  This article considers the problem of estimating a multivariate probit model
in a panel data setting with emphasis on sampling a high-dimensional
correlation matrix and improving the overall efficiency of the data
augmentation approach. We reparameterise the correlation matrix in a principled
way and then carry out efficient Bayesian inference using Hamiltonian Monte
Carlo. We also propose a novel antithetic variable method to generate samples
from the posterior distribution of the random effects and regression
coefficients, resulting in significant gains in efficiency. We apply the
methodology by analysing stated preference data obtained from Australian
general practitioners evaluating alternative contraceptive products. Our
analysis suggests that the joint probability of discussing combinations of
contraceptive products with a patient shows medical practice variation among
the general practitioners, which indicates some resistance to even discuss
these products, let alone recommend them.
