Formally, in binary logistic regression there is a single binary dependent variable, coded by an indicator variable, where the two values are labeled "0" ...
Logisticregression
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Statisticalmodelforabinarydependentvariable
"Logitmodel"redirectshere.NottobeconfusedwithLogitfunction.
Examplegraphofalogisticregressioncurvefittedtodata.Thecurveshowstheprobabilityofpassinganexam(binarydependentvariable)versushoursstudying(scalarindependentvariable).See§ Exampleforworkeddetails.
Instatistics,thelogisticmodel(orlogitmodel)isastatisticalmodelthatmodelstheprobabilityofaneventtakingplacebyhavingthelog-oddsfortheeventbealinearcombinationofoneormoreindependentvariables.Inregressionanalysis,logisticregression[1](orlogitregression)isestimatingtheparametersofalogisticmodel(thecoefficientsinthelinearcombination).Formally,inbinarylogisticregressionthereisasinglebinarydependentvariable,codedbyanindicatorvariable,wherethetwovaluesarelabeled"0"and"1",whiletheindependentvariablescaneachbeabinaryvariable(twoclasses,codedbyanindicatorvariable)oracontinuousvariable(anyrealvalue).Thecorrespondingprobabilityofthevaluelabeled"1"canvarybetween0(certainlythevalue"0")and1(certainlythevalue"1"),hencethelabeling;[2]thefunctionthatconvertslog-oddstoprobabilityisthelogisticfunction,hencethename.Theunitofmeasurementforthelog-oddsscaleiscalledalogit,fromlogisticunit,hencethealternativenames.See§ Backgroundand§ Definitionforformalmathematics,and§ Exampleforaworkedexample.
Binaryvariablesarewidelyusedinstatisticstomodeltheprobabilityofacertainclassoreventtakingplace,suchastheprobabilityofateamwinning,ofapatientbeinghealthy,etc.(see§ Applications),andthelogisticmodelhasbeenthemostcommonlyusedmodelforbinaryregressionsinceabout1970.[3]Binaryvariablescanbegeneralizedtocategoricalvariableswhentherearemorethantwopossiblevalues(e.g.whetheranimageisofacat,dog,lion,etc.),andthebinarylogisticregressiongeneralizedtomultinomiallogisticregression.Ifthemultiplecategoriesareordered,onecanusetheordinallogisticregression(forexampletheproportionaloddsordinallogisticmodel[4]).See§ Extensionsforfurtherextensions.Thelogisticregressionmodelitselfsimplymodelsprobabilityofoutputintermsofinputanddoesnotperformstatisticalclassification(itisnotaclassifier),thoughitcanbeusedtomakeaclassifier,forinstancebychoosingacutoffvalueandclassifyinginputswithprobabilitygreaterthanthecutoffasoneclass,belowthecutoffastheother;thisisacommonwaytomakeabinaryclassifier.
Analogouslinearmodelsforbinaryvariableswithadifferentsigmoidfunctioninsteadofthelogisticfunction(toconvertthelinearcombinationtoaprobability)canalsobeused,mostnotablytheprobitmodel;see§ Alternatives.Thedefiningcharacteristicofthelogisticmodelisthatincreasingoneoftheindependentvariablesmultiplicativelyscalestheoddsofthegivenoutcomeataconstantrate,witheachindependentvariablehavingitsownparameter;forabinarydependentvariablethisgeneralizestheoddsratio.Moreabstractly,thelogisticfunctionisthenaturalparameterfortheBernoullidistribution,andinthissenseisthe"simplest"waytoconvertarealnumbertoaprobability.Inparticular,itmaximizesentropy(minimizesaddedinformation),andinthissensemakesthefewestassumptionsofthedatabeingmodeled;see§ Maximumentropy.
Theparametersofalogisticregressionaremostcommonlyestimatedbymaximum-likelihoodestimation(MLE).Thisdoesnothaveaclosed-formexpression,unlikelinearleastsquares;see§ Modelfitting.LogisticregressionbyMLEplaysasimilarlybasicroleforbinaryorcategoricalresponsesaslinearregressionbyordinaryleastsquares(OLS)playsforscalarresponses:itisasimple,well-analyzedbaselinemodel;see§ Comparisonwithlinearregressionfordiscussion.ThelogisticregressionasageneralstatisticalmodelwasoriginallydevelopedandpopularizedprimarilybyJosephBerkson,[5]beginninginBerkson(1944)harvtxterror:notarget:CITEREFBerkson1944(help),wherehecoined"logit";see§ History.
