Paleobiology, 42(4), 2016, pp. 707–729 DOI: 10.1017/pab.2016.24


On the measurement of occupancy in ecology and paleontology Michael Foote


Abstract.—Occupancy statistics in ecology and paleontology are biased upward by the fact that we generally do not have solid data on species that exist but are not found. The magnitude of this bias increases as the average occupancy probability decreases and as the number of sites sampled decreases. A maximum-likelihood method is developed to estimate the underlying distribution of occupancy probabilities of all species based only on the sample of observed species with nonzero occupancy. The method is based on determining the probability that the number of occupied sites will take on any specific value for a given occupancy probability, integrated over the entire distribution of occupancy probabilities. If the shape of the underlying distribution is well modeled, the resulting occupancy estimates circumvent the bias inherent in failing to observe some species and the fact that this bias depends on the number of sites. For occupancy data on marine animal genera drawn from the Paleo- biology Database, the underlying distribution is reasonably approximated as a right-truncated log- normal, but the methods developed can be extended to any distribution. Examples are presented to illustrate some observations that are robust and others that need to be revised in light of this bias correction. The method is compared to a recently developed, distribution-free approach to the same problem.


Michael Foote. Department of the Geophysical Sciences, University of Chicago, Chicago, Illinois 60637, U.S.A. E-mail:mfoote@uchicago.edu.


Accepted: 9 June 2016 Published online: 23 August 2016 Data available from the Dryad Digital Repository: http://dx.doi.org/10.5061/dryad.5sr24


Estimating the occupancy of a species or


higher taxon—the number or proportion of sites, collections, geographic areas, or other sampling units in which it is found—has many applications in ecology and paleontology. Here I will briefly discuss two facets of a bias in the measurement of occupancy and then suggest a solution that allows estimation of the prob- ability distribution of occupancy for a set of species, including unsampled species. For simplicity I will generally refer to species and sites, but the discussion applies to biological units at all hierarchical levels and to collections, quadrats, and so on. For simplicity I will not


consider problems raised by species misidenti- fications (cf. Ferguson et al. 2015). Statement of the Problem


Suppose there are n sites and S observed


species. Any number of species can occupy a given site, and a species can occupy any number of sites up to n. Let ki be the number of sites known to be occupied by species i.


© 2016 The Paleontological Society. All rights reserved. B0 k; n; p ðÞ¼


We would like to estimate the value of p, the probability of site occupancy. It would seem straightforward to estimate pi for species i as ki/n, and themean probability for a collection of


species as k=n ¼Pki=n=S. But we generally do not have a reliable roster of species with k=0, meaning that the species reallywere present in a specified area at a particular time but were not found at any sampled sites. Therefore our observed values of k are necessarily greater than zero and k/n is biased upward (Fig. 1). (This situationiscommonin ecology,inpartbecause so many species are rare [Fisher et al. 1943; Preston 1948; Rabinowitz 1981], and is not merely a paleontologist’s headache reflecting the incompleteness of the fossil record.) Specifi- cally, if p is a constant, the probability that k sites will be occupied, given that k is greater than zero, is equal to a conditional binomial:


pk 1p


n k


ðÞ nk 1 1p ðÞ n ; (1)


where the denominator is the probability that k>0, and the expected number of sites occupied


0094-8373/16


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