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How to reduce bias in the estimates of count data regression? Ashwini Joshi Sumit Singh PhUSE 2015, Vienna
Howtoreducebiasintheestimatesofcountdataregression?
AshwiniJoshiSumitSingh
PhUSE2015,Vienna
13-Oct-15 2PhUSE2015:SP03
PrecisionProblem
bias
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• CountData• PoissonRegression• MaximumLikelihoodEstimateandBiasReductionMethod
• ProfileLikelihoodBasedConfidenceInterval• NegativeBinomialRegression
Agenda
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• Numberofadverseeventsoccurringduringafollowupperiod
• Numberoflesionsornumberofrelapsesinmultiplesclerosispatients
• Numberofhospitalizations• Numberofseizuresinepileptics• Andmanymore…
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Count Data in Clinical Trials
• ParameterEstimation• MaximumLikelihoodEstimate(MLE)
– Biasinthecaseofsmallsampledata– AsymmetricWaldConfidenceInterval
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Poisson Regression
• BiasCorrection:MLEcorrectedforbias
• BiasReduction:Likelihoodfunctionorscoreequationsaremodified
• Firth(1993):penalizationofthelikelihoodorscorefunctioninoppositedirectionofbias
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Bias correction and Reduction Methods
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Firth Method
:FisherInformation
• ForSingleCovariate
FindCIlimitswherelikelihoodsatisfy
• Incaseofmultiplecovariates– Fixparameterfordesiredcovariate– maximizethelikelihoodfunctionoverallotherparameters– Checktheabovecondition– Keepchangingparametervaluetillaboveconditionissatisfied
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Profile likelihood based Confidence Interval
21,1max0 5.0 αχ −−= ll
maxl
0l0l
• Aspectsofbiasreductionmethodforsmallsampledata
• ExistenceinthecaseofQuasi-separation
• Performanceincaseoflargedata
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Scenarios
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Example 1: Multiple Sclerosis• CommonlyusedEndPoints:
– NumberofRelapses– Annualizedrelapserate(ARR)
• Relapseistheappearanceofanewneurologicalabnormality
FitPoissonModel:numberofrelapses~BaselineEDSSEDSS:ExpandedDisabilityStatusScore
data MultScler;inputSubID EDSSAgeSexCode RaceCode ArmCountTime;cards;1 1 19 2 1 1 0 7582 1 54 1 2 2 0 57...31 2 53 2 1 1 0 75132 2 46 1 1 0 1 449;PROC LOGXACT DATA=MultScler;MODELCount=EDSS/link=Poisson;ES/ASEDSS;RATETime;RUN;
PROC LOGXACT DATA=MultScler;MODELCount=EDSS/link=Poisson;ES/ASPMLEEDSS;RATETime;RUN;
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Example 1 (cont.):Proc LogXact
TheSASSystem16:15Saturday,August1,20154
Outputfrom LogXact(r)(v11.0)PROCs _SAS9_1Copyright(c)1997-2015 CytelInc.,Cambridge,MA,USA.
--------------------------------------------------------------------------------------------COUNTREGRESSION--------------------------------------------------------------------------------------------BASICINFORMATION:Datafile:MULTSCLERModel:Count=Intercept+EDSSLinktype:PoissonRateMultiplier:TimeStratumvariable:<Unstratified>Analysistype:Estimate::AsymptoticNumber oftermsinmodel:2Number ofterm(s)dropped: 0Sumof RateMultiplier:20654Number ofrecordsrejected:0Number ofgroups:32--------------------------------------------------------------------------------------------SUMMARYSTATISTICS:--------------------------------------------------------------------------------------------StatisticValueDFP-valueDeviance32.8859300.3275--------------------------------------------------------------------------------------------PARAMETERESTIMATE:--------------------------------------------------------------------------------------------
PointEstimateConfidenceIntervalandPValueforBetaModelTermTypeBetaSEType95.0%C.I.P-Value
LowerUpper2*1-sided--------------------------------------------------------------------------------------------InterceptMLE-5.23581.1002Asymptotic-7.3922-3.07930.0000EDSSMLE-1.53661.0260Asymptotic-3.54750.47430.1342--------------------------------------------------------------------------------------------AnalysisTime=00:00:00
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Example 1 (cont.): Output
PARAMETERESTIMATE:--------------------------------------------------------------------------------------------
PointEstimate ConfidenceIntervalandPValueforBetaModelTermType BetaSEType 95.0%C.I.P-Value
LowerUpper2*1-sided-------------------------------------------------------------------------------------------------------------------------------InterceptMLE-5.23 Asymptotic -7.3922-3.07930.0000EDSSMLE-1.54 Asymptotic -3.54750.47430.1342-------------------------------------------------------------------------------------------------------------------------------
PARAMETERESTIMATE:-------------------------------------------------------------------------------------------------------------------------------
PointEstimate ConfidenceIntervalandPValueforBetaModelTermType BetaSEType 95.0%C.I.P-Value
LowerUpper2*1-sided-------------------------------------------------------------------------------------------------------------------------------InterceptPMLE-5.58 Asymptotic -7.4193-3.75932.706e-009EDSSPMLE-1.16 Asymptotic -2.81780.50360.1721-------------------------------------------------------------------------------------------------------------------------------
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Example 1 (cont.): Comparison
1.10
0.93
1.03
0.85
1.10
0.930.85
1.03
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Example 1 (cont.)
