Interpolating, extrapolating, and comparing incidence-based species accumulation curves

A general binomial mixture model is proposed for the species accumulation function based on presence-absence (incidence) of species in a sample of quadrats or other sampling units. The model covers interpolation between zero and the observed number of samples, as well as extrapolation beyond the observed sample set. For interpolation (sample-based rarefaction), easily calculated, closed-form expressions for both expected richness and its confidence limits are developed (using the method of moments) that completely eliminate the need for resampling methods and permit direct statistical comparison of richness between sample sets. An incidence-based form of the Coleman (random-placement) model is developed and compared with the moment-based interpolation method. For extrapolation beyond the empirical sample set (and simultaneously, as an alternative method of interpolation), a likelihood-based estimator with a bootstrap confidence interval is described that relies on a sequential, AIC-guided algorithm to fit the mixture model parameters. Both the moment-based and likelihood-based estimators are illustrated with data sets for temperate birds and tropical seeds, ants, and trees. The moment-based estimator is confidently recommended for interpolation (sample-based rarefaction). For extrapolation, the likelihood-based estimator performs well for doubling or tripling the number of empirical samples, but it is not reliable for estimating the richness asymptote. The sensitivity of individual-based and sample-based rarefaction to spatial (or temporal) patchiness is discussed.