How to reduce bias in the estimates of count data regression? · 2017. 10. 8. · • Count Data...

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…


Count Data in Clinical Trials

• ParameterEstimation• MaximumLikelihoodEstimate(MLE)

– Biasinthecaseofsmallsampledata– AsymmetricWaldConfidenceInterval


Poisson Regression

• BiasCorrection:MLEcorrectedforbias

• BiasReduction:Likelihoodfunctionorscoreequationsaremodified

• Firth(1993):penalizationofthelikelihoodorscorefunctioninoppositedirectionofbias


Bias correction and Reduction Methods


Firth Method

:FisherInformation

• ForSingleCovariate

FindCIlimitswherelikelihoodsatisfy

• Incaseofmultiplecovariates– Fixparameterfordesiredcovariate– maximizethelikelihoodfunctionoverallotherparameters– Checktheabovecondition– Keepchangingparametervaluetillaboveconditionissatisfied


Profile likelihood based Confidence Interval

21,1max0 5.0 αχ −−= ll

maxl

0l0l

• Aspectsofbiasreductionmethodforsmallsampledata

• ExistenceinthecaseofQuasi-separation

• Performanceincaseoflargedata


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

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Thankyou

[email protected]@cytel.com

How to reduce bias in the estimates of count data regression? · 2017. 10. 8. · • Count Data...

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