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2. 2. Copyright © 2008, SAS Institute Inc. All rights reserved. DEMA 2008, August 2008, Cambridge 2  Respondents indicate the alternative they prefer most in each choice set  Alternatives are called profiles  Each profile is a combination of attribute levels  Choice sets typically consist of two, three or four profiles Discrete Choice Experiment Setup
3. 3. Copyright © 2008, SAS Institute Inc. All rights reserved. Example: Marketing a new laptop computer Attributes Levels Hard Drive 40 GB 80 GB Speed 1.5 GHz 2.0 GHz Battery Life 4 hours6 hours Price \$1,000 \$1,200 \$1,500 DEMA 2008, August 2008, Cambridge 3
4. 4. Copyright © 2008, SAS Institute Inc. All rights reserved. Sample Choice Set Hard Disk Speed Battery Price 40Gig 1.5GHz 6hours \$1,000 40Gig 2.0GHz 4hours \$1,500 Check the box for the laptop you prefer. Profile 1 Profile 2 4DEMA 2008, August 2008, Cambridge
5. 5. Copyright © 2008, SAS Institute Inc. All rights reserved. 5 multinomial logit model based on the random utilities model where xjs represents the attribute levels and β is the set of parameter values probability of choosing alternative j in choice set s Statistical model ε′= +js js jsU xβ =1 option chosen in choice set js ts js J t j e p s e ′ ′   = ÷   ∑ xβ xβ DEMA 2008, August 2008, Cambridge
6. 6. Copyright © 2008, SAS Institute Inc. All rights reserved. D criterion - minimize the determinant of the variance matrix of the estimators: ( )( )1 det − M Xβ, Design optimality criterion DEMA 2008, August 2008, Cambridge 6 Equivalently – maximize the determinant of the information matrix, M.
7. 7. Copyright © 2008, SAS Institute Inc. All rights reserved. 7 Bayesian optimal designs: • construct a prior distribution for the parameters • find design that performs best on average • Sándor & Wedel (2001, 2002, 2005) Dependence on the unknown parameter, β ( ) ( ) ( ) ( )( ) 1 , S s s s s s s= ′ ′= −∑M Xβ X P β p β p β X DEMA 2008, August 2008, Cambridge
8. 8. Copyright © 2008, SAS Institute Inc. All rights reserved. Design for Nonlinear Models To design an informative experiment ….. You need to know something about the response function ….. And about the parameter values. 8DEMA 2008, August 2008, Cambridge
9. 9. Copyright © 2008, SAS Institute Inc. All rights reserved. Bayesian D-Optimal Design Bayesian ideas are natural to cope with the fact that the information matrix, M, depends on β. Chaloner and Larntz (1986) developed a Bayesian D- Optimality criterion: Φ(d) = ∫ log det [M(β;d)] p(β) dβ 9DEMA 2008, August 2008, Cambridge
10. 10. Copyright © 2008, SAS Institute Inc. All rights reserved. Computing Bayesian D-Optimal Designs A major impediment to Bayesian D-optimal design has been COMPUTATIONAL. The integral over β can be VERY SLOW. It must be computed MANY TIMES in the course of finding an optimal design. 10DEMA 2008, August 2008, Cambridge
11. 11. Copyright © 2008, SAS Institute Inc. All rights reserved. Bayesian Computations Gotwalt, Jones and Steinberg (2007) use a quadrature method, due to Mysovskikh. This method is guaranteed to exactly integrate all polynomials up to 5th degree and all odd-degree monomials. With p parameters, it requires just O(p2) function evaluations. 11DEMA 2008, August 2008, Cambridge
12. 12. Copyright © 2008, SAS Institute Inc. All rights reserved. Mysovskikh quadrature Assume a normal prior with independence. • Center the integral about the prior mean. • Scale each variable by its standard deviation. • Integrate over distance from the prior mean and, at each distance, over a spherical shell. 12DEMA 2008, August 2008, Cambridge
14. 14. Copyright © 2008, SAS Institute Inc. All rights reserved. Spherical integral A simplex, its edge midpoints on the sphere, and the inverses of all of these points. Simplex point weights: p(7-p)/2(p+1)2(p+2). Mid-point weights: 2(p-1)2/p(p+1)2(p+2). 14DEMA 2008, August 2008, Cambridge
15. 15. Copyright © 2008, SAS Institute Inc. All rights reserved. Simplex point Mid-point Each point is both a simplex point and a mid-point. All weights equal1/6. Quadrature points in two dimensions 15DEMA 2008, August 2008, Cambridge
16. 16. Copyright © 2008, SAS Institute Inc. All rights reserved. Study Description 16 Respondents – 8 developers 8 sales & marketing 9 Male 7 Female 2 Surveys with 6 choice sets in each Respondents were assigned randomly to surveys blocked by job function 16DEMA 2008, August 2008, Cambridge