[Optimal design in conjoint analysis] [Elektronische Ressource] / [Habib Jafari]

Description

Contents1 INTRODUCTION 72 MODEL SPECIFICATION 112.1 Multinomial Logit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.1 The Generalized Extreme Value(GEV) . . . . . . . . . . . . . . . . . . . 142.1.2 Consistency with Random Utility Maximization (RUM) . . . . . . . . . . 162.1.3 Independence from Irrelevant Alternative (IIA) . . . . . . . . . . . . . . 172.1.4 Likelihood Function and Parameters Estimator . . . . . . . . . . . . . . 182.2 Nested Multinomial Logit Model (NMNL) . . . . . . . . . . . . . . . . . . . . . 192.2.1 Two-Level Nested MNL Model . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Three-Level MNL Model . . . . . . . . . . . . . . . . . . . . . . . 293 OPTIMAL DESIGN 353.1 The General Equivalence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Other Optimality Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3 The General Properties of D-Optimal Designs . . . . . . . . . . . . . . . . . . . 423.4 The Properties of Information Matrices . . . . . . . . . . . . . . . . . . . . . . . 433.5 Bayesian Optimal Design Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 443.6 Support Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.7 Optimal Design for Logit Models . . . . . . . . . . . . . . . . . . . . . . . . . . 473.7.1 Optimal Design in MNL Model . . . . . . . . . . . . . . . . . . . . . . . 563.7.2 Maximin E cient Designs . . . . .

Sujets

• Management

Informations

• Publié par: otto-von-guericke-universitat_magdeburg
• Publié le: 01 janvier 2010
• Nombre de lectures: 114
• Poids de l'ouvrage: 1 Mo

Extrait

Contents

INTRODUCTION1

7

11TIONSPECIFICAMODEL22.1MultinomialLogitModel...............................11
2.1.1TheGeneralizedExtremeValue(GEV)...................14
2.1.2ConsistencywithRandomUtilityMaximization(RUM)..........16
2.1.3IndependencefromIrrelevantAlternative(IIA)..............17
2.1.4LikelihoodFunctionandParametersEstimator..............18
2.2NestedMultinomialLogitModel(NMNL).....................19
2.2.1Two-LevelNestedMNLModel.......................20
2.2.2Three-LevelNestedMNLModel.......................29

35DESIGNOPTIMAL33.1TheGeneralEquivalenceTheorem.........................37
3.2OtherOptimalityCriteria..............................40
3.3TheGeneralPropertiesofD-OptimalDesigns...................42
3.4ThePropertiesofInformationMatrices.......................43
3.5BayesianOptimalDesignTheory..........................44
3.6SupportPoints....................................45
3.7OptimalDesignforLogitModels..........................47
3.7.1OptimalDesigninMNLModel.......................56
3.7.2MaximinEfficientDesigns..........................67
3.7.3Invariance...................................69

4OPTIMALDESIGNINTWO-LEVELNMNLMODELS75
4.1ModelSpecifications.................................75
4.2InformationMatrixforTheNMNLModel.....................76
4.3D-OptimalDesignforNMNLModel........................81
4.4Example........................................85

5OPTIMALDESIGNINATHREE-LEVELNMNLMODEL99
5.1ModelSpecifications.................................100
5.2InformationMatrix..................................102
5.3D-OptimalCriterion.................................111

1

tstenCon

6

7

A

2

5.4Example.....................................

OPTIMALDESIGNINTHERANK-ORDERTWO-LEVELNMNLMODEL
6.1Rank-OrderMNL(RO.MNL)Model.....................
6.2Rank-OrderTwo-LevelNestedMNL(RO.NMNL)Models.........
6.2.1InformationMatrixforRO.NMNLModel..............
6.2.2D-OptimalDesign...........................
6.2.3Example.................................

EXTENSIONSandDISCUSSION7.1DiscussaboutIIA................................
7.2MoreAboutNMNLModels..........................
7.3AboutOptimalDesign.............................

NOMENCLATURE

.

.....

...

.

.....

...

121.

121.122126.130.133.134.

149150.154.155.

157

ablesTofList

3.1MNLModel:Discretechoiceexperimentwithfourchoicesets,C1,...,C4each
withthreealternatives(Js=3;∀s∈S),therearetwoattributeseachwithtwo
levels;ajsdenotesthejthalternativeofthechoicesetCs.............
3.2MNLmodelwithtwoattributeseachwithtwolevels:LocallyD-optimaldesignbasedon
experiments2×2/3/4(Ψ1(ξ1,β),Fourchoicesetseachwiththreealternatives)and2×
2/2/6(Ψ2(ξ2,β),Sixchoicesteseachwithtwoalternatives)andlocallyD-optimalcriterion
Ψr(ξr,β)=(detM(ξr,β))−1;r=1,2............................

