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The geographical relative risk function is a useful tool for investigating the spatial distribution of disease based on case and control data. The most common way of estimating this function is using the ratio of bivariate kernel density estimates constructed from the locations of cases and controls, respectively. An alternative is to use a local-linear (LL) estimator of the log-relative risk function. In both cases, the choice of bandwidth is critical. In this article, we examine the relative performance of the two estimation techniques using a variety of data-driven bandwidth selection methods, including likelihood cross-validation (CV), least-squares CV, rule-of-thumb reference methods, and a new approximate plug-in (PI) bandwidth for the LL estimator. Our analysis includes the comparison of asymptotic results; a simulation study; and application of the estimators on two real data sets. Our findings suggest that the density ratio method implemented with the least-squares CV bandwidth selector is generally best, with the LL estimator with PI bandwidth being competitive in applications with strong large-scale trends but much worse in situations with elliptical clusters. [ABSTRACT FROM AUTHOR]