A Markov random field model with cumulative logistic functions for spatially dependent ordinal data

Abstract

This paper presents a class of regression models with cumulative logistic functions that are chiefly designed to analyse spatially dependent ordinal data. In contrast to previous works, the proposed model requires neither the sites to be regularly spaced nor the assumption of an underlying continuous variable. It belongs to a more general class of Markov random field models, and can be considered an extension of the ordinal regression model with the proportional odds link function. Our proposed model allows practitioners to interpret the model parameters using odds ratios. Apart from the theoretical developments, this work also highlights the practical aspects of model fitting, including parameterisation, selection of neighbourhood, and calculation of standard errors. Simulation studies with regularly and irregularly spaced sites were conducted. Modelling strategies including pseudo-likelihood methods were found to be useful in both settings. The proposed model and the non-spatial counterpart were applied to the daily air quality index measured in the United Kingdom. The results indicate the presence of spatial effects and the incorporation of spatial effects led to better model performance in terms of various goodness-of-fit measures.

KEYWORDS:

• Auto-model
• irregularly spaced data
• neighbourhood
• ordinal regression
• pseudo-likelihood

Acknowledgments

The authors thank two reviewers and an associate editor for their constructive comments which have improved the earlier version of this work.

Data availability statement

The data that support the findings of this study are openly available on the website of the Department for Environment Food & Rural Affairs (Defra), UK, at https://uk-air.defra.gov.uk/air-pollution/.

Disclosure statement

No potential conflict of interest was reported by the authors.