedistributionofthenalteststatistic.ForaxedN+=n+,thenalvarianceestimate^2canbepartitionedintotwosources.Oneisfromthestage1(internalpilot)sample,theotherisfromtheorthogonalcomplementofthestage1sample^2=^21+(^2)]TJ /F1 11.955 Tf 12.57 0 Td[(^21)+^2 2=1^21 2++^2)]TJ /F3 11.955 Tf 11.95 0 Td[(1^21 2SSE+ 2=SSE1 2+SSE+)]TJ /F1 11.955 Tf 11.96 0 Td[(SSE1 2: (2{19)Thersttermontheright-handside,SSE1=2,isadoubly-truncatedchi-squarevariablewithdegreesoffreedom1,lowerandupperlimitsqLandqU,whichisdenotedasSSE1=22T(1;qL;qU).Theauthorsproved(SSE+)]TJ /F1 11.955 Tf 12.46 0 Td[(SSE1)=22(n2),andthesetwosourcesofvarianceareindependent.Hence,conditiononN+=n+,theconditionalprobabilityofnotrejectingthenull,denotedbyI(n+),canbewrittenasI(n+)=PrfFCMfcrit(n+)N+=n+g=PrSSH+=a SSE+=+fcrit(n+)qL
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asaweightedsumofindependentchi-squarevariablesI(n+)=Pr(SSH+=a2 SSE1+(SSE+)]TJ /F1 11.955 Tf 11.96 0 Td[(SSE1)=(+2)fcrit(n+)q(n+))=Pr(SSH+=a2 [q(n+)+SSE+)]TJ /F4 7.97 Tf 6.59 0 Td[(SSE1 2]=+fcrit(n+)q(n+))=Pr+ afcrit(n+)SSH+ 2)]TJ /F1 11.955 Tf 13.15 8.08 Td[(SSE+)]TJ /F1 11.955 Tf 11.96 0 Td[(SSE1 2q(n+)q(n+): (2{21)Themarginal(unconditional)distributionoftheteststatisticFCMcanthenbeobtainedbyapplyingadoubleintegration.TheinnerintegralimplementedDavies'salgorithm[ 32 ]tocalculatetheCDFofweightedsumofindependentnon-centralchi-squarevariablesconditioningonthedoubly-truncatedchi-squarevariable.TheouterintegralintegratedoverallpossiblerealizationsofN+.CoeyandMuller[ 18 ]comparedresultscomputedfromtheanalyticalmethodstothesimulationsusingtheexampleconsideredbyWittesandBrittain[ 2 ].ThecloseagreementsuggestedtheproposedanalyticalmethodshavegoodaccuracyincomputingTypeIerrorrateandpowerforaninternalpilotdesignwithunivariateoutcomes.Extensiveanalyticalresultswerepresentedtoexaminetheperformanceofinternalpilotdesigninawiderangeofdesignspecications.Theinternalpilotdesignhasaconsiderableimprovementinpowerperformancecomparedwiththexeddesignforallthescenariosconsidered.However,theinationofTypeIerrorratewasobservedfortheunrestrictedrulewhenthepriorguessoverestimatesthetruevariance.Forsmallsamples(n0=20),bothrestrictedandunrestrictedruleleadtoTypeIerrorrateabovethetargetlevel,buttheinationismoreprominentfortheunrestrictedrule.Inaddition,theauthorsassessedtheeectofstage1samplesizebychoosingdierentproportions,i.e.,=0:25;0:5and0:75.Ingeneral,therestrictedrulecontrolstheTypeIerrorratewellfordierentstage1samplesizes.TheauthorsalsoarguedagainstBirkettandDay[ 3 ]forkeeping20degreesoffreedominthestage1sampletoprotectlevel.Simulationresultsshownevenforamoderatesamplesize(n0=46;n1=24),theTypeIerrorratecouldbeashighas0.057(targeted 30

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levelis0.05)whenadoptingtheunrestrictedrule.Finally,practicalresourcelimitationswereconsideredbyimposingamaximumallowablesamplesizeforthestudy.Theresultssuggestedestablishinganupperboundofthesamplesizewouldreducepower,especiallywhenthepriorguessunderestimatesthetruevariance.CoeyandMuller[ 31 ]extendedtheirworkandshowedFCMisthelikelihoodratioteststatisticfortheinternalpilotdesign.However,FCMdoesnotfollowanFdistributiondueto^2systematicallyunderestimatethetruevariance2,justasinthetwo-samplet-testsituation.Thenalvarianceestimate^2isaweightedaverageoftwounbiased2estimators ^2=1 +^21+N2 +^2;(2{22)where ^2=SSE+ +;^21=SSE1 1;and^2=SSE+)]TJ /F1 11.955 Tf 11.95 0 Td[(SSE1 N2:(2{23)Iftheweightsarexed,^2wouldalsobeanunbiasedestimateof2.Buttherstweight1=+dependsonthevalueof^21.Thelargerthevalueof^21,thelargerthenalsamplesizen+(andthecorrespondingdegreesoffreedom+=n+)]TJ /F3 11.955 Tf 12.79 0 Td[(r)andthelessweight.Sincelargevaluesof^2aredownweighted,thetrue2isunderestimated.InordertoalleviatetheTypeIerrorrateinationduetothebiasedestimator^2,CoeyandMuller[ 19 ]proposedusingtheboundingapproach.TheboundingapproachadoptsthesameteststatisticFCM,buthasanincreasedcriticalvaluesuchthatthemaximumTypeIerrorrateiscontrolledatorbelowthetargetlevel.Theauthorsprovidedevidencetosubstantiatethehypothesisthatthereisauniquemaximumoffunctionallydependentontheratioofactualandpriorguessofthevariance,i.e.,2=20.Theadjustedlevel(adj)isestablishedtoboundmaxatthetargetleveloft.Theboundingapproachisconsideredconservativebecauseitonlyfocusesonthecontrolofmax,whichmayleadtotheactualTypeIerrorratesmallerthanthetarget.Accordingtothesimulationresults,theboundingapproachachievestightcontroloftheTypeIerrorratewhilemaintainingthegoodpowerperformanceofaninternalpilotdesign.Coeyet 31

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al.[ 20 ]providedfaster,morestableandaccuratecomputationmethodsfortheboundingapproachbasedonsimpliedformulasfortheexactTypeIerrorrateandpower. 2.4InternalPilotDesignforRepeatedMeasuresResultsfortheinternalpilotdesignarescarceintheliteraturewhenrepeatedmeasuresarepresent.ZuckerandDenne[ 33 ]developedaprocedureforthetwo-armrepeatedmeasuresstudiesunderthegenerallinearmixedmodelframework.Theirresultsarerestrictedtohighlystructuredcovariancepatterns.CoeyandMuller[ 4 ]examinedcovariancemisspecicationforxedandinternalpilotdesignsforthegenerallinearmultivariatemodel.Theyshowedthattheinternalpilotdesignpreservespowerbetterthanthexeddesignwhenthecovariancematrixismisspecied.TheyalsopointedoutthattheinternalpilotdesignwithrepeatedmeasurescanleadtotheTypeIerrorrateination,ashasbeenobservedintheunivariatesetting.AllinternalpilotresultsreportedbyCoeyandMuller[ 4 ]werebasedonsimulations.Usingsimulationsfortheinternalpilotpowercalculationscanbecomputationallyintensiveduetonestednatureofthepoweranalyses.Specically,thesecondpoweranalysisisperformedbasedonthedatacollectedattheinternalpilotstage,whosesizewascalculatedintheinitialpoweranalysis.Thecomputationalburdenincreaseswhenmakinginferenceaboutrepeatedmeasures.Theworkherellsthegapintheliteraturebyprovidinganalyticalandnumericalresultsfortheinternalpilotdesignwithrepeatedmeasures.InChapter 3 ,theinternalpilotdesignisextendedtostudieswithrepeatedlymeasuredGaussianoutcomes.ThedistributionsofthenalsamplesizeandtheUnivariateApproachtoRepeatedMeasures(UNIREP)teststatisticarederived.InChapter 4 ,theboundingapproachproposedbyCoeyandMuller[ 19 ]isextendedtolongitudinalinternalpilotstudiestocontroltheinationoftheTypeIerrorrate.InChapter 5 ,wediscussstrengthsandlimitationsofthecurrentworkandprovideareasoffurtherresearch.Mathematicalproofs,supplementaryresults,anddetailsofsimulationsandmotivatingexampleareincludedintheAppendix. 32

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CHAPTER3INTERNALPILOTDESIGNFORREPEATEDMEASURES 3.1MotivatingExampleHumanimmunodeciencyvirustype1(HIV-1)infectioncausesdepletedCD4+Tcellsandalteredimmunefunction.Withtheadventofantiretroviraltherapy(ART),HIV-1infectionhasbecomeamanageablechronicdisease.ApplicablethresholdoftheCD4+countinstartingtheARTinyouthlivingwithHIV(YLWH)hasgraduallyincreasedinthelastveyears[ 34 ].AlthoughthebenetsofearlyARThasbeenestablished,byseveralrandomizedstudies,inreducinglong-termHIV-relatedevents[ 35 ],theimpactofearlytreatmentoninammatorybiomarkersremainsunclear.Astudy,sponsoredbytheAdolescentTrialsNetwork(ATN)forHIV/AIDS,wasconductedtoperiodicallymonitortheclinicalandimmuneresponsesoftheYLWHoverthreeyears.Onehundredadolescentsaged18-24yearswithbehaviorallyacquiredHIV-1infectionandCD4cellcountabove350cells/mm3wereenrolled.A3:1randomizationwasusedtoallocate75HIV-infectedparticipantstostartARTimmediately(earlytreatment),andtheremaining25participantstodeferARTuntiltheirCD4cellcountdroppedbelow350cells/mm3orwhentheydevelopedconditionsrequiredtherapy(standardofcare).Arecentmanuscriptinpreparationreportedacross-sectionalanalysisofasetof19inammatorybiomarkersattheendofthestudy(year3).ResultsshowedbiomarkerssolubleCD27,CD14andCD163areimportantcontributorstodistinguishHIV-infectedoutcomegroupsseparatedbyARTandvirus.AreplicationstudyisplannedtoinvestigatetheimpactofearlytreatmentonthesethreebiomarkerswithinYLWH.Toplanareplicationstudy,informationaboutmeandierencesandtheircorrelationsarerequired.Figure 3-1 displaystheobservedmeansandcorrelationsofthethreecontributingbiomarkersfor76YLWHwithcompletedatafromtheexistingstudy.Theexistingdatawillbeusedtoguidethestudyplanningandpoweranalysis.Theobservedmeans(solidlines,Figure 3-1 A)areusedtodeterminethetreatmentdierences 33

