Conjoint Utilities and Covariance Matrices: Estimating Respondent Deviations from The Mean Using Reduced Factors

The standard HB estimation of conjoint respondent utilities assumes a multivariate normal prior with unknown mean vector µ and covariance Σ as additional parameters.  We describe a low-rank approximation Σ ~ LR*LR’:

• Σ is dimension PxP

• LR is reduced dimension PxP_Fac (recommending P_Fac<=10)

• For each respondent only P_Fac z-values are directly estimated, computing full P utilities as µ + LR*z.

Compared with estimations of full Σ using Inverse Wishart, Wishart, or LKJ, reduced factor parameterization decreases noise and improves speed, convergence, and holdout fit -- especially for large parameter spaces.