A spatial model for binary data - meeting report

The Northern Ireland local group of the RSS held an online meeting using MS Teams on Wednesday, 27 October 2021, at 1pm. The speaker was Dr. Gabrielle Kelly, UCD, Dublin, Ireland.

Gabrielle was well-known to RSSNI audiences. She had spoken previously to the group about modelling the spread of bovine TB in Ireland and the role of badgers. This issue was of concern to the farming community and to animal rights (anti-cull) activists. Today, Gabrielle's talk was about some of the technical issues inherent in the spatial model used in her work.

Spatial binary data, where the correlations satisfy the Frechet-Hoeffding bounds, are modelled by clipping an underlying latent Gaussian spatial process. The binary variables are then related to latent variables that have an assumed Matern spatial correlation matrix via a multivariate probit model. Both the marginal means of the binary variables and their spatial distances contribute to their correlations.

The model yields a non-trivial log-likelihood for the regression and spatial parameters involving incomplete integrals that can be approximated using the numerical methods of Genz-Bretz or Madsen. This log likelihood is then optimised.

Simulations demonstrate that the Madsen method is biased upward for both regression and scale parameters for small values of the Matern shape parameter, but not for larger values. The Genz-Bretz method performs well for the regression parameters, but does not provide reasonable estimates for the scale parameter. These issues required further investigation. Gabrielle mentioned the considerable literature on Bayesian models and methodology for binary point-referenced spatial data, but noted that the likelihood approach adopted here is less well developed.

In this study the goal was not prediction, but rather, determination of significant predictors of the mean and estimation of the scale of spatial correlation. The model was fitted to bovine TB (bTB) infection data in cattle herds and wildlife badgers in Ireland. Minimum and maximum values of the binary correlations were estimated and associated Euclidean distances recorded. The findings suggested that cattle herds infect neighbouring cattle while badgers roam over a larger area and thus have a greater infectivity range.

This was a very good talk on a difficult area. It was well-received by an audience of c30 participants attending online. Gabrielle was asked several questions about the computational difficulties. One participant debated the appropriateness of the estimation methods adopted for the model. The issues involved were too technical for a general audience and the debate was curtailed. The meeting thanked the speaker and the discussants for a stimulating talk in the usual way.

Written by Gilbert MacKenzie, RSSNI secretary, on 7 November 2021.

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