Characterization and Optimal Designs for Discrete Choice Experiments

In discrete choice experiments, a choice design involves n attributes (factors) with i-th attribute at l(i) levels, and there are N choice sets each of size in. Demirkale, Donovan and Street (2013) considered the setup of symmetric factorials (l(i) = l) and obtained D-optimal choice designs under main effects model in the absence of two or higher order interaction effects. They provide some sufficient conditions for a design to be D-optimal. In this paper, we first derive a modified Information matrix of a choice design for estimating the factorial effects of a l(1) x l(2) x ... x l(n) choice experiment. For a 2(n) choice experiment, following Singh, Chai and Das (2015), under the broader main effects model (both in the presence and in the absence of two-factor interactions) we give a simple necessary and sufficient condition for the Information matrix to be diagonal. Furthermore, we characterize the structure of the choice sets which gives maximum trace of the Information matrix. Our characterization of such an Information matrix facilitates construction of universally optimal choice designs for estimating main effects, both in the presence and in the absence of two-factor interactions but, in the absence of three or higher order interaction effects.