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ofinterest(dashedlines).Theobservedcorrelationmatrices(Figure 3-1 B)areusedtopermitexaminingtheeectsofcovariancemisspecication.Wewillrevisitthisexampleinsection 3.5 ,toplananinternalpilotstudyinvestigatingtheeectsofearlyARTonthelevelsofinammatorybiomarkers. 3.2ExistingResultsfortheUNIREPTests 3.2.1NotationWefollowthenotationusedinMullerandStewart[ 6 ]todenethegenerallinearmultivariatemodelasY(Np)=XB(Nqp)+E(Np),withsecondaryparameters(ab)=C(aq)B(qp)U(pb).DimensionsofYindicatethereareNsubjects(orindependentsamplingunits)andprepeatedmeasuresforeachsubject.Modeldegreesoffreedomaredenotedbye=N)]TJ /F1 11.955 Tf 12.06 0 Td[(rank(X).EachrowofEisassumedtoindependentlyandidenticallydistributedwithrowi(E)0Np(0;).Subsequently,YfollowsamatrixGaussiandistributionwithYNN;p(XB;IN;)[ 6 ].MatricesCandUcontainthebetween-subjectandwithin-subjectcontrasts,respectively.EachrowofCcorrespondstoacontrastamongthepredictorsinX.EachcolumnofUcorrespondstoacontrastamongtheprepeatedmeasures,whichcollectivelydenesthetransformedresponsesYU=YU,withYUNN;b(XBU;IN;)and=U0U.ThegenerallinearhypothesisassociatedwiththegenerallinearmultivariatemodelisH0:=CBU=0.AtestablehypothesisrequiresbothCandUtobefullrank,i.e.,rank(C)=aqandrank(U)=bp,aswellasC=C(X0X))]TJ /F1 11.955 Tf 7.09 -4.34 Td[((X0X).Withoutlossofgenerality,0=0canbeassumed[ 4 ].Leastsquareestimators~B=(X0X))]TJ /F5 11.955 Tf 7.09 -4.34 Td[(X0Yand^=fY0[IN)]TJ /F5 11.955 Tf 12.62 0 Td[(X(X0X))]TJ /F5 11.955 Tf 7.08 -4.34 Td[(X0]Yg=eallowthecalculationsofthehypothesissumofsquaresmatrixSh=^=(^)]TJ /F5 11.955 Tf 12.73 0 Td[(0)0M)]TJ /F4 7.97 Tf 6.59 0 Td[(1(^)]TJ /F5 11.955 Tf 12.74 0 Td[(0)anderrorsumofsquaresmatrixSe=e^=eU0^U.Here^=C~BU,andthemiddlematrixisM=C(X0X))]TJ /F5 11.955 Tf 7.08 -4.33 Td[(C0.Theunivariateapproachtorepeatedmeasures(UNIREP)teststatisticisaratiooftraces,i.e.,Tu=[tr(Sh)=a][tr(Se)=e].ThereareveUNIREPtests:Box,Geisser-Greenhouse(GG),Chi-Muller(CM),Hunyh-Feldt(HF)anduncorrected(UN).All 34

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Figure3-1. DataandparametersrequiredtoplanalongitudinalstudyinvestigatingtheeectofearlytreatmentontheconcentrationsofsolubleCD27,CD14andCD163.A)Blacksolidlinesrepresenttheobservedmeanpatterns,andreddashedlinesrepresentthetreatmenteectsofinterestforplanning.B)heatmapoftheobservedcorrelationmatrices.C)heatmapofthecorrelationpatternsassumedtobetrue. 35

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veusetheteststatistic,Tu.ThedistributionofTudependsonthesphericityparameter=tr2()[btr(2)],whichmeasuresthespreadofeigenvalues.Whenalltheeigenvaluesofareequal,achievesitsupperbound(=1),andtheassumptionofsphericityismet.Conversely,minimumsphericityoccurswhenthereisonlyonenonzeroeigenvalue.Inthiscase,achievesitslowerbound(=1=b).Withsphericity(=1),teststatisticTufollowsanFdistributionexactly,TuF)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ab;be;wT,withthenoncentralityparameterwT=tr()=tr()and=()]TJ /F5 11.955 Tf 12.28 0 Td[(0)0M)]TJ /F4 7.97 Tf 6.58 0 Td[(1()]TJ /F5 11.955 Tf 12.28 0 Td[(0).Whenthesphericityassumptionisnotmet(6=1),theteststatisticTufollowsanFdistributionapproximatelywithTu:F[ab;be;wT].TheveUNIREPtestsdierbytheirapproachestoestimate,whichresultindierencesintheTypeIerrorrateandpower.TheBoxconservativetestusesthelowerboundof=1=b,whiletheUNtestusestheupperboundof=1.Whensphericity(=1)ispresent,theUNtestissizeanduniformlymostpowerful(amongsimilarlyinvarianttests).TheGG,HFandCMtestsestimatethesphericityparameterfromthedata.TheGGtestusestheMLE,^=tr2(^)=[btr(^2)].TheoriginalHFtestusesaratioofunbiasedestimators,~=(Nb^)]TJ /F1 11.955 Tf 12.8 0 Td[(2)=[b(e)]TJ /F3 11.955 Tf 12.8 0 Td[(b^)].Gribbinetal.[ 36 ]provedtheunbiasednessholdsonlywhenrank(X)=1andproposedusingtherank-adjustedestimator~r=[(e+1)b^)]TJ /F1 11.955 Tf 11.17 0 Td[(2]=[b(e)]TJ /F3 11.955 Tf 11.17 0 Td[(b^)]toapplytotherank(X)1cases.Inthecurrentchapter,theHFtestisreferredtotherank-adjustedHFtestusing~r.TheCMtestusesamultipleof~rwith~CM=~r(a)]TJ /F1 11.955 Tf 12.04 0 Td[(2)(a)]TJ /F1 11.955 Tf 12.03 0 Td[(4)=2aanda=(e)]TJ /F1 11.955 Tf 12.04 0 Td[(1)+e(e)]TJ /F1 11.955 Tf 12.04 0 Td[(1)=2.TheCMtestwasdevelopedtoextendtheUNIREPapproachtothehighdimension,lowsamplesizesetting.Theveestimatesareordered:1=b^~CM~r1.Consequently,theTypeIerrorrateandpowerareinthesameorder(BoxGGCMHFUN). 3.2.2PowerApproximationsforFixedDesignAtraditionalxeddesignhasonepowercalculation,whileaninternalpilotdesignhastwopowercalculations.Gribbinetal.[ 36 ]approximatedpoweroftheUNIREPtestsfor 36

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thexeddesignwhenthecovarianceisknown: P1)]TJ /F3 11.955 Tf 11.95 0 Td[(FFF)]TJ /F4 7.97 Tf 6.59 0 Td[(1F(1)]TJ /F3 11.955 Tf 11.96 0 Td[(;abe;bee);abn;be;w:(3{1)Here,andPdenotetheTypeIerrorrateandpower.TheveUNIREPtestshavedierentcriticalvaluesduetothedierentchoicesofthemultipliere,i.e.,e(Box)=1=b,e(GG)=E(^),e(HF)=E(~r),e(CM)=E(~CM),ande(UN)=1.Mulleretal.[ 5 ]proposedusingtherst-orderTaylorseriestoapproximatetheexpectationsofestimators.TheyprovedE(^)Etr2(^)bEtr(^2),withEtr2(^)=(1=e)[2tr(2)+etr2()]andEtr(^2)=(1=e)[(e+1)tr(2)+tr2()].ThesameTaylorseriesapproximationcanbeusedtoobtainE(~r)(e+1)Etr2(^))]TJ /F1 11.955 Tf 9.9 0 Td[(2Etr(^2)beEtr(^2))]TJ /F3 11.955 Tf 9.89 0 Td[(bEtr2(^),andE(~CM)=[(a)]TJ /F1 11.955 Tf 11.96 0 Td[(2)(a)]TJ /F1 11.955 Tf 11.96 0 Td[(4)=2a]E(~r).InEquation( 3{1 ),thenumeratordegreeoffreedom(abn)isafunctionoftheparametern==.Here=tr()=bistheaverageeigenvalueof,andisdenedas=[tr(2)+2tr(=a)][tr()+2tr(=a)].Thenoncentralityw=tr()=isafunctionofknowncovarianceandmeans. 3.2.3TheGeneralLinearMultivariateModelforanInternalPilotDesignThegenerallinearmultivariatemodelforaninternalpilotdesigncanbeformulatedtoreectitstwo-stagenature.Table 3-1 summarizesthenotation.Inthestudyplanningphase,researchersspecifyasetofmeansandabestguessofthecovariancematrix0(pp)tocalculatetheinitialsamplesizen0.Afractionoftheplannedtotalnumberofobservations(n1=n0;0<<1)isthencollectedtoformthestage1sample.Anestimateofthecovariancematrix^1isobtainedfromthestage1sampleandusedtoadjustthenalsamplesize.Equation( 3{2 )expressesthemodelwithsubscripts1,2and 37

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+,representingstage1,2andnalsample,respectively.266666664Y1(n1p)Y2(N2p)377777775=266666664X1(n1q)X2(N2q)377777775B++266666664E1(n1p)E2(N2p)377777775Y+(N+p)=X+B+(N+qp)+E+(N+p): (3{2)Similartotheunivariatesetting,weassumethedesignmatrixindierentstagesofaninternalpilotdesignhasthesamerank,i.e.,rank(X1)=rank(X2)=rank(X+)=r.Consequently,Es(X1)=Es(X2)=Es(X+),wheretheessencematrixEs(X+)isdenedbydeletinganyduplicaterowsfromthedesignmatrixX+.FollowingCoeyandMuller[ 18 ],werestrictattentiontothenalsamplesizebeinganintegermultipleofthenumberofrowsintheessencematrix.WithN+=n+,X0+X+=n+[Es(X+)0W+Es(X+)]andtr(+)=n+tr(Es).Here,Es(X+)0W+Es(X+)andEsarefunctionallyindependentofn+,withEs=()]TJ /F5 11.955 Tf 12.26 0 Td[(0)0fC[Es(X+)0W+Es(X+)])]TJ /F5 11.955 Tf 7.08 -4.34 Td[(C0g)]TJ /F4 7.97 Tf 6.59 0 Td[(1()]TJ /F5 11.955 Tf 12.25 0 Td[(0).GivenN+=n+,W+(rr)isthediagonalweightmatrixwithelements(n+;g=n+;g=1;:::;r),theproportionsofthesubgroupsamplesizestothetotalsamplesize.Forabalanceddesign,W+=(1=r)Ir.Wealsoassumethesubgroupsamplesizeproportionsarethesamebetweenstage1and2samples,i.e.,W1=W2=W+. 3.3NewApproximationsforanInternalPilotDesign 3.3.1DistributionoftheFinalSampleSizeInthissection,werstprovideanapproximationforthedistributionofthenalsamplesizeN+.Then,wederivetheapproximateconditionaldistributionofthenalteststatisticforaparticularrealizationofN+=n+.Finally,weobtainthemarginaldistributionofthenalteststatisticbyusingthelawoftotalprobability.AssummarizedinTable 3-1 ,estimatesfromstage1,2andnalsamplearedenotedbysubscript1,2and+,respectively.AllproofsareincludedintheAppendix. 38

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Table3-1. Notationforthegenerallinearmultivariatemodelwithinternalpilotdesign.Elementsaregroupedinto5categories-Design,Knownconstants,Samplesize,CovarianceandHypothesis. CategorySymbolDimensionDenition Design11Proportionforthestage1samplet11TargetTypeIerrorratePt11Targetpowert=CBtUabClinicallyrelevanteectsEs(X1)=Es(X2)=Es(X+)rqEssencematrixr11RankofdesignmatrixKnownCaqBetween-subjectcontrastmatrixconstantsUpbWithin-subjectcontrastmatrixa11RankofCb11RankofUSamplesizen011Initialsamplesizen1=n011Stage1samplesize1=n1)]TJ /F13 9.963 Tf 9.96 0 Td[(r11Stage1sampledegreesoffreedomN+11Finalsamplesize+=N+)]TJ /F13 9.963 Tf 9.96 0 Td[(r11FinalsampledegreesoffreedomN2=N+)]TJ /F13 9.963 Tf 9.97 0 Td[(n111Stage2samplesizeCovarianceppOutcomecovariancematrix0ppPriorguessofcovariancematrix^1(^+)ppStage1(nal)covariancematrixestimates=U0UbbHypothesiscovariancematrix^1(^+)bbStage1(nal)hypothesiscovariancematrixestimatesb1EigenvaluevectorofbbEigenvectormatrixof=tr2()[btr(2)]11Sphericityparameter^1(^+)11Stage1(nal)estimatesforGGtest~r1(~r+)11Stage1(nal)estimatesforrank-adjustedHFtest~CM1(~CM+)11Stage1(nal)estimatesforCMtestHypothesisBqpPrimaryregressionparameters~B1(~B+)qpStage1(nal)primaryparameterestimates=CBUabSecondaryparameters0abSecondaryparametersunderthenull^1(^+)abStage1(nal)secondaryparameterestimatesEsbbEssenceunscalednoncentrality^1(^+)bbStage1(nal)unscalednoncentralityestimatesSh1(Sh+)bbStage1(nal)hypothesissumofsquaresmatricesSe1(Se+)bbStage1(nal)errorsumofsquaresmatrices 39

