Accounting for imperfect detection in Hill numbers for biodiversity studies

Hill numbers unify biodiversity metrics by combining several into one expression. For example, species richness, Shannon's diversity index and the Gini-Simpson index are a few of the most used diversity measures, and they can be expressed as Hill numbers. Traditionally, Hill numbers have been calculated from relative abundance data, but the expression has been modified to use incidence data as well. We demonstrate an approach for estimating Hill numbers using an occupancy modelling framework that accounts for imperfect detection. We alter the Hill numbers formula to use occupancy probabilities as opposed to the incidence probabilities that have been used previously and to calculate its summations from the modelled species richness. After introducing the occupancy-based Hill numbers, we demonstrate the differences between them and the incidence-based Hill numbers previously used through a simulation study and two applications. In the simulation study and the two examples using real data, the occupancy-based Hill numbers were larger than the incidence-based Hill numbers, although species richness was estimated similarly using both methods. The occupancy-based Hill number estimators are always at their asymptotic values (i.e. as if an infinite number of samples have been taken for the study region), therefore making it easy to compare biodiversity between different assemblages. In addition, the Hill numbers are computed as derived quantities within a Bayesian hierarchical model, allowing for straightforward inference.