How should we estimate diversity in the fossil record? Testing richness estimators using sampling-standardised discovery curves

1. To infer genuine patterns of biodiversity change in the fossil record, we must be able to accurately estimate relative differences in numbers of taxa (richness) despite considerable variation in sampling between time intervals. Popular subsampling (=interpolation) methods aim to standardise diversity samples by rarefying the data to equal sample size or equal sample completeness (=coverage). Standardising by sample size is misleading because it compresses richness ratios, thereby flattening diversity curves. However, standardising by coverage reconstructs relative richness ratios with high accuracy. Asymptotic richness extrapolators are widely used in ecology, but rarely applied to fossil data. However, a recently developed parametric extrapolation method, TRiPS (True Richness estimation using Poisson Sampling), specifically aims to estimate the true richness of fossil assemblages. 2. Here, we examine the suitability of a range of richness estimators (both interpolators and extrapolators) for fossil datasets, using simulations and a novel method for comparing the performance of richness estimators with empirical data. We constructed sampling-standardised discovery curves (SSDCs) for two datasets, each spanning 150 years of palaeontological research: Mesozoic dinosaurs at global scale, and Mesozoic-early Cenozoic tetrapods from North America. These approaches reveal how each richness estimator responds to both simulated best-case and empirical real-world accumulation of fossil occurrences. 3. We find that extrapolators can only truly standardise diversity data once sampling is sufficient for richness estimates to have asymptoted. Below this point, directly comparing extrapolated estimates derived from samples of different sizes may not accurately reconstruct relative richness ratios. When abundance distributions are not perfectly flat and sampling is moderate to good, but not perfect, TRiPS does not extrapolate, because it overestimates binomial sampling probabilities. Coverage-based interpolators, by contrast, generally yield more stable subsampled diversity estimates, even in the face of dramatic increases in face-value counts of species richness. Richness estimators that standardise by coverage are among the best currently available methods for reconstructing deep-time biodiversity patterns. However, we recommend the use of sampling-standardised discovery curves to understand how biased reporting of fossil occurrences may affect sampling-standardised diversity estimates.