4.1NMNLModel:ThetotalnumberofalternativeshasbeendividedintoMnests
eachwithJmalternatives;a˜jmdenotesalternativejinnestm..........
4.2NMNLModelwithtwonestsandtheChoicesetsrelatedtoExample4.1:There
arethree(N=3)classes,J11s=J12s=2;∀s∈S1,J21s=1,J22s=3;∀s∈S2
andJ31s=3,J32s=1;∀s∈S3(ajnmsdenotesjthalternativebyclassnfrom
nestmw.r.tchoicesets.)..............................
4.3NMNLmodel,β=0,λ1=λ2=λ(twonests)withtwoattributes,oneofthem
withthreelevelsandtheotherwithtwo:LocallyD-optimaldesignforCns;n=
1,s=1,2,...,9andCns;n=2,3,s=1,2,3,where3w1+6w2+6w3=1,with
LocalD-optimalitycriterion:Ψ(ξ,θ0)=ln(det(I(ξ,θ0)))forExample4.1(with
initialvaluew1=0.1,w2=0.1,w3=1sothatallofresultsconverge),Here
w1∗,w2∗,w3∗havebeenroundedtofour60digits....................
4.4NMNLmodel,β=0,λ1=λ2(twonests)withtwoattributes,oneofthem
withthreelevelsandtheotherwithtwo:LocallyD-optimaldesignforCns;n=
1,s=1,2,...,9andCns;n=2,3,s=1,2,3,where3w1+6w2+3w3+3w4=1,
withLocallyD-optimalcriterion:Ψ(ξ,θ0)=ln(det(I(ξ,θ0)))forExample4.1
(withinitialvalue∗w1∗=0.∗1,w∗2=601,w3=0.1,w4=0.1sothatallofresultsare
converge),Herew1,w2,w3,w4havebeenroundedtoourdigits...........
4.5NMNLModel(twonests):Therearetwoclasses(N=2),eachclasswithtwo
choicesets(Cns;n=1,2,s=1,2)whichincludethreealternativesintwonests,
whereajnmsdenotethejthalternativeofthemthnestinchoicesetsbyclassn.
4.6NMNLmodel(twonests),β2=0,λ1=λ2=λ:LocallyD-optimalweightsw1∗,w2∗,w3∗
andw4∗,accordingtolocalD-optimalitycriterionΨ(ξ,θ)=ln(det(I(ξ,θ)))andw.r.tβ1∈
(−1,1),λ∈(0,1](basedonRUMconditions),fordesignξExample4.2withinitialvalues
w1=w2=w3=0.2,w4=0.4(allofresultsareconverge).................

6166

818789919393

3

ablesTofList

4

4.74.85.15.25.35.45.55.65.75.85.96.1

NMNLModelwithtwonestsandtheChoicesetsrelatedtoExample4.2:There
aretwo(N=2)thclasses,J11s=2,J12s=1;∀s∈S1,J21s=1,J22s=2;∀s∈S2
(ajnmsdenotesjalternativebyclassnfromnestmw.r.tchoicesets)......94
NMNLmodel(twonests),β2=0,λ1=λ2=λ:Locallyoptimalweightw∗,w.r.t
localD-optimalitycriterionΨ(ξ,θ)=ln(det(I(ξ,θ)))fordesign(4.6),where
β1∈(−1,1),λ∈(0,1](basedonRUMconditions)................96
NMNLModel:Therearetwonests,thefirstnestincludestwosub-nests(with
J11andJ21alternatives)andtheseconddoesnothaveanysub-nestwithJ2
alternatives,wherea˜jhmdenotesthejthalternativeinsub-nesthofnestm....108
NMNLModel:Therearetwonests,thefirstnestincludestwosub-nesteach
withtwoalternativesandthesecondnestdoesnotsub-nestandcontainstwo
alternatives.......................................112
NMNLModel(twonests,thefirstnestwithtwosub-nestsandtheseconddoes
notsub-nest):Therearethreeclasses,N=3,eachwithtwochoicesets(Cns=
Cs;∀n∈N,s∈Sn,s=1,...,6),basedonajnhms,whichdenotesthejth
alternativeinclassnofthesub-nesthofthenestminchoicesets........113
MNLModel(twonests,thefirstnestwithtwosub-nestsandtheseconddoesnot
sub-nest):Thecharacterizesofthreeattributeseachwithtwolevels;considering
sixchoicesetseachwithfivealternatives,Cns=Cs;∀n∈N,s∈Sn,s=1....,6.114
NMNLModel,µ1=2λ,µ2=4λ(twonests,firstnestwithtwosub-nestsandthe
seconddoesnotsub-nest):LocallyD-optimaldesignwhen0<λ≤0.25(based
onRUMconditions)withinitialvaluesw1=0.1,w2=w3=0.2andw.r.tlocal
D-optimalitycriterionΨ(ξ,θ0)=(det(I(ξ,θ0)))−1.................116
NMNLModel,µ1=0.15,µ2=0.25andλ1=0.1(twonests,firstnestwithtwo
sub-nestsandtheseconddoesnotsub-nest):LocallyD-optimaldesignwhen
0<λ2≤0.150(basedonRUMconditions)withinitialvaluesw1=0.1,w2=
w3=0.2andw.r.tlocalD-optimalitycriterionΨ(ξ,θ0)=(det(I(ξ,θ0)))−1...117
NMNLModel,µ1=µ2=λ1=λ2=λ(twonests,firstnestwithtwosub-nests
andtheseconddoesnotsub-nest):LocallyD-optimaldesignwhen0<λ≤1
(basedonRUMconditions)withinitialvaluesw1=0.1,w2=w3=0.2andw.r.t
localD-optimalitycriterionΨ(ξ,θ0)=(det(I(ξ,θ0)))−1..............117
NMNLModel,µ1=0.1,λ1=λ2=0.08(twonests,firstnestwithtwosub-nests
andtheseconddoesnotsub-nest):LocallyD-optimaldesignwhen0.1≤µ2≤1
(basedonRUMconditions)withinitialvaluesw1=0.1,w2=w3=0.2andw.r.t
localD-optimalitycriterionΨ(ξ,θ0)=(det(I(ξ,θ0)))−1..............119
NMNLModel,µ2=0.5,λ1=0.1,λ2=0.2(twonests,firstnestwithtwosub-
nestsandtheseconddoesnotsub-nest):LocallyD-optimaldesignwhen0.2≤
µ1≤0.5(basedonRUMconditions)withinitialvaluesw1=0.1,w2=w3=0.2
andw.r.tlocalD-optimalitycriterionΨ(ξ,θ0)=(det(I(ξ,θ0)))−1........119
NMNLModel:TherearetwoNestseachwithJ1andJ2alternatives(Chapter4)134