PartofaseriesonRegressionanalysis
Models
Linearregression
Simpleregression
Polynomialregression
Generallinearmodel
Generalizedlinearmodel
Vectorgeneralizedlinearmodel
Discretechoice
Binomialregression
Binaryregression
Logisticregression
Multinomiallogisticregression
Mixedlogit
Probit
Multinomialprobit
Orderedlogit
Orderedprobit
Poisson
Multilevelmodel
Fixedeffects
Randomeffects
Linearmixed-effectsmodel
Nonlinearmixed-effectsmodel
Nonlinearregression
Supportvectorregression
Nonparametric
Semiparametric
Robust
Quantile
Isotonic
Principalcomponents
Leastangle
Local
Segmented
Errors-in-variables
Estimation
Leastsquares
Linear
Non-linear
Ordinary
Weighted
Generalized
Generalizedestimatingequation
Partial
Total
Non-negative
Ridgeregression
Regularized
Leastabsolutedeviations
Iterativelyreweighted
Bayesian
Bayesianmultivariate
Least-squaresspectralanalysis
HeteroscedasticityConsistentRegressionStandardErrors
HeteroscedasticityandAutocorrelationConsistentRegressionStandardErrors
Background
Regressionvalidation
Meanandpredictedresponse
Errorsandresiduals
Goodnessoffit
Studentizedresidual
Gauss–Markovtheorem
Mathematicsportalvte
Contents
1Applications
2Example
2.1Problem
2.2Model
2.3Fit
2.4Parameterestimation
2.5Predictions
2.6Modelevaluation
2.7Generalizations
3Background
3.1Definitionofthelogisticfunction
3.2Definitionoftheinverseofthelogisticfunction
3.3Interpretationoftheseterms
3.4Definitionoftheodds
3.5Theoddsratio
3.6Multipleexplanatoryvariables
4Definition
4.1Manyexplanatoryvariables,twocategories
4.2Multinomiallogisticregression:Manyexplanatoryvariablesandmanycategories
5Interpretations
5.1Asageneralizedlinearmodel
5.2Asalatent-variablemodel
5.3Two-waylatent-variablemodel
5.3.1Example
5.4Asa"log-linear"model
5.5Asasingle-layerperceptron
5.6Intermsofbinomialdata
6Modelfitting
6.1Maximumlikelihoodestimation(MLE)
6.2Iterativelyreweightedleastsquares(IRLS)
6.3Bayesian
6.4"Ruleoften"
7Errorandsignificanceoffit
7.1Devianceandlikelihoodratiotest─asimplecase
7.2Goodnessoffitsummary
7.2.1Devianceandlikelihoodratiotests
7.2.2Pseudo-R-squared
7.2.3Hosmer–Lemeshowtest
7.3Coefficientsignificance
7.3.1Likelihoodratiotest
7.3.2Waldstatistic
7.3.3Case-controlsampling
8Discussion
9Maximumentropy
9.1Proof
9.2Otherapproaches
10Comparisonwithlinearregression
11Alternatives
12History
13Extensions
14Software
15Seealso
16References
17Furtherreading
18Externallinks
Applications[edit]
Logisticregressionisusedinvariousfields,includingmachinelearning,mostmedicalfields,andsocialsciences.Forexample,theTraumaandInjurySeverityScore(TRISS),whichiswidelyusedtopredictmortalityininjuredpatients,wasoriginallydevelopedbyBoydetal.usinglogisticregression.[6]Manyothermedicalscalesusedtoassessseverityofapatienthavebeendevelopedusinglogisticregression.[7][8][9][10]Logisticregressionmaybeusedtopredicttheriskofdevelopingagivendisease(e.g.diabetes;coronaryheartdisease),basedonobservedcharacteristicsofthepatient(age,sex,bodymassindex,resultsofvariousbloodtests,etc.).[11][12]AnotherexamplemightbetopredictwhetheraNepalesevoterwillvoteNepaliCongressorCommunistPartyofNepalorAnyOtherParty,basedonage,income,sex,race,stateofresidence,votesinpreviouselections,etc.[13]Thetechniquecanalsobeusedinengineering,especiallyforpredictingtheprobabilityoffailureofagivenprocess,systemorproduct.[14][15]Itisalsousedinmarketingapplicationssuchaspredictionofacustomer'spropensitytopurchaseaproductorhaltasubscription,etc.[16]Ineconomicsitcanbeusedtopredictthelikelihoodofapersonendingupinthelaborforce,andabusinessapplicationwouldbetopredictthelikelihoodofahomeownerdefaultingonamortgage.Conditionalrandomfields,anextensionoflogisticregressiontosequentialdata,areusedinnaturallanguageprocessing.
