利用逻辑斯回归分析.docx
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利用逻辑斯回归分析
利用逻辑斯回归分析
第12週講義（12-1）
利用邏輯斯迴歸分析（LogisticRegressionAnalysis）作分類分析
BinaryResponsewithoneexplanatoryvariable:
:
ResponseofthethsamplingunitiYi
=Y0or1i
:
predictor（explanatoryvariable）forthethsamplingunitiXi
,，xeP=P（Y=1|x）=x,，,x1，e
Px,，,xloglogitP==x1,Px
,,,x若令P=P（Y=0|x）,則logitP=。
xx
,，xeP（y=1|x）=之圖形,，,x1，e
,0,,,1,,0,,,,1
估計法:
最大概似估計量或加權最小平方法。
對X=x有n個中c個反應變數為1。
iii
:
A:
最大概似估計法
kn,,n,ciciii,,，，概似函數p1,p,xx,,iic1i,,i,
c,nci,,，xiikin,,,,e1,,i,,,,=，，,L,,,,,,，，,,,,xx,,,,iic1e1e，，,,,1i,i,,,
,,找出使得以上之機率為最大。
:
B:
WeightedLSE（leastsquareestimator）
cciin,,
（1）np（1,p）c之Var（c）=用來估計iiiixxiinnii
1給予權數:
加權數,加重數:
。
Varci
第12週講義（12-2）
BinaryResponsewithseveralexplanatoryvariables:
1,:
theresponseofthindividual,binaryresponsei,YY,ii0,
:
p–dimvectorofcovariateXX,（X,X,...,X）ii1i2ipi~~
P,P（Y,1|X）iii~
LogisticRegressionmodel
,,logitP,，X，...，X011iippi
PlogitP,log1,P
,,，，...，XX011ippiei.e.P,i,，,，...，,XX011ippi1，e
,logitP（Y,1|X,r，1）,logitP（Y,1|X,r）iiiii
P（Y,1|X,r，1）/P（Y,0|X,r，1）iiii,lnP（Y,1|X,r）/P（Y,0|X,r）iiii
:
logoddsratiocorrespondingtoaunitchangeinpredictorwhenallotherareX,Xjii
holdconstants.
,,2,G:
H:
,...,,0（model0）p012,,,
H:
logitP,，X，...，Xiippi1011
maxLikelihoodLH00LR,,,,2logLR,,2（logL,logL）,，2logL,2logLmm00maxLikelihoodLm,,22G,,2ln（LR）~p
故其值其值,以表示所選擇變數之相關重要性
,,2,D:
（deviance）H:
logitP,，X，...，Xpp0011
H:
saturatedmodel（models）1
2logLR,，2logL,2logLsm
22,D,,2ln（LR）~,,np1
故其值其值,表示fit好
22G，D,2logL,2logLs0
H:
logitP,,0022,,G，D,constant,之,2ln（LR）,H:
saturated1,
第12週講義（12-3）
LogisticRegressionwithOneContinuousCovariate
TheSASSystem程式:
TheLOGISTICProcedureoptionsnodatenonotes;DataSet:
WORK.P261datap261;
ResponseVariable（Events）:
CResponseProfileinputloadnc;r=c/n;ResponseVariable（Trials）:
NOrderedBinarycards;NumberofObservations:
10ValueOutcomeCount25005010LinkFunction:
Logit1EVENT33727007017
2NOEVENT353290010030ModelFittingInformationandTesting31006021
GlobalNullHypothesisBETA=033004018Intercept35008543
Interceptand37009054CriterionOnlyCovariatesChi-SquareforCovariates39005033AIC958.172847.712.41008060SC962.709856.785.43006551
-2LOGL956.172843.712112.460with1DF（p=0.0001）;Score..107.066with1DF（p=0.0001）proclogistic;
modelc/n=load;AnalysisofMaximumLikelihoodEstimatesoutputout=a
ParameterStandardWaldPr>pred=pred;VariableDFEstimateErrorChi-SquareChi-Squarerun;INTERCPT1-5.33970.545795.75000.0001..procprint;LOAD10.001550.00015896.60850.0001run;
OddsRatioEstimates
Point95%Wald
EffectEstimateConfidenceLimits
load1.0021.0011.002
AssociationofPredictedProbabilitiesandObservedResponses
PercentConcordant68.0Somers'D0.452
PercentDiscordant22.9Gamma0.497
PercentTied9.1Tau-a0.226
Pairs118961c0.726
AssociationofPredictedProbabilitiesandObservedResponses
Concordant=68.0%Somers'D=0.452
Discordant=22.9%Gamma=0.497
Tied=9.1%Tau-a=0.226
（118961pairs）c=0.726
OBSLOADNCRPRED
1250050100.200000.18715
2270070170.242860.23886
32900100300.300000.29959
4310060210.350000.36829
5330040180.450000.44278
6350085430.505880.51994
7370090540.600000.59616
8390050330.660000.66801
9410080600.750000.73280
10430065510.784620.78894
第12週講義（12-4）
LogisticRegressionwithMixedCovariate
Datacrab;
inputcolorspinewidthsatellweight;
ifsatell>0theny=1;ifsatell=0theny=0;n=1;
weight=weight/1000;color=color-1;
cards;
3328.383050
4322.501550
2126.092300
.
