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IncrementalResponseModelingwithSASEM(14.1)AlsocalledNetLiftModeling

BruceLundMagnifyAnalyticSolutions,DivisionofMarketingAssociates

MSUG Meeting Feb 11, 2016

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AnIncrementalResponsebyaprospectisaresponsethatwouldnothaveoccurredwithoutthetreatment(i.e.themarketingcampaign).

IncrementalResponseModel(IRM)predictstherateatwhichprospectswillbeincrementalduetothetreatment.Forexample,anincrementalresponsemodelrateof0.02meansthat2%ofsuchprospectswillmakeanincrementalresponseduetothetreatment.Note:AnIRMratecanbenegative.

DifferentfromaPropensityModel(PM).PMassignsaprobabilitytoaprospectgivingthelikelihoodthattheprospectwillrespond(duringafixedfuturetimeperiod).Untilrecentyears,PMswereusedinlieuofIRMs.

AIRMisfittoacampaignwithrandomlyselectedtreatedandcontrolgroups.Withexpectationthatfuturecampaignwillhavesametreatmentandsimilarprospects.Mustbeabletotrackresponsesbycontrolgroup.IRMisrelativelyrecent.Anearlypaperonthistopicdatesto2002.

WhatisanIncrementalResponseModel?(akaNetLiftModel)

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FromLee,etal.IncrementalResponseModelingusingSAS(R)EnterpriseMiner(2013),SGFPaper0962013

Aschematic

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SomeresponderswithintheTreatmentGroupareincrementalwhileotherresponderswouldhaverespondedwithouttheoffer.IRMtriestopredictwhicharewhich.

AfterconsideringincrementalResponses thenextstepistoestimateincrementalrevenueorprofitfromaprospect.SASusesOutcometorefertotherevenueorprofit.

ThisextrameasurementisneededwhentheOutcomefromaResponse willvaryaccordingtotheresponse.

GenerallytheOutcomewillvary

buyacar(profitvariesgreatly),

openabankaccount(initialdepositwillvary),

makeadonation(amountvaries).

Outcome:IncrementalRevenueorProfit

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ForeachProspect:Treatment: 1=treatedor0=controlResponse: 1=respondedor0=notrespondedPredictorsforResponse: XiOptional:Outcome (RevenueorProfit): Usuallydollaramount

Outcome equalsmissingfornonrespondersOptional:PredictorsforOutcome: Zi

XiandZimayoverlap.

InputDataforIncrementalResponseModeling

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Lo,V.(2002),TheTrueLiftModel:ANovelDataMiningApproachtoResponseModelinginDatabaseMarketing.ACMSIGKDDExplorationsNewsletter4:7886.

Zhong,Jun(2009),VPTargetingandAnalytics,CardServicesCustomerMarketing,WellsFargointhepresentation:PredictiveModeling&TodaysGrowingDataChallengesatPredictiveAnalyticsWorldinSanFrancisco,CA2009.

KimLarsen(2009)introducesNetLiftModelingtoSASatM2009,12th AnnualDataMiningConference,LasVegas.

Larsen,Kim.(2010),NetLiftModels:OptimizingtheImpactofYourMarketingEfforts.SASCourseNotes.Cary,NC:SASInstituteInc. SASTrainingClassonNetLiftModel

Lund,Bruce(2012)DirectMarketingProfitModel,MWSUG2012,paperCI04

LeeT.,ZhangR.,MengX.,andRyanL.(2013),IncrementalResponseModelingUsingSASEnterpriseMiner,ProceedingsoftheSASGlobalForum2013Conference,Paper0962013.(Seeotherreferencesgiveninthispaper.)

HistoryandReferences

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Alinkfunction isafunctionthattransformstheexpectedresponsetoalinearcombina onofpredictorsj*Xj.Thatis,LINK()=j*Xj =X

Twoimportantlinkfunctioninthecaseofabinaryresponsearethelogit linkfunction,andtheprobit linkfunction.Let betheexpectedresponse.

LogitLink: log( /(1 ))=XEquivalently: =P(Y=1)= X 1 X

ProbitLink: 1()= Xwhere1 istheinverseofthestandardnormalcumulativedistribution.

Equivalently: =P(Y=1)=

exp 0.5 X 2 dz

BothLogitandProbitlinkshavearoleintheSASEMIncrementalResponseNodeaswillbediscussed.

