Confidence intervals for ratios of proportions: implications for selection ratios

The selection ratio (proportional resource use divided by proportional availability) is often used by ecologists to measure the degree to which individuals and populations are selective in their food sources and habitats. Yet, confidence interval approaches for this metric are scarce and poorly evaluated. In this paper, we compare 13 methods that can be used to construct simultaneous confidence intervals for selection ratios. Seven of the methods are applicable when availabilities are unknown. These are bootstrapping and six methods adapted from relative risk (Katz-log, adjusted-log, Bailey, inverse hyperbolic sine, Koopman and Noether). The other six approaches are applicable when availabilities are known. These are bootstrapping, two existing methods for relative risk (Wald-adjusted, Noether-fixed) and three straightforward new methods (fixed-log, Agresti-Coull-adjusted and Bayes-beta). None of the 13 approaches have been previously evaluated in the context of selection ratios. In simulations with unknown availabilities, the Koopman method performed best, and the currently recommended Noether method performed worst. In simulations with fixed availabilities, the new Agresti-Coull-adjusted, fixed-log, Bayes-beta methods all outperformed the currently recommended Wald-adjusted method. In the context of real ecological data sets, the Noether and Wald-adjusted methods produced anomalous results that putatively would alter management decisions. We note that our findings, including those for new methods, are directly conferrable to relative risk, allowing extension of our work to the many branches of biology that rely on this measure. The poor performance of the Noether and Wald-adjusted methods is troubling because these are currently the most widely used procedures for calculating confidence intervals for type I resource-selection designs. Based on our findings, we recommend that the Noether method be replaced with the Koopman method, and the Wald-adjusted method be replaced with the Agresti-Coull-adjusted method.