Generalizing the spatial relative risk function

• Spatial relative risk function generalized to show changes with time or covariates.
• Kernel estimates proposed and their properties studied.
• Bandwidth selection and construction of tolerance contours discussed.
• Application to 2001 outbreak of foot and mouth disease in the U.K.
• Effective tool for visualizing case-control disease data.

The spatial relative risk function is defined as the ratio of densities describing respectively the spatial distribution of cases and controls. It has proven to be an effective tool for visualizing spatial variation in risk in many epidemiological applications over the past 20 years. We discuss the generalization of this function to spatio-temporal case-control data, and also to situations where there are covariates available that may affect the spatial patterns of disease. We examine estimation of the generalized relative risk functions using kernel smoothing, including asymptotic theory and data-driven bandwidth selection. We also consider construction of tolerance contours. Our methods are illustrated on spatio-temporal data describing the 2001 outbreak of foot-and-mouth disease in the United Kingdom, with farm size as a covariate.