Background:LinkFunctions

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TheIncrementalResponseNodeinSASEM

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DataNode:Yourdata(herecalledA_SIM).

DataPartition:DividesYourdataintoTrainingandValidation.StratifybyTreatmentandTargettoavoidthepossibilityofabiasedsample.ThisNodecanbeomitted.

IncrementalResponseNode:ThisNodeoffersthechoiceoftwomodels:CombinedModelDifferenceModel

Bothmodelsinclude:(1)ResponseModelforbinarytarget(2)OutcomeModelforintervaltarget(optional)e.g.Revenue,Profit

Ifboth(1)and(2),thenthereisafinalcomputationtogiveexpectedincrementalOutcome(revenue,profit,etc.)perprospect.

A. Treatedandcontrolgroupsareappended.B. IndicatorvariableTshowsifprospectistreated(T=1)orcontrol(T=0).C. Supposethereisone predictorX.TheninteractionwithTisX*TD. FittoResponseY.TheCombinedModelislogistic:Y =P(Y=1)=exp( +*X+*T+*(X*T))/(1+exp( +*X+*T+*(X*T)))

E. Afterfitting,assume=1,=2,=3,and=4.Nowforeachprospectcompute2probabilities:1. Y _ =P(Y=1|T=1)=exp(1+2*X+3+4*X)/(1+exp(1+2*X+3+4*X))

=exp(4+6*X)/(1+exp(4+6*X))

2. Y _ =P(Y=1|T=0)=exp(1+2*X)/(1+exp(1+2*X))

Thedifferencescore(incrementalrate)is: _ _

CombinedModel IllustratedStepbyStep

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2

A

C

IfthereisOutcomeTarget: LinearRegressionModelpredictstheOutcome:

SupposethereisasinglepredictorZ.ThenZ*Tistheinteractionwithtreatment

Modelis:Y =*Z+*T+*(Z*T)FitforOutcome.(Responders)

SetT=1toobtainY _ andT=0forY _

Finally,thereisEXPECTEDincrementalOutcomeforaprospect:

R _ =Y _ * Y _ Y _ *Y _ FixedCost

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T = 1

Incr.

CombinedModel Illustrated

(Estimatedoutcomeforatreatedresponder)X (LikelihoodofResponseifTreated)

(Estimatedoutcomeforacontrolresponder)X (LikelihoodofResponseifControl)

Y =*Z+*T+*(Z*T)ModelisfitwhenOutcome.(Responders)

CanwesetOutcome=0whenOutcome=.andrunregressiononallprospects?ThiswouldnotaccomplishthepurposeoftheOutcomeModelwhichis:

Topredicttheexpectedoutcome forTreated(Control)foranewProspectListGiventhataprospectisaresponder,wewanttherespondersOutcometobeestimatedfromrealOutcomes.Thentheexpectedoutcome foranewprospectistheEstimatedOutcome XProbabilityofResponse.(UsingzerosintheOutcomeModelwoulddrasticallyreduceexpectedoutcome.)ButwhataboutSelectionBias?CanaregressionmodelthatisfitonlyonRespondersbeusedtoestimateoutcomeforeveryoneinanewlist?MaybethereissomedefectamongthenonrespondersthatwouldaffecttheirOutcomeifinfacttheyresponded?

ThereisnocorrectioninCombinedModelforselectionbias.Butseenextslides

WhynotsetOutcometo0fornonResponders?

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Supposethereisone predictorX.

FortreatedT:AmodelisfittoresponseY=1vsY=0.LetY _ =P(Y=1|T)IfnoOutcometarget,thenlogisticisused:

Y _ = T+TX 1 T+TX

IfthereisOutcometarget,thenprobitmodelisused(why?Seenextslides)

LikewiseforthecontrolgroupC:Y _ =P(Y=1|C)

Thedifferencescore(incrementalrate)is: _ _

DifferenceModel(=differenceoftwoModels)

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T

C

DS

Ofcourse,thepredictorsforTreatedModelandControlModelcanbecompletelydifferent.

IfthereisOutcomeTarget:LinearRegressionpredictstheOutcome.ModelisfitwhenOutcome.(Responders)

IfT,thenY _ =TZ +bT*M

Thisadded termbT*M correctsforselectionbias.

