Robust conjoint analysis by controlling outlier sparsity

Preference measurement (PM) has a long history in marketing, healthcare, and the biobehavioral sciences, where conjoint analysis is commonly used. The goal of PM is to learn the utility function of an individual or a group of individuals from expressed preference data (buying patterns, surveys, ratings), possibly contaminated with outliers. For metric conjoint data, a robust partworth estimator is developed on the basis of a neat connection between ℓ0-(pseudo)norm-regularized regression, and the least-trimmed squared estimator. This connection suggests efficient solvers based on convex relaxation, which lead naturally to a family of robust estimators subsuming Huber's optimal M-class. Outliers are identified by tuning a regularization parameter, which amounts to controlling the sparsity of an outlier vector along the entire robustification path of least-absolute shrinkage and selection operator solutions. For choice-based conjoint analysis, a novel classifier is developed that is capable of attaining desirable tradeoffs between model fit and complexity, while at the same time controlling robustness and revealing the outliers present. Variants accounting for nonlinear utilities and consumer heterogeneity are also investigated.