718


MICHAEL FOOTE


Comparison with Coverage-based Estimation Coverage methods are based on estimating


the proportion of species abundances or occur- rences represented by a sample (Good 1953); the most notable paleontological application is Alroy’s (2010a,b) shareholder quorum sub- sampling algorithm. Chao et al. (2015a: pp. 1198–1199) used a generalization of cover- age theory to adjust k/n for individual species. They fit a two-parameter correction factor that, intuitively, assumes a declining exponential function, so that the larger the value of k, the smaller the correction. The parameters can be estimated, because coverage theory predicts the relationship between (1) the sum of occupancy probabilities of the sampled spe- cies, raised to an integral power defining the order of coverage; and (2) the complete suite of values of k/n (Chao et al. 2015a: eqs. D.6, D.7). The first-order coverage is the proportional sum of occupancy probabilities attributable to the sampled species. The complementary pro- portion represents the unsampled species, and the number of these unsampled species— actually a lower bound—can also be estimated (Chao 1987). In this way, the true mean occupancy probability can be estimated for the sampled species, the unsampled species, and therefore the combination of the two. The estimation of coverage does not hinge on assuming any particular distribution. The coverage-estimated distribution of occupancy probabilities for the sampled species differs fromthat developed herein insofar as, with the coverage method, all sampled species with the same value of k are inferred to have the same, constant occupancy probability (cf. Fig. 5D); thus, the inferred probability distribution for sampled species is discrete. For the sake of illustration, Chao et al. (2015a,b) assume that the unsampled species follow a geometric distribution, but this assumption is not crucial to the method and has no bearing on the estimated mean occupancy probability of sampled or unsampled species. Figure 12 compares the occupancy probability distributions estimated with the two methods, applied to the data of Figure 5 (Chao et al. 2015b; Supplementary Data). The observed sumPk=n is equal to 37.46. This provides an estimate of the


(A) 1.0 0.8 All taxa


Coverage theory (unsampled taxa) Coverage theory (sampled taxa) Log-normal fit


0.6


0.4


0.2


0.0 (B) 1.0 0.8 Sampled taxa


0.6


0.4


0.2


0.0


0.00 0.05 0.10 0.15 0.20 0.25 0.30 Occupancy probability


FIGURE 12. Estimated distribution of occupancy


probabilities for Sandbian genera (see Fig. 5), based on log- normal fit (dotted curve) and coverage-based analysis (bars). For sampled genera, height ofbarsisproportionaltothe number of genera with the corresponding occupancy probability. Unsampled genera are represented by a distribution of occupancy probabilities; in this instance they range only from 0.01260 to 0.01263 and so are depicted as a single, dashed bar in A. In this case, the coverage-estimated mean occupancy probability of unsampled genera is nearly thesameasthatofgenerawith k=1 (left-most solid bar), but this is not a general feature of the method. Other than one being discrete and the other continuous, the two methods yield similar distributions for sampled genera (B), whereas the log-normal fit yields lower mean probabilities for unsampled genera and therefore for all genera combined.


sum of occupancy probabilities, including both sampled and unsampled taxa (Chao et al. 2015a: Appendix D); thus, the key issue is how that


Density or probability (scaled to maximum)


Density or probability (scaled to maximum)


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