ISYE 6414 – Final COMPLETELY SOLVED
GRADED A 2024
Logistic iRegression i- i(correct ianswer) i-Commonly iused ifor imodeling ibinary iresponse idata. iThe
iresponse ivariable iis ia ibinary ivariable, iand ithus, inot inormally idistributed. i
In ilogistic iregression, iwe imodel ithe iprobability iof ia isuccess, inot ithe iresponse ivariable. iIn ithis
imodel, iwe ido inot ihave ian ierror iterm
g-function i- i(correct ianswer) i-We ilink ithe iprobability iof isuccess ito ithe ipredicting ivariables iusing ithe
ig ilink ifunction. iThe ig ifunction iis ithe is-shape ifunction ithat imodels ithe iprobability iof isuccess iwith
irespect ito ithe ipredicting ivariables
The ilink ifunction ig iis ithe ilog iof ithe iratio iof ip iover ione iminus ip, iwhere ip iagain iis ithe iprobability iof
isuccess
Logit ifunction i(log iodds ifunction) iof ithe iprobability iof isuccess iis ia ilinear imodel iin ithe ipredicting
ivariables
The iprobability iof isuccess iis iequal ito ithe iratio ibetween ithe iexponential iof ithe ilinear icombination iof
ithe ipredicting ivariables iover i1 iplus ithis isame iexponential
Odds iof ia isuccess i- i(correct ianswer) i-This iis ithe iexponential iof ithe iLogit ifunction
Logistic iRegression iAssumptions i- i(correct ianswer) i-Linearity: iThe irelationship ibetween ithe ig iof ithe
iprobability iof isuccess iand ithe ipredicted ivariable, iis ia ilinear ifunction. i
,Independence: iThe iresponse ibinary ivariables iare iindependently iobserved
Logit: iThe ilogistic iregression imodel iassumes ithat ithe ilink ifunction ig iis ia ilogit ifunction
Linearity iAssumption i- i(correct ianswer) i-The iLogit itransformation iof ithe iprobability iof isuccess iis ia
ilinear icombination iof ithe ipredicting ivariables. iThe irelationship imay inot ibe ilinear, ihowever, iand
itransformation imay iimprove ithe ifit
The ilinearity iassumption ican ibe ievaluated iby iplotting ithe ilogit iof ithe isuccess irate iversus ithe
ipredicting ivariables. i
If ithere's ia icurvature ior isome inon-linear ipattern, iit imay ibe ian iindication ithat ithe ilack iof ifit imay ibe
idue ito ithe inon-linearity iwith irespect ito isome iof ithe ipredicting ivariables
Logistic iRegression iCoefficient i- i(correct ianswer) i-We iinterpret ithe iregression icoefficient ibeta ias ithe
ilog iof ithe iodds iratio ifor ian iincrease iof ione iunit iin ithe ipredicting ivariable
We ido inot iinterpret ibeta iwith irespect ito ithe iresponse ivariable ibut iwith irespect ito ithe iodds iof
isuccess
The iestimators ifor ithe iregression icoefficients iin ilogistic iregression iare iunbiased iand ithus ithe imean
iof ithe iapproximate inormal idistribution iis ibeta. iThe ivariance iof ithe iestimator idoes inot ihave ia
iclosed iform iexpression
Model iparameters i- i(correct ianswer) i-The imodel iparameters iare ithe iregression icoefficients. i
There iis ino iadditional iparameter ito imodel ithe ivariance isince ithere's ino ierror iterm. i
For iP ipredictors, iwe ihave iP i+ i1 iregression icoefficients ifor ia imodel iwith iintercept i(beta i0).
We iestimate ithe imodel iparameters iusing ithe imaximum ilikelihood iestimation iapproach
, Response ivariable i- i(correct ianswer) i-The iresponse idata iare iBernoulli ior ibinomial iwith ione itrial iwith
iprobability iof isuccess
MLE i- i(correct ianswer) i-The iresulting ilog-likelihood ifunction ito ibe imaximized, iis ivery icomplicated
iand iit iis inon-linear iin ithe iregression icoefficients ibeta i0, ibeta i1, iand ibeta ip
MLE ihas igood istatistical iproperties iunder ithe iassumption iof ia ilarge isample isize ii.e. ilarge iN
For ilarge iN, ithe isampling idistribution iof iMLEs ican ibe iapproximated iby ia inormal idistribution
The ileast isquare iestimation ifor ithe istandard iregression imodel iis iequivalent iwith iMLE, iunder ithe
iassumption iof inormality.