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Inaninternalpilotstudy,observationsinthestage1sampleprovideanestimateofthecovariancematrix^1=U0^1U.Afterstage1,thesecondpoweranalysisisconductedtodeterminethenalsamplesizeN+.ForaparticularrealizationofN+=n+,wecanndthenoncentralityw(n+)whichsatises Pt1)]TJ /F3 11.955 Tf 11.95 0 Td[(FFF)]TJ /F4 7.97 Tf 6.59 0 Td[(1F(1)]TJ /F3 11.955 Tf 11.96 0 Td[(t;abe1;b+e1);abn;b+;w(n+);(3{3)withe1denedasinSection 3.2.2 butwithdegreesoffreedom1=n1)]TJ /F3 11.955 Tf 13.12 0 Td[(r.Here,tandPtdenotethetargetTypeIerrorrateandtargetpower.SincethecumulativedistributionfunctionofanFrandomvariableismonotonicallydecreasingintermsofthenoncentrality,auniquesolutionofw(n+)canbeobtainedgivenN+=n+.Inturn,(n+),afunctionofw(n+),canbedeterminedby (n+)=tr() w(n+)=n+tr(Es) w(n+):(3{4)WeproposeusingthedistributionoftheestimatortoapproximatethedistributionofN+.ThisapproachisinparalleltotheunivariatesettingpresentedbyCoeyandMuller[ 18 ],whichleadstoPrfN+n+g=Prf^212(n+)g.Intheunivariatetesting(b=1),(n+)=2(n+).ResultsinGribbinetal.[ 36 ]leadtousing~1=^1=~n1,aratioobtainedfromthestage1sample,toestimate==n.Thenumerator^1=tr(^1)=bisanunbiasedestimatorof,whilethedenominator~n1=[tr2(^1)+2tr(^1)tr(=a)]b[tr(^21)+2tr(^1=a)]isabiasedestimatorofn.Thedistributionof~1canbeapproximatedbyachi-squaredistributionwith~1~=2(~)usingtheSatterthwaiteapproximationanddegreesoffreedom~=(b1)^1=~n1.Forpowercalculation,weconsiderapproximating~by=(b1)E[^1]=E[~n1].AsdiscussedinSection( 3.2.2 ),E(^1)Etr2(^1)=fbEtr(^21)g.Wepropose 40

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approximatingE[~n1]by E[~n1]Etr2(^1)+2Etr(^1)tr(=a) bfEtr(^21)+2Etr(^1=a)g=Etr2(^1)+2tr()tr(=a) bfEtr(^21)+2tr(=a)g:(3{5)TheequalityinEquation( 3{5 )holdsbecauseGribbinetal.[ 36 ]provedtr(^1)andtr(^1)areunbiasedestimatorsoftr()andtr().Inaninternalpilotstudy,then1observationsofstage1sampleareusedtoobtainthecovarianceestimate^1,andsubsequentlycompute~1=^1=~n1.Ifthevalueof~1islessorequalto(n+),thenalsamplesizewouldbenogreaterthann+.HencethedistributionofN+canbeapproximatedbythedistributionof~1,namely PrfN+n+gPrf~1(n+)gPr2() n+tr(Es) w(n+):(3{6)ThediscretenatureofN+leadstoPrfN+=n+g=PrfN+n+g)]TJ /F1 11.955 Tf 20.59 0 Td[(PrfN+n+)]TJ /F1 11.955 Tf 11.96 0 Td[(1gPrf(n+)]TJ /F1 11.955 Tf 11.95 0 Td[(1)<~1(n+)g: (3{7)Consideringaspecialcasewhenthetestingisunivariate(b=1),weproveLemma 1 toshowEquation( 3{6 )reducestoEquation(17)inCoeyandMuller[ 18 ].TheresultshowsthattheproposedmethodsreducetotheexactdistributionderivedbyCoeyandMullerfortheunivariatecase. Lemma1. Forunivariatetesting,i.e.,rank(U)=b=1,=2,(n+)=2(n+),and=1=n1)]TJ /F3 11.955 Tf 11.96 0 Td[(r. 3.3.2ConditionalDistributionoftheFinalTestStatisticTheresultsthroughoutthissubsectionarederivedundertheconditionthatN+=n+,i.e.,whenthenalsamplesizeequalsaparticularrealization.LetI(n+)representthe 41

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probabilityoffailingtorejectH0givenN+=n+:I(n+)=PrfTufcrit(n+)jN+=n+g=Prtr(Sh+)=a tr(Se+)=+fcrit(n+)jN+=n+: (3{8)Herefcrit(n+)=F)]TJ /F4 7.97 Tf 6.59 0 Td[(1F)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 12.26 0 Td[(t;abe+;b+e+)isthecriticalvalueand+=n+)]TJ /F3 11.955 Tf 12.26 0 Td[(risthedegreesoffreedomofthenalsample.Forthexeddesign,Chietal.[ 14 ]provedthehypothesissumofsquaresmatrixShanderrorsumofsquaresmatrixSeareindependent.WeproveLemma 2 toshowavarietyofindependentpropertiesformatricesinaninternalpilotstudy. Lemma2. Foraninternalpilotdesign,givenN+=n+,thefollowingpairsofmatrices(andtheirtraces)areindependent:Sh+andSe+;Sh+andSe1;Sh+and(Se+)]TJ /F5 11.955 Tf 12.32 0 Td[(Se1);Se1and(Se+)]TJ /F5 11.955 Tf 11.95 0 Td[(Se1).Forthexeddesign,Mulleretal.[ 5 ]reportedaccurateapproximationsoftr(Sh)andtr(Se)with(n=)tr(Sh)2(abn;w)and(=)tr(Se)2(be).Thetwo-stagenatureofaninternalpilotdesignimposesaconstraintonthestage1errorsumofsquaresmatrixSe1whenN+=n+.Lemma 3 providesapproximatedistributionsfortr(Sh+),tr(Se+)]TJ /F5 11.955 Tf 11.95 0 Td[(Se1),andtr(Se1). Lemma3. Foraninternalpilotdesign,givenN+=n+,tr(Sh+)andtr(Se+)]TJ /F5 11.955 Tf 12.39 0 Td[(Se1)canbeapproximatedbyasinglechi-squarevariablewith(n=)tr(Sh+)2(abn;w)and(=)tr(Se+)]TJ /F5 11.955 Tf 12.22 0 Td[(Se1)2(bn2,wheren2=n+)]TJ /F3 11.955 Tf 12.22 0 Td[(n1.Inaddition,(=)tr(Se1)2(b1)withalowerboundqL(n+)n(b1=)(n+)]TJ /F1 11.955 Tf 12.88 0 Td[(1)andanupperboundqU(n+)n(b1=)(n+).FromLemma 3 ,(=)tr(Se1)followsadoublytruncatedchi-squaredistributionwhenN+=n+.Coeyetal.[ 20 ]derivedthedensityofthesumoftwoindependentrandomvariables,withonefollowingadoublytruncatedchi-squaredistribution,andtheotherfollowingastandardchi-squaredistribution,withdierentdegreesoffreedom.Usingthe 42

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result,wecanapproximatethedistributionofZ=(=)tr(Se+)forz>qL(n+)as fZ(z)f2(z;b+)FqU(n+)=z;b1=2;bn2=2)]TJ /F3 11.955 Tf 11.95 0 Td[(FqL(n+)=z;b1=2;bn2=2 F2[qU(n+);b1])]TJ /F3 11.955 Tf 11.95 0 Td[(F2[qL(n+);b1]:(3{9)WhenzqL(n+),fZ(z)=0.FromEquation( 3{8 ),theconditionaldistributionoftheteststatisticcanthenbeexpressedasI(n+)=Prtr(Sh+)afcrit(n+) +tr(Se+)qL(n+)<(=)tr(Se1)qU(n+)Pr2(abn;w)Z c(n+)z>qL(n+)Z1qL(n+)F2z c(n+);abn;wfZ(z)dz; (3{10)wherec(n+)=(+)=[afcrit(n+)n].Ifthenalsamplesizeisallowedtobesmallerthann0(unrestrictedrule),thenthesizeofthestage2samplecouldbezero.Whenn2=0,anyvalueof(=)tr(Se1)intheinterval(0;qU(n1)]wouldleadtochoiceofanalsamplesizeofn1.Inthiscase,Z=(=)Se1isadoublytruncatedchi-squarevariablewithalowerbound0andanupperboundqU(n1).TheconditionalprobabilityI(n+)thenreducestoI(n1)inEquation( 3{11 ),wherec(n1)=(1)=[afcrit(n1)n]. I(n1)ZqU(n1)0F2z c(n1);abn;wf2(z;b1) F2[qU(n1);b1]dz:(3{11) 3.3.3MarginalDistributionoftheFinalTestStatisticLetnLrepresentthesmallestpossiblevalueofN+greaterthann1.Themarginal(unconditional)probabilityoffailingtorejectH0canbederivedbyusingthelawoftotalprobability PrfTufcritg=I(n1)PrfN+=n1g+1Xn+=nLI(n+)PrfN+=n+g:(3{12)Thersttermontheright-handsideofEquation( 3{12 )correspondswithN+=n1.Thetermwillonlyappearwiththeunrestrictedrule,i.e.,allowingthenalsamplesizetodecreaseandbesmallerthantheinitialn0.Iftherestrictedruleisused,thersttermwill 43

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disappearandnL=n0.Theoretically,thereisnoupperlimitforthenalsamplesize.Practically,resourcelimitationgenerallyimposesamaximumfeasiblesamplesize. 3.4Simulations 3.4.1SimulationOverviewTwosetsofsimulationswereconducted:onewithaone-groupmultivariatedesignandtheotherwithatwo-grouprepeated-measuresdesign.Thesimulationsexaminedtheimpactofcovariancemisspecication,andevaluatedtheperformanceofthenewapproximationovermultipleplausiblecovariancepatterns.Theinitialsamplesizen0waschosentoachievetargetxed-designpowerfortheCMtest.ObservedTypeIerrorratesandpowervalueswerereportedforbothaone-stagexeddesignandatwo-stageinternalpilotdesign.Theinternalpilotcalculationsusedhalfoftheinitialobservations(i.e.,=0:5,n1=0:5n0,roundedtointeger)asthestage1sampletoinitiatesamplesizere-calculation.Theunrestrictedrule(i.e.,N+n1)wasusedtoallowthenalsamplesizetodecreasefromtheinitialsamplesize.SAS/IMLR(SAS9.4,SASInstituteInc.)wasusedforallcalculations.ThesoftwarePOWERLIBversion2.2 hhttp://samplesizeshop.org/software-downloads/other/i wasusedtodeterminethenalsamplesizeN+attheconclusionoftheinternalpilotphase.Allsimulatedresultsweresummarizedbasedon10,000replications.InSections 3.4.2 and 3.4.3 ,wepresenttheresultsfortheGG,CM,and(rank-adjusted)HFtests.ResultsfortheuncorrectedandBoxconservativetestsshowsimilarpatterns,andareomittedforpresentation.TheuncorrectedtesthastheexactcontroloftheTypeIerrorratewhen=1(alleigenvaluesareequal).TheBoxconservativetestprovidesgoodcontroloftheTypeIerrorrateonlywhen=1=b. 3.4.2Simulation1:One-GroupMultivariateDesignDataweregeneratedassumingfouroutcomevariables(p=4),withfourdistinctcovariancepatternsfor=U0U.TheobjectivewastomakeinferenceaboutthemeansoftheoutcomeswithH0:1=2=3=4=0.Specically,wehaveC=1, 44

