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• Discrete Choice Experiment Setup  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 DEMA 2008, August 2008, 2 Copyright © 2008, SAS Institute Inc. All rights reserved. Cambridge
• Example: Marketing a new laptop computer Attributes Levels Hard Drive 40 GB 80 GB Speed 1.5 GHz 2.0 GHz Battery Life 4 hours 6 hours Price \$1,000 \$1,200 \$1,500 DEMA 2008, August 2008, 3 Copyright © 2008, SAS Institute Inc. All rights reserved. Cambridge
• Sample Choice Set Check the box for the laptop you prefer. Hard Disk Speed Battery Price 40Gig 1.5GHz 6hours \$1,000 Profile 1 40Gig 2.0GHz 4hours \$1,500 Profile 2 DEMA 2008, August 2008, 4 Copyright © 2008, SAS Institute Inc. All rights reserved. Cambridge
• Statistical model multinomial logit model based on the random utilities model U js  xjsβ   js where xjs represents the attribute levels and β is the set of parameter values probability of choosing alternative j in choice set s x jsβ  option j chosen  e p js    t =1  J  in choice set s  e xtsβ DEMA 2008, August 2008, 5 Copyright © 2008, SAS Institute Inc. All rights reserved. Cambridge
• Design optimality criterion D criterion - minimize the determinant of the variance matrix of the estimators: det M 1  X,β   Equivalently – maximize the determinant of the information matrix, M. DEMA 2008, August 2008, 6 Copyright © 2008, SAS Institute Inc. All rights reserved. Cambridge
• Dependence on the unknown parameter, β S M  X,β    X Ps β   ps β  p β   X s s s s1 Bayesian optimal designs: • construct a prior distribution for the parameters • find design that performs best on average • Sándor & Wedel (2001, 2002, 2005) DEMA 2008, August 2008, 7 Copyright © 2008, SAS Institute Inc. All rights reserved. Cambridge
• Design for Nonlinear Models To design an informative experiment ….. You need to know something about the response function ….. And about the parameter values. DEMA 2008, August 2008, 8 Copyright © 2008, SAS Institute Inc. All rights reserved. Cambridge
• Bayesian D-Optimal Design Bayesian ideas are natural to cope with the fact that the information matrix, M, depends on b. Chaloner and Larntz (1986) developed a Bayesian D- Optimality criterion: Φ(d) = ∫ log det [M(b;d)] p(b) db DEMA 2008, August 2008, 9 Copyright © 2008, SAS Institute Inc. All rights reserved. Cambridge
• Computing Bayesian D-Optimal Designs A major impediment to Bayesian D-optimal design has been COMPUTATIONAL. The integral over b can be VERY SLOW. It must be computed MANY TIMES in the course of finding an optimal design. DEMA 2008, August 2008, 10 Copyright © 2008, SAS Institute Inc. All rights reserved. Cambridge
• 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. DEMA 2008, August 2008, 11 Copyright © 2008, SAS Institute Inc. All rights reserved. Cambridge
• 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. DEMA 2008, August 2008, 12 Copyright © 2008, SAS Institute Inc. All rights reserved. Cambridge