  In this work we present a statistical approach to distinguish and interpret
the complex relationship between several predictors and a response variable at
the small area level, in the presence of i) high correlation between the
predictors and ii) spatial correlation for the response. Covariates which are
highly correlated create collinearity problems when used in a standard multiple
regression model. Many methods have been proposed in the literature to address
this issue. A very common approach is to create an index which aggregates all
the highly correlated variables of interest. For example, it is well known that
there is a relationship between social deprivation measured through the
Multiple Deprivation Index (IMD) and air pollution; this index is then used as
a confounder in assessing the effect of air pollution on health outcomes (e.g.
respiratory hospital admissions or mortality). However it would be more
informative to look specifically at each domain of the IMD and at its
relationship with air pollution to better understand its role as a confounder
in the epidemiological analyses. In this paper we illustrate how the complex
relationships between the domains of IMD and air pollution can be deconstructed
and analysed using profile regression, a Bayesian non-parametric model for
clustering responses and covariates simultaneously. Moreover, we include an
intrinsic spatial conditional autoregressive (ICAR) term to account for the
spatial correlation of the response variable.
