On the analysis of accumulation curves

Identifying and counting the total number of biological species observed, when plotted against a measure of the effort used to record them, gives rise to a species accumulation curve. We investigate estimation of the total number of species and other relevant properties of accumulation by elaborating on the multinomial model (Nakamura and Peraza, 1998, Journal of Agricultural, Biological, and Environmental Statistics, 3, 17-36) that includes specification of a beta density for recording probabilities. We consider a unified description including complete and incomplete (aggregated) curves, a more general scheme for recording, and a Bayesian framework to allow for inclusion of the biologist's knowledge with regard to typical recording probabilities and the total number of species. The beta distribution is used as a prior, but recording probabilities are not restrained to be beta distributed. The methods yield either closed analytical expressions for inference or relatively simple numerical procedures. Predictive distributions of future recordings that would eventually lead to an optimal decision-theoretical rule for stopping collection effort may be easily calculated. A case study regarding species of bats is considered, including some guidelines for elicitation of an informative prior.