Example[edit]
Problem[edit]
Theimagerepresentswhatisincludedinlogisticregression,includinganexploratoryvariable,event,andtwopossibleoutcomes.Theexploratoryvariableisunderlinedintheexampleabove,theeventistheexam,whiletheoutcomesareeitherpassorfail.Notethattheexploratoryvariable,event,andoutcomescanchangebasedonthelogisticregressionyouchoosetoconduct.Besidesexams,forexample,eventscanalsoincludeinterventions,treatments,gatherings,etc.,
Asasimpleexample,wecanusealogisticregressionwithoneexplanatoryvariableandtwocategoriestoanswerthefollowingquestion:
Agroupof20studentsspendsbetween0and6hoursstudyingforanexam.Howdoesthenumberofhoursspentstudyingaffecttheprobabilityofthestudentpassingtheexam?
Thereasonforusinglogisticregressionforthisproblemisthatthevaluesofthedependentvariable,passandfail,whilerepresentedby"1"and"0",arenotcardinalnumbers.Iftheproblemwaschangedsothatpass/failwasreplacedwiththegrade0–100(cardinalnumbers),thensimpleregressionanalysiscouldbeused.
Thetableshowsthenumberofhourseachstudentspentstudying,andwhethertheypassed(1)orfailed(0).
Hours(xk)
0.50
0.75
1.00
1.25
1.50
1.75
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
4.00
4.25
4.50
4.75
5.00
5.50
Pass(yk)
0
0
0
0
0
0
1
0
1
0
1
0
1
0
1
1
1
1
1
1
Wewishtofitalogisticfunctiontothedataconsistingofthehoursstudied(xk)andtheoutcomeofthetest(yk =1forpass,0forfail).Thedatapointsareindexedbythesubscriptkwhichrunsfrom
k
=
1
{\displaystylek=1}
to
k
=
K
=
20
{\displaystylek=K=20}
.Thexvariableiscalledthe"explanatoryvariable",andtheyvariableiscalledthe"categoricalvariable"consistingoftwocategories:"pass"or"fail"correspondingtothecategoricalvalues1and0respectively.
Model[edit]
Graphofalogisticregressioncurvefittedtothe(xm,ym)data.Thecurveshowstheprobabilityofpassinganexamversushoursstudying.
Thelogisticfunctionisoftheform:
p
(
x
)
=
1
1
+
e
−
(
x
−
μ
)
/
s
{\displaystylep(x)={\frac{1}{1+e^{-(x-\mu)/s}}}}
whereμisalocationparameter(themidpointofthecurve,where
p
(
μ
)
=
1
/
2
{\displaystylep(\mu)=1/2}
)andsisascaleparameter.Thisexpressionmayberewrittenas:
p
(
x
)
=
1
1
+
e
−
(
β
0
+
β
1
x
)
{\displaystylep(x)={\frac{1}{1+e^{-(\beta_{0}+\beta_{1}x)}}}}
where
β
0
=
−
μ
/
s
{\displaystyle\beta_{0}=-\mu/s}
andisknownastheintercept(itistheverticalinterceptory-interceptoftheline
y
=
β
0
+
β
1
x
{\displaystyley=\beta_{0}+\beta_{1}x}
),and
β
1
=
1
/
s
{\displaystyle\beta_{1}=1/s}
(inversescaleparameterorrateparameter):thesearethey-interceptandslopeofthelog-oddsasafunctionofx.Conversely,
μ
=
−
β
0
/
β
1
{\displaystyle\mu=-\beta_{0}/\beta_{1}}
and
s
=
1
/
β
1
{\displaystyles=1/\beta_{1}}
.
Fit[edit]
Theusualmeasureofgoodnessoffitforalogisticregressionuseslogisticloss(orlogloss),thenegativelog-likelihood.Foragivenxkandyk,write
p
k
=
p
(
x
k
)
{\displaystylep_{k}=p(x_{k})}
.The
p
k
{\displaystylep_{k}}
aretheprobabilitiesthatthecorresponding
y
k
{\displaystyley_{k}}
willbeunityand
1
−
p
k
{\displaystyle1-p_{k}}
aretheprobabilitiesthattheywillbezero(seeBernoullidistribution).Wewishtofindthevaluesof
β
0
{\displaystyle\beta_{0}}
and
β
1
{\displaystyle\beta_{1}}
whichgivethe"bestfit"tothedata.Inthecaseoflinearregression,thesumofthesquareddeviationsofthefitfromthedatapoints(yk),thesquarederrorloss,istakenasameasureofthegoodnessoffit,andthebestfitisobtainedwhenthatfunctionisminimized.