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U=I4andEs(X)=1.OrthonormalinvarianceofTuallowsusingdiagonal,i.e.,=diag(k)fork=1;2;3;4.FollowingthedesigninCoeyandMuller(TableII,condition5-8)[ 4 ],fkgwerechosentohaveavarieddegreeofdeviationfromasphericalstructure,and2f0:282;0:505;0:72;1g.ThetargetTypeIerrorrateandpowerweret=0:04andPt=0:9,withtheeectsofinterestB=0:25[0:51)]TJ /F1 11.955 Tf 9.3 0 Td[(10:5].Toexaminetheimpactofcovariancemisspecication,weassumed0(priorguessoffortheinitialpowercalculation)tobeofthestructureoffkgwithwhich0=0:72.Subsequently,when6=0:72,thepoweranalysiswouldbebasedonanincorrectlyspeciedcovariancematrix.Weconsideredn0=17,whichresultedfromachievingthetargetedpowerPtusingtheCMtestatthetlevelwhenwascorrectlyspecied.Inturn,n1=9.Figure 3-2 displaystheobservedandpredictedpowerandTypeIerrorratefortheGG,CMandHFtests.Thecloseagreementbetweenpredictedandobservedpowervaluesforallthreetestssuggestsourproposedmethodsperformedwellinapproximatingpowerforaninternalpilotdesign.Moreover,aninternalpilotdesignyieldspowermuchclosertothetargetthanthexeddesignwhencovariancestructureismisspecied(i.e.,when2f0:282;0:505;1g).Whenmisspeciedcovarianceleadstothexeddesignbeingunderpowered,aninternalpilotdesignisabletoincreasethenalsamplesizeandguaranteethetargetedpower.Ontheotherhand,whenmisspeciedcovarianceresultsinthexeddesignbeingoverpowered,aninternalpilotdesignisabletoreducethesamplesizewhilestillensuringthetargetedpower.AninternalpilotdesignprotectspoweragainstcovariancemisspecicationwithariskofinatingtheTypeIerrorrate.TheCMtestwasshowntoprovideaccuratep-valueswithoutbeingeithertooliberalortooconservativeforxeddesignsetting[ 14 ],butitsgoodcontroloftheTypeIerrorratewasnotpreservedinaninternalpilotdesign.Evenwhenthecovariancematrixwascorrectlyspecied(=0:72),theobservedTypeIerrorrateof0.051wasgreaterthanthetargeted0.04.TheconservativeGGtestcontrolledthe 45

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TypeIerrorratewhenthecovariancewascorrectlyspecied,butwasoverlyconservativewhenthesphericityassumptionwasmet(=1).ThesamephenomenonwasobservedbyCoeyandMuller[ 4 ].WenoticedthelargestdierencebetweentheobservedandpredictedTypeIerrorratewhen=1.Whenstrongevidencesuggestsasphericalunderlyingcovariancestructure,forexample,exchangeabilityofobservationsispresentandveriable,noneofthethreeUNIREPtestsshouldbeused.Instead,theuncorrectedtestisthenaturalchoiceanduniformlymostpowerful. Figure3-2. Simulation1(one-groupmultivariatedesign)results.Blackandgraybarsrepresentobservedvaluesbasedon10,000replicationsforthexedandaninternalpilotdesign,respectively.Redbarsrepresentpredictedvaluesforaninternalpilotdesigncalculatedfromourproposedapproximation.DashedlinesindicatethetargetpowerPtandTypeIerrorratet.TheaveragenalsamplesizeofaninternalpilotdesignisdenotedbyE(N+). 3.4.3Simulation2:Two-GroupRepeated-MeasuresDesignDataweregeneratedassumingverepeatedmeasures(p=5)attime0,0.25,0.5,0.75,1.0.Theobjectivewastocomparethedierenceinchangesovertimebetweentwogroups.Thus,wehaveEs(X)=I2,C=[1)]TJ /F1 11.955 Tf 9.3 0 Td[(1],andUa54matrixwithcolumnsfororthonormalpolynomialcontrastsofthevetimepoints.ThetargetType 46

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Ierrorrateandpowerweret=0:05andPt=0:9,withtheeectsofinterest=[)]TJ /F1 11.955 Tf 9.3 0 Td[(1:13)]TJ /F1 11.955 Tf 9.3 0 Td[(0:050:350:22].Weconsideredcombinationsofcovariancestructurescommonlyusedinlongitudinalstudies:compoundsymmetry(CS)andrst-orderautoregressive(AR(1)).Fourcovariancestructureswerespecied: 1,IID: randominterceptandi.i.d.within-subjecterrors(=1); 1,AR: randominterceptandAR(1)within-subjecterrors(=0:94); 2,IID: randominterceptandslope(withcorrelation=0:25)andi.i.d.within-subjecterrors(=0:95); 2,AR: randominterceptandslope(withcorrelation=0:25)andAR(1)within-subjecterrors(=0:83).Thevariancesoftherandomintercept,randomslope,andtheerrortermwere2,1,and1,respectively.ThecorrelationforAR(1)was0.25.Toexaminetheeectofmisspecifyingthecovariancestructure,weassumed0(priorguessof)tobecompoundsymmetric(1,IID,0=1).Consequently,when6=1,thepoweranalysiswasconductedusingamisspeciedcovariancematrix.Aninitialsamplesizeofn0=44(n1=22)isrequiredtoachievethetargetedpowerPtatthetlevelusingtheCMtest.Figure 3-3 displaystheobservedandpredictedpowerandTypeIerrorratefortheGG,CMandHFtests.WeobservedsimilarpatternsasshownintherstsimulationandFigure 3-2 .Aninternalpilotdesignpreservespowerbetterthanthexeddesign,particularlywhenthecovariancematrixismisspecied(i.e.,2f0:94;0:95;0:83gcorrespondsto1,AR,2,IID,and2,AR,respectively),buthasadisadvantageofpotentiallyinatingtheTypeIerrorrate.InFigure 3-3 ,aninternalpilotdesignincreasedthesamplesizeandmaintainedtargetedpowerwhenthecovariancematrixunderspeciedthetruestructure(2,IIDand2,AR),wherethexeddesignledtopowerlowerthanthetarget. 47

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Figure3-3. Simulation2(two-grouprepeated-measuresdesign)results.Blackandgraybarsrepresentobservedvaluesbasedon10,000replicationsforthexedandaninternalpilotdesign,respectively.Redbarsrepresentpredictedvaluesforaninternalpilotdesigncalculatedfromourproposedapproximation.DashedlinesindicatethetargetpowerPtandTypeIerrorratet.TheaveragenalsamplesizeofaninternalpilotdesignisdenotedbyE(N+). TheTypeIerrorrateinationinSimulation2ismuchsmallerthanwhatweobservedinSimulation1.TheresultappearstobeduetothelessextremecovariancemisspecicationinSimulation2.Specically,thecovariancemisspecicationledtousingaspherical0(0=1)whenthetruehas2f0:94;0:95;0:83g.InSimulation1however,thepopulationrangesfrom0.282to1,whilethecorrectcovariancematrixhasthestructurewith=0:72.Inadditiontothemisspecication,thelargesamplesizecouldalsoalleviatetheTypeIerrorrateination.InFigure 3-4 ,weexaminedtheeectofinternalpilotsamplesizeusing=0:25(n1=10),=0:75(n1=32),and=1(n1=44).Thechoiceofhaslittleimpactonpower,butdoesaecttheTypeIerrorrate,withgreaterinationwhenn1issmall.Insummary,theTypeIerrorrateinationofaninternalpilotdesignwasreducedwhenthesamplesizewaslargeorthedeviationbetweenthetrueandplannedcovariancematrixwassmall. 48

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Figure3-4. Simulation2(two-grouprepeated-measuresdesign)resultsfordierentsizesofthestage1sample,i.e.,2f0:25;0:75;1gcorrespondston1=10,32,and44,respectively.Graybarsrepresentobservedvaluesbasedon10,000replicationsforaninternalpilotdesign,redbarsrepresentpredictedvaluescalculatedfromourproposedapproximationforaninternalpilotdesign.DashedlinesindicatethetargetpowerPtandTypeIerrorratet. 3.5ApplicationtotheMotivatingExampleTheproposedmethodswereappliedtodesignalongitudinalstudyinvestigatingtheimpactofearlytreatmentoninammatorybiomarkersovertimeforYLWH.ConcentrationsofthreecontributingbiomarkerssolubleCD27,CD14andCD163,previouslyidentiedasimportant,willbemeasuredtocharacterizetheimmunefunctionforeachstudyparticipant.Datawillbecollectedatstudyentry,year0.5,year1andyear3(endofstudy).A3:1randomizationschemewillbeusedtoallocate75%ofthestudyparticipantstoearlytreatmentarmand25%ofthestudyparticipantstostandardofcare.ThetargetTypeIerrorratewast=0:05=3(withBonferronicorrection),andthetargetpowerwasPt=0:9.Theunderlyingpopulationcorrelationstructure(Figure 3-1 C)wasassumedtobeasmoothedversionoftheobservedcorrelationmatrix(Figure 3-1 B),i.e.,AR(1)forsolubleCD27andCD14with=0:7and0.5,respectively,and 49

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compoundsymmetryforsolubleCD163with=0:62.Correlationvaluesarebasedonsix-monthlags.Themeanobservedvarianceswereassumedtobethetruevariances.TheCMtestwasusedtodeterminetheinitialsamplesizen0inachievingthetargetpowerforthexeddesign.Inturn,n0=284,n0=256andn0=136forsolubleCD27,CD14andCD163,respectively.Weusedhalfoftheinitialsampletoformthestage1sample.The3:1randomizationratioledtoconsideringn1tobeamultipleof4,i.e.,n1=144,128and68.Afterdeterminingthestage1samplesizen1,theprobabilityofeverypossiblerealizationofN+willbeevaluated.Followinganunrestrictedrule,thesmallestpossibleN+hasthevalueofn1.Figure 3-5 showsthedistributionsofthenalsamplesizeN+ofthreebiomarkersusingtheCMtest.TheexpectednalsamplesizesforsolubleCD27,CD14andCD163areE(N+)=303,E(N+)=137andE(N+)=131,respectively.Aninternalpilotdesignprotectspowertoreachthetargeted0.9forallthreebiomarkers.WenotetheexpectednalsamplesizeofsolubleCD14ismuchsmallerthantheinitialsamplesize,i.e.,E(N+)n0.Whenthetruecovarianceisconsidered,usingtotaln0=284observationswillresultinapowerof0.88,smallerthanthetargeted0.9.Therefore,aninternalpilotdesigncanpreventunderpoweredstudiescausedbyanundersizedinitialsample.Forthestudywithmultiplebiomarkers,wechoosethemaximumcalculatedsamplesizetoensurethetargetpowerforallthreebiomarkersinvestigated. 3.6DiscussionAninternalpilotdesignhastheadvantageofprotectingagainstapoorguessoferrorcovariance.Theadvantagebecomesmoreimportantinplanningastudywithrepeatedmeasureswhenspecicationsofmultiplevarianceandcovarianceparametersarerequired. 50

