710 (A) 0.06 0.05 0.04 0.03 0.02


MICHAEL FOOTE rs = 0.842 (B) 0.040 0.035 0.030 0.025 0.020 0.015


Modeling and Fitting the Distribution of Occupancy Probabilities


(C)


0.005 0.010 0.015 0.020 0.025 0.030 0.035


20 50 100 Number of sites (n) 200


Standardized quota of sites (m=21) rs = 0.874


I will illustrate the modeling of occupancy probabilities for the data on Phanerozoic marine animal genera using a log-normal distribution of p. This distribution has two parameters, μlog (the mean of the logarithm of p) and σlog (the standard deviation of the logarithm of p). Because the log-normal distribution allows for values up to +∞,we must modify it if we use it to model occupancy probabilities. I have therefore fitted a trunca- tion point, pmax, the maximumvalue that p can take on, as a third parameter. I will hereinafter refer to the resulting distribution as the


FIGURE 3. Analysis of data on marine animal genera, where k is the number of equal-area cells (2×105km2) occupied by a genus and n is the total number of cells with any data in a given stage. Each point denotes a single stage. A, Mean raw proportional occupancy, k=n.B, Haldane– Fisher maximum-likelihood estimate of occupancy probability, assuming p is constant for each stage. C, Maximum-likelihood estimate of p, assuming p is constant, when data are first subsampled, so each stage is represented by the same number of randomly chosen cells (equal to n = 21, the minimum among all stages), based on the mean of 100 replicates. Bias in k=n is expected even if p is constant. Bias in maximum-likelihood estimate is interpreted in part to reflect the fact that p is in fact a probability distribution for each stage and that higher values of p are sampled preferentially at lower values of n.


Full quota of sites rs = 0.671


occupancy probabilities, as we sample fewer sites we preferentially sample the species with higher occupancy probabilities, that is, the more common species. The shape of the distribution is crucial, because the probability that k>0 depends on it. Figure 4A shows examples for four different distributions that all have the samemean occupancy probability, p. Although the mean is the same, the prob- ability that k>0 varies substantially and is lower in each instance than it would be if p were constant (Fig. 4B,C). The Haldane– Fisher maximum-likelihood approach can easily be modified to encompass variability in p if we can either assume the distribution of p or fit it to the data. The essence lies in determining the probability that k out of n sites will be occupied for a given value of p and integrating this over the entire distribution of p (Appendix).


p (Fisher maximum-likelihood estimate) ^


p (Fisher maximum-likelihood estimate) ^


k n


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