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Copyright © 2008, SAS Institute Inc. All rights reserved.
Practical Design for
Discrete Choice
Experiments
Bradley Jones, SAS Institute
August 13, 2008

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

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
Cambridge
3

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
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
Cambridge

D criterion - minimize the determinant of the variance matrix of
the estimators:
( )( )1
det −
M Xβ,
Design optimality criterion
Cambridge
6
Equivalently – maximize the determinant of the
information matrix, M.

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
Cambridge

Design for Nonlinear Models
To design an informative experiment …..
You need to know something about the response function …..
And about the parameter values.
Cambridge

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
Cambridge

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.
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.
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.
Cambridge

Mysovskikh quadrature continued…
Radial integral: Generalized Gauss-LaGuerre
quadrature, with an extra point at the origin.
Spherical integrals: The Mysovskikh quadrature
scheme.
Cambridge

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).
Cambridge

Simplex point
Mid-point
Each point is both a simplex
point and a mid-point.
All weights equal1/6.
Quadrature points in two dimensions
Cambridge

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
Cambridge

Software Demonstration
Cambridge

Conclusions
1) Discrete Choice Conjoint Experiments require design
methods for nonlinear models.
2) D-Optimal Bayesian designs reduce the dependence
of the design on the unknown parameters.
3) New quadrature methods make computation of these
designs much faster.
4) Commercial software makes carrying out such
studies simple and efficient.
Cambridge

References
Atkinson, A. C. and Donev, A. N. (1992). Optimum Experimental Designs, Oxford U.K.: Clarendon Press.
Cassity C.R., (1965) “Abscissas, Coefficients, and Error Term for the Generalized Gauss-Laguerre Quadrature Formula Using the Zero
Ordinate,” Mathematics ofComputation, 19, 287-296.
Chaloner, K. and Verdinelli, I. (1995). Bayesian experimental design: a re-view, Statistical Science 10: 273-304.
Grossmann, H., Holling, H. and Schwabe, R. (2002). Advances in optimum experimental design for conjoint analysis and discrete
choice models, in Advances in Econometrics, Econometric Models in Marketing, Vol. 16, Franses, P. H. and Montgomery, A. L.,
eds. Amsterdam: JAI Press, 93-117.
Gotwalt, C., Jones, B. and Steinberg, D. (2009) Fast Computation of Designs Robust to Parameter Uncertainty for Nonlinear Settings
accepted at Technometrics.
Huber, J. and Zwerina, K. (1996). The importance of utility balance in efficient choice designs, Journal of Marketing Research 33: 307-
317.
McFadden, D. (1974). Conditional logit analysis of qualitative choice behavior, in Frontiers in Econometrics, Zarembka, P., ed. New
York: Academic Press, 105-142.
Meyer, R. K. and Nachtsheim, C. J. (1995). The coordinate-exchange algorithm for constructing exact optimal experimental designs,
Technometrics 37: 60-69.
Monahan, J. and Genz, A. (1997). Spherical-radial integration rules for Bayesian computation, Journal of the American Statistical
Association 92: 664-674.
Sandor, Z. and Wedel, M. (2001). Designing conjoint choice experiments using managers' prior beliefs, Journal of Marketing Research
38: 430-444.
Cambridge

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