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Figure3-5. DistributionofnalsamplesizefortheCMtestwhencovarianceismisspecied.Blackdashedlinesrepresentinitialsamplesizen0.ReddashedlinesrepresentexpectednalsamplesizeE(N+). However,thebenetmaybereducedduetothepotentialinationoftheTypeIerrorrate.Althoughlargesamplesizecouldminimizeoreveneliminatetheination,wesuggestalwaysusingtheproposedapproximationtoexaminearangeofplausiblecovariancepatternsanddocumentthepossibleTypeIerrorrateinationbeforestartinganinternalpilotstudy.SimulationscouldalsoprovideobservedvaluesoftheTypeIerrorrateandpower,butsimulationsofaninternalpilotdesignarecomputationallyintensivebyrequiringnestedanalyses.Ourproposedmethodsareeasytoimplementforpoweranalysisandsamplesizedeterminationinreallongitudinalstudiesasdemonstratedinthemotivatingexample.Thecloseagreementbetweenanalyticalandsimulatedresultssupporttheaccuracyofourmethodsinpowerapproximations.However,ourmethodsaredevelopedunderthegenerallinearmultivariatemodelframework,whichdoesnotpermitmissingdataorrepeatedcovariates.DevelopingmethodstotoleratemissingdataorprovidetightcontroloftheTypeIerrorratewhilemaintainingthebenetsofaninternalpilotdesigncanbeafutureresearcharea. 51

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CHAPTER4AVOIDINGBIASINLONGITUDINALINTERNALPILOTSTUDIES 4.1MotivationPlanningaclinicaltrialorepidemiologicalstudywithlongitudinalGaussianoutcomesrequiresproperchoiceoftheerrorcovariancematrix.Accuratedeterminationofthenuisanceparametersintheerrorcovariancematrixisimportanttoensureadequatepoweranalysisandsamplesizeselection.Aninternalpilotdesignaccountsfortheuncertaintyinspecifyingthecovariancematrixbyusingafractionofthedatatore-estimatethenuisanceparametersandadjustthenalsamplesize.InChapter 3 ,aninternalpilotdesignwasextendedtotherepeatedmeasuressetting,thedistributionsofthenalsamplesizeandtheUNIREPteststatisticwerederived.ThecloseagreementbetweenanalyticalresultsandsimulationresultsshowstheproposedmethodshavegoodaccuracyinapproximatingtheTypeIerrorrateandpowerforaninternalpilotdesign.Althoughaninternalpilotdesignhastheadvantageofprotectingagainstpoorguessoferrorcovariancematrixandmaintainingtargetedpower,thisbenetcanbeosetduetotheriskoftheTypeIerrorrateination,whichhasbeenobservedinbothunivariateandrepeatedmeasuressetting[ 4 18 ].TheneedofcontrollingtheTypeIerrorratewhilemaintainingthebenetsofaninternalpilotdesignmotivatedthiswork.ForunivariateGaussianoutcomes,CoeyandMuller[ 19 ]proposedusingtheunadjustedteststatistic(asforthexeddesign),butmodifyingthecriticalvaluetoensurethemaximumpossibleTypeIerrorrateofaninternalpilotstudydoesnotexceedthetargetlevel.ThisboundingapproachmaybeconservativeandhaveobservedTypeIerrorrateslowerthanthetarget,sinceitonlycontrolstheupperboundoftheTypeIerrorrate[ 19 20 ].Inthischapter,theboundingapproachisextendedtoaninternalpilotstudywithlongitudinalGaussianoutcomes.Theextensionappliestoaplausibleclassofcovariancestructures,proposedbySimpsonetal.[ 37 ],forthelongitudinaldata.Italsoappliestothemostgeneralmodelwhenthecovariancematrixisunstructured. 52

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4.2PowerConsiderationsforLongitudinalStudies 4.2.1InputsforSampleSizeDeterminationInordertoobtainanappropriatesamplesizeforalongitudinalstudywithGaussianoutcomes,researchersneedtospecifythefollowinginputs:targetTypeIerrorratet,targetpowerPt,theessencedesignmatrixEs(X),between-subjectcontrastmatrixC,within-subjectcontrastmatrixU,clinicallyrelevanteectsBt(andthecorrespondingt=CBtU),thetreatmenteectsunderthenull0,andthecovariancematrixusedforstudyplanning0.TheessencematrixEs(X)iscreatedbydeletinganyduplicaterowsfromthedesignmatrixX.CoeyandMuller[ 4 ]provedthesimplied0=0canbeconsideredfortestingH0:=CBU=0withoutlossofgenerality.Inpractice,investigatorsusuallyarecomfortableaboutdeterminingtheclinicallyrelevanteects,butaccuratelyspecifyingthenuisanceparametersintheerrorcovariancematrix(pp)couldbeadiculttask.Severalcovariancepatternmodelshavebeenproposedtocapturecommonlyencounteredstructuresinlongitudinaldata.Gurkaetal.[ 38 ]pointedoutresearchersconductingmedicalstudiesoftenchoosecompoundsymmetrycovariancestructure(i.e.,exchangeabilityacrossobservations).However,theconstantcorrelationassumptionofcompoundsymmetryisgenerallyinappropriateforlongitudinalanalysis.Inrepeatedmeasuressetting,Simpsonetal.[ 37 ]notedtherst-orderautoregressive,writtenasAR(1),isthemostcommonlyusedstructuretomodelthecorrelationthatdeclineexponentiallywithtimeordistance.Inspiteofitswideuse,theAR(1)structuredisallowsalterationofthecorrelationdecayrate.Amoreexiblecharacterizationofthecovariancematrixisdesiredtoaccommodatelongitudinaldatawithslowerorfasterdecayrates. 4.2.2LEARStructureThecovariancepatternmodelproposedbySimpsonetal.[ 37 ]allowsaexiblespecicationofthecovariancestructure,whileretainingthebenetsofparsimoniousparameterization.Inaddition,Simpsonetal.[ 37 ]demonstratedtheproposedcovariance 53

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modelhasappealingstatisticalproperties,suchasaccurateinferenceforxedeectsandgoodconvergencebehavior.Thecovariancemodelrequiresthefollowingfeatures:(1)ahomogeneousvariancecomponent,(2)alinearexponentautoregressive(LEAR)correlationpattern.Throughoutthischapter,weusetheterm\LEARstructure"torepresentthecovariance(insteadofthecorrelation)structurethatsatisfytheabovetwoconditions.TheLEAR(covariance)structurehasthefollowingform hijk=28>><>>:dmin+[(djk)]TJ /F6 7.97 Tf 6.58 0 Td[(dmin)=(dmax)]TJ /F6 7.97 Tf 6.59 0 Td[(dmin)];j6=k;1;j=k:(4{1)Here,djkisthedistancebetweenmeasurementsjandk,dmin=minjdmax)]TJ /F3 11.955 Tf 12.17 0 Td[(dmin,despitealackofconclusiveproof,extensivecomputationresultsensurethecovariancematrixisalwayspositivedenitewhentimesofmeasurementareequallyspacedwithdmin2.Inthischapter,weset5timepointsequallyspacedfrom0to8(dmin=2,dmax=8)forallanalysestoeliminateanyissuesof 54

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non-positivedenitecovariancematrix.Withdmin=2,2isthewithin-subjectcorrelationbetweenanymeasurementstaken2timeunitsapart. 4.2.3LongitudinalInternalPilotStudyPlanningalongitudinalinternalpilotstudyrequiresthesameinputsasdiscussedinSection 4.2.1 .Weassumetheessencematricesindierentstagesoftheinternalpilotdesignarethesame,i.e.,Es(X1)=Es(X2)=Es(X+).ThemethodsofinternalpilotdesignwithrepeatedmeasureswedevelopedinChapter 3 ,whichrequireaknowncovariancematrix,canbeappliedtodeterminethesamplesize.Whentheunderlyingcovariancematrix(pp)isassumedtohaveaLEARstructure,weuse=2=20(>0)tocharacterizethedeviationbetweentheinitialguessofthevariance20andthetrueunderlyingvariance2.Inturn,thetruevariancecanbeexpressedas2=20.Inpractice,themeanofdiagonalelementsofanobservedcovariancematrixisusuallytakenas20.Hence,whentheLEARstructureisimposed,weonlyneedtospecifythreeparametersf;;gtofullycharacterizethecovariance,whiletotalp(p+1)=2parametersneedtobespeciedwhentheunderlyingcovariancematrixisassumedtobeunstructured. 4.3ControllingTypeIErrorRateforLongitudinalInternalPilotStudiesConductinginternalpilotstudieswithoutadjustingtherandomnessinthenalsamplesizecouldcausetheinationoftheTypeIerrorrate.WeextendtheboundingapproachproposedbyCoeyandMuller[ 19 ],whichcontrolstheTypeIerrorrateofaninternalpilotdesigninagenerallinearunivariatemodel,tolongitudinalinternalpilotstudies.TheextendedboundingapproachappliestoagenerallinearmultivariatemodelwithGaussianerrors.InChapter 3 ,wediscussedforaparticularrealizationofN+=n+,theUNIREPteststatisticisTu=[tr(Sh+)=a][tr(Se+)=+].ThenullhypothesisisrejectedifTu(n+)>fcrit(n+;t),wherefcrit(n+;t)=F)]TJ /F4 7.97 Tf 6.59 0 Td[(1F)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 12.49 0 Td[(t;abe+;b+e+)isthecriticalvalue.WeadoptthesameteststatisticTufortheboundingapproach,butusingamodiedcritical 55

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valuefcrit(n+;adj)=F)]TJ /F4 7.97 Tf 6.58 0 Td[(1F)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F3 11.955 Tf 12.38 0 Td[(adj;abe+;b+e+).Themodiedcriticalvalue,whichisdeterminedbytheadjustedleveladj,ensuresthemaximumpossibleTypeIerrorrateofalongitudinalinternalpilotstudyisnogreaterthanthetargetlevelt.Sincetheonlydierencebetweentheboundingapproachandtheinternalpilotdesignwithoutadjustmentisthecriticalvalueforhypothesistesting,thesetwomethodsresultinthesameexpectednalsamplesizeE(N+).Thedeterminationoftheadjustedleveladjdependsontheunderlyingcovariancestructure. 4.3.1BoundingApproachforLEARStructureForacovariancematrixofLEARstructure,weonlyneedthreeparametersf;;gtofullyspecifytheerrorcovariance.Theadjustedlevelcanbedeterminedasfollows.Foranygivenvalueof2(0;t],wecancomputethecorrespondingcriticalvaluefcrit(n+;)=F)]TJ /F4 7.97 Tf 6.59 0 Td[(1F)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F3 11.955 Tf 11.99 0 Td[(;abe+;b+e+).ThenwecanndthemaximumpossibleTypeIerrorrateacrossf;;gusingthecriticalvaluefcrit(n+;).ThemaximumpossibleTypeIerrorratecanbeobtainedusingnonlinearoptimizationbyNelder-Meadsimplexmethod(SAS/IMLR,SAS9.4,SASInstituteInc.,NLPNMSsubroutine).Nelder-Meadsimplexmethodisanon-derivativebasedapproachonlyrequiringcontinuousobjectivefunction.FollowingSections 4.2.2 and 4.2.3 ,theoptimizationsubjectstothreeconstraints:(1)=2=20>0,(2)0<1,and(3)0.Byrepeatingtheoptimizationacrosstheentirerangeof2(0;t],wecanndthelargesttohavethemaximumTypeIerrorrateatorbelowthetargetlevelt.Thislargestischosenastheadjustedleveladjfortheboundingapproach. 4.3.2BoundingApproachforUnstructuredCovarianceMatrixForanunstructuredcovariancematrix,althoughaparsimoniousspecicationisnolongerattainable,wecanobtainathreeparametersetforpowerapproximationasdetailedinLemma 4 .ThesamestepsforLEARstructurecanbefollowedtodetermineadj,excepttheoptimizationisbasedonthenewparametersetanditsconstraints. 56

