第六章用于多項式分布Logit模型

MulticategoryLogitModels6.1名義因變量logit模型6.2有序因變量的累積logit模型6.3配對分類有序logit模型6.4檢驗條件獨立性設,()是因變量的J個類別的結果發生的概率J個類別分布是多項分布

6.1名義因變量logit模型

LOGITMODELSFORNOMINALRESPONSESBaseline-CategoryLogitsModel當把第J類j結果作為Baseline(基線),Logits模型有J-1個方程利用軟件可同時給出這J-1個方程的參數估計，和其它統計量對結果落在a類和b類的概率的比也是線性形式,與基線的選取無關其中例Y三種基本食物：F,I,O,x:長度

F=fishI=InvertebratesO=otherAnalysisofMaximumLikelihoodEstimatesFunctionStandardChi-ParameterNumberEstimateErrorSquarePr>ChiSq-------------------------------------------------------------------Intercept11.61771.30731.530.215925.69741.793710.090.0015length1-0.11010.51710.050.83142-2.46540.89967.510.0061

即結論：1.檢驗：食物的選擇與鱷魚長度無關假設檢驗表6.2,p值0.0002，拒絕零假設.2.在魚和無脊椎動物之間,大鱷魚易選前者.X增加1單位,勝算比為：exp(2.355)=10.5EstimatingResponseProbabilities計算各類概率它滿足例中其中，1可以看為：對每個給定x，如最長鱷而，即此鱷吃魚的可能性最大0.76，吃無脊椎動物的可能性是0.005,吃其它的可能性為0.23

dataAlligator;inputsizetype$@@;cards;1.24I1.30I1.30I1.32F1.32F1.40F1.42I1.42F1.45I1.45O1.47I1.47F1.50I1.52I1.55I1.60I1.63I1.65O1.65I1.65F1.65F1.68F1.70I1.73O1.78I1.78I1.78O1.80I1.80F1.85F1.88I1.93I1.98I2.03F2.03F2.16F2.26F2.31F2.31F2.36F2.36F2.39F2.41F2.44F2.46F2.56O2.67F2.72I2.79F2.84F3.25O3.28O3.33F3.56F3.58F3.66F3.68O3.71F3.89F;proclogistic;modeltype(ref='O')=size/link=glogitexpb;run;SAS程序TheLOGISTICProcedureType3AnalysisofEffectsWaldEffectDFChi-SquarePr>ChiSqsize28.93600.0115AnalysisofMaximumLikelihoodEstimatesStandardWaldParametertypeDFEstimateErrorChi-SquarePr>ChiSqExp(Est)InterceptF11.61771.30731.53140.21595.042InterceptI15.69741.793810.08810.0015298.104sizeF1-0.11010.51710.04530.83140.896sizeI1-2.46540.89977.51010.00610.085OddsRatioEstimatesPoint95%WaldEffecttypeEstimateConfidenceLimitssizeF0.8960.3252.468sizeI0.0850.0150.496

點擊gator.sav選Analyze–regression–MultinomialLogisticRegression將food放到dependent，選定referencecategory(last),把length放到CovariateSPSS程序例2：Y信仰(是、不確定、否);X1性別;X2種族具體模型為（最終模型與前面相同）：你有可能得到下面的SPSS輸出具體模型為：模型檢驗：

如去掉genderDeviance:8.0-0.8=7.2（df=2）

P值0.03檢驗：如果去掉raceDeviance:

2.8-0.8=2（df=2）

P值0.368各類人beliefinafterlife的概率datacatbelief;inputrace$gender$belief$count@@;cards;WhiteFY371WhiteFU49WhiteFN74WhiteMY250WhiteMU45WhiteMN71BlackFY64BlackFU9BlackFN15BlackMY25BlackMU5BlackMN13;proc

logistic;freqcount;classgender(ref='M')race(ref='Black')/param=ref;modelbelief(ref='N')=genderrace/link=glogitexpb;run;SAS程序TestingGlobalNullHypothesis:BETA=0TestChi-SquareDFPr>ChiSqLikelihoodRatio8.743740.0678Score8.849840.0650Wald8.781840.0668Type3AnalysisofEffectsWaldEffectDFChi-SquarePr>ChiSqgender27.20740.0272race22.08240.3530AnalysisofMaximumLikelihoodEstimates

