Empirical Bayes Estimation of Proportions with Application to Cowbird Parasitism Rates

Bayesian models provide a structure for studying collections of parameters such are considered in the investigation of communities, ecosystems, and landscapes. This structure allows for improved estimation of individual parameters by considering them in the context of a group of related parameters. Individual estimates are differntially adjusted toward in overall mean, with the magnitude of their adjustment based on their precision. Consequently, Bayesian estimation allows for a more reliable ranking of parameters and, in particular, a more credible identification of extreme values from a collection of estimates. In Bayesian models, individual parameters are regarded as values sampled from a specified probability distribution, called a prior. The requirements that the prior be known is often regarded as an unattractive feature of Bayesian analysis and may be the reason Bayesian analyses are not frequently applied in ecological studies. Empirical Bayes methods provide an alternative approach that incorporates the structural advantages of Bayesian models while requirng a less stringent specification of prior knowledge. Empirical Bayes methods require only that the prior be in a certain family of distributions, indexed by hyperparameters that can be estimated from the available data. This structur is of interest per se, in addition to its value in allowing for improved estimation of individual parameters; for example, hypothese regarding the existence of distinct subgroups in a collection of paramet ers can be considered under the empirical Bayes framework by allowing the hyperparameters to vary among subgroups. We describe the empirical Bayes approach in application to estimation of proportions, using data obtained in a community—wide study Brown—headed Cowbird paratism rates for illustration. Empirical Bayes estimates identify those species for which there is the greatest evidence of extreme parasitism rates. Subgroup analysis of our data on cowbird parasitism rates indicates that parasitisms rates for neotropical migrants as a group are no greater than those of resident/short—distance migrant in this forest community. Our data and analyses demonstrate that the parasitism rates for certain neotropical migrant species (Wood Thrush and Rose—breasted Grosbeak) are remarkably low while those for others (Ovenbird and Red—eyed Vireo) are remarkably high.