Logistic regression (with R).pdf

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1、Logisticregression(withR)ChristopherManning4November20071TheoryWecantransformtheoutputofalinearregressiontobesuitableforprobabilitiesbyusingalogitlinkfunctiononthelhsasfollows:plogitp=logo=log=β0+β1x1+β2x2+···+βkxk(1)1−pTheoddscanvaryonascaleof(0,∞),s

2、othelogoddscanvaryonthescaleof(−∞,∞)–preciselywhatwegetfromtherhsofthelinearmodel.Forareal-valuedexplanatoryvariablexi,theintuitionhereisthataunitadditivechangeinthevalueofthevariableshouldchangetheoddsbyaconstantmultiplicativeamount.Exponentiating,th

3、isisequivalentto:1elogitp=eβ0+β1x1+β2x2+···+βkxk(2)o=p=eβ0eβ1x1eβ2x2···eβkxk(3)1−pTheinverseofthelogitfunctionisthelogisticfunction.Iflogit(π)=z,thenezπ=1+ezThelogisticfunctionwillmapanyvalueoftherighthandside(z)toaproportionvaluebetween0and1,asshowni

4、nfigure1.Noteacommoncasewithcategoricaldata:Ifourexplanatoryvariablesxiareallbinary,thenfortheonesthatarefalse(0),wegete0=1andthetermdisappears.Similarly,ifx=1,eβixi=eβi.Soweareileftwithtermsforonlythexithataretrue(1).Forinstance,ifx3,x4,x7=1only,wehav

5、e:logitp=β0+β3+β4+β7(4)o=eβ0eβ3eβ4eβ7(5)TheintuitionhereisthatifIknowthatacertainfactistrueofadatapoint,thenthatwillproduceaconstantchangeintheoddsoftheoutcome(“Ifhe’sEuropean,thatdoublestheoddsthathesmokes”).LetL=L(D;B)bethelikelihoodofthedataDgivent

6、hemodel,whereB={β0,...,βk}aretheparametersofthemodel.Theparametersareestimatedbytheprincipleofmaximumlikelihood.Technicalpoint:thereisnoerrorterminalogisticregression,unlikeinlinearregressions.1Notethatwecanconvertfreelybetweenaprobabilitypandoddsofor

7、aneventversusitscomplement:poo=p=1−po+11Logisticfunction0.00.20.40.60.81.0-6-4-20246Figure1:Thelogisticfunction2BasicRlogisticregressionmodelsWewillillustratewiththeCedegrendatasetonthewebsite.cedegren<-read.table("cedegren.txt",header=T)Youneedtocrea

8、teatwo-columnmatrixofsuccess/failurecountsforyourresponsevariable.Youcannotjustusepercentages.(Youcangivepercentagesbutthenweightthembyacountofsuccess+failures.)attach(cedegren)ced.del<-cbind(sDel,sNoDel)Makethelogisticregressionmodel.Theshort