IMPACTOFLUMBARPUNCTUREONSURVIVALOFCOMATOSEMALAWIANCHILDREN:APROPENSITY-SCORE-BASEDANALYSISByJung-EunLeeATHESISSubmittedtoMichiganStateUniversityinpartialentoftherequirementsforthedegreeofBiostatistics-MasterofScience2016ABSTRACTIMPACTOFLUMBARPUNCTUREONSURVIVALOFCOMATOSEMALAWIANCHILDREN:APROPENSITY-SCORE-BASEDANALYSISByJung-EunLeeComaisafrequentclinicalpresentationofseverelyillchildreninsub-SaharanAfrica.Itmayhaveanumberofinfectiousandnon-infectiousetiologiesincludingcerebralmalaria,viralencephalitis,andbacterialandtuberculousmeningitis[7].Duetoitshighratesofmortalityandmorbidity,rapiddiagnosisandtargetedinterventionstooptimizeoutcomesarecritical.However,clinicalassessmentalonecannotdistinguishbetweentheseetiologies,identifyingcomaetiologiesbylumbarpuncture(LP)isimportant.LPisaclinicalprocedurethatisusedtocollectandexaminethecerebrospinalsurroundingthebrainandspinalcord.Ithasbeenwidelyutilizedtodiagnosesymptomsandsignscausedbyinfection,on,cancer,orbleedinginthecentralnervoussystem.LPisanessential,simple,andwidelyavailable,procedurethatisgenerallytheonlywaytoelyidentifyunderlyinginfectiouscomaetiologies.DespitetheclearofLP,cliniciansmaybereluctanttoperformtheprocedureinacomatosechild,duetoconcernsthattheproceduremaybringoutcerebralherniationanddeath[4].Inthisthesis,weaimtoassesstheimpactofLPonthesurvivalofcomatosechildren.WeperformedaretrospectivecohortstudyonsurvivalofcomatoseMalawianpediatricinpatientsrecruitedoverconsecutiverainyseasonsfrom1997-2013.Duetothelackofrandomnessinbeingtreated(LP)anduntreated(Non-LP)groups,baselinecharacteristicsarenotbalanced.Weappliedpropensityscoremethodstocompensatetheimbalance.OuranalysisresultsshowednoimpactindeathrateassociatedwithLP.ACKNOWLEDGMENTSFirstandaboveall,IpraiseandthanktotheGod,theAlmighty,forHisshowersofblessingsthroughoutmyresearchworkandgratingmethecapabilitytoproceedsuccessfully.Iwouldliketoexpressmydeepandsinceregratitudetomyresearchsupervisor,Dr.ChenxiLiforgivingmetheopportunitytoworkonthisresearchtopicandprovidinginvalu-ableguidancethroughouttheresearch.Hisvisionandmotivationhavedeeplyinspiredme.Hehastaughtmethemethodologytocarryouttheresearchandtopresenttheresearchworksasclearlyaspossible.Itwasagreatprivilegeandhonortoworkandstudyunderhisguidance.IwouldalsoliketothankprofessorJosephGardiner,andprofessorZhehuiLuoforservingasmycommitteemembers,especiallyforlettingmydefensebeanenjoyablemoment,andfortheirbrilliantcommentsandsuggestions.Iamextremelythankfultomyfamilyfortheirlove,understanding,prayersandcontinu-ingsupport.Myspecialthanksgoestomyfriends,SangInChung,MiranKim,andSunnieOhfortheirprayersandlove.iiiTABLEOFCONTENTSLISTOFTABLES....................................vLISTOFFIGURES...................................viiChapter1Introduction...............................11.1CerebralMalariaandLumbarPuncture.....................11.2SurvivalDataandSurvivalAnalysis.......................31.2.1LumbarPunctureData..........................41.2.1.1TreatmentsandOutcomes...................51.2.1.2CharacteristicVariables....................51.2.1.3TimeVariables.........................81.2.1.4SelectionBias..........................8Chapter2Methods..................................112.1SurvivalAnalysis.................................112.1.1TheHazardFunction...........................122.1.2CompetingRisks.............................142.1.3CoxProportionalHazardsModel....................152.1.4Log-RankTest..............................162.2PropensityScore.................................182.2.1InverseProbabilityofTreatmentWeighting..............192.2.2...............................202.3SASProcedures..................................20Chapter3Results...................................223.1DataExploration.................................223.2PropensityScoreEstimation...........................223.3ImpactofLPonIn-HospitalDeathRate....................273.3.1ResultsfromInverseProbabilityofTreatmentWeighting.......273.3.2Resultsfrom........................303.3.3ComparisonofDeathRatesovererentTimeWindows......323.3.4SubgroupAnalyses............................353.3.4.1ofPapilledema....................353.3.4.2ImpactofLPonsurvivalofchildrenwithincreasedbrainvolume.............................363.3.5ValidationofAssumption........................38Chapter4ConclusionandDiscussion......................40BIBLIOGRAPHY....................................42ivLISTOFTABLESTable1.1:DescriptionsofCategoricalVariablesfromtheOriginalData.TheTotalNumberofsubjectsintheoriginaldatasetis2,399............7Table1.2:DescriptionsofContinuousVariablesfromtheOriginalData.TheTotalNumberofsubjectsintheoriginaldatasetis2,399............8Table1.3:DistributionsofTimetoEvent.......................8Table3.1:BaselineCharacteristicsoftheChildrenbeforePropensityScoreAdjust-ment(AnalysisincludingPapilledema)...................24Table3.2:BaselineCharacteristicsoftheChildrenbeforePropensityScoreAdjust-ment(AnalysisexcludingPapilledema)...................24Table3.3:SummaryStatisticsofPropensityScoreinLPandNon-LPgroups.Pa-pilledemawasincludedinPSestimation.Thenumberofsubjectsis1,010......................................25Table3.4:SummaryStatisticsofPropensityScoreinLPandNon-LPgroups.Pa-pilledemawasexcludedinPSestimation.Thenumberofsubjectsis1,772......................................26Table3.5:NumberofSubjectsinEachStratum....................32Table3.6:ResultsofLog-RankTestsforofTreatmentoverSpTimeWindows...................................35Table3.7:ResultsofAnalysisofMaximumLikelihoodEstimatesfortheIPTWCoxModelConsideringtheMobyPapilledema........36Table3.8:DistributionsofLPandNon-LPgroupswithinPositiveandNegativePapilledemaGroups.............................37Table3.9:DistributionofSubjectswhohaveEdemascore..............38Table3.10:ResultsofAnalysisofMaximumLikelihoodEstimatesfortheIPTWCoxModelConsideringtheMoofEdema...........38Table3.11:ResultsofLinearHypothesesTestingResultsforProportionality:Pa-pilledemaSubgroupAnalysis........................39vTable3.12:ResultsofLinearHypothesesTestingResultsforProportionality:EdemaSubgroupAnalysis..............................39viLISTOFFIGURESFigure1.1:DistributionsofTimetoDeathforLP(LP=1)andNon-LP(LP=0)Groups.....................