Theloglossforthek-thpointis:
{
−
ln
p
k
if
y
k
=
1
,
−
ln
(
1
−
p
k
)
if
y
k
=
0.
{\displaystyle{\begin{cases}-\lnp_{k}&{\text{if}}y_{k}=1,\\-\ln(1-p_{k})&{\text{if}}y_{k}=0.\end{cases}}}
Theloglosscanbeinterpretedasthe"surprisal"oftheactualoutcome
y
k
{\displaystyley_{k}}
relativetotheprediction
p
k
{\displaystylep_{k}}
,andisameasureofinformationcontent.Notethatloglossisalwaysgreaterthanorequalto0,equals0onlyincaseofaperfectprediction(i.e.,when
p
k
=
1
{\displaystylep_{k}=1}
and
y
k
=
1
{\displaystyley_{k}=1}
,or
p
k
=
0
{\displaystylep_{k}=0}
and
y
k
=
0
{\displaystyley_{k}=0}
),andapproachesinfinityasthepredictiongetsworse(i.e.,when
y
k
=
1
{\displaystyley_{k}=1}
and
p
k
→
0
{\displaystylep_{k}\to0}
or
y
k
=
0
{\displaystyley_{k}=0}
and
p
k
→
1
{\displaystylep_{k}\to1}
),meaningtheactualoutcomeis"moresurprising".Sincethevalueofthelogisticfunctionisalwaysstrictlybetweenzeroandone,theloglossisalwaysgreaterthanzeroandlessthaninfinity.Notethatunlikeinalinearregression,wherethemodelcanhavezerolossatapointbypassingthroughadatapoint(andzerolossoverallifallpointsareonaline),inalogisticregressionitisnotpossibletohavezerolossatanypoints,since
y
k
{\displaystyley_{k}}
iseither0or1,but
0
<
p
k
<
1
{\displaystyle0
0
i.e.
−
ε
i
<
β
⋅
X
i
,
0
otherwise.
{\displaystyleY_{i}={\begin{cases}1&{\text{if}}Y_{i}^{\ast}>0\{\text{i.e.}}-\varepsilon_{i}
0
∣
X
i
)
=
Pr
(
β
⋅
X
i
+
ε
i
>
0
)
=
Pr
(
ε
i
>
−
β
⋅
X
i
)
=
Pr
(
ε
i
<
β
⋅
X
i
)
(becausethelogisticdistributionissymmetric)
=
logit
−
1
(
β
⋅
X
i
)
=
p
i
(seeabove)
{\displaystyle{\begin{aligned}\Pr(Y_{i}=1\mid\mathbf{X}_{i})&=\Pr(Y_{i}^{\ast}>0\mid\mathbf{X}_{i})\\[5pt]&=\Pr({\boldsymbol{\beta}}\cdot\mathbf{X}_{i}+\varepsilon_{i}>0)\\[5pt]&=\Pr(\varepsilon_{i}>-{\boldsymbol{\beta}}\cdot\mathbf{X}_{i})\\[5pt]&=\Pr(\varepsilon_{i}
Y
i
0
∗
,
0
otherwise.
{\displaystyleY_{i}={\begin{cases}1&{\text{if}}Y_{i}^{1\ast}>Y_{i}^{0\ast},\\0&{\text{otherwise.}}\end{cases}}}
Thismodelhasaseparatelatentvariableandaseparatesetofregressioncoefficientsforeachpossibleoutcomeofthedependentvariable.Thereasonforthisseparationisthatitmakesiteasytoextendlogisticregressiontomulti-outcomecategoricalvariables,asinthemultinomiallogitmodel.Insuchamodel,itisnaturaltomodeleachpossibleoutcomeusingadifferentsetofregressioncoefficients.Itisalsopossibletomotivateeachoftheseparatelatentvariablesasthetheoreticalutilityassociatedwithmakingtheassociatedchoice,andthusmotivatelogisticregressionintermsofutilitytheory.(Intermsofutilitytheory,arationalactoralwayschoosesthechoicewiththegreatestassociatedutility.)Thisistheapproachtakenbyeconomistswhenformulatingdiscretechoicemodels,becauseitbothprovidesatheoreticallystrongfoundationandfacilitatesintuitionsaboutthemodel,whichinturnmakesiteasytoconsidervarioussortsofextensions.(Seetheexamplebelow.)