MLE iis ithe imost iapplied iestimation iapproach
Parameter iestimation i- i(correct ianswer) i-Maximizing ithe ilog ilikelihood ifunction iwith irespect ito
ibeta0, ibeta1 ietc iin iclosed i(exact) iform iexpression iis inot ipossible ibecause ithe ilog ilikelihood ifunction
iis ia inon-linear ifunction iin ithe imodel iparameters ii.e. iwe icannot iderive ithe iestimated iregression
icoefficients iin ian iexact iform
Use inumerical ialgorithm ito iestimate ibetas i(maximize ithe ilog ilikelihood ifunction). iThe iestimated
iparameters iand itheir istandard ierrors iare iapproximate iestimates
Binomial iData i- i(correct ianswer) i-This iis ibinary idata iwith irepititions
Marginal iRelationship i- i(correct ianswer) i-Capturing ithe iassociation iof ia ipredicting ivariable ito ithe
iresponse ivariable iwithout iconsideration iof iother ifactors
Conditional iRelationship i- i(correct ianswer) i-Capturing ithe iassociation ioof ia ipredicting ivariable ito ithe
iresponse ivariable iconditional iof iother ipredicting ivariables iin ithe imodel
Simpson's iparadox i- i(correct ianswer) i-This iis iwhen ithe iaddition iof ia ipredictive ivariable ireverses ithe
isign ion ithe icoefficients iof ian iexisting iparameter
It irefers ito ireversal iof ian iassociation iwhen ilooking iat ia imarginal irelationship iversus ia ipartial ior
iconditional ione. iThis iis ia isituation iwhere ithe imarginal irelationship iadds ia iwrong isign
GRADED A 2024
Logistic iRegression i- i(correct ianswer) i-Commonly iused ifor imodeling ibinary iresponse idata. iThe
iresponse ivariable iis ia ibinary ivariable, iand ithus, inot inormally idistributed. i
In ilogistic iregression, iwe imodel ithe iprobability iof ia isuccess, inot ithe iresponse ivariable. iIn ithis
imodel, iwe ido inot ihave ian ierror iterm
g-function i- i(correct ianswer) i-We ilink ithe iprobability iof isuccess ito ithe ipredicting ivariables iusing ithe
ig ilink ifunction. iThe ig ifunction iis ithe is-shape ifunction ithat imodels ithe iprobability iof isuccess iwith
irespect ito ithe ipredicting ivariables
The ilink ifunction ig iis ithe ilog iof ithe iratio iof ip iover ione iminus ip, iwhere ip iagain iis ithe iprobability iof
isuccess
Logit ifunction i(log iodds ifunction) iof ithe iprobability iof isuccess iis ia ilinear imodel iin ithe ipredicting
ivariables
The iprobability iof isuccess iis iequal ito ithe iratio ibetween ithe iexponential iof ithe ilinear icombination iof
ithe ipredicting ivariables iover i1 iplus ithis isame iexponential
Odds iof ia isuccess i- i(correct ianswer) i-This iis ithe iexponential iof ithe iLogit ifunction
Logistic iRegression iAssumptions i- i(correct ianswer) i-Linearity: iThe irelationship ibetween ithe ig iof ithe
iprobability iof isuccess iand ithe ipredicted ivariable, iis ia ilinear ifunction. i
,Independence: iThe iresponse ibinary ivariables iare iindependently iobserved
Logit: iThe ilogistic iregression imodel iassumes ithat ithe ilink ifunction ig iis ia ilogit ifunction
Linearity iAssumption i- i(correct ianswer) i-The iLogit itransformation iof ithe iprobability iof isuccess iis ia
ilinear icombination iof ithe ipredicting ivariables. iThe irelationship imay inot ibe ilinear, ihowever, iand
itransformation imay iimprove ithe ifit
The ilinearity iassumption ican ibe ievaluated iby iplotting ithe ilogit iof ithe isuccess irate iversus ithe