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Lemma4. Exceptforknownconstants,i.e.,rank(C)=a,rank(U)=b,andrank(X1)=rank(X2)=rank(X+)=r,theminimum,sucientparametersetforpowerapproximationofalongitudinalinternalpilotdesignincludesthreeparameters,namely=tr()=tr(0),==0,and=tr(Es)=tr(0Es).Thehypothesiscovariancematrix=U0U,andtheessenceunscalednoncentralityEs=()]TJ /F5 11.955 Tf 12.56 0 Td[(0)0fC[Es(X+)0W+Es(X+)])]TJ /F5 11.955 Tf 7.09 -4.33 Td[(C0g)]TJ /F4 7.97 Tf 6.59 0 Td[(1()]TJ /F5 11.955 Tf 12.57 0 Td[(0),whereW+isthediagonalweightmatrixofthesubgroupsamplesizeproportions.Therstparametergeneralizes=2=20with=tr()=tr(0)==0(>0).Intheunivariatetesting(b=1),=.Thesecondparameterisaratioofsphericityparameterwith==0.Sincethesphericityparameter1=b1,wehave1=(b0)1=0.Thersttwoparametersrepresentthedeviationsofaverageeigenvalues,andthespreadofeigenvaluesbetweenthetrueandplannedcovariancematrix.ThesetwoparametershavealsobeenproposedbyCoeyandMuller[ 4 ]toapproximatethemisspecicationof.Weintroducethethirdparameter=tr(Es)=tr(0Es)tocapturethediscrepancyofinteractionbetweentreatmenteectsandtruecovariance,andinteractionbetweentreatmenteectsandplannedcovariance.Thematrices,0,andEsareallpositivesemidenite,forthesakeofpracticality,weonlyconsider>0.Inaddition,Coope[ 41 ]provedforpositivesemidenitematrices,tr(Es)tr()tr(Es).Consequently,0<[tr(0)tr(Es)]=tr(0Es). 4.4NumericalResultsWeuseatwo-groupcomparisonstudytoillustratetheboundingapproach.Theobjectiveofthestudywastocomparethegroupdierenceinchangesoververepeatedlymeasuredoutcomesattime0,2,4,6,8.Acompoundsymmetrycovariancematrixwasusedforstudyplanningwithvariance20=3andconstantcorrelation20=2=3.ThetargetedTypeIerrorrateandpowerweret=0:05andPt=0:9,withtheclinicallyrelevanteects=[)]TJ /F1 11.955 Tf 9.29 0 Td[(1:13)]TJ /F1 11.955 Tf 9.3 0 Td[(0:050:350:22].Otherrequiredinputswere:Es(X)=I(2),C=[1)]TJ /F1 11.955 Tf 9.3 0 Td[(1],andUa54matrixoforthonormalpolynomialcontrasts 57

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forthevetimepoints.TheunderlyingcovariancematrixwasassumedtohaveaLEARstructure,sodmin=2,dmax=8,and2wasthecorrelationbetweenmeasurementstaken2timeunitsapart.AlltheresultspresentedinSections 4.4.1 and 4.4.2 werebasedontheChi-Muller(CM)test.Theinitialsamplesizen0=44wasdeterminedbyaxeddesignpowercalculationtoensurethetargetpowerPtatthetlevelfortheeectsofinterest.Aninternalpilotdesignusedhalfoftheinitialsampletoformthestage1sample(i.e.,=0:5;n1=0:5n0=22).Anunrestrictedrulewasadoptedtoallowthenalsamplesizetodecreaseandbesmallerthantheinitialsamplesize(i.e.,N+n1).Weconsideredtohaveadj=0:0462,whichresultedfromcontrollingthemaximumpossibleTypeIerrorratenogreaterthanthetargetlevel(t=0:05)basedontheboundingapproach. 4.4.1EnumerationsFigure 4-1 and 4-2 displaythepredictedTypeIerrorrateandpowerasafunctionof=2=20inlog2scale.ThepredictedvalueswerecomputedusingthemethodswedevelopedinChapter 3 .Weconsidered=0.2,0.5and0.8tomodellow,moderateandhighwithin-subjectcorrelations,respectively.SincelongitudinalepidemiologicalstudiesoftenpresentaslowerdecayrateascomparedwiththeAR(1)structure[ 39 40 ],wefocusedon06.Specically,2f0;2;4;6gwasplottedwithcompoundsymmetry(=0)andAR(1)structure(=6)asspecialcases.InFigure 4-1 ,thexeddesign(dottedlines)alwaysyieldstheTypeIerrorrateatthetargetedlevel,regardlessofspecicationofthethreeparameters.Theboundingapproach(solidlines)controlstheTypeIerrorrateinationinducedbytheuseofaninternalpilotdesign(dashedlines).InFigure 4-2 ,asopposedtothexeddesign(dottedlines),aninternalpilotdesignalwayspreservespowerclosetothetarget,especiallywhenthespecicationerrorintermsofthevariancecomponentislarge.Inaddition,thepowerdierencesbetweenaninternalpilotdesignwithboundingapproach(solidlines)andaninternalpilotdesignwithoutadjustment(dashedlines)arenegligible.Hence,adopting 58

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theboundingapproachwhenplanningalongitudinalinternalpilotstudycancontroltheTypeIerrorratenotexceedtothetargetlevelwithverylittlepowerloss.ThisdesirablepropertyoftheboundingapproachhasalsobeennoticedintheunivariatesettingbyCoeyetal.[ 20 ]. Figure4-1. PlotsofpredictedTypeIerrorrateasafunctionof=2=20(log2scale)forCMtest.Solidlinesrepresentinternalpilotdesignwithboundingapproach(adj=0:0462),dashedlinesrepresentinternalpilotdesignwithoutadjustment(adj=t=0:05),dottedlinesrepresentxeddesignwithn0=44.TargetTypeIerrorratet=0:05. 59

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Figure4-2. Plotsofpredictedpowerasafunctionof=2=20(log2scale)forCMtest.Solidlinesrepresentinternalpilotdesignwithboundingapproach(adj=0:0462),dashedlinesrepresentinternalpilotdesignwithoutadjustment(adj=t=0:05),dottedlinesrepresentxeddesignwithn0=44.TargetpowerPt=0:9. 60

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4.4.2SimulationsToverifytheaccuracyoftheboundingapproach,weconductedasetofsimulationsforthetwo-groupcomparisonstudy.Theconditionsweexaminedprovideacross-classicationwiththethreeparametersf;;g,2f0:125;0:25;0:5;1;2;4;8g,2f0:2;0:5;0:8g,and2f0;2;4;6g.Figure 4-3 and 4-4 displaytheempiricalTypeIerrorandpowerasafunctionof=2=20inlog2scale.Eachcondition(eachdotintheplot)wasbasedon10,000replications.InFigure 4-3 ,weobservedasimilarpatternasthepredictedTypeIerrorrate(Figure 4-1 ).Adoptingtheboundingapproach(solidlines)inaninternalpilotstudycontrolstheTypeIerrorrateinationingeneral.ThecontroloftheTypeIerrorrateisparticularlyachievedwhenalowormoderatewithin-subjectcorrelationispresented(=0:2and0.5).Whenmeasurementsarehighlycorrelatedwithinsubjects(=0:8),weobservedsomeempiricalTypeIerrorratesareabovethetargetlevel,especiallywhentheinitialguessofthevarianceoverestimatesthetruevariance,i.e.,=2=20<1,log2()<0.TheempiricalpowervaluesasshowninFigure 4-4 arealsoingoodagreementwiththepredictedpowervaluesshowninFigure 4-2 .Theboundingapproachhasnearlythesamepowerasaninternalpilotdesignwithoutadjustment.ThepatternsweobservedinbothenumerationsandsimulationssuggestusingtheboundingapproachtoplanalongitudinalinternalpilotstudycancontroltheTypeIerrorratewhilemaintainingthegoodpowerperformance.ExaminingplotsofempiricalTypeIerrorrateoverarangeofpossibledesignparameters,suchasFigure 4-3 ,couldbeanalternativeapproachtondtheadjustedleveladjtoavoidbias.However,simulationsforinternalpilotdesignwithrepeatedmeasuresaretimeconsuming.Forexample,generatingFigure 4-3 requirescomputingtimeof47hoursand58minutes,whilecomputingthesamesetofconditionsusingourproposedapproximationonlytakes56seconds.Thecloseagreementbetweenenumerationsandsimulationsshowsourproposedmethodsprovideaccurateapproximationsofthe 61

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TypeIerrorrateandpowervaluesforaninternalpilotdesign.ThetimeadvantageofourproposedmethodsallowsvisuallyillustratingtheTypeIerrorrateandpoweroverwiderangesofdesignspecicationsinatimelymanner,whichhasbeenshowntohavegreatpracticalvalue[ 20 ]. Figure4-3. EmpiricalTypeIerrorrateforCMtest.Solidlinesrepresentinternalpilotdesignwithboundingapproach(adj=0:0462),dashedlinesrepresentinternalpilotdesignwithoutadjustment(adj=t=0:05).Eachdotintheplotisbasedon10,000replicationswitht=0:05,Pt=0:9,n0=44,andn1=22. 62

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Figure4-4. EmpiricalpowerforCMtest.Solidlinesrepresentinternalpilotdesignwithboundingapproach(adj=0:0462),dashedlinesrepresentinternalpilotdesignwithoutadjustment(adj=t=0:05).Eachdotintheplotisbasedon10,000replicationswitht=0:05,Pt=0:9,n0=44,andn1=22. 4.5DiscussionAninternalpilotdesigncanprotectagainstunderpoweredstudiescausedbyanincorrectlyspeciedcovariancematrix.However,thisprotectionbringsthedisadvantageofpotentialTypeIerrorrateination.Inthischapter,weadoptthedistributionofUNIREPteststatisticderivedinChapter 3 ,andextendtheboundingapproachtolongitudinalinternalpilotstudieswithGaussianoutcomes.Whenacovariancepattern 63

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modelimposesstructure,suchasLEARstructure,ontheunderlyingcovariancematrix,ourresultssuggestthattheboundingapproachcontrolstheTypeIerrorratenogreaterthanthetargetlevelingeneralwithnegligiblepowerloss.WerecommendalwaysusingtheboundingapproachtoreducetheriskofTypeIerrorrateinationwhenplanningalongitudinalinternalpilotstudy.Althoughconductingasetofsimulationscanalsondthemodiedcriticalvaluetoavoidbias,basedontheempiricalTypeIerrorrate,ourproposedmethodsprovideaccurateresultswithmuchfasterspeed.Theboundingapproachcanalsoapplytothemostgeneralunstructuredcovariancematrix.Forthetwo-groupcomparisonstudyweconsidered,whenassuminganunstructuredcovariance,theboundingapproachleadstoadj=0:027.Theadjustedlevelismuchsmallerthantheadj=0:0462whenassumingtheLEARcovariancestructure.Thegeneralityoftheunstructuredcovariancematrixcomesatacostofover-correctingcovariancepatternsthatareunlikelytoappearinlongitudinaldata.Developingnewanalyticalresultsfortheminimum,sucientparametersetofpowerapproximationtoalleviatetheover-correctingissue,andexploringadditionalimprovementsoftheboundingapproachcouldbeafuturedirectionofresearch. 64

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CHAPTER5CONCLUSIONS 5.1SummaryThisdissertationdevelopedanalyticalmethodstoapproximatetheTypeIerrorrateandpowerforaninternalpilotdesignwithrepeatedmeasures,andextendedtheboundingapproachtocontrolthepotentialinationoftheTypeIerrorrateinlongitudinalinternalpilotstudies.ThemethodsdevelopedinthisdissertationareforagenerallinearmultivariatemodelwithGaussianoutcomesusingunivariateapproachtorepeatedmeasures(UNIREP)techniques.Poweranalysisandsamplesizedeterminationiscriticalinplanningaclinicaltrialorepidemiologicalstudy.Thevaluesofthenuisanceparametersintheerrorcovariancematrixareusuallyuncertaininthestudyplanningphaseduetolimitedinformation.Theuncertaintyhasbeenahugebarrierfortheplannedstudytoachieveadequatepowerandobtainappropriatesamplesize.Themisspeciedcovariancematrixcouldleadtounderpoweredstudieswithlittlechanceofdetectingclinicallyrelevanteects,oroverpoweredstudieswhichwastetimeandresources.Toavoidsuchuncertainty,aninternalpilotdesignwasintroduced,whichusesafractionoftheobservationstore-estimatethenuisanceparametersandadjustthesamplesize.Mostworkofinternalpilotdesignhasbeenfocusedonthetwo-samplettestorunivariateoutcomes.InChapter 3 ,weextendedaninternalpilotdesigntorepeatedmeasuressetting,andderivedtheapproximatedistributionsofthenalsamplesize,andtheUNIREPteststatistic.Theaccuracyoftheanalyticalresultscomputedfromourproposedapproximationisveriedbycomparingwiththesimulationresults.Theproposedmethodsareeasytoimplementforrepeatedmeasuresstudiesinclinicalresearch.AlongitudinalstudyinvestigatingtheeectofearlyantiretroviraltherapyoninammatorybiomarkersforyouthlivingwithHIVdemonstratestheimplementationoftheproposedmethods. 65

