The use of Hill numbers—or effective numbers—to partition the total diversity (gamma) of species within a region into within- (alpha) vs. among-sample (beta) components is becoming increasingly popular for describing community diversity as well as multivariate traits. Using this approach, the effective number of species is dependent on how differential abundances are weighted. However, one limitation is that, until recently, most applications of of Hill numbers relied on point estimates. This introduces bias because finite censusing of individuals within a sample (e.g., transect, quadrat) can result in the underestimation of diversity for low diversity orders such as species richness. Similarly, bias can occur when comparing diversities among multiple communities or regions, each with different sample sizes. As more individuals of rare species are observed, naive estimates of alpha diversity within a community can be too low while naive estimates of beta diversity inflate turnover.
Here we apply a hierarchical Bayesian approach to the estimation of species abundances within and among samples, communities, and regions. We then use posterior predictive samples and the underlying parameter estimates of the abundances themselves to partition diversity for any order of q.
In our Bayesian approach, we explicitly model sampling the uncertainty by assuming that the observed species abundances are themselves random variables. Analyses of real and simulated datasets show that our method largely eliminates bias due to finite sampling of individuals, removes the bias associated with turnover, and results in estimates of diversity that are on par with other methods of bias correction. Finally, our method provides intuitive measures of error for any order of diversity, which then facilitates explicit statistical comparisons among different study areas.