Ellipsoidal Methods for Adaptive Choice-Based Conjoint Analysis

Questionnaires for adaptive choice-based conjoint analysis aim at minimizing some measure of the uncertainty associated with estimates of preference parameters (e.g., partworths). Bayesian approaches to conjoint analysis quantify this uncertainty with a multivariate distribution that is updated after the respondent answers. Unfortunately, this update often requires multidimensional integration, which effectively reduces the adaptive selection of questions to impractical enumeration. An alternative approach is the polyhedral method for adaptive conjoint analysis, which quantifies the uncertainty through a (convex) polyhedron. The approach has a simple geometric interpretation and allows for quick credibility-region updates and effective optimization-based heuristics for adaptive question selection. However, its performance deteriorates with high response-error rates. Available adaptations to this method do not preserve all of the geometric simplicity and interpretability of the original approach. We show how, by using normal approximations to posterior distributions, one can include response error in an approximate Bayesian approach that is as intuitive as the polyhedral approach and allows the use of effective optimization-based techniques for adaptive question selection. This ellipsoidal approach extends the effectiveness of the polyhedral approach to the high response-error setting and provides a simple geometric interpretation (from which the method derives its name) of Bayesian approaches. Our results precisely quantify the relationship between the most popular efficiency criterion and heuristic guidelines promoted in extant work. We illustrate the superiority of the ellipsoidal method through extensive numerical experiments.