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Aninternalpilotdesignpreservespowerclosetothetargetevenforsmallsamples,andisrobustagainstcovariancemisspecication.However,ignoringtherandomnessinthenalsamplesizeofaninternalpilotstudyislikelytoinducetheTypeIerrorrateination.Toavoidthebiasofaninternalpilotdesign,theboundingapproachwasproposedbyCoeyandMuller[ 19 ]foragenerallinearunivariatemodel.Theboundingapproachusestheunadjustedteststatistic(asforthexeddesign),butmodiesthecriticalvalueforhypothesistestingtocontrolthemaximumTypeIerrorratenogreaterthanthetargetlevel.InChapter 4 ,theboundingapproachwasextendedtolongitudinalinternalpilotstudies.Weconsideredacovariancepatternmodelcommonlypresentedinlongitudinaldata,i.e.,theLEARstructure.Enumerationandsimulationresultsshowverysimilarpatterns.BothsuggesttheextendedboundingapproachcancontroltheinationoftheTypeIerrorratewhilehavingnegligiblepowerlossascomparedtoaninternalpilotdesignwithoutadjustment.Theextendedboundingapproachcanalsoapplytothemostgeneralunstructuredcovariancematrix.Aminimum,sucientthree-parametersetforpowerapproximationhasbeenproposedtoenablethecomputationofmaximumTypeIerrorrateandthedeterminationoftheadjustedlevel(andthecorrespondingmodiedcriticalvalue). 5.2FutureResearchThemethodspresentedinthisdissertationfocusonthegenerallinearmultivariatemodelwithUNIREPtests.Thisframeworkassumescompletedatawithnotime-varyingcovariates.CatellierandMuller[ 42 ]generalizedtheUNIREPprocedurestoallowmissingdata.Consideringsuchgeneralizationinaninternalpilotdesignhasgreatappeal,especiallyforclinicalresearchstudies.InChapter 4 ,severalchallengesshouldbeovercometobetterapplytheboundingapproachtounstructuredcovariancematrix.AsopposedtotheLEARstructure,thethree-parametersetfortheunstructuredcovarianceissucientforpowerapproximation,butnotsucientforfullyspecifyingtheerrorcovariancematrix.ThemaximumType 66

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Ierrorratedeterminedbythisparametersetmayleadtoover-correctingcovariancepatternsthatareunlikelytoappearinlongitudinaldatasetting.Inturn,theadjustedleveladjcouldbetoosmalltobeofanypracticalvalue.Furtherexaminationofthisparametersetisnecessarytoensuretheproperuseoftheboundingapproachforthemostgeneralcovariancestructure. 67

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APPENDIXAMATHEMATICALPROOFS A.1ProofofLemma 1 Proof. CoeyandMuller[ 18 ]derivedthedistributionofthenalsamplesizeforaninternalpilotdesignwithunivariateoutcomes.Theyadoptedthegenerallinearunivariatemodelframeworkwithy+(N+1)=X++(N+q1)+e+(N+1)ande+NN+(0;2I).TheassociatedgenerallinearhypothesisisH0:(a1)=C(aq1)=0.Formultivariatemodelwithunivariatetesting,i.e.,b=rank(U)=1,thetransformedresponseYU+=Y+Uisunivariate,andtheunderlyingcovariancematrixreducestotheunderlyingvariance,i.e.,=U0U=2.Equivalently,theestimatedcovariancematrixfromthestage1samplereducestotheestimatedvariancefromthestage1sample,i.e.,^1=U0^1U=^21.CoeyandMuller[ 18 ]usedSSHEstodenotetheessencehypothesissumsofsquareswithSSHEs=()]TJ /F11 11.955 Tf 13 0 Td[(0)0fC[Es(X+)0W+Es(X+)])]TJ /F5 11.955 Tf 7.08 -4.34 Td[(C0g)]TJ /F4 7.97 Tf 6.59 0 Td[(1()]TJ /F11 11.955 Tf 13 0 Td[(0),wheretheessencematrixEs(X+)andthediagonalweightmatrixW+havethesamedenition,regardlessofwhethertheoutcomeisunivariateormultivariate.Consequently,tr(Es)=Es=SSHEswhenb=1.Inturn,reducesto2 =tr(2)+2n+tr(Es=a) tr()+2n+tr(Es=a)=(2)2+2n+2(SSHEs=a) 2+2n+(SSHEs=a)=2:(A{1)Todemonstrate(n+)=n+tr(Es)=w(n+)reducesto2(n+)=n+SSHEs=w(n+)fortheunivariatetesting,weneedtoprovethatthenoncentralityw(n+)reducestow(n+).GivenN+=n+,w(n+)needstosatisfyEquation( 3{3 ).Whenb=1,thestage1estimatorsforveUNIREPtestsareallequalto1.Inturn,theexpectationsoftheseestimatorsareallequalto1,i.e.,themultipliere1=1.Furthermore,=tr2()=fbtr(2)g=(2)2=(2)2=1,=tr()=b=2,andn===1.Equation( 3{3 )reducestoPt1)]TJ /F3 11.955 Tf 12.39 0 Td[(FFF)]TJ /F4 7.97 Tf 6.59 0 Td[(1F(1)]TJ /F3 11.955 Tf 11.96 0 Td[(t;a;+);a;+;w(n+),whichistheconditionw(n+)needstosatisfyintheunivariatesettingwhenN+=n+,i.e.,Equation(13)in 68

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CoeyandMuller[ 18 ].Hence,whenb=1,(n+)=2(n+).Inaddition, ~n1=tr2(^1)+2n+tr(^1)tr(Es=a) bftr(^21)+2n+tr(^1Es=a)g=(^21)2+2n+^21(SSHEs=a) (^21)2+2n+^21(SSHEs=a)=1;(A{2)whichleadstoE[~n1]=1,and=(b1)E[^1]=E[~n1]=1.Equation( 3{6 )reducesto PrfN+n+gPr2(1)1 2n+SSHEs w(n+);(A{3)whichisthedistributionofthenalsamplesizeintheunivariatesetting,i.e.,Equation(17)inCoeyandMuller[ 18 ].Insum,whenthetestingisunivariate(b=1),thedistributionofthenalsamplesizedeterminedbythemultivariatemodelisthesameasCoeyandMullerderivedfortheunivariateoutcome. A.2ProofofLemma 2 Proof. LetYU+=Y+Urepresentthetransformedresponse,whichfollowsamatrixGaussiandistributionwithYU+NN+;b(X+B+U;IN+;).Withoutlossofgenerality,weconsider0=0fortestingH0:=0[ 4 ].ToprovethematrixindependenceswhenN+=n+,werstshowSh+,Se+,Se1,andSe+)]TJ /F5 11.955 Tf 12.2 0 Td[(Se1canallbeexpressedasmultivariateGaussianquadraticformsinYU+,i.e.,Sh+=Y0U+AhYU+,Se+=Y0U+AeYU+,Se1=Y0U+A1YU+,andSe+)]TJ /F5 11.955 Tf 11.99 0 Td[(Se1=Y0U+(Ae)]TJ /F5 11.955 Tf 11.99 0 Td[(A1)YU+.ThenweproveAh,Ae,A1,and(Ae)]TJ /F5 11.955 Tf -457.42 -23.91 Td[(A1)areallidempotentmatrices.Ifanypairsoftheseidempotentmatriceshasaproductof0,wecanapplytheorem10.8inMullerandStewart[ 6 ]toshowtheindependenceofthecorrespondingmultivariateGaussianquadraticforms(andtheirtraces).Here,Ah=X+(X0+X+))]TJ /F5 11.955 Tf 7.09 -4.34 Td[(C0M)]TJ /F4 7.97 Tf 6.59 0 Td[(1+C(X0+X+))]TJ /F5 11.955 Tf 7.09 -4.34 Td[(X0+isidempotentofranka,andAe=IN+)]TJ /F5 11.955 Tf 12.58 0 Td[(X+(X0+X+))]TJ /F5 11.955 Tf 7.09 -4.34 Td[(X0+isidempotentofrank+.WedeneA1asthefollowingform.TheidempotencyofX1(X01X1))]TJ /F5 11.955 Tf 7.09 -4.33 Td[(X01indicatesthatA1isanidempotentmatrixwithrank1. A1(N+N+)=2664In1)]TJ /F5 11.955 Tf 11.95 0 Td[(X1(X01X1))]TJ /F5 11.955 Tf 7.09 -4.34 Td[(X01(n1n1)0(n1N2)0(N2n1)0(N2N2)3775(A{4) 69

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Toprove(Ae)]TJ /F5 11.955 Tf 11.95 0 Td[(A1)isanidempotentmatrixwithrankn2,werstshow X0+A1=X01X02264In1)]TJ /F5 11.955 Tf 11.95 0 Td[(X1(X01X1))]TJ /F5 11.955 Tf 7.08 -4.34 Td[(X01000375=264X01)]TJ /F5 11.955 Tf 11.95 0 Td[(X01X1(X01X1))]TJ /F5 11.955 Tf 7.09 -4.34 Td[(X01000375=0(A{5)Inturn,AeA1=[IN+)]TJ /F5 11.955 Tf 12.64 0 Td[(X+(X0+X+))]TJ /F5 11.955 Tf 7.09 -4.34 Td[(X0+]A1=A1andalsoAeA1issymmetric.Hence,(Ae)]TJ /F5 11.955 Tf 12.5 0 Td[(A1)(Ae)]TJ /F5 11.955 Tf 12.5 0 Td[(A1)=Ae)]TJ /F5 11.955 Tf 12.5 0 Td[(A1Ae)]TJ /F5 11.955 Tf 12.5 0 Td[(AeA1+A1=(Ae)]TJ /F5 11.955 Tf 12.5 0 Td[(A1).Furthermore,basedonmatrixalgebra,wehaveAhAe=AhA1=Ah(Ae)]TJ /F5 11.955 Tf 12.33 0 Td[(A1)=A1(Ae)]TJ /F5 11.955 Tf 12.33 0 Td[(A1)=0.Consequently,Sh+andSe+,Sh+andSe1,Sh+andSe+)]TJ /F5 11.955 Tf 12.92 0 Td[(Se1,Se1andSe+)]TJ /F5 11.955 Tf 12.93 0 Td[(Se1areindependent,soastheirtraces[ 6 ]. A.3ProofofLemma 3 Proof. GivenN+=n+,Sh+followsanon-centralWishartdistribution,whileSe1andSe+)]TJ /F5 11.955 Tf 12.81 0 Td[(Se1followacentralWishartdistributionwithSh+Wb(a;;)]TJ /F4 7.97 Tf 6.58 0 Td[(1),Se1Wb(1;;0),andSe+)]TJ /F5 11.955 Tf 12.16 0 Td[(Se1Wb(n2;;0),respectively.GlueckandMuller[ 43 ]provedtraceofaWishartequalspositivelyweightedsumofindependentchi-squarerandomvariables.Kimetal.[ 44 ]proposedmatchingtwomomentsandusingasinglenon-centralchi-squarevariabletoapproximatetheweightedsumofindependentchi-squarerandomvariables.Applyingtheirresults,tr(Sh+)(=n)2(abn;w),tr(Se1)(=)2(b1)andtr(Se+)]TJ /F5 11.955 Tf 11.96 0 Td[(Se1)(=)2(bn2).Equation( 3{7 )implies~1=^1=~n1hasalowerbound(n+)]TJ /F1 11.955 Tf 12.62 0 Td[(1),andanupperbound(n+)whenN+=n+.Accordingly,wecanapproximatethelowerboundqL(n+)andupperboundqU(n+)of(=)tr(Se1)=(=)b1^1(=)b1n~1withqL(n+)n(b1=)(n+)]TJ /F1 11.955 Tf 11.95 0 Td[(1)andqU(n+)n(b1=)(n+). A.4ProofofLemma 4 Proof. Bydeningthethreeparameters,and,theunderlyingpopulationquantitiescanbeexpressedastr()=tr(0),=0,tr(Es)=tr(0Es),and 70

