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ISYE 6414 FINAL EXAM QUESTIONS AND VERIFIED ANSWERS 2024
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ISYE 6414 FINAL EXAM QUESTIONS AND VERIFIED ANSWERS 2024
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ISYE 6414 - FINALLogisticdRegressiond-
dans✔✔Commonlyduseddfordmodelingdbinarydresponseddata.dThedresponsedvariabledisdad
binarydvariable,danddthus,dnotdnormallyddistributed.d
Indlogisticdregression,dwedmodeldthedprobabilitydofdadsuccess,dnotdthedresponsedvariable.dI
ndthisdmodel,dweddodnotdhavedanderrordterm
g-functiond-
dans✔✔Wedlinkdthedprobabilitydofdsuccessdtodthedpredictingdvariablesdusingdthedgdlinkdfun
ction.dThedgdfunctiondisdtheds-
shapedfunctiondthatdmodelsdthedprobabilitydofdsuccessdwithdrespectdtodthedpredictingdvaria
bles
Thedlinkdfunctiondgdisdthedlogdofdthedratiodofdpdoverdonedminusdp,dwheredpdagaindisdthedpro
babilitydofdsuccess
Logitdfunctiond(logdoddsdfunction)dofdthedprobabilitydofdsuccessdisdadlineardmodeldindthedpr
edictingdvariables
Thedprobabilitydofdsuccessdisdequaldtodthedratiodbetweendthedexponentialdofdthedlineardco
mbinationdofdthedpredictingdvariablesdoverd1dplusdthisdsamedexponential
Oddsdofdadsuccessd-dans✔✔ThisdisdthedexponentialdofdthedLogitdfunction
LogisticdRegressiondAssumptionsd-
dans✔✔Linearity:dThedrelationshipdbetweendthedgdofdthedprobabilitydofdsuccessdanddthedpr
edicteddvariable,disdadlineardfunction.d
Independence:dThedresponsedbinarydvariablesdaredindependentlydobserved
Logit:dThedlogisticdregressiondmodeldassumesdthatdthedlinkdfunctiondgdisdadlogitdfunction
,LinearitydAssumptiond-
dans✔✔ThedLogitdtransformationdofdthedprobabilitydofdsuccessdisdadlineardcombinationdofdt
hedpredictingdvariables.dThedrelationshipdmaydnotdbedlinear,dhowever,danddtransformation
dmaydimprovedthedfit
Thedlinearitydassumptiondcandbedevaluateddbydplottingdthedlogitdofdthedsuccessdratedversu
sdthedpredictingdvariables.d
Ifdthere'sdadcurvaturedordsomednon-
lineardpattern,ditdmaydbedandindicationdthatdthedlackdofdfitdmaydbedduedtodthednon-
linearitydwithdrespectdtodsomedofdthedpredictingdvariables
LogisticdRegressiondCoefficientd-
dans✔✔Wedinterpretdthedregressiondcoefficientdbetadasdthedlogdofdthedoddsdratiodfordandin
creasedofdonedunitdindthedpredictingdvariable
Weddodnotdinterpretdbetadwithdrespectdtodthedresponsedvariabledbutdwithdrespectdtodthedod
dsdofdsuccess
Thedestimatorsdfordthedregressiondcoefficientsdindlogisticdregressiondaredunbiaseddanddthu
sdthedmeandofdthedapproximatednormalddistributiondisdbeta.dThedvariancedofdthedestimatord
doesdnotdhavedadcloseddformdexpression
Modeldparametersd-dans✔✔Thedmodeldparametersdaredthedregressiondcoefficients.d
Theredisdnodadditionaldparameterdtodmodeldthedvariancedsincedthere'sdnoderrordterm.d
FordPdpredictors,dwedhavedPd+d1dregressiondcoefficientsdfordadmodeldwithdinterceptd(betad
0).
Wedestimatedthedmodeldparametersdusingdthedmaximumdlikelihooddestimationdapproach
Responsedvariabled-
dans✔✔ThedresponseddatadaredBernoullidordbinomialdwithdonedtrialdwithdprobabilitydofdsuc
cess
MLEd-dans✔✔Thedresultingdlog-
likelihooddfunctiondtodbedmaximized,disdverydcomplicateddandditdisdnon-
lineardindthedregressiondcoefficientsdbetad0,dbetad1,danddbetadp
MLEdhasdgooddstatisticaldpropertiesdunderdthedassumptiondofdadlargedsampledsizedi.e.dlar
gedN
FordlargedN,dthedsamplingddistributiondofdMLEsdcandbedapproximateddbydadnormalddistribu
tion
, ThedleastdsquaredestimationdfordthedstandarddregressiondmodeldisdequivalentdwithdMLE,du
nderdthedassumptiondofdnormality.
MLEdisdthedmostdapplieddestimationdapproach
Parameterdestimationd-
dans✔✔Maximizingdthedlogdlikelihooddfunctiondwithdrespectdtodbeta0,dbeta1detcdindclosedd
(exact)dformdexpressiondisdnotdpossibledbecausedthedlogdlikelihooddfunctiondisdadnon-
lineardfunctiondindthedmodeldparametersdi.e.dwedcannotdderivedthedestimateddregressiondc
oefficientsdindandexactdform
Usednumericaldalgorithmdtodestimatedbetasd(maximizedthedlogdlikelihooddfunction).dThedes
timateddparametersdanddtheirdstandardderrorsdaredapproximatedestimates
BinomialdDatad-dans✔✔Thisdisdbinaryddatadwithdrepititions
MarginaldRelationshipd-
dans✔✔Capturingdthedassociationdofdadpredictingdvariabledtodthedresponsedvariabledwitho
utdconsiderationdofdotherdfactors
ConditionaldRelationshipd-
dans✔✔Capturingdthedassociationdoofdadpredictingdvariabledtodthedresponsedvariabledcon
ditionaldofdotherdpredictingdvariablesdindthedmodel
Simpson'sdparadoxd-
dans✔✔Thisdisdwhendthedadditiondofdadpredictivedvariabledreversesdthedsigndondthedcoeffic
ientsdofdandexistingdparameter
Itdrefersdtodreversaldofdandassociationdwhendlookingdatdadmarginaldrelationshipdversusdadp
artialdordconditionaldone.dThisdisdadsituationdwheredthedmarginaldrelationshipdaddsdadwron
gdsign
Thisdhappensdwhendthed2dvariablesdaredcorrelated
NormaldDistributiond-
dans✔✔Normalddistributiondreliesdondadlargedsampledofddata.dUsingdthisdapproximatednor
malddistributiondwedcandfurtherdderivedconfidencedintervals.d
Sincedtheddistributiondisdnormal,dthedconfidencedintervaldisdthedz-interval
**AppliesdfordLogisticd&dPoissondRegression
HypothesisdTestingd(coefficientd==d0)d-
dans✔✔Todperformdhypothesisdtesting,dwedcandusedthedapproximatednormaldsamplingddis
tribution.d
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