Thechoiceofthetype-1extremevaluedistributionseemsfairlyarbitrary,butitmakesthemathematicsworkout,anditmaybepossibletojustifyitsusethroughrationalchoicetheory.
Itturnsoutthatthismodelisequivalenttothepreviousmodel,althoughthisseemsnon-obvious,sincetherearenowtwosetsofregressioncoefficientsanderrorvariables,andtheerrorvariableshaveadifferentdistribution.Infact,thismodelreducesdirectlytothepreviousonewiththefollowingsubstitutions:
β
=
β
1
−
β
0
{\displaystyle{\boldsymbol{\beta}}={\boldsymbol{\beta}}_{1}-{\boldsymbol{\beta}}_{0}}
ε
=
ε
1
−
ε
0
{\displaystyle\varepsilon=\varepsilon_{1}-\varepsilon_{0}}
Anintuitionforthiscomesfromthefactthat,sincewechoosebasedonthemaximumoftwovalues,onlytheirdifferencematters,nottheexactvalues—andthiseffectivelyremovesonedegreeoffreedom.Anothercriticalfactisthatthedifferenceoftwotype-1extreme-value-distributedvariablesisalogisticdistribution,i.e.
ε
=
ε
1
−
ε
0
∼
Logistic
(
0
,
1
)
.
{\displaystyle\varepsilon=\varepsilon_{1}-\varepsilon_{0}\sim\operatorname{Logistic}(0,1).}
Wecandemonstratetheequivalentasfollows:
Pr
(
Y
i
=
1
∣
X
i
)
=
Pr
(
Y
i
1
∗
>
Y
i
0
∗
∣
X
i
)
=
Pr
(
Y
i
1
∗
−
Y
i
0
∗
>
0
∣
X
i
)
=
Pr
(
β
1
⋅
X
i
+
ε
1
−
(
β
0
⋅
X
i
+
ε
0
)
>
0
)
=
Pr
(
(
β
1
⋅
X
i
−
β
0
⋅
X
i
)
+
(
ε
1
−
ε
0
)
>
0
)
=
Pr
(
(
β
1
−
β
0
)
⋅
X
i
+
(
ε
1
−
ε
0
)
>
0
)
=
Pr
(
(
β
1
−
β
0
)
⋅
X
i
+
ε
>
0
)
(substitute
ε
asabove)
=
Pr
(
β
⋅
X
i
+
ε
>
0
)
(substitute
β
asabove)
=
Pr
(
ε
>
−
β
⋅
X
i
)
(now,sameasabovemodel)
=
Pr
(
ε
<
β
⋅
X
i
)
=
logit
−
1
(
β
⋅
X
i
)
=
p
i
{\displaystyle{\begin{aligned}\Pr(Y_{i}=1\mid\mathbf{X}_{i})={}&\Pr\left(Y_{i}^{1\ast}>Y_{i}^{0\ast}\mid\mathbf{X}_{i}\right)&\\[5pt]={}&\Pr\left(Y_{i}^{1\ast}-Y_{i}^{0\ast}>0\mid\mathbf{X}_{i}\right)&\\[5pt]={}&\Pr\left({\boldsymbol{\beta}}_{1}\cdot\mathbf{X}_{i}+\varepsilon_{1}-\left({\boldsymbol{\beta}}_{0}\cdot\mathbf{X}_{i}+\varepsilon_{0}\right)>0\right)&\\[5pt]={}&\Pr\left(({\boldsymbol{\beta}}_{1}\cdot\mathbf{X}_{i}-{\boldsymbol{\beta}}_{0}\cdot\mathbf{X}_{i})+(\varepsilon_{1}-\varepsilon_{0})>0\right)&\\[5pt]={}&\Pr(({\boldsymbol{\beta}}_{1}-{\boldsymbol{\beta}}_{0})\cdot\mathbf{X}_{i}+(\varepsilon_{1}-\varepsilon_{0})>0)&\\[5pt]={}&\Pr(({\boldsymbol{\beta}}_{1}-{\boldsymbol{\beta}}_{0})\cdot\mathbf{X}_{i}+\varepsilon>0)&&{\text{(substitute}}\varepsilon{\text{asabove)}}\\[5pt]={}&\Pr({\boldsymbol{\beta}}\cdot\mathbf{X}_{i}+\varepsilon>0)&&{\text{(substitute}}{\boldsymbol{\beta}}{\text{asabove)}}\\[5pt]={}&\Pr(\varepsilon>-{\boldsymbol{\beta}}\cdot\mathbf{X}_{i})&&{\text{(now,sameasabovemodel)}}\\[5pt]={}&\Pr(\varepsilon3.3.co;2-f.PMID 9160492.
