Disease mapping via negative binomial M-quantile regression

A new approach to ecological regression for disease mapping is introduced, based on semi- parametric M-quantile regression models. In particular, we define a Negative Binomial M-quantile model as an alternative to Empirical Bayes or fully Bayesian approaches to disease mapping. The area-level covariates used in ecological regression are usually measured with error, and the pro- posed M-quantile modelling approach is easily made robust against outlying data in the model covariates. Differences between the M-quantile model and the usual random effects models are discussed, and these alternative approaches are compared using the well-known Scottish Lip cancer data and a simulation experiment. The Lip Cancer data example shows that the Negative Binomial M-quantile model confirms results obtained by other methods, but also seems to have less shrinkage than the Empirical Bayes method, so reducing the problem of oversmoothing. The simulation experiment suggests that the new model leads to estimates with smaller mean square error. We also show how the Negative Binomial M-quantile can be extended to account for spatial correlation between areas using a Geographically Weighted Regression strategy.