Bayesian D-optimal designs for rank-order conjoint ...

Bayesian D-optimal designs for rank-order conjoint choice experiments Bart Vermeulen Katholieke Universiteit Leuven, Faculty of Economics and Applied Economics

Peter Goos University of Antwerp, Faculty of Applied Economics

Martina Vandebroek Katholieke Universiteit Leuven Faculty of Economics and Applied Economics University Center of Statistics

Outline

1. Introduction  Rank-order conjoint choice experiments  Optimal Design for rank-order conjoint choice experiments

2. Rank-order multinomial logit model 3. Bayesian D-optimal designs for rank-order conjoint choice experiments 4. Performance of the Bayesian D-optimal designs 5. Improvement in the accuracy of the estimates and predictions

Introduction: rank-order conjoint choice experiment

What is a rank-order conjoint choice experiment?  A rank-order conjoint choice experiment aims to estimate the

values respondents attach to the features of a product...  ... by asking him to rank a number of alternatives of several

choice sets instead of to choose the most preferred one as is done in a classical choice experiment

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Introduction: rank-order conjoint choice experiment

Example: Rank the following laptops in decreasing order of preference:

Screen:15.4 inch

Screen:17 inch

Screen:15.4 inch

Screen:17 inch

HD:80 GB

HD:60 GB

HD:120 GB

HD:80 GB

Battery:5h

Battery:6h

Battery:4h

Battery:6h

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Introduction: Optimal design for rank-order conjoint choice experiments Goal: Performing a rank-order conjoint choice experiment in a statistically efficient way: with a small number of choice sets obtaining a maximum amount of information →Leads to precise estimates of the parameters with a minimum variance

⇓ Problem: A large number of possible candidate alternatives to include in a ranking experiment and how to group them into choice sets

⇓ Solution: Constructing D-optimal design for rank-order conjoint choice experiments

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Introduction: Optimal design for rank-order conjoint choice experiments

Questions  How to construct D-optimal designs for rank-order conjoint

choice experiments? ⇒ Development of an expression for the information matrix for rank-order conjoint choice experiments  Do these designs outperform other benchmark designs?  What is the improvement in estimation and prediction accuracy

if we include extra ranking steps in an experiment?

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The rank-ordered multinomial logit model The rank-ordered multinomial logit model (MLM) = extension of the multinomial logit model (Begs et al., 1981) ,→ Ranking alternatives = sequential and conditional choice task

⇒ If ranking alternative i, i’ and i”, the probability of assigning rank 1 to alternative i in this choice set is: 0

Pik1

exp(xik β) P = 0 exp(x j∈{i,i0 ,i00 } jk β)

(1)

⇒ Probability of assigning rank 2 to alternative i’ is then: 0

Pi0 k2

exp(xi0 k β) =P 0 exp(x j∈{i0 ,i00 } jk β)

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(2)

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The rank-ordered multinomial logit model

⇒ The joint probability of ranking alternative i first, alternative i’ second and alternative i” third is: 0

0

exp(xi0 k β) exp(xik β) P · 0 0 exp(x β) exp(x 0 00 0 00 j∈{i,i ,i } j∈{i ,i } jk jk β)

Pii0 i00 k = P

(3)

⇒ This leads to the following log-likelihood function for one choice set and for one respondent: ln(L) = Yii0 i00 k ln(Pii0 i00 k ) + Yii00 i0 k ln(Pii00 i0 k ) + Yi0 ii00 k ln(Pi0 ii00 k ) + Yi0 i00 ik ln(Pi0 i00 ik ) + Yi00 ii0 k ln(Pi00 ii0 k ) + Yi00 i0 ik ln(Pi00 i0 ik )

Bayesian D-optimal designs for rank-order conjoint choice experiments

(4)

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Bayesian D-optimal design for the rank-ordered MLM Aim: maximize information coming from the experiment 7→ Bayesian D-optimality criterion: maximizes the expected determinant of the Fisher Information matrix over the prior distribution of the unknown parameters

Db − error =

−1/p

R