Bayesian estimation of order-restricted and unrestricted association models

Abstract

a b s t r a c tAssociation 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 threeway 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.

Bayesian approaches to the weighted kappa-like inter-rater agreement measures

“…Goodman 29 mentions difficulties of score estimation because the order-restricted RC model is not a log-linear model when the scores are not fixed. Demirhan 15 proposes Bayesian approaches for the estimation of scores for both unrestricted and order-restricted cases. Under the unrestricted case, the prior distribution of scores is derived from the generalized multivariate log-gamma (G-MVLG) distribution proposed by Demirhan 30 .…”

“…We utilize this correlation structure in the estimation of scores from the agreement table. When there is no order restriction on the scores, following Demirhan 15 , we induce a G-MVLG prior on scores (see Demirhan 30 for the details of this multivariate distribution). Due to the independence of the main effect (u • ) and association (β 12 ) parameters, we set independent log-gamma priors on the main effect and association parameters.…”

“…Each of ρ i(jk) is defined in terms of γ i(ts) to link the prior information on the correlation between the scores to the covariance matrix Σ i as in equation ( 13) 15 :…”

“…Demirhan 15 derives the full conditional posterior distribution of δ i given g = (x i , h i , y) as follows:…”

“…With this approach, we attempt to improve the accuracy of the weighted inter-rater agreement measures against anomalies in the agreement table by using Bayesian approaches. Kateri et al 12 , Iliopoulos et al 13,14 and Demirhan 15 have demonstrated that the Bayesian estimation of scores for ordinal variables from data instead of using successive integers is beneficial in improving the accuracy of the models for contingency tables depending on the score values. Following their results, we propose a new Bayesian approach for the estimation of the degree of agreement for ordinal tables based on the estimation of scores used to compute weights from the data.…”