Accelerating Monte Carlo Markov chain convergence for cumulative-link generalized linear models

Abstract

The ordinal probit, univariate or multivariate, is a generalized linear model (GLM) structure that arises frequently in such disparate areas of statistical applications as medicine and econometrics. Despite the straightforwardness of its implementation using the Gibbs sampler, the ordinal probit may present challenges in obtaining satisfactory convergence.We present a multivariate Hastings-within-Gibbs update step for generating latent data and bin boundary parameters jointly, instead of individually from their respective full conditionals. When the latent data are parameters of interest, this algorithm substantially improves Gibbs sampler convergence for large datasets. We also discuss Monte Carlo Markov chain (MCMC) implementation of cumulative logit (proportional odds) and cumulative complementary log-log (proportional hazards) models with latent data.

Clustering Ordinal Data via Latent Variable Models

“…The algorithm used here is similar to the algorithm proposed in [3]. A Gibbs sampler is used to sample all latent variables and parameters, except the threshold parameters, γ j .…”

“…The algorithm therefore converges slowly. This difficulty is overcome by sampling from the posterior of the threshold parameters us-ing a Metropolis-Hastings step, as in [3,11]. Candidate values v j,k are proposed…”

Semi-parametric analysis of multi-rater data

“…There are two schemes that could be adopted to overcome this problem -fixing the values of selected parameters to force the likelihood to be identifiable (often described as strong identifiability) or placing proper priors on the parameters that are sufficiently strong to overcome parameter coupling (weak identifiability). For the former case, identifiability in single rater ordinal models can be ensured by fixing one of the threshold values (see, for example [6]). …”

“…Following [6] we generated N = 200 datapoints x n from N (x n |0, 1). True latent trait values for each of L = 4 raters were then generated according to m nl = 1 − 2x n + nl where nl ∼ N (0, α −1 l ) and the precision values for the four raters were α l = [1, 0.5, 2, 4].…”

“…Strong or weak identifiability alone ('fixed' and 'priors') offer similar performance and large improvements are seen when they are combined. The autocorrelation is still rather high, suggesting thinning of the output samples is required -a characteristic of models of this type - [6] investigates methods for speeding up sampling for a linear model and investigating similar techniques for the GP prior is an interesting avenue for future investigation.…”