712
MICHAEL FOOTE Two points to bear in mind before proceed-
ing: (1) The goal is to characterize and estimate the distribution of p (e.g., Fig. 4A) that under- lies the distribution of k (e.g., Fig. 4B), not the distribution of k itself (cf. Buzas et al. 1982). The shape of the distribution of k may resemble that of p, but clearly it need not do so. As an obvious example, if p is constant, k will have a binomial
(A) 400 Observed Predicted from fit 60 300 50 40 200 30 20 100 10 0 0 (C) 30 100 25 80 20 15 10 5 0 0.00 0.02 0.04 0.06 Occupancy probability (p) 0.08 0.10 60 40 20 0 0.00 0.02 0.04 0.06 Occupancy probability (p)
FIGURE 5. Example of fitting truncated log-normal distribution of occupancy probabilities to occupancy data for the Sandbian stage of the Late Ordovician. A, Observed frequency distribution of k and distribution predicted from best-fit log-normal distribution (B). C, Conditional probability distribution, corresponding to the best-fit curve in B, for the subset of genera with k>0. D, Conditional probability distributions for subsets of genera having specified values of k. Genera with higher values of k preferentially sample higher values of p, but even those with k=0 are inferred to represent a part of the probability distribution above zero.
0.08 0.10 5 10 15 Number of equal-area cells (k) Taxa with k > 0 (D) 120
k = 0 k = 1 k = 2 k = 3 k = 4 k = 5
20 0 0.00 0.02 0.04 0.06 Occupancy probability (p) 0.08 0.10
distribution. (2) As with the Haldane–Fisher approach, it will not be possible to estimate occupancy probability if all species are known from a single site. Figure 5A shows the frequency distribution
of k for an arbitrarily chosen time interval, the Sandbian stage of the Ordovician. There are 69 cells (n), the maximum value of k is 22, the
(B) All taxa (implicitly including those with k = 0)
Probability density
Number of genera
Probability density
Probability density
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