^Harrell,FrankE.(2010).RegressionModelingStrategies:WithApplicationstoLinearModels,LogisticRegression,andSurvivalAnalysis.NewYork:Springer.ISBN 978-1-4419-2918-1.[page needed]
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^GarethJames;DanielaWitten;TrevorHastie;RobertTibshirani(2013).AnIntroductiontoStatisticalLearning.Springer.p. 6.
^Pohar,Maja;Blas,Mateja;Turk,Sandra(2004)."ComparisonofLogisticRegressionandLinearDiscriminantAnalysis:ASimulationStudy".MetodološkiZvezki.1(1).
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^Cramer2002,p. 4.
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^Cramer2002,p. 7–9.
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^Cramer2002,p. 11.
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Furtherreading[edit]
Cox,DavidR.(1958)."Theregressionanalysisofbinarysequences(withdiscussion)".JRStatSocB.20(2):215–242.JSTOR 2983890.
Cox,DavidR.(1966)."Someproceduresconnectedwiththelogisticqualitativeresponsecurve".InF.N.David(1966)(ed.).ResearchPapersinProbabilityandStatistics(FestschriftforJ.Neyman).London:Wiley.pp. 55–71.
Cramer,J.S.(2002).Theoriginsoflogisticregression(PDF)(Technicalreport).Vol. 119.TinbergenInstitute.pp. 167–178.doi:10.2139/ssrn.360300.
Publishedin:Cramer,J.S.(2004)."Theearlyoriginsofthelogitmodel".StudiesinHistoryandPhilosophyofSciencePartC:StudiesinHistoryandPhilosophyofBiologicalandBiomedicalSciences.35(4):613–626.doi:10.1016/j.shpsc.2004.09.003.
Thiel,Henri(1969)."AMultinomialExtensionoftheLinearLogitModel".InternationalEconomicReview.10(3):251–59.doi:10.2307/2525642.JSTOR 2525642.
Wilson,E.B.;Worcester,J.(1943)."TheDeterminationofL.D.50andItsSamplingErrorinBio-Assay".ProceedingsoftheNationalAcademyofSciencesoftheUnitedStatesofAmerica.29(2):79–85.Bibcode:1943PNAS...29...79W.doi:10.1073/pnas.29.2.79.PMC 1078563.PMID 16588606.
Agresti,Alan.(2002).CategoricalDataAnalysis.NewYork:Wiley-Interscience.ISBN 978-0-471-36093-3.
Amemiya,Takeshi(1985)."QualitativeResponseModels".AdvancedEconometrics.Oxford:BasilBlackwell.pp. 267–359.ISBN 978-0-631-13345-2.
Balakrishnan,N.(1991).HandbookoftheLogisticDistribution.MarcelDekker,Inc.ISBN 978-0-8247-8587-1.
Gouriéroux,Christian(2000)."TheSimpleDichotomy".EconometricsofQualitativeDependentVariables.NewYork:CambridgeUniversityPress.pp. 6–37.ISBN 978-0-521-58985-7.
Greene,WilliamH.(2003).EconometricAnalysis,fifthedition.PrenticeHall.ISBN 978-0-13-066189-0.
Hilbe,JosephM.(2009).LogisticRegressionModels.Chapman&Hall/CRCPress.ISBN 978-1-4200-7575-5.
Hosmer,David(2013).Appliedlogisticregression.Hoboken,NewJersey:Wiley.ISBN 978-0470582473.
Howell,DavidC.(2010).StatisticalMethodsforPsychology,7thed.Belmont,CA;ThomsonWadsworth.ISBN 978-0-495-59786-5.
Peduzzi,P.;J.Concato;E.Kemper;T.R.Holford;A.R.Feinstein(1996)."Asimulationstudyofthenumberofeventspervariableinlogisticregressionanalysis".JournalofClinicalEpidemiology.49(12):1373–1379.doi:10.1016/s0895-4356(96)00236-3.PMID 8970487.
Berry,MichaelJ.A.;Linoff,Gordon(1997).DataMiningTechniquesForMarketing,SalesandCustomerSupport.Wiley.
Externallinks[edit]
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