  We introduce a semi-parametric approach to ecological regression for disease
mapping, based on modelling the regression M-quantiles of a Negative Binomial
variable. The proposed method is robust to outliers in the model covariates,
including those due to measurement error, and can account for both spatial
heterogeneity and spatial clustering. A simulation experiment based on the
well-known Scottish lip cancer data set is used to compare the M-quantile
modelling approach and a random effects modelling approach for disease mapping.
This suggests that the M-quantile approach leads to predicted relative risks
with smaller root mean square error than standard disease mapping methods. The
paper concludes with an illustrative application of the M-quantile approach,
mapping low birth weight incidence data for English Local Authority Districts
for the years 2005-2010.
