Bayesian estimation of order-restricted and unrestricted association models

Association models include score parameters to multiplicatively represent the hierarchy between the levels of the considered ordinal factor. If order restrictions are placed on the scores, an estimation problem becomes a non-linear and restricted estimation, which is somewhat problematic with respect to the classical approaches. In this article, we consider the Bayesian estimation of the scores and other parameters of an association model both with and without order restrictions. We propose the use of a previously introduced multivariate prior in the unrestricted case and an order statistics approach in the order-restricted case. The advantages of using these prior structures are that we are able to consider the correlation patterns arising from the hierarchy between the levels of ordinal factors, there is no violation of the exchangeability assumption, the approaches are general for any size of contingency table, and the posterior inferences are easily derived. The proposed approaches are applied to both a previously analyzed popular two-way contingency table and a three-way contingency table. Smaller standard deviations than those of previous analyses are obtained, and a new best-fitting model is identified for the two-way table.