Bayesian Spatially Varying Coefficient Models in the Presence of Collinearity

The belief that relationships between explanatory variables and a response variable in a regression model may vary within a study area has lead to the development of Bayesian regression models that allow for spatially varying coefficients (Gelfand et al., 2003). In the typical application of these spatially varying coefficient process (SVCP) models, marginal inference on the spatial pattern of regression coefficients is of central interest. In light of this, there is a need to assess the validity of these marginal posterior inferences, since these inferences can be misleading in the presence of explanatory variable correlation (i.e. collinearity). We present the results of a simulation study designed to assess the sensitivity of spatially varying coefficients to a range of levels of collinearity. The results show that the SVCP model is overall fairly robust to moderate levels of collinearity in terms of marginal coefficient inference, but degrades in coefficient accuracy with strong collinearity. We also illustrate that the posterior mean of the SVCP model coefficients can be viewed as ridge regression solutions with the amount of coefficient penalization controlled by numerous model parameters. Finally, we present an application of the spatially varying coefficient model to a cancer dataset, where the relationship between cancer rates and some explanatory variables is suspected to vary spatially.