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Appendix: Estimating Occupancy Probabilities


The goal is to find the occupancy probability,


p, that is, the probability that a species will occupy a site, if p is constant, or the probability distribution of p, f(p), if p varies. This distribu- tion is distinct from the frequency distribution of the number of sites occupied, k. For example, if p is constant, k follows a binomial distribu- tion. See Figure 4 for other examples. The number of sites is denoted n and the observed number of species S. For a given n and p, the probability that


exactly k sites will be occupied is given by the binomial:


B k; n; p ðÞ¼ pkð1pÞ


n k


nk: (A1)


The probability that no sites will be occupied is equal to 1 − (1 − p)n. Therefore, the condi- tional probability that exactly k sites will be occupied, given that k>0, is given by


B0 k; n; p ðÞ¼ 1 1p B k; n; p


ðÞ ðÞ


n : (A2)


Therefore, if p is a constant, the likelihood of p, given a set of ki and ni (i=1, …, S)is proportional to


L p; k; n


ðÞ¼Y S


pki i¼1 1 1p


ðÞ ; ðÞ


1p


niki ni


(A3)


omitting the constant factors of ni ki


ðÞ¼Y n


"#Nk pk 1p


ðÞ k¼1 1 1p ðÞ


nk n


; : If n is


the same for all species, this simplifies to L p; k; n


(A4) where Nk is the number of species occupying


exactly k sites. This likelihood is maximized by finding the value of p such that k ¼


np 1 1p ðÞ n ; (A5)


where k ¼Pki=S is the mean value of k (Fisher 1936: eq. 2). See Figures 1 and 2.


If p varies, then the probability that exactly


k out of n sites will be occupied is obtained as the binomial probability for a given value of p, weighted by the probability density of p and integrated over all values of p:


P k; n;Θ Ð pmax ðÞ¼


0 fp ðÞdp Ð pmax


ðÞB k; n; p 0 fp


ðÞdp ; (A6)


where Θ denotes the parameters that describe the density function f(p), and pmax is the maximum value of p over which the prob- ability density is integrated. This can be prescribed (e.g., as 1.0) or treated as a free parameter to be estimated. The denominator in the foregoing expression is necessary in cases in which we integrate over a range of values of p that does not sum to unity. For example, a log-normal distribution extends to +∞, but we cannot integrate p past 1.0. The probability that no sites will be occupied is P(0; n), the probability that at least one site will be occupied is 1 − P(0; n) (Fig. 4), and thus, the conditional probability that exactly k of n sites will be occupied, given that k>0, is given by


P0 k; n;Θ ðÞ¼ 1P0;n;Θ P k; n;Θ


ðÞ : ðÞ


(A7)


In the case of the log-normal, the parameters are μlog, the mean of the logarithm of p; σlog, the standard deviation of the logarithm of p; and pmax. The likelihood ofΘgiven a set of ki and ni, (i=1,…, S) is given by


L Θ; k; n


ðÞ¼Y S


i¼1 P0 Θ; ki; niðÞ: (A8)


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