ThevariableM iscalledtheinverseMillsratioanditarisesfromProbitModelM=(p)/(p)wherepistheresponseprobabilityfromProbitModelisstdnormaldensityandisstdnormalcum.distribution

ThecoefficientbT isfitbytheregressionTheoryofusingInverseMillsratioisdiscussedinAppendixA.[ UsingLogisticinsteadofProbitinM wouldgiveverysimilarresults.]

LikewiseforC togiveY _ =CZ +bC*M

Finally,thereistheexpectedincrementalOutcomeforaprospect:

R _ =Y _ *Y _ Y _ *Y _ FixedCost

DifferenceModelandOutcomes

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CombinedModelandinverseMillsratio?

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SASEMdocumentation:Thedefaultoutcomemodel(DifferenceModel)isatwostagemodelthatusestheinverseMillsratio.TheCombinedModelusesseparateregressionsforthebinaryandintervaltarget.

Thisstatementsays(Ithink)thatinverseMillsratioisneverusedfortheCombinedOutcomeModel.MyConjecture:Althoughtheremaystillbeaselectionbias,theSelectionBiascorrectionusinginverseMillsratiodoesnotapplytotheCombinedOutputmodelsincethisapproachdoesnotaccountforatreatmentvariable.Theoriginalreferenceis:Heckman,J.(1979).SampleSelectionBiasasaSpecificationError.Econometrica47:153161.

WeightofEvidence(WOE)forX(fortreated)

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YX 0 1 PT(X=i|Y=0) PT(X=i|Y=1) WOE

1 800 200 16.0% 23.5% 0.38566

2 1200 170 24.0% 20.0% 0.182323 900 160 18.0% 18.8% 0.044744 1000 160 20.0% 18.8% 0.060625 1100 160 22.0% 18.8% 0.15593

SUM 5000 850

800/5000=16%and200/850=23.5%WOE=LOG(16%/23.5%)=0.38566

SASEM(IRMNode)canprescreenpredictorsusingNetInformationValue(NIV).ThenextslidesexplainNIV

First,hereisthedefinitionofWeightofEvidence ofapredictorX

NetWeightofEvidence(NWOE)forX

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TreatedY

X 0 1 PT(X=i|Y=0) PT(X=i|Y=1) WOE1 800 200 16.0% 23.5% 0.385662 1200 170 24.0% 20.0% 0.182323 900 160 18.0% 18.8% 0.044744 1000 160 20.0% 18.8% 0.060625 1100 160 22.0% 18.8% 0.15593

ControlY

X 0 1 PC(X=i|Y=0) PC(X=i|Y=1) WOE1 1000 80 13.5% 24.2% 0.584412 1300 70 17.6% 21.2% 0.188523 1500 60 20.3% 18.2% 0.108734 1700 60 23.0% 18.2% 0.233905 1900 60 25.7% 18.2% 0.34512

X NWOE

1 0.19882 0.37083 0.15354 0.17335 0.1892

T C=

ThisconceptwaspresentedbyKLarsen(2009)

IfTreatedWOEisbig+andControlWOEisbigORConversely,thenAbs(NWOE)islarge

WeightingNWOEtocomputeNetInformationValue(NIV)

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Treated

X PT(X=i|Y=0) PT(X=i|Y=1) WOE1 16.0% 23.5% 0.385662 24.0% 20.0% 0.182323 18.0% 18.8% 0.044744 20.0% 18.8% 0.060625 22.0% 18.8% 0.15593

Control

X PC(X=i|Y=0) PC(X=i|Y=1) WOE1 13.5% 24.2% 0.584412 17.6% 21.2% 0.188523 20.3% 18.2% 0.108734 23.0% 18.2% 0.233905 25.7% 18.2% 0.34512

X NWOE(a) Weight (b) = (a)*(b)1 0.1988 0.0070 0.00142 0.3708 0.0158 0.00583 0.1535 0.0054 0.00084 0.1733 0.0069 0.00125 0.1892 0.0083 0.0016

NIV= Sum *1000= 108.4

PT(X=i|Y=1)*PC(X=i|Y=0) PT(X=i|Y=0)*PC(X=i|Y=1)

Y=0 Y=1T A CC B D

= B*C A*D

IfTreatmentisstrong