• ProblemofSeparation(PerfectFit)occurswhen– Responsevaluedividessetofcovariateorcombinationsofcovariates
– Acovariatevaluepredictsvalueoftheresponse
ForAge<60,Events=0:Separation
ForGender:Quasi-separation
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Separation, Quasi-separation
Age Gender Eventsold Male 1old Male 1old Female 1
young Female 0young Female 0young Female 0
AnimalData:DatafromHeinze andPuhr (2010).Study:Toinvestigateeffectofheparinized vs non-heparinized,
vascularsubstituteinratsonaneurysmformation.
Theevent ofinterestisaneurysmformation.Thecovariate isnon-heparinized implant.Stratumisfollowuptime.
• Model:Aneur ~Nohep• CaseofQuasi-separation• MLEcannotbeobtained
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Example 2
--------------------------------------------------------------------------------------------PARAMETERESTIMATE:--------------------------------------------------------------------------------------------
PointEstimateModelTerm Type Beta SE--------------------------------------------------------------------------------------------NoHep MLE ? ?--------------------------------------------------------------------------------------------NoHep PMLE 2.1970 1.6665--------------------------------------------------------------------------------------------
?:non-convergence
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Example 2 (cont.)
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Example 2 (cont.)
• Poisson:Variance=mean
• OverdispersioninPoisson:Variance>mean
• NB-2variance=μ +a.μ2
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Negative Binomial Regression
• Data:FromtheUSnationalMedicareinpatienthospitaldatabase(Medpar)forthe1991MedicarefilesforthestateofArizona(Hilbe2011).
• RESPONSE
los:lengthofstayinthehospital
• PREDICTORS
hmo:PatientbelongstoaHealthMaintenanceOrganization(1),orprivatepay(0)
white:PatientidentifiesthemselvesasprimarilyCaucasian(1)incomparisontonon-white(0)
type:Athree-levelfactorpredictorrelatedtothetypeofadmission.1=elective(referent),2=urgent,3=emergency
• FitNB-2 model:los~factor(type)13-Oct-15 20PhUSE2015:SP03
Example 3 – Medpar Data
---------------------------------------------------------------------------------------------------------------------------------------PARAMETERESTIMATE:---------------------------------------------------------------------------------------------------------------------------------------
PointEstimate ConfidenceIntervalandPValueforBetaModelTermType BetaSE Type 95.0%C.I. P-Value
LowerUpper2*1-sided---------------------------------------------------------------------------------------------------------------------------------------
InterceptMLE2.1783 Asymptotic2.13472.22190.0000type_2MLE0.2379 Asymptotic0.13940.33630.0000type_3MLE0.7252 Asymptotic0.57680.87362.04e-019------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------InterceptPMLE2.1781 Asymptotic2.13452.22170.0000type_2PMLE0.2368Asymptotic0.13840.33520.0000type_3PMLE0.7235 Asymptotic0.57520.87172.278e-019---------------------------------------------------------------------------------------------------------------------------------------
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Example 3 (cont.)
0.02220.0502
0.0757
0.02220.0502
0.0756
• PropertiesofBiasreducedEstimatesforsmallsampledata:– Smallerstandarderrors– Shorterconfidenceintervals– Existenceinthecaseofquasi-separation
• ProfileLikelihoodbasedCI– betterthanWaldCIforasymmetriclikelihoodfunction
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Concluding remarks
• FirthD.(1993).“Biasreductionofmaximumlikelihoodestimates”.Biometrika (1993),80,1,pp.27-38
• Heinze G.,Puhr R.(2010).“Bias-reducedandseparation-proofconditionallogisticregressionwithsmallorsparsedatasets”.StatisticsinMedicine
• HilbeJ.(2011).“NegativeBinomialRegression”.CambridgeUniversityPress
• Kosmidis I.(2007).“BiasReductioninExponentialFamilyNonlinearModels”.
• Kosmidis I.(2010).“Agenericalgorithmforreducingbiasinparametricestimation”.ElectronicJournalofStatistics.Vol.4(2010)1097–1112
• LogXact11SoftwareandUserManual13-Oct-15 23PhUSE2015:SP03
References