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tr(2)=[2tr2(0)]=(b0).ForaparticularrealizationofN+=n+,wecanwrite=tr(2)+2tr(=a) tr()+2tr(=a)=tr(2)+2n+tr(Es=a) tr()+2n+tr(Es=a)n= =tr2()+2n+tr()tr(Es=a) b[tr(2)+2n+tr(Es=a)]=(b1)E[^1] E[~n1](b1)Etr2(^1)bEtr(^21) Etr2(^1)+2n+tr()tr(Es=a)bEtr(^21)+(2bn+)tr(Es=a): (A{6)Mulleretal.[ 5 ]provedthatEtr2(^1)=(1=1)[2tr(2)+1tr2()]andEtr(^21)=(1=1)[(1+1)tr(2)+tr2()].Hence,foralongitudinalinternalpilotstudy,thedistributionoftheUNIREPteststatisticwederivedinChapter 3 ,canbefullycharacterizedbythethreeparameters,and(withknownconstantsa,bandr). 71

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APPENDIXBSUPPLEMENTALRESULTSFigure B-1 and B-2 displaythedistributionsofthenalsamplesizeN+forsimulation1(one-groupmultivariatedesign)andsimulation2(two-grouprepeated-measuresdesign).ThegoodagreementbetweenthepredictedandempiricalprobabilitysuggestsourproposedmethodperformedwellinapproximatingthedistributionofN+foraninternalpilotdesign. 72

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FigureB-1. Simulation1(one-groupmultivariatedesign)results:DistributionofN+forn0=17(n1=9).Blackverticallinesrepresentempiricalprobabilitybasedon10,000replications.Redcurvesrepresentdistributionofthenalsamplesizecalculatedfromourproposedapproximation. 73

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FigureB-2. Simulation2(two-grouprepeated-measuresdesign)results:DistributionofN+forn0=44(n1=22).Blackverticallinesrepresentempiricalprobabilitybasedon10,000replications.Redcurvesrepresentdistributionofthenalsamplesizecalculatedfromourproposedapproximation. 74

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APPENDIXCSIMULATIONDETAILS Simulation1Simulation1hasinitialparametersEs(X)=1,C=1andU=I4.Fourcovariancestructuresareconsideredusingconditions5-8inTableIIofCoeyandMuller[ 4 ]01=[0:479600:010000:010000:01000](=0:282)02=[0:345550:061230:055610:04721](=0:505)03=[0:235550:171230:055610:04721](=0:720)04=[0:127400:127400:127400:12740](=1):FollowingCoeyandMuller[ 4 ],theclinicallyrelevanteectsB=0:25[0:51)]TJ /F1 11.955 Tf 9.3 0 Td[(10:5]and=CBU=B.Thethirdcovariancestructureconditionisusedforstudyplanning.Hence,00=[0:235550:171230:055610:04721],0=diag(0).PopulationparametersunderH0:=0,andunderHA:=0:25[0:51)]TJ /F1 11.955 Tf 9.3 0 Td[(10:5]. Simulation2Simulation2hasinitialparametersEs(X)=I2,andC=[1)]TJ /F1 11.955 Tf 9.3 0 Td[(1].Within-subjectcontrastmatrixUiscreatedwithorthogonalpolynomialcontrastsforvetimepoints:f0,0.25,0.5,0.75,1.0gU=266666666664)]TJ /F1 11.955 Tf 9.3 0 Td[(0:630:53)]TJ /F1 11.955 Tf 9.3 0 Td[(0:320:12)]TJ /F1 11.955 Tf 9.3 0 Td[(0:32)]TJ /F1 11.955 Tf 9.3 0 Td[(0:270:63)]TJ /F1 11.955 Tf 9.3 0 Td[(0:480)]TJ /F1 11.955 Tf 9.3 0 Td[(0:5300:720:32)]TJ /F1 11.955 Tf 9.3 0 Td[(0:27)]TJ /F1 11.955 Tf 9.3 0 Td[(0:63)]TJ /F1 11.955 Tf 9.3 0 Td[(0:480:630:530:320:12377777777775:NotethattheUmatrixisthesameforvetimepointsf0,0.25,0.5,0.75,1.0gconsideredinChapter 3 ,andvetimepointsf0,2,4,6,8gconsideredinChapter 4 .WeadoptthesamecovariancestructuresconsideredbyGurkaetal.[ 38 ].Thevariances 75

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ofrandominterceptandrandomslopeare2I=2and2S=1,respectively.Theerrorvarianceis2=1,andthecorrelationforAR(1)structureis=0:25. (1) 1,IID:randominterceptandi.i.d.within-subjecterrors(=1)=2I110+2I5=2110+1I5=2666666666643222223222223222223222223377777777775: (2) 1,AR:randominterceptandAR(1)within-subjecterrors(=0:94)=2I110+226666666666412341232123214321377777777775=26666666666432:252:062:0222:2532:252:062:022:062:2532:252:062:022:062:2532:2522:022:062:253377777777775: 76

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(3) 2,IID:randominterceptandslope(assumecorrelatedIS=0:25),andi.i.d.within-subjecterrors(=0:95)=2666666666641010:2510:510:75113777777777752642IISISISIS2S3752641111100:250:50:751375+2I5=26666666666432:092:182:272:352:093:242:392:542:692:182:393:602:823:032:272:542:824:093:372:352:693:033:374:71377777777775: (4) 2,AR:randominterceptandslope(assumecorrelatedIS=0:25),andAR(1)within-subjecterrors(=0:83)=2666666666641010:2510:510:75113777777777752642IISISISIS2S3752641111100:250:50:751375+226666666666412341232123214321377777777775=26666666666432:342:242:282:362:343:242:642:602:712:242:643:603:073:092:282:603:074:093:622:362:713:093:624:71377777777775: 77

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PopulationparametersunderH0:=0.WeconsiderclinicallyrelevanteectsB=2640000000:110:421:271:2375.Inturn,populationparametersunderHA:=CBU=[)]TJ /F1 11.955 Tf 9.3 0 Td[(1:13)]TJ /F1 11.955 Tf 9.3 0 Td[(0:050:350:22]: 78

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APPENDIXDPARAMETERSOFTHEMOTIVATINGEXAMPLETheprimarypurposeofthereplicationstudyistoinvestigatetheeectsofearlytreatmentonthelevelsofinammatorybiomarkerssolubleCD27,CD14andCD163.ThestudyhastargetTypeIerrorratet=0:05=3(withBonferronicorrection),andtargetpoweratPt=0:9.Sincea3:1randomizationisadoptedtoassign75%ofthestudyparticipantstoearlytreatmentarmand25%ofthestudyparticipantstostandardofcarearm,theessencexdesignmatrixisEs(X)=I2withdiagonalweightmatrixW=2643=4001=4375:Thebetween-subjectcontrastmatrixisC=[1)]TJ /F1 11.955 Tf 9.29 0 Td[(1]tocompareearlytreatmentandstandardofcarearms.Dataarecollectedatfourtimepoints:yearf0,0.5,1,3gtotrackthelongitudinalpatternofbiomarkers.Thewithin-subjectcontrastmatrixiscreatedwithorthogonalpolynomialcontrastsforthesefourtimepointsU=266666664)]TJ /F1 11.955 Tf 9.3 0 Td[(0:490:63)]TJ /F1 11.955 Tf 9.29 0 Td[(0:33)]TJ /F1 11.955 Tf 9.3 0 Td[(0:27)]TJ /F1 11.955 Tf 9.3 0 Td[(0:190:80)]TJ /F1 11.955 Tf 9.3 0 Td[(0:05)]TJ /F1 11.955 Tf 9.3 0 Td[(0:71)]TJ /F1 11.955 Tf 9.29 0 Td[(0:500:820:270:03377777775:AsshowninFigure 3-1 (A),theobservedmeanpatternsofthreebiomarkersintheexistingstudyareusedtoguidethespecicationoftheclinicallyrelevanteects.ThetreatmentdierencesofinterestforsolubleCD27,CD14andCD163aredenotedasBt1, 79

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Bt2andBt3,respectively.Bt1=2645753666868625860375;Bt2=26412501300135015501200170012701350375;Bt3=264680570500480680770630550375:Theobservedcovariancematricesofthethreebiomarkersareusedforstudyplanning.Weuse01,02and03todenotetheplannedcovariancestructureforsolubleCD27,CD14andCD163,respectively.01=2666666641280:941028:361175:78)]TJ /F1 11.955 Tf 9.29 0 Td[(39:851028:361332:841177:91138:501175:781177:911979:82171:11)]TJ /F1 11.955 Tf 9.3 0 Td[(39:85138:50171:111125:13377777775;02=266666664107148:045652:5359553:8778362:135652:53950897:1761968:3026216:5359553:8761968:30196608:8497181:9978362:1326216:5397181:99204235:50377777775;03=26666666487453:4752877:3448149:7046114:9452877:3482420:6547700:1747819:6948149:7047700:1758801:9441990:6446114:9447819:6941990:6480491:93377777775:Hence,theobservedvariancesforsolubleCD27,CD14andCD163are1429.7,364722.4and77292,respectively.Weassumetheobservedvariancesarethetruevalues.Inaddition,assumingthepopulationcorrelationstructuresaresmoothedversionsofthe 80

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observedcorrelations,i.e.,AR(1)forsolubleCD27andCD14,compoundsymmetryforsolubleCD163withhalf-yearcorrelation0.7,0.5and0.62,respectively.ThepopulationcorrelationmatricesforthethreebiomarkersareshowninFigure 3-1 (C).Inturn,theunderlyingcovariancepatternsforsolubleCD27,CD14andCD163aredenotedas1,2and3,respectively.1=2666666641429:681000:78700:54168:201000:781429:681000:78240:29700:541000:781429:68343:27168:20240:29343:271429:68377777775;2=266666664364722:39182361:1991180:605698:79182361:19364722:39182361:1911397:5891180:60182361:19364722:3922795:155698:7911397:5822795:15364722:39377777775;3=26666666477291:99647921:03847921:03847921:03847921:03877291:99647921:03847921:03847921:03847921:03877291:99647921:03847921:03847921:03847921:03877291:996377777775: 81

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BIOGRAPHICALSKETCHXinruiZhangwasbornandgrewupinChina.ShereceivedherBachelorofScienceinstatisticsfromZhongnanUniversityofEconomicsandLaw,andBachelorofScienceinnance(doubledegree)fromWuhanUniversityin2011.From2011to2013,sheattendedUniversityofFloridaandreceivedthedegreeofMasterofScienceinbiostatistics.ShehadbeenaPh.D.studentintheDepartmentofBiostatisticsatUniversityofFloridasinceAugust2013.Inthesummerof2016,shereceivedherPh.D.degreeinbiostatistics. 86