................9Figure1.2:DistributionsofTimetoDischargeforLP(LP=1)andNon-LP(LP=0)Groups.....................................10Figure3.1:StudyPopulationsintheLPdataset....................23Figure3.2:DistributionofPropensityScoreacrossLPandNon-LPgroups.Pa-pilledemainformationwasincludedinpropensityscoreestimation....25Figure3.3:DistributionofPropensityScoreacrossLPandNon-LPgroups.Propen-sityscoreswereestimatedwithoutPapilledemainformation.......26Figure3.4:ComparisonofHazardfunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithPapilledema.Thevalueoftheband-widthwassetwithdefaultvalue.......................28Figure3.5:ComparisonofHazardfunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithPapilledema.Thevalueoftheband-widthwassetas8..............................28Figure3.6:ComparisonofHazardfunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithPapilledema.Thevalueoftheband-widthwassetas10..............................29Figure3.7:ComparisonofHazardfunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithPapilledema.Thevalueoftheband-widthwassetas12..............................29Figure3.8:ComparisonofHazardfunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithoutPapilledema.Thevalueofthebandwidthwassetwithdefaultvalue....................30Figure3.9:ComparisonofHazardfunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithoutPapilledema.Thevalueofthebandwidthwassetas8............................31viiFigure3.10:ComparisonofHazardfunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithoutPapilledema.Thevalueofthebandwidthwassetas10...........................31Figure3.11:ComparisonofHazardfunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithoutPapilledema.Thevalueofthebandwidthwassetas12...........................32Figure3.12:ComparisonofCumulativeIncidenceFunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithPapilledema....33Figure3.13:ComparisonofCumulativeIncidenceFunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithoutPapilledema..34Figure3.14:ComparisonofDistributionsofTimetoDeathbetweenPositive(PAP=1)andNegative(PAP=0)PapilledemaGroups................36Figure3.15:ComparisonofDistributionsofTimetoDischargebetweenPositive(PAP=1)andNegative(PAP=0)PapilledemaGroups................37viiiChapter1Introduction1.1CerebralMalariaandLumbarPunctureInlowandmiddle-incomecountries,comaisafrequentclinicalpresentationofseverelyillchildren.Themostcommoncauseofthesecomatosepatientsiscerebralmalaria(CM).Theglobalannualincidenceofseveremalariacanbeestimatedatapproximatelytwomillioncasesandabout90%oftheworld'ssevereandfatalmalariaoccurstoyoungchildreninsub-SaharanAfrica[15].TheWorldHealthOrganization(WHO)cerebralmalariaasaclinicalsyndromecharacterizedofcoma(inabilitytolocalizeapainfulstimulus)atleastonehouraftertermi-nationofaseizureorcorrectionofhypoglycemia,asexualformsofPlasmodiumfalciparumparasitesonperipheralbloodsmears,andexclusionofothercausesofencephalopathy(e.g.viralencephalitis,poisoning,andmetabolicdisease)[10].However,thisisnotwellobservedinpractice.Patientswhosecomaiscausedbyotherencephalopathiesorpreviouslyunrecognizedneurologicalabnormalitiesbuthaveincidentalparasitemiamaybeincluded.Duetothelackofspity,usingclinicalevaluationbyitselfcannottiatebetweentheseetiologies[18].Byreasonsoftheamountofriskandthelowfeasibilityofexperimen-tallytreatingallcasesofinfectiouscomaetiologiesinsuspectedchildrenwithCM,utilizinglumbarpuncture(LP)increasesthechanceforcorrecttreatmentandaccuratediagnoses.Lumbarpuncture(LP)isaclinicalprocedurethatisusedtocollectandexaminethe1cerebrospinal(CSF)surroundingthebrainandspinalcord.Ithasbeenwidelyutilizedtodiagnosesymptomsandsignscausedbyinfection,cancer,orbleedinginthecentralnervoussystem.LPisalsousedtomeasuretheCSFpressurewithintheepiduralspace[9].Inparticular,LPhasbeenavaluableandgenerallytheonlyavailabletoolforidentifyingtheetiologyofCM.Despiteits,clinicianswholackaccesstosuchresourceslikepre-proceduralneuro-imagingmaybehesitanttoperformit.ThisisbecauseperformingLPoncomatosepatientsmayincurcerebralherniationanddeathiftheabsenceofbrainshiftorincreasedintracranialpressure(ICP)isnotverthroughneuroimagingormoleculartesting[19,1].Moxonetal.examinedsafetyofLPincomatoseAfricanchildrenwithclinicalfeaturesofCM[2].TheyfoundnoevidencethatundergoingLPincreasesmortalityincomatosechildrenwithsuspectedCM.Thiswasalsotrueinchildrenwithmagneticresonanceimaging(MRI)evidenceofseverebrainswelling.Inaddition,thestudyprovidedevidencethatLPdoesnotplayacausalroleinfatalherniationinthecontextofincreasedICP.TheyconjecturethatLPdoesnotexacerbateherniationinCMbecause,duringLP,theCSFpressureisabletorapidlyequilibrate.Inthisstudy,weextendtheMoxon'sworktosurvivalanalysis.WeconductstatisticalanalysistoassesstheimpactofLPonthedeathrateofcomatosechildren.Inparticular,weexaminewhetheri)thetemporalassociationbetweenLPanddeathimpliescausation;ii)LPcontributestomortalityinchildrenwithCMwhohaveincreasedICPbycomparingthehospitaldeathratesbetweenthetreatmentandcontrolgroups;andiii)ofLPondeathinCMchildrenwithandwithoutPapilledema.21.2SurvivalDataandSurvivalAnalysisSurvivaldataareintheformoftimefromaworiginuntiltheoccurrenceofsomeparticulareventorend-pointsuchasdeath,diseaseonset,machinefailure,automobileaccidents,promotions,orendofmarriage[3].Survivaldataarenotamenabletoconventionalstatisticalproceduresbecauseoftheirspecialfeature,censoring.Censoringisatypicalcharacteristicinsurvivalanalysis,representingaparticulartypeofcoarseneddata.Itoccurswhentheend-pointofinteresthasnotbeenobservedforasubjectduetoendofinvestigation,drop-outofsubjects,ortheexperimentdesignwiththresholdforthetimewindow.Theinformationofsuchcensoredobservationsisthereforeincomplete.Inaddition,survivaldataarealsogenerallynotsymmetricallydistributedbutpositivelyskewed[3].Thesurvivaltimesusuallyhavespecializednon-normaldistributions,suchastheexponential,Weibull,andlog-normal.Hence,conventionalstatisticalanalysismethodsarelimitedfordealingwithsurvivaldata.Unlikeconventionalstatisticalmethods,survivalanalysiscorrectlyincorporatesinfor-mationfrombothcensoredanduncensoredobservationsinestimatingimportantmodelparameters.Twokeyelementsofsurvivaldataarei)atimetoevent/censoringindicatinghowlonguntilthesubjecteitherexperiencedtheeventorwascensored,andii)acensoringindicatordenotingwhetheranobservationaleventwasexperiencedorcensored.Basedonthetwoelements,thesurvivalandhazardfunctionsareestimatedfordescribingthedistri-butionofeventtimes.Thesurvivalfunctionrepresentstheprobabilitythatanindividualsurvives(ordoesnotexperiencetheevent)beyondanygiventime.Thehazardfunctiongivesthepotentialriskorhazardofaneventatthesptime.Itisgenerallyofinterestinsurvivalanalysistodescribetherelationshipofafactorofinterest(e.g.treatment)tothe3timetoevent,inthepresenceofseveralcovariates(e.g.age,gender,race,etc.)Anumberofmodelsfromparametric,nonparametricandsemi-parametricapproachesareavailabletoanalyzetherelationshipofasetofpredictorvariableswiththesurvivaltime.Inoursurvivalanalysis,weexaminethedeathhazardratebetweenpatientswithLP,(i.e.thetreatmentgroup)andwithoutLP(i.e.thecontrolgroup.)Dischargefromthemedicalinstitutionisregardedasacompetingendpoint,whichisactuallydependentontimetodeath.Allthesurvivalanalysesinthisstudyarethereforeforcompetingrisks,i.e.estimatingcause-sphazardofdeath.1.2.1LumbarPunctureDataWeconductaretrospectivecohortstudyofpediatricinpatientsrecruitedfrom1997to2015atthePediatricResearchWardatQueenElizabethCentralHospital(QECH)inBlantyre,Malawi.Thestudypopulationcontains1;772comatosechildrenagedtwomonthstofourteenyears,including1;442patientsinthetreatmentgroupand350patientsinthecontrolgroup.PatientsinthetreatmentgroupreceivedLPsinoneoftwoways.IfthepatientsweredirectlyadmittedtoQECH,theprocedurewasdoneaspartoftheadmissionprocess.Ontheotherhand,ifpatientswerereferredtoQECHfromotheraccidentandemergencydepartments,theLPsweredoneatthoseinstitutions.LPswerenotdoneinthecontrolgroupbecausecliniciansdeterminedthatthechildrenwerenotstableenoughtotoleratetheprocedure.Thedatasetathandiscomprisedofcharacteristicvariablesontheadmissiontothemedicalinstitutionthataclinician'sdecisiontoperformaLPorwereindependentlyassociatedwithdeath.Severaltimevariablessuchastimeofadmission,death,hospitaldischarge,andLPoperationarealsoincludedinthedataset.41.2.1.1TreatmentsandOutcomesThetreatmentgroupcontains81:4%andthecontrolgroupcontains18:6%ofcomatosechildren.Thetotalnumberofsubjectis1;772.ThepatientsinthecontrolgrouphavenotreceivedLPbecauseofoneormoreofvereasonsbelow:1.TheclinicianwasconcernedthatthechildwasnotstableenoughtotolerateanLPbasedonshock,severerespiratorydistressorintractableseizures.2.TheclinicianidenPapilledemaonretinalexamination.3.LPwasattemptedbutfailedfortechnicalreason(s).4.TheparentdidnotconsenttoLP.5.ThechildregainedconsciousnessbeforeLPwasperformed.Foroutcome,3,6,12-hourmortalityratesandoverallmortalityrateduringhospitaliza-tionwereusedtoevaluatethetreatment1.2.1.2CharacteristicVariablesThestatisticalanalysisincludesfactorswhichaphysician'sdecisiontoperformLPandothersobtainedfrompreviousstudiesassociatedwithafataloutcome.Depthofcoma(BlantyreComaScore)SystolicbloodpressureforageWeight-for-heightz-score(nutritionalstatus)Respiratorydistressoracidoticbreathing5PulserateCardiovascularsystemexamination(signsofheartfailure)GenderAdmissionbloodglucoseconcentrationPeripheralparasitedensityHematocritMalariaretinopathystatusPapilledemaWeight-for-heightz-scorewascreatedbyweightandheightaccordingtotheWHOChildGrowthStandards[8].Respiratorydistressoracidoticbreathingwasdeterminedfromsignofgrunting,deepbreathandnormalchestexam:ifanyofthethreeisabnormal,thenitsuggeststhatrespiratorydiseaseispresent.Inthedata,256patientsdidnothavemalariaretinopathystatusonadmission.Asaresultweimputedthevalueofthisstatusutilizinglogisticregressionbasedon:plateletcount,hematocrit,andglucose.Wheretheprobabilityofretinopathyisgreaterthan50%,wesetthisvalueaspositive,ifnotnegative.ThethresholdfortheprobabilitywassettobeconsistentwithMoxon's[2].Papilledemawasconsideredanimportantfactorforthestatisticalanalysis.However,atportionofchildren(i.e.43%)havemissinginformationofPapilledemadetermination.Wethereforeperformedsub-analysiswithoutPapilledemafortroleofPapilledemainstatisticalanalysis.Anysubjectswithmissingdata,otherthanretinopathyorPapilledema,6wereexcludedfromthestatisticalanalysis.Tables1.1and1.2showsexplanatoryvariablesfromtheoriginaldatasetandtheirdescriptions.Table1.1:DescriptionsofCategoricalVariablesfromtheOriginalData.TheTotalNumberofsubjectsintheoriginaldatasetis2,399.VariableDescriptionUnitsCountCOMASCDepthofcomma(Blantyrecomascore:2=unrousablecoma)0=Mostsevere1=Severe2=LessSevereNo.ofMissing3839491,0670BPSTATSystolicbloodpressureforage0=Low1=Normal2=HighNo.ofMissing282,07189211WHWeight-for-HeightZ-score(Nutritionalstatus)0=Normal1=LowNo.ofMissing2,067211121RESPDISRespiratorydistress0=NotPresent1=PresentNo.ofMissing1,228821350PULSESTATPulserate0=Low1=Normal2=HighNo.ofMissing439251,299132NHEARTCardiovascularsystemexamination(Normalhearttest)0=Normal1=AbnormalNo.ofMissing1352,23628SEXGender0=Male1=FemalNo.ofMissing1,1431,2524ADMRETINMalariaretinopathystatus0=NotPresent1=PresentNo.ofMissing8221,321256PAPPapilledema0=NotPresent1=PresentNo.ofMissing1,0332411,1257Table1.2:DescriptionsofContinuousVariablesfromtheOriginalData.TheTotalNumberofsubjectsintheoriginaldatasetis2,399.VariableNo.ofMissingDescriptionLPMeanSDMinMaxADMHCT122HematocritNon-LP21.348.386.0059.00LP23.758.072.0046.90ADMGLUC10AdmissionbloodglucoseNon-LP6.293.980.529.60LP6.784.050.333.00LOGADMPTA189PeripheralparasiteDensityNon-LP10.223.32014.29LP9.224.27015.021.2.1.3TimeVariablesSeveraltimevariablessuchastimesofadmission,deathandhospitaldischargeaswellasLPoperationareincludedinthedataset.ThetimeoriginisthetimeofLPperformedforthetreatmentgrouporthetimeofadmissionforthecontrolgroup.Timeofdeathisregardedastheendpointwhiledischargefromthemedicalinstitutionisregardedasacompetingendpoint.Table1.3comparesthedistributionsoftimetoevent(i.e.deathordischarge)betweenLPandNon-LPgroups.Figures1.1and1.2showdistributionsoftimetoeventinLPandNon-LPgroups.Table1.3:DistributionsofTimetoEvent.LPOutcomeNo.ofSubjectsMeanSDMinMaxNon-LPDischarge2593.5552.340.50026.79Death910.7981.380.01411.17LPDischarge12063.7912.980.20833.77Death2161.4113.270.02131.111.2.1.4SelectionBiasAswementionedabove,decisionsaboutwhetherLPsweremedicallycontraindicatedweremadebytadmittingclinicians,resultinginnon-randomvariationinseverityofillnessamongchildrenwhodidanddidnotreceivedLPs.Duetothelackofrandomnessintreated8Figure1.1:DistributionsofTimetoDeathforLP(LP=1)andNon-LP(LP=0)Groups.