ipredicting ivariables. i
If ithere's ia icurvature ior isome inon-linear ipattern, iit imay ibe ian iindication ithat ithe ilack iof ifit imay ibe
idue ito ithe inon-linearity iwith irespect ito isome iof ithe ipredicting ivariables
Logistic iRegression iCoefficient i- i(correct ianswer) i-We iinterpret ithe iregression icoefficient ibeta ias ithe
ilog iof ithe iodds iratio ifor ian iincrease iof ione iunit iin ithe ipredicting ivariable
We ido inot iinterpret ibeta iwith irespect ito ithe iresponse ivariable ibut iwith irespect ito ithe iodds iof
isuccess
The iestimators ifor ithe iregression icoefficients iin ilogistic iregression iare iunbiased iand ithus ithe imean
iof ithe iapproximate inormal idistribution iis ibeta. iThe ivariance iof ithe iestimator idoes inot ihave ia
iclosed iform iexpression
Model iparameters i- i(correct ianswer) i-The imodel iparameters iare ithe iregression icoefficients. i
There iis ino iadditional iparameter ito imodel ithe ivariance isince ithere's ino ierror iterm. i
For iP ipredictors, iwe ihave iP i+ i1 iregression icoefficients ifor ia imodel iwith iintercept i(beta i0).
We iestimate ithe imodel iparameters iusing ithe imaximum ilikelihood iestimation iapproach
, Response ivariable i- i(correct ianswer) i-The iresponse idata iare iBernoulli ior ibinomial iwith ione itrial iwith
iprobability iof isuccess
MLE i- i(correct ianswer) i-The iresulting ilog-likelihood ifunction ito ibe imaximized, iis ivery icomplicated
iand iit iis inon-linear iin ithe iregression icoefficients ibeta i0, ibeta i1, iand ibeta ip
MLE ihas igood istatistical iproperties iunder ithe iassumption iof ia ilarge isample isize ii.e. ilarge iN
For ilarge iN, ithe isampling idistribution iof iMLEs ican ibe iapproximated iby ia inormal idistribution
The ileast isquare iestimation ifor ithe istandard iregression imodel iis iequivalent iwith iMLE, iunder ithe
iassumption iof inormality.
MLE iis ithe imost iapplied iestimation iapproach
Parameter iestimation i- i(correct ianswer) i-Maximizing ithe ilog ilikelihood ifunction iwith irespect ito
ibeta0, ibeta1 ietc iin iclosed i(exact) iform iexpression iis inot ipossible ibecause ithe ilog ilikelihood ifunction
iis ia inon-linear ifunction iin ithe imodel iparameters ii.e. iwe icannot iderive ithe iestimated iregression
icoefficients iin ian iexact iform
Use inumerical ialgorithm ito iestimate ibetas i(maximize ithe ilog ilikelihood ifunction). iThe iestimated
iparameters iand itheir istandard ierrors iare iapproximate iestimates
Binomial iData i- i(correct ianswer) i-This iis ibinary idata iwith irepititions
Marginal iRelationship i- i(correct ianswer) i-Capturing ithe iassociation iof ia ipredicting ivariable ito ithe
iresponse ivariable iwithout iconsideration iof iother ifactors
Conditional iRelationship i- i(correct ianswer) i-Capturing ithe iassociation ioof ia ipredicting ivariable ito ithe
iresponse ivariable iconditional iof iother ipredicting ivariables iin ithe imodel
Simpson's iparadox i- i(correct ianswer) i-This iis iwhen ithe iaddition iof ia ipredictive ivariable ireverses ithe
isign ion ithe icoefficients iof ian iexisting iparameter
It irefers ito ireversal iof ian iassociation iwhen ilooking iat ia imarginal irelationship iversus ia ipartial ior
iconditional ione. iThis iis ia isituation iwhere ithe imarginal irelationship iadds ia iwrong isign