StandardWaldParameterbeliefDFEstimateErrorChi-SquarePr>ChiSqExp(Est)InterceptU1-0.75820.36144.40310.03590.468InterceptY10.88280.242613.23900.00032.418genderFU10.10510.24650.18170.66991.111genderFY10.41860.17135.97370.01451.520raceWhiteU10.27120.35410.58630.44381.311raceWhiteY10.34200.23702.08140.14911.408OddsRatioEstimatesPoint95%WaldEffectbeliefEstimateConfidenceLimitsgenderFvsMU1.1110.6851.801genderFvsMY1.5201.0862.126raceWhitevsBlackU1.3110.6552.6256.2有序因變量的累積logit模型CUMULATIVELOGITMODELSFORORDINALRESPONSES模型：

X軸是否出錯了？概率：

對所有j上述似然比的log值

log似然比為例1：

x=1是demo模型統計推斷的95%置信區間即:=模型擬合Noevidenceoflackoffit擬合不充分時,可把原模型中用代替datapoliti;inputgenderpartybeliefcount@@;cards;00144002470031180042300532011180122801386014390154810136102341035310418105231111211218113621144511551;proclogistic;freqcount;classparty(param=refref='1');modelbelief=party;run;SAS程序TheLOGISTICProcedureModelFitStatisticsInterceptInterceptandCriterionOnlyCovariatesAIC2541.6302484.985SC2560.5402508.622-2LogL2533.6302474.985TestingGlobalNullHypothesis:BETA=0TestChi-SquareDFPr>ChiSqLikelihoodRatio58.64511<.0001Score57.24481<.0001Wald57.01821<.0001

Type3AnalysisofEffectsWaldEffectDFChi-SquarePr>ChiSqparty157.0182<.0001AnalysisofMaximumLikelihoodEstimatesStandardWaldParameterDFEstimateErrorChi-SquarePr>ChiSqIntercept11-2.46900.1318350.8122<.0001Intercept21-1.47450.1091182.7151<.0001Intercept310.23710.09486.24970.0124Intercept411.06950.1046104.6082<.0001

party010.97450.129157.0182<.0001OddsRatioEstimatesPoint95%WaldEffectEstimateConfidenceLimitsparty0vs12.6502.0583.412

SPSS輸出例2：SES:(1high,0low);lifeevents:過去三年Birthofchild,newjob,divorce,deathoffamily模型：給定lifeevents值，對高SES水平的人得輕一級別精神損傷的勝算是e(1.111）=3對給定SES,Lifeevents的數目多的人易得更嚴重的精神病

用更復雜的模型：發現交叉項不顯著AnalysisofMaximumLikelihoodEstimatesStandardWaldParameterDFEstimateErrorChi-SquarePr>ChiSqIntercept110.09810.81100.01460.9037Intercept211.59250.83723.61860.0571Intercept312.60660.90978.21110.0042life1-0.42040.19034.88110.0272SES10.37091.13020.10770.7428life*SES10.18130.23610.58960.4426datamental;inputnummentalSESlife@@;cards;11112121921192220331142321341132421151022530061102631471012730381132830991132931610117303041110131303121023241813215334121420634417152133540516201364041721837404182123841819205394082021540409;proclogistic;modelmental=lifeses;run;proclogistic;modelmental=lifeseslife*ses;run;SAS程序TheLOGISTICProcedureTestingGlobalNullHypothesis:BETA=0TestChi-SquareDFPr>ChiSqLikelihoodRatio9.944220.0069Score9.143120.0103Wald8.501820.0143AnalysisofMaximumLikelihoodEstimatesStandardWaldParameterDFEstimateErrorChi-SquarePr>ChiSqIntercept11-0.28180.6231