(LP)anduntreated(non-LP)groups,anumberofbaselinecharacteristicsofthetwogroupsarenotbalanced.Asaresult,asimplecomparisonofmortalityratesbetweenthetreatedanduntreatedgroupswouldbebiased[12].BecauselessillchildrenwouldbemorelikelytoundergoanLP,resultinginasurvivalbiasaccruingtothatgroup.Toaddressthisbias,weapplypropensityscore(PS)methodstoalleviatetheimbalancebetweenthetwogroupsinthesurvivalanalysis.PSmethodsallowustoreducetheconfoundingthatcanoccurduetoncesinthedistributionsofbaselinecharacteristicsbetweenthegroups.Similartorandomization,PSmethodscompareoutcomesinthetreatedanduntreatedsubjectswhohaveasimilardistributionofmeasuredbaselinecovariates.WediscusspropensityscoremethodsinSection2:2indetail.9Figure1.2:DistributionsofTimetoDischargeforLP(LP=1)andNon-LP(LP=0)Groups.10Chapter2MethodsWecomputetwonon-parametricestimatorsofthehospitaldeathratesfortheLPandnon-LPgroupsrespectively:i)inverseprobabilityoftreatmentweighted(IPTW)kernel-smoothedhazard,andii)kernel-smoothedhazard.WethenperformaPS-adjustedlog-ranktest[7]andalog-ranktesttocomparethecause-sphazardofdeathbetweenLPandnon-LPsubjects.Wefoundthattheproportionalhazardsassumptionholdsaccordingtotheestimateddeathhazardfunction,sothatourmethodsareappropriate.Inadditiontotheprimaryanalysis,wealsoperformsub-analysesrestrictedtopatientsi)withPapilledemaandii)withhighcerebralvolumescores(edema).BytwoIPTWCoxmodelstothesubgroups,wewillexaminewhetherthepresenceofPapilledemaorsevereedemamothecausalofLPontimetodeathandinferthecausalofLPinthesubgroupsofPapilledema(severeedema)andnoPapilledema(nosevereedema).Wediscussfundamentalconceptsofthehazardfunction,competingrisk,Coxproportionalhazardmodelandlog-ranktestthatarerelevanttothisstudy.Finally,weintroduceSASproceduresthatwereutilizedforthedataanalysis.2.1SurvivalAnalysisSurvivaldataaregenerallysummarizedbythesurvivorfunction,thehazardfunction,andthecumulativehazardfunction.LetTbeanon-negativerandomvariablerepresentingthe11timeuntiltheoccurrenceofanevent.WeassumethatTisacontinuousrandomvariablewithprobabilitydensityfunctionf(t)andcumulativedistributionfunctionF(t)=P(T<t)=Rt0f(u)durepresentingtheprobabilitythattheeventhasoccurredbydurationt.SurvivalfunctionS(t)isthentobetheprobabilitythattheeventofinteresthasnotoccurredbydurationt.S(t)=PfTtg=1F(t)=Z1tf(u)du(2.1)2.1.1TheHazardFunctionThehazardfunctionisawidelyusedfunctiontodeterminetheriskorprobabilityofaneventsuchasthelossoflifeatacertaintimet.Thisfunctionisconditionedonthesubjecthavingsurviveduntiltimet,anditisobtainedfromtheprobabilityofthepatient'sdeathattimet.TheTliessomewherebetweentandt+t,conditionalonTbeingequalorgreaterthant,P(tT<t+tjTt).Therateisgivenbytheconditionalprobabilitybeingexpressedwhentheprobabilityperunittimeisdividedbythetimeintervalrepresentingbyt.Theresultingfunctionisnowthelimitingvalue,astgoestozero,i.e.,thehazardfunctionbelow:h(t)=limt!0ˆP(tT<t+tjTt)t˙(2.2)Equation(2.2)therateoftheeventattimet,aslongastheeventhasnotoccurredbeforetheselectedtimet.Ifthesurvivaltimeismeasuredindays,h(t)istheapproximateprobabilitythatanindividual,whoisatriskoftheeventoccurringatthebeginningofdayt,experiencesthateventduringthatday.Therefore,thehazardfunctionatdaytcanberegardedastheexpectednumberofeventsexperiencedbyanindividualintheday,giventhattheeventhasnotoccurredbeforethedayt.12TheconditionalprobabilityinthenumeratorinEquation(2.2)canberepresentedastheratioofthejointprobabilitythatTisintheinterval[t;t+dt)andTttotheprobabilityoftheconditionTt.TheprobabilityP([t;t+dt))canbewrittenasf(t)dtforsmalldt,whiletheprobabilityP(Tt)isS(t)byition.Dividingbydtandpassingthelimitgivestheresult,h(t)=f(t)S(t)(2.3)FromEquation(2.3),itfollowsthath(t)=ddtflogS(t)g;(2.4)andsoS(t)=expH(t)g;(2.5)whereH(t)=Zt0h(u)du:(2.6)ThefunctionH(t)isthecumulativehazardfunction.FromEquation(2.5),thecumulativehazardfunctioncanalsobeobtainedfromthesurvivorfunction,H(t)=logS(t):(2.7)Aninstinctivewaytoestimatethehazardfunctionistocomparethenumberofdeathsandthenumberofindividualatriskatthattime.Iftheassumptionismadethatthehazardfunctionisconstantovertimeperiod,thenthehazardperunittimecanbefoundbyfurtherdividingbythetimeinterval.Inotherwords,ifthenumberofdeathsbythejthdeathtime,13t(j),j=1;2;:::;r,isdjandnjatriskattimet(j),thehazardfunctionintheintervalfromt(j)tot(j+1)canbeestimatedby^h(t)=djnj˝j;(2.8)fort(j)t<t(j+1),where˝j=t(j+1)t(j).Inpractice,thehazardfunctionestimatein(2.8)isnotconsistentandtendtobeirregular.Asaresulttheplotsofthehazardfunctionaremademoreclearby`smoothing'.Thehazardfunctioncanbesmoothedthroughvariousmethods,whichbringaboutaweightedaverageofthevaluesattimeofdeathclosetotestimatedbyhazard^h(t).Anexampleisthekernelsmoothedapproximation(usingtheEpanechnikovkernel)ofthehazardfunction,establishedbytherordereddeathtimes,t(1);t(2);:::;t(r),withdjdeathsandnjatriskattimet(j),asshowninEquation(2.9).hy(t)=1brXj=134(1ttjb2)djnj;(2.9)wherethevalueofbandwidthbneedstobechosen.Theintervalfrombtot(r)beveryvalueoftinhy(t)andt(r)isthelargesttimeofdeath.Foranyvalueoftinthisinterval,thedeathtimeintheinterval(tb;t+b)willcontributetotheweightedaverage.Thebandwidthbisthecontrollingfactorfortheshapeoftheplot.Thelargerbgetsthemore`smooth',orclearthesmoothedcurvebecomes.2.1.2CompetingRisksAcompetingriskssituationoccurswhenboththeeventtimeTanditscauseJaretakenintoconsiderationwherethecausesofeventaremutuallyexclusive.Inourstudy,therearetwotypesofcausesfortheterminalstage,i.e.deathanddischargefromthehospital.The14probabilityofeventbytimetfromcausejisbythecumulativeincidencefunction(CIF)forcausejisFj(t)=P[Tt;J=j].Thecause-spehazardfunctionisthenhj(t)=limdt!0fP(tTt+dt;J=jjTt)dtg(2.10)andthecumulativecause-spehazardfunctionisHj(t)=Zt0hj(u)du(2.11)Therefore,theCIFcanbeexpressedintermsofthehazardsbyFj(t)=Rt0hj(u)S(u)du,j=1;2;:::;mwheremissizeofJ.2.1.3CoxProportionalHazardsModelTheCoxmodelisasemi-parametricmodelinwhichthehazardfunctionofthesurvivaltimeisgivenbyh(t;X)=h0(t)e0X(t);t>0(2.12)whereh0(t)isanarbitraryandunspbaselinehazardfunction,X(t)isavectoroftime-dependentcovariates,andisavectorofunknownregressionparametersfortheexplanatoryvariables[6].Whenusingacovariateoftheform=expf0+1xg(2.13)0isincorporatedintothebaselinehazardfunctionh0(t).Whenxischanged,thehazardfunctionsproportionallychangewithoneanother.Hazardfunctionsforanypairoft15covariatevaluesxiandXjcanbecomparedusinghazardratio:HazardRatio=h0(t)expfxigh0(t)expfxjg=expf(xixj)g;i6=j(2.14)Therefore,thehazardratioisaconstantproportionandthismodelisaproportionalhazardsmodel.Thereasonthatthemodelisreferredtoasasemi-parametricmodelisbecausepartofthemodelinvolvestheunspbaselinefunctionovertime(whichisindimensional)andtheotherpartinvolvesanumberofregressionparameters.Toestimated,Cox[6]introducedthepartiallikelihoodfunction,whicheliminatestheunknownbaselinehazardfunctionh0(t)andaccountsforcensoredsurvivaltimes.ThepartiallikelihoodoftheCoxmodelalsoallowstime-dependentcovariates.Anexplanatoryvariableistime-dependentifitsvalueforanygivenindividualcanchangeovertime.Thevalidityoftheproportionalhazardsmodelcanbetestedbytestingforinteractionbetweentime-dependentcovariatesandtheresponsetime.2.1.4Log-RankTestLog-ranktestisoneofthemostpopularmethodsofcomparingthesurvivalofgroups.Intuitively,onemaycomparetheproportionsofsurvivingatanysptime,butthisapproachdoesnotprovideacomparisonofthetotalsurvivalinformation.Itonlyprovidesacomparisonatsomearbitrarytimepoints.Ontheotherhand,thelog-ranktesttakesthewholefollow-upperiodintoconsiderationwhileitdoesnotrequireinformationoftheshapeofthesurvivalcurvenorthedistributionofsurvivaltimes[5].Thelog-ranktestisusedtotestthenullhypothesisthatthereisnobetween16thegroupsintheprobabilityofaneventatanytimepoint,i.e.thetwogroupshavingidenticalsurvivalorhazardfunctions.Foreacheventtimeineachgroup,thetestcalculatestheobservednumberofeventsandthenumberofexpectedeventsunderthenullofnobetweengroups.Incaseofcensoredsubjects,theindividualsareconsideredtobeatriskoftheeventatthetimeofcensoring,butnotinthesubsequenttimepoint.Letj=1;:::;Jbethedistincttimesofobservedeventsineithergroup.Foreachtimej,letN1jandN2jbethenumberofsubjectsatriskatthestartofperiodjinthetwogroups,respectively.LetNj=N1j+N2j.LetO1jandO2jbetheobservednumberofeventsinthegroupsattimej,andOj=O1j+O2j.GiventhatOjeventshappenedacrossbothgroupsattimej,underthenullhypothesis,O1jhasthehypergeometricdistributionwithparametersNj,N1j,andOj.ThisdistributionhasexpectedvalueE1j=OjNjN1j(2.15)andvarianceVj=Oj(N1j=Nj)(1N1j=Nj)(NjOj)Nj1:(2.16)Thelog-rankstatisticcompareseachO1jtoitsexpectationE1junderthenullhypothesisandisasZ=PJj=1(O1jE1j)qPJj=1Vj:(2.17)Thelog-ranktestismostlikelytodetectabetweengroupswhenthehazardofaneventisconsistentlygreaterforonegroupthananotherovertime,butitisunlikelytodetectawhensurvivalcurvescross[5].Inaddition,thelog-ranktestisatestofsothatitdoesnotprovidethesizeofthebetweenthegroups.17Inthestatisticalanalysis,weapplyinverseprobabilityoftreatmentweightedlog-ranktesttocomparethecause-sphazardofdeathfortheLPandNon-LPgroupsbytreatingthefailuretimesfromcausesotherthanthecauseofinterestascensoredobservations.2.2PropensityScoreAllocationtoLPwasnon-randomandwasassociatedwithseverityofillness.Weconductpropensityscore-basedanalysistoreduceforthisbiasandassesstheimpactonLPonthesurvivalofthepatients.Propensityscore(PS)istheprobabilityoftreatmentassignmentconditionalonthegivenvectorofobservedcovariates[14].PScanbeviewedasabalancingscorebecausethedis-tributionofobservedcharacteristiccovariateswillbesimilarbetweencontrolandtreatmentgroupsbasedonthepropensityscore.Hence,propensityscoreallowsonetoanalyzeanon-randomizedobservationalstudysothatitmimicssomeofthecharacteristicsofarandomizedcontrolledtrial.PSestimationmethodisespeciallyusefulifadatasetcontainsanumberofvariables,possiblycontinuous,becauseitwillbehardtoadjustforsuchhigh-dimensionalconfounderswithcommontechniques.ThepropensityscorewasbyRosenbaum&Rubin[14].LetZibeanindicatorvariabledenotingthetreatmentreceived(Zi=0forcontrolgroupvs.Zi=1fortreatmentgroup)andXibethecovariatesofsubjecti.Then,thepropensityscoreforsubjecti,ei,canbeasei=Pr(Zi=1jXi)(2.18)suchthatZiandXiareindependentgivenei.Consequently,alargenumberofcovariatescanbereducedtoanumberbetween0and1.18Propensityscoresarecommonlyestimatedbyregressionmethodssuchaslogisticregres-sionandprobitregressionofZonX.WeapplylogisticregressionmodeltoestimatethepropensityscoresoftheLPdatagivenalinearcombinationofallthecovariates,asshownin(2.19)lnei1ei=lnP(Zi=1jXi)1P(Zi=1jXi)=0+Xi(2.19)TherearethreecommonwaysofutilizingtheestimatedPS:i)PSisusedasacovariateinadditiontothetreatmentindicatorinamultivariableregressionfortheoutcomeofinterest,ii)subjectsareintobinsoftheestimatedPS,andiii)atreatedsubjectismatchedtooneormorecomparisonsubject(s)basedontheestimatedPS[11].Inthisstudy,weutilizedthetwoPSmethods,i.e.inverseprobabilityoftreatmentweightingand2.2.1InverseProbabilityofTreatmentWeightingInverseprobabilityoftreatmentweighting(IPTW)usesthepropensityscoretoconstructapseudo-populationforestimatingthecausalparametersofinterest.Then,thedistribu-tionofmeasuredbaselinecovariatesisindependentoftreatmentassignmentinthepseudo-population[16].Asweearlier,letZibeanindicatordenotingtreatmentassignmentandeibethepropensityscoreforithsubject.Weightforsubjecti,i.e.wi,canbeaswi=Ziei+(1Zi)1ei:(2.20)Thepseudo-populationcreatedbyIPTWconsistsofwicopiesofeachsubjectiandtheindividual'sweightisequaltotheinverseoftheprobabilityofreceivingthetreatmentthatthesubjectactuallyreceived.BecausethedistributionofebetweenZ=0andZ=1arethesameintheweightedpseudo-population,theconnectionbetweenZandeisthenremoved.192.2.2involvespartitioningsubjectsintomutuallyexclusivesubsetsbasedontheestimatedpropensityscore.Subjectsarerankedaccordingtotheirestimatedpropensityscoreandthenintosubsetsbasedonthresholds.Acommonapproachistostratifysubjectsintoeequal-sizestrataofthepropensityscores.RosenbaumandRubin[17]claimedthatstratifyingonthequintilesofthepropensityscoreeliminatesapproximately90%ofthebiasduetomeasuredconfounderswhenestimatingalineartreatmentAnimprovementinbiasreductionshouldappearwithincreasingnumberoftotalstrata.Withineachpropensityscorestratum,treatedanduntreatedsubjectshavenearlysimilarvaluesofthepropensityscore.Therefore,thedistributionofcovariateswillbeapproximatelysimilarbetweentreatedanduntreatedgroupsinthesamestratumifthepropensityscorehasbeencorrectlysped.Then,thetreatmentareestimatedineachstratumwithaweightedaverageoftheewillgiveanoverallestimateofthetreatment2.3SASProceduresSASVer.9.4software1wasusedforthecurrentdataanalysis.WeappliedtwoSASproce-dures:i)PROCLIFETEST,anon-parametricprocedureforestimatingthesurvivorfunc-tion,andii)PROCPHREG,asemi-parametricprocedurethattheCoxproportionalhazardsmodel,inthestatisticalanalysisoftheLPdataset.TheLIFETESTprocedurecanbeusedtocomputenonparametricestimatesofthesur-vivorfunctioneitherbytheproduct-limitmethod(alsocalledtheKaplan-Meiermethod)orbythelifetablemethod.Theprocedureproducesthesurvivaldistributionfunction(SDF),1Copyrightc2015SASInstituteInc.SASandallotherSASInstituteInc.productorservicenamesareregisteredtrademarksortrademarksofSASInstituteInc.,Cary,NC,USA.20thecumulativedistributionfunction(CDF),theprobabilitydensityfunction(PDF),andthehazardfunction.WecomputeIPTWkernel-smoothedhazardfunction,kernel-smoothedhazardfunction,andtheAalen-Johansenestimatorofcumu-lativeincidencesfunction(CIF)ofdeath.NotethatweonlycomputeCIFforhazardsinceIPTWCIFiscurrentlynotavailableinSASsoftware.PROCLIFETESTpro-videsseveralrankteststoevaluateofbetweentreatedanduntreatedgroups.Weadoptthelog-ranktesttocomparethehospitaldeathrateforthetreatedanduntreatedgroups.TheLIFETESTprocedureisalsoappliedformakingplotstocomparehazardcurvesamonggroups.ThePHREGprocedureperformssubgroupanalysesoftheLPdatabasedontheCoxproportionalhazardsmodel.Theprocedureisalsoappliedtotestproportionalhazardassumption.21Chapter3Results3.1DataExplorationTheLPdatahaveinformationofcomatosechildren(Blantyrecomascore2)agedtwomonthsto14years.Itoriginallycontains2;399observations.Wediscardedabout26%oftheindividuals(i.e.629subjects)afterretinopathyimputationduetomissingexplanatoryvariablesbecausetheycausedmissingvaluesinpropensityscoreestimation.Thereareninediscretevariables(suchasdepthofcoma,gender,andbloodpressure),andthreecontin-uousvariables(suchasadmission,bloodglucoseconcentration,andhematocrit).WhenPapilledemaisconsideredasacovariate,theLPdatacontains1;010totalnumberofsub-jectsintheLPdataincluding810treatedand200untreatedindividuals.InthecaseofwherePapilledemaisexcludedinpropensityscoreestimation,thereare1;772totalnumberofsubjectsinthedataincluding1;442treatedand350untreatedindividuals.ThestudypopulationintheLPdatasetisshowninFigure3.1,Table3.1,andTable3.2.3.2PropensityScoreEstimationPropensityscoresareusedtoreduceconfoundingandthusincludevariablesthoughttoberelatedtobothtreatmentandoutcome.Toestimateapropensityscore,acommonstepistousealogitorprobitregressionwithtreatment(i.e.indicationofLPperformedinour22Figure3.1:StudyPopulationsintheLPdataset.23Table3.1:BaselineCharacteristicsoftheChildrenbeforePropensityScoreAdjustment(AnalysisincludingPapilledema).VariableLP(N=810)Non-LP(N=200)MeanSDMeanSDCOMASC1.310.711.290.74BPSTAT1.010.191.030.27WH0.090.290.120.33RESPDIS0.340.470.420.49PULSESTAT1.570.531.560.54NHEART0.970.180.930.26SEX0.540.500.520.50ADMRETIN0.590.490.740.44PAP0.140.350.420.49ADMHCT23.868.1721.278.58ADMGLUC6.864.086.283.31LOGADMPTA8.964.4610.173.59Table3.2:BaselineCharacteristicsoftheChildrenbeforePropensityScoreAdjustment(AnalysisexcludingPapilledema).VariableLP(N=1,422)Non-LP(N=350)MeanSDMeanSDCOMASC1.300.711.210.76BPSTAT1.030.221.010.26WH0.080.280.100.30RESPDIS0.370.480.480.50PULSESTAT1.570.531.550.54NHEART0.960.190.930.26SEX0.530.500.540.50ADMRETIN0.580.490.750.44ADMHCT23.958.0721.348.38ADMGLUC6.764.066.263.99LOGADMPTA9.224.2610.223.2324Figure3.2:DistributionofPropensityScoreacrossLPandNon-LPgroups.Papilledemainformationwasincludedinpropensityscoreestimation.Table3.3:SummaryStatisticsofPropensityScoreinLPandNon-LPgroups.PapilledemawasincludedinPSestimation.Thenumberofsubjectsis1,010.TreatmentNo.ofSubjectsMeanSDMinimumMaximumLP8100.8240.1120.3180.967Non-LP2000.7130.1630.2680.968data)astheoutcomevariableandtheexplanatoryvariables.Sinceourdependentvariableisbinary,weappliedlogisticregressionwiththe12explanatoryvariables.WeestimatedtwosetsofPSwithandwithoutPapilledemainformation.Onceapropensityscorehasbeenestimatedforeachobservation,wemustensurethatthereisoverlapintherangeofpropensityscoresacrossLPandNon-LPgroups.Noinfer-encesabouttreatmentcanbemadeforatreatedindividualforwhomthereisnotacomparisonindividualwithasimilarpropensityscore.Commonsupportissubjectivelyassessedbyexaminingagraphofpropensityscoresacrosstreatmentandcontrolgroups.TheoverlapofthedistributionofthepropensityscoresacrossLPandNon-LPgroupsisdis-playedinFigures3.2and3.3.ThepropensityscoresofchildreninLPandNon-LPgroups25Figure3.3:DistributionofPropensityScoreacrossLPandNon-LPgroups.PropensityscoreswereestimatedwithoutPapilledemainformation.Table3.4:SummaryStatisticsofPropensityScoreinLPandNon-LPgroups.PapilledemawasexcludedinPSestimation.Thenumberofsubjectsis1,772.TreatmentNo.ofSubjectsMeanSDMinimumMaximumLP1,4220.8120.0770.4310.958Non-LP3500.7660.0980.3930.928overlappedtlyindicatingthatpropensityscorematchinganalysiswasfeasible.Inadditiontooverlapping,thepropensityscoreshouldhaveasimilardistribution(i.e.balanceddistribution)inthetreatedandcomparisongroups.Aroughestimateofthepropen-sityscore'sdistributioncanbeobtainedbydescriptivestatisticssuchasmeanandstandarddeviation(seeTables3.3and3.4).ThemeanpropensityscorewithPapilledemaintreatedis0.824withastandarddeviation(SD)of0.122andinuntreated0.713,withSD0.168.WhenexcludingPapilledemainPSestimation,themeanpropensityscoreintreatedis0.81withSD,0.077andinuntreated0.766withSD0.09.Balanceon12covariateswascheckedbasedonstandardizedasatestmeasurement.Ithasbeensuggestedthatifthestandardizedisgreaterthan10%,thereisameaningfulimbalanceincovariatesin26twogroups[13].Wefoundthatonlyfourvariables,(i.e.comascore,bloodpressure,pulserate,andgender),andthreevariables,(i.e.weight/heightscore,pulserate,andgender)werebalancedinthedatasetwithandwithoutPapilledema,respectively.AfterPSadjustment,balancesinall12covariates(11covariatesexcludingPapailledema)wereachievedacrossLPandNon-LPgroups.3.3ImpactofLPonIn-HospitalDeathRateBasedonthepropensityscoreadjustedanalysis,weassessedimpactoflumbarpunctureonsurvivalofcomatosechildren.WefoundthattherewasnotinhospitaldeathratesbetweentreatedanduntreatedgroupswithPapilledemainformation.However,whenPapilledemainformationwasexcludedinthePSestimation,wefoundatbetweenthetwogroups.Theseresultswerethecaseregardlessofthepropensityscoreadjustmentmethods.3.3.1ResultsfromInverseProbabilityofTreatmentWeightingAccordingtotheIPTWmethod,thehospitaldeathratewasnottlytinchildrenwhounderwentLPcomparedtothosewhodidnotifthePapilledemainformationwasincludedinthepropensityscoreestimation(p-valuefromalog-ranktest=0.775).Hazardfunctionswithtbandwidthvalues,(i.e.defaultvalueoptimizedbySAS,8,10and12)areshowninFigures3.4,3.5,3.6,and3.7,respectively.Ontheotherhand,wefoundatbetweenthegroupswherePa-pilledemainformationwasexcludedinthePSestimation.Thep-valuefromalog-rankis0.009.HazardfunctionsbasedonexcludingPapilledemaareshowninFigures3.4,3.9,3.10,27Figure3.4:ComparisonofHazardfunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithPapilledema.Thevalueofthebandwidthwassetwithdefaultvalue.Figure3.5:ComparisonofHazardfunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithPapilledema.Thevalueofthebandwidthwassetas8.28Figure3.6:ComparisonofHazardfunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithPapilledema.Thevalueofthebandwidthwassetas10.Figure3.7:ComparisonofHazardfunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithPapilledema.Thevalueofthebandwidthwassetas12.29Figure3.8:ComparisonofHazardfunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithoutPapilledema.Thevalueofthebandwidthwassetwithdefaultvalue.and3.11,withtbandwidthvalues(i.e.defaultvalueoptimizedbySAS,8,10and12),respectively.3.3.2ResultsfromForbasedanalysis,wethePSintoestratacontainingbothpatientsinthetreatmentgroupandthecontrolgroup.EachstratumhassimilardistributionsofPSfortreatedanduntreatedsubjectsandthesamplesizesinestrataaresimilarwitheachother(Table3.5).WefoundconsistentresultswiththatfromtheIPTWmethod.WiththePSadjustmentincludingPapilledema,wefoundnotrenceinhospitaldeathratebetweentreatedanduntreatedgroups.Thep-valuefromalog-ranktestis0:309.ThePS-adjustedcumulativeincidencefunctions(CIFs)alsofoundnocebetweenthegroups.Thep-valuefromGray'stestforequalityofCIFsis0:3118(Figure3.12).Onthe30Figure3.9:ComparisonofHazardfunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithoutPapilledema.Thevalueofthebandwidthwassetas8.Figure3.10:ComparisonofHazardfunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithoutPapilledema.Thevalueofthebandwidthwassetas10.31Figure3.11:ComparisonofHazardfunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithoutPapilledema.Thevalueofthebandwidthwassetas12.Table3.5:NumberofSubjectsinEachStratum.StratumWithPapilledemaWithoutPapilledemaLP(%)Non-LP(%)LP(%)Non-LP(%)1112(55.45)90(44.55)236(66.67)118(33.33)2164(81.19)38(18.81)290(81.69)65(18.31)3172(85.15)30(14.85)274(77.40)80(22.60)4181(89.60)21(10.40)310(87.32)45(12.68)5181(89.60)21(10.40)312(88.14)42(11.86)otherhand,withthePSadjustmentexcludingPapilledema,thereistbetweenthetreatedanduntreatedgroups,i.e.p-valuefromalog-ranktestis0:0001.Thep-valuefromGray'stestforequalityofCIFsis<0:0001(Figure3.13).3.3.3ComparisonofDeathRatesovertTimeWindowsWecomparedthetimespdeathratesoverthree,sixand12hoursfromthetimeorigin(i.e.thetimeofLPperformedfortreatedandthetimeofadmissionforuntreatedsubjects).32Figure3.12:ComparisonofCumulativeIncidenceFunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithPapilledema.33Figure3.13:ComparisonofCumulativeIncidenceFunctionsbetweenLP(LP=1)andNon-LP(LP=0)groups.ThePSwasestimatedwithoutPapilledema.34Table3.6:ResultsofLog-RankTestsforofTreatmentoverSpTimeWindows.TimeChi-SquarePr>ChiSq3Hour1.0290.3106Hour0.9250.33612Hour2.8710.090IPTWlog-ranktestsbasedontheestimatedPSwithPapilledemawereusedtocomparetheofLPondeathrateoversptimewindows.Wefoundnottofthetreatment,i.e.p-valueof0.310forthreehourfromthetimeorigin,0.336forsixhour,and0.09for12hourfromlog-ranktests,respectively(Table3.6).3.3.4SubgroupAnalyses3.3.4.1ofPapilledemaToexaminewhetherthepresenceofPapilledemachangestheimpactofLPonthesurvivalofthepatients,weassessedtheinteractionbetweenPapilledemaandLPintheIPTWweightedCoxregressionmodel.WefoundthattheofLPonsurvivalwasnott(p-value=0.279)betweenchildrenwithpositiveandnegativePapilledema(Table3.7).Inaddition,wefoundnotofLPineachofthetwogroups,i.e.p-valuesare=0.8963withpositiveand0.167withnegativePapilledemagroups.WefoundthatPapilledemaisaconfounder.ThisisbecausePapilledemaisacommoncausetoLPoperationandtheoutcome(i.e.deathordischarge).IfachildwassuspectedtohavePapilledemabyafunduscopicexamination,LPwasnotperformed.IntheLPdataset,wefoundonly58%ofsubjectswithpositivePapilledemahadLPwhile85%ofpatientswithnegativePapilledemahadLP(Table3.8).Inaddition,childrenwithnegativePapilledematendtohavelongertimetodeath/discharge(Figures3.14and3.15).Therefore,Papilledemashouldbeaccountedforaconfounderintheanalysistocorrectlyestimatetherelationship35Table3.7:ResultsofAnalysisofMaximumLikelihoodEstimatesfortheIPTWCoxModelConsideringtheectMobyPapilledema.ParameterEstimateErrorChi-Squarep-valueLP0.0390.3020.0170.896PAP1.1240.34710.4870.001LP*PAP-0.4450.4121.1680.279Figure3.14:ComparisonofDistributionsofTimetoDeathbetweenPositive(PAP=1)andNegative(PAP=0)PapilledemaGroups.betweendependentandindependentvariables.3.3.4.2ImpactofLPonsurvivalofchildrenwithincreasedbrainvolumeIntheLPdata,thevariableEdemacontainsinformationofincreasedbrainvolumeexaminedbythebrainMRIscan.Itisacategoricalvariablewith9tvaluesfrom0to8wherehighernumbermeansworsecondition.131childrenunderwentMRIscanswith101treatedand30untreatedsubjects(Table3.9)wherethePSwereestimatedwithPapilledema36Table3.8:DistributionsofLPandNon-LPgroupswithinPositiveandNegativePapilledemaGroups.PapilledemaNo.ofSubjectsTotalLPNon-LPPositive11483197Negative696117818Total8102001,010Figure3.15:ComparisonofDistributionsofTimetoDischargebetweenPositive(PAP=1)andNegative(PAP=0)PapilledemaGroups.37Table3.9:DistributionofSubjectswhohaveEdemascore.LPSeverityTotal012345678Performed2068121725265101Not-Performed012312114530Total218121319363010131Table3.10:ResultsofAnalysisofMaximumLikelihoodEstimatesfortheIPTWCoxModelConsideringtheectMoofEdema.ParameterEstimateErrorChi-Squarep-valueLP1.4351.2210.0780.780EDEMA2.7931.0946.5190.011LP*EDEMA-0.6861.3650.2450.620information.WeredtheEdemavariableasbinarywithnon-severe(i.e.edema<7)andsevere(i.e.edema7)basedonphysician'srecommendationandtheIPTWCoxmodel.Asaresult,wefoundnocantectmobyEdemaonthesurvival(p-value=0:620)(Table3.10).Inaddition,wefoundnotofLPineachofthetwogroups,i.e.p-valuesare0.700withnon-severeEdemaand0.680withsevereEdemagroups.3.3.5ValidationofAssumptionWhenmodelingaCoxproportionalhazardmodel,akeyassumptionisproportionalhazards.Tovalidatetheassumption,weincludedtimedependentcovariatesintheCoxmodelbycreatingproductsofthecovariatesandafunctionoftime.Inthisstudyweappliedthelogfunctionofsurvivaltime.Ifanyofthetimedependentcovariatesaretitindicatesthatthecovariateisnotproportional.InSAS,itispossibletocreateallthetimedependentvariableinsidePROCPHREG.ByusingtheTESTstatement,wetestedallthetimedependentcovariatesallatonce.WethattheproportionalassumptionisvalidforvariablesLP(p-value=0.895)andPapilledema(p-value=0.539),andalsoforthe38interactionbetweenLPandPapilledema(p-value=0.981)fromthesubgroupanalysisofPapilledema(Table3.11).TheassumptionisalsovalidinthesubgroupanalysisofEdema.Thep-valuesare0.380forLP,0.359forEdema,and0.266fortheinteractionbetweenLPandEdema(Table3.12).Table3.11:ResultsofLinearHypothesesTestingResultsforProportionality:PapilledemaSubgroupAnalysis.VariableWaldChi-Squarep-valueLP0.0170.895PAP0.3770.539LP*PAP0.0010.981Table3.12:ResultsofLinearHypothesesTestingResultsforProportionality:EdemaSub-groupAnalysis.VariableWaldChi-Squarep-valueLP0.7700.380EDEMA0.8390.359LP*EDEMA1.2360.26639Chapter4ConclusionandDiscussionInthisstudy,weconductedaretrospectiveanalysistoassesstheimpactoflumbarpuncture(LP)onsurvivalofcomatoseMalawianchildren.Overall,ouranalysisresultsshowednoimpactonsurvivalofthepatientsassociatedwithLP.Wefoundthatafterbalancingthetreated(LP=1)anduntreated(LP=0)groupsusingPapilledemiainformation,itdidnotre-sultinatinthehospitaldeathrates.AlthoughexclusionofPapilledemainformationfromthestatisticalanalysisresultedintheoppositeconclusion(i.e.thereisatinthedeathrates),weconsideredPapilledemaasanimportantcovari-ateintheanalysisbecausetheinformationistheprimaryelementinmakingthedecisionforperformingLP.WealsothattstatusinPapilledemaandEdemadoesnothaveaterenceinthehazardofdeathbetweenbothtreatedanduntreatedpatients.TheLPdatahaveafundamentallimitation,i.e.lackofrandomnessinthetreatedandcontrolgroups,bythenatureofobservationalstudy.Wecompensatedthislimitationbyuti-lizingpropensityscore(PS)methods.Thepropensityscoreswereestimatedbyusingalinearcombinationof12characteristiccovariates.ThroughthePSmethod,wecouldobtainedbal-anceincovariatesbetweentreatmentandcontrolgroups.Weutilizedthepropensityscore,anaveragetreatmentforeachsubject,intwoways,i)inverseprobabilityoftreatmentweighting(IPTW)andii)Wefoundthesameresults,i.e.notevi-denceofLPimpactonsurvivalofcomatosepatientswithPapilledemainformation,through40bothmethods.AlthoughPSmethodovercamethelackofrandomnessintheLPdata,wewerealsoconfrontedbyitsownlimitation.Becausethepropensityscoreisbasedonobserveddataandclinicians'experience,itispossibletohaveunmeasuredandunobservedconfounderswhichcannotbecontrolled.Therefore,covariatesintheLPdatamaycausebiasontheoutcomeandtheremightbeotherunobservedfactorsthatwouldthedecisiontoundergotheLPoperation.Inaddition,thePSmethodonlyusesobservedcovariatessothatweneededtodiscardalargeportionofthedatasetwithmissingcovariateinformation.Asaresult,weexcluded57:9%and26:1%oftheoriginaldataintheanalysiswithPapilledemaandwithoutPapilledema,respectively.Theexclusionmayincurdecreaseinpoweraswellaslossofusefulinformation.41BIBLIOGRAPHY42BIBLIOGRAPHY[1]eA.R.Lumbarpunctureandbrainherniationinacutebacterialmeningitis:areview.JournalofIntensiveCareMedicine,22:194{207,2007.[2]MoxonC.A.,ZhaoL.,ChenxiL.,etal.Safetyoflumbarpunctureincomatosechildrenwithclinicalfeaturesofcerebralmalaria.Neurology,2016.[3]D.Collett.ModellingSurvivalDatainMedicalResearch,SecondEdition.Chapman&Hall/CRCTextsinStatisticalScience.Taylor&Francis,2003.[4]NewtonC.R.,KirkhamF.J.,WinstanleyP.A.,etal.Intracranialpressureinafricanchildrenwithcerebralmalaria.TheLancet,337:573{576,1991.[5]HarringtonD.LinearRankTestsinSurvivalAnalysis.JohnWiley&Sons,Ltd,2005.[6]CoxD.R.Regressionmodelsandlifetables.JournaloftheRoyalStatisticalSociety,34:187{220,1972.[7]AkpedeG.O.,AbiodunP.O.,andAmbeJ.P.Etiologicalconsiderationsinthefebrileunconsciouschildintherainforestandaridregionsofnigeria.EastAfricanMedicalJournal,73:245{250,1996.[8]WorldHealthOrganizationMulticentreGrowthReferenceStudyGroup.WHOChildGrowthStandards:Methodsanddevelopment.Geneva:WorldHealthOrganization,2006.[9]JohnsHopkinsMedicine.Lumbarpuncture.Accessed:2016-02-06.[10]WorldHealthOrganization.Severefalciparummalaria.worldhealthorganization,com-municablediseasescluster.SocietyofTropicalMedicineandHygiene,94:S1{90,2000.[11]AustinP.C.Anintroductiontopropensityscoremethodsforreducingtheofconfoundinginobservationalstudies.multivariatebehavioralresearch.MultivariateBehavioralResearch,46:399{424,2011.43[12]AustinP.C.Theuseofpropensityscoremethodswithsurvivalortime-to-eventout-comes:reportingmeasuresofsimilartothoseusedinrandomizedexperiments.StatisticsinMedicine,33:1242{1258,2014.[13]AustinP.C.,GrootendorstP.,andAndersonG.M.Acomparisonoftheabilityoftpropensityscoremodelstobalancemeasuredvariablesbetweentreatedanduntreatedsubjects:amontecarlostudy.StatisticsinMedicine,26:734{753,2007.[14]RosenbaumP.R.andRubinD.B.Thecentralroleofthepropensityscoreinobserva-tionalstudiesforcausalBiometrika,70:41{55,1983.[15]BlackR.E.,CousensS.,JohnsonH.L.,etal.Global,regional,andnationalcausesofchildmortalityin2008:asystematicanalysis.Lancet,375:1969{1987,2010.[16]P.R.Rosenbaum.Model-baseddirectadjustment.TheJournaloftheAmericanStatis-tician,82:387{394,1987.[17]P.R.RosenbaumandD.B.Rubin.Reducingbiasinobservationalstudiesusingsub-onthepropensityscore.JournaloftheAmericanStatisticalAssociation,79:516{524,1984.[18]TaylorT.E.,FuW.J.,CarrR.A.,etal.tiatingthepathologiesofcerebralmalariabypostmortemparasitecounts.NatureMedicine,10:143{145,2004.[19]vanCrevelH.,HijdraA.,andJ.deGans.Lumbarpunctureandtheriskofherniation:whenshouldweperformct?JournalofNeurology,249:129{137,2002.44