Fair Sampling of Taxonomic Richness and Unbiased Estimation of Origination and Extinction Rates

Abstract

Paleobiologists are reaching a consensus that biases in diversity curves, origination rates, and extinction rates need to be removed using statistical estimation methods. Diversity estimates are biased both by methods of counting and by variation in the amount of fossil data. Traditional counts are essentially tallies of age ranges. Because these counts are distorted by interrelated factors such as the Pull of the Recent and the Signor-Lipps effect, counts of taxa actually sampled within intervals should be used instead. Sampling intensity biases can be addressed with randomized subsampling of data records such as individual taxonomic occurrences or entire fossil collections. Fair subsampling would yield taxon counts that track changes in the species pool size, i.e., the diversity of all taxa that could ever be sampled. Most of the literature has overlooked this point, having instead focused on making sample sizes uniform through methods such as rarefaction. These methods flatten the data, undersampling when true diversity is high. A good solution to this problem involves the concept of frequency distribution coverage: a taxon's underlying frequency is said to be “covered” when it is represented by at least one fossil in a data set. A fair subsample, but not a uniform one, can be created by drawing collections until estimated coverage reaches a fixed target (i.e., until a “shareholder quorum” is attained). Origination and extinction rates present other challenges. For many years they were thought of in terms of simple counts or ratios, but they are now treated as exponential decay coefficients of the kind featuring in simple birth-death models. Unfortunately, these instantaneous rates also suffer from counting method biases (e.g., the Pull of the Recent). Such biases can be removed by only examining taxa sampled twice consecutively, three times consecutively, or in the first and third of three intervals but not the second (i.e., two timers, three timers, and part timers). Two similar equations involving these counts can be used. Alternative methods of estimating diversity and turnover through extrapolation share some of the advantages of quorum subsampling and two-timer family equations, but it remains to be shown whether they produce precise and accurate estimates when applied to fossil data.