..
5327.002625
3224.502000
;
procgenmod;classcolor;
modely/n=colorwidth/dist=binlink=logit;
proclogistic;
modely=colorweightwidth/selection=backward;
run;
PARTOFOUTPUT:
TheGENMODProcedure
ModelInformation
DataSetWORK.CRAB
DistributionBinomial
LinkFunctionLogit
ResponseVariable（Events）y
ResponseVariable（Trials）n
ObservationsUsed173
NumberOfEvents111
NumberOfTrials173
ClassLevelInformation
ClassLevelsValues
color41234
CriteriaForAssessingGoodnessOfFit
CriterionDFValueValue/DF
Deviance168187.45701.1158
ScaledDeviance168187.45701.1158
PearsonChi-Square168168.65901.0039
ScaledPearsonX2168168.65901.0039
LogLikelihood-93.7285
Algorithmconverged.
AnalysisOfParameterEstimates
StandardWald95%ConfidenceChi-ParameterDFEstimateErrorLimitsSquarePr>ChiSq
Intercept1-12.71512.7618-18.1281-7.302121.20<.0001
color111.32990.8525-0.34103.00082.430.1188
color211.40230.54840.32742.47736.540.0106
color311.10610.5921-0.05432.26663.490.0617
color400.00000.00000.00000.0000..width10.46800.10550.26110.674819.66<.0001
Scale01.00000.00001.00001.0000NOTE:
Thescaleparameterwasheldfixed.
第12週講義（12-5）
TheLOGISTICProcedure
ModelInformation
DataSetWORK.CRAB
ResponseVariabley
NumberofResponseLevels2
NumberofObservations173
Modelbinarylogit
OptimizationTechniqueFisher'sscoring
ResponseProfile
OrderedTotal
ValueyFrequency
1062
21111
Probabilitymodeledisy=0.
BackwardEliminationProcedure
Step0.Thefollowingeffectswereentered:
Interceptcolorweightwidth
ModelConvergenceStatus
Convergencecriterion（GCONV=1E-8）satisfied.
ModelFitStatistics
Intercept
Interceptand
CriterionOnlyCovariates
AIC227.759195.925
SC230.912208.538
-2LogL225.759187.925
TestingGlobalNullHypothesis:
BETA=0
TestChi-SquareDFPr>ChiSq
LikelihoodRatio37.83363<.0001
Score33.38873<.0001
Wald27.88553<.0001
第12週講義（12-6）
TheLOGISTICProcedure
Step1.Effectweightisremoved:
ModelFitStatistics
Intercept
Interceptand
CriterionOnlyCovariates
AIC227.759195.121
SC230.912204.581
-2LogL225.759189.121
TestingGlobalNullHypothesis:
BETA=0
TestChi-SquareDFPr>ChiSq
LikelihoodRatio36.63732<.0001
Score32.73582<.0001
Wald27.06092<.0001
ResidualChi-SquareTest
Chi-SquareDFPr>ChiSq
1.203110.2727
NOTE:
No（additional）effectsmetthe0.05significancelevelforremovalfromthemodel.
SummaryofBackwardElimination
EffectNumberWald
StepRemovedDFInChi-SquarePr>ChiSq
1weight121.17780.2778
TheLOGISTICProcedure
AnalysisofMaximumLikelihoodEstimates
StandardWald
ParameterDFEstimateErrorChi-SquarePr>ChiSq
Intercept110.07082.806912.87330.0003
color10.50900.22375.17910.0229
width1-0.45830.104019.4129<.0001
OddsRatioEstimates
Point95%Wald
EffectEstimateConfidenceLimits
color1.6641.0732.579
width0.6320.5160.775
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