708 (A)


0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14


(B) 20 10 p constant


p =0.1 p =0.05 p =0.02 p =0.01 p =0.005


MICHAEL FOOTE The bias caused by filtering cases where


k=0 is exacerbated by variation in the number of sites sampled, n.As n becomes smaller, the difference between k/n and the underlying occupancy probability increases (Fig. 1), to the point where k/n=1 when n=1, regardless of the true occupancy probability. If, for instance, we wanted to construct a time series of average taxonomic occupancy to assess the extent to which genera tend to be more broadly distributed in the aftermaths of mass extinc- tions than during other times (Schubert and Bottjer 1995; Hallam and Wignall 1997; Erwin 2001; Miller and Foote 2003), the time series could potentially be strongly and inversely correlated with the number of sites per time interval. Strictly speaking, the occupancy probability,


5 2 1 10 20 Number of sites (n)


FIGURE 1. Dependence of mean proportional occupancy (k=n) on occupancy probability (p) and number of sites (n), when p is constant and k>0. A, Mean occupancy. B, Ratio of mean occupancy to true occupancy probability. Bias inherent in k=n increases as p decreases and as n decreases.


per species is equal to np/[1− (1−p)n](see Appendix). Therefore, the estimate k=n is higher than p by a factor of 1/[1−(1−p)n]. The bias resulting from this kind of data censoring was discussed by Haldane (1932, 1938) and Fisher (1936) in the context of estimating frequencies of genetic conditions in human populations (in which families with no instances of the condition may go unreported). Finney (1949) and Rider (1955) also discussed the more general problem of truncated binomial distributions where k>1, k>2, and so on.


50 100


p, is the joint probability that a species actually lived at a site and that it was detected. It is possible in principle to estimate detection probabilities separately given data on repeated surveys of the same sites within a time span sufficiently short that what lives where has not changed (MacKenzie et al. 2002, 2003, 2006; Royle and Nichols 2003; Dorazio et al. 2006). In contrast to these studies, I amassuming data in which we have but one sample, possibly aggregated over several collections, for each site. For simplicity I will continue to use occupancy probability as the probability that a species is found at a site; this is also referred to as incidence (e.g., Colwell et al. 2012; Chao et al. 2014, 2015a).


Haldane and Fisher’s Solution In the problem facing Haldane and Fisher, ki


is the number of instances of a genetic condition in family i; ni is the size of family i;and S is the observed number of families. (Haldane and Fisher in fact used different notation.) For most of this discussion, I will assume that n,the number of potential sites, is the same for all species, but this assumption is not essential (Appendix). The maximum-likelihood solution used byHaldane and by Fisher depends only on k and n, and simply requires finding, by numerical methods, the value of p such that k=n is equal to p/[1 − (1 − p)n], but of course the solution is more tightly constrained as


k n p =


1 1 (1 p)n


k n =


p 1 (1 p)n


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