6.26.36.46.56.66.7

ablesTofList

Two-levelNestedMNLmodel:Fourchoicesets,C;s=1,2,3,4,eachwiththree
sthalternatives(adenotethejalternativeofthenestminchoicesets)....137
msjRO.NMNLmodelλ1=.6,λ2=.4(Twonests):LocallyD-optimaldesignfor
Design(6.9),wheretherearefourchoicesetseachwiththreealternatives;w.r.t
1−localD-optimalcriterion,ΨR(ξ,θ0)=(det(IR(NMNL)(ξ,θ0)))..........142
C.NMNLmodel,λ=.6,λ=.4(twonests):LocallyD-optimaldesignfor
21Design(6.9),wheretherearefourchoicesetseachwiththreealternatives;w.r.t
1−localD-optimalcriterion,Ψ(ξ,θ)=(det(I(ξ,θ)))(Comparingto
R0R(NMNL)0
Table6.3).......................................143
RO.NMNLmodel,β=β=0.0(twonests):LocallyD-optimaldesignfor
21Design(6.9),wheretherearefourchoicesetseachwiththreealternatives;w.r.t
1−localD-optimalcriterion,ΨR(ξ,θ0)=(det(IR(NMNL)(ξ,θ0)))..........144
RO.NMNLmodel,β2=0,λ1=λ2=λ(twonests):LocallyD-optimaldesign,
∗w,withrespecttoDesign(6.11)..........................146
1∗RO.NMNLmodel,β=0:LocallyD-optimaldesign,w,withrespecttoDesign
(6.12).........................................147

5

List

6

of

Tables

FiguresofList

3.13.23.33.44.14.25.15.26.1

Partitionofthe(β,β)-planeintoregions,wheredifferenttypesofdesignsare
21optimal,fordifferentvaluesofβ,forexample,a)β=1,b)β=0.5,andsoon50
333∗∗∗∗β=0:optimalweight(a)w=w,(b)w=wbasedon(β,β)andtwo
3123412
1212specialcases;(c)β=βand(d)β=−β,where(e)isoverlappedFigurew.r.t
Figures(c),(d)....................................52
MNLModel:Partitionofthe(β,β)-planeintoregions,wheredifferenttype
21designsareoptimal..................................60
22MNLModel:LocallyD-optimaldesign;(a);β=0.5and(b);β=−0.5,based
onthree-point(ξ)optimaldesigns(w.r.tExample3.2)andFigure3.3)....64
mr(a):TheNMNLModelwithMnestseachwithJalternatives(w.r.tChoiceSet
msC)and(b):whenM=2(Lemma4.1)......................77
sNMNLModel(twonests),β=0,λ=λ=λ(0<λ<1):Optimalweights
21∗∗w(dottedline)andw(solid)............................89
32NMNLModel:ThereareMnestseachwithH;m=1,2,...,Msub-nestsand
m