MEASURING OCCUPANCY
species with k=1, which often account for a plurality of data. Second, we could fit the distribution of occupancy probabilities to all species, extract the part of the distribution corresponding to taxa with any given value of k (including k=0) (Fig. 5D), and use themean or median of this subdistribution as a point estimate. The latter approach is preferable, but it is
worth remembering that the species actually sampled represent an upwardly biased subset of all species, and that, for a fixed underlying distribution of occupancy probabilities, the sampled species progressively represent the higher part of this distribution as the number of sampled sites decreases (Fig. 15). For example, using the distribution of Figure 5, sampled species have a mean occupancy
probability of pk>0 ¼ 0:0356 if n=50 and 0.0301 if n=100 (eq. A14). This fact may seem like a nuisance, but point estimates for indivi- dual species should vary with n, because, all else being equal, species with a given value of k do in truth have a higher occupancy probability, on average, if n is lower. The situation with point estimates is ana-
logous to that with the mean duration of a distribution of species versus the duration of a single species. Modeling the form of the true duration distribution and how it is affected by incomplete sampling allows an unbiased estimate of this distribution, even though observed stratigraphic ranges are truncated and some species are not recorded at all (Foote and Raup 1996; Foote 1997; Solow and Smith 1997). However, individual species that are sampled represent an upwardly biased subset of the entire distribution (Foote and Raup 1996: Fig. 2). Extinction selectivity is another area in
which revised occupancy estimates may make a difference. In particular, bias correction could affect the details of how much the odds of survival increase per unit increase in occu- pancy (Payne and Finnegan 2007). Moreover, the prospect of estimating p for species that are known to exist but happen not to be sampled could provide an alternative to assigning arbitrary minimum values (Finnegan et al. 2008) or ignoring them altogether (Foote and Miller 2013).
0.20
721
µlog = 4.55 log = 1.24
pmax = 0.275 0.10
0.05
0.02
k > 0 k = 1 k = 2 k = 3 k = 4 k = 5
0.01 2 5 10 20 Number of sites (n)
FIGURE 15. Predicted mean occupancy probability for subsets of genera with selected values of k, as a function of number of sites, n. Curves are calibrated to the parameters fit to the Sandbian stage (see Fig. 5). Even if
the underlying distribution of occupancy probabilities is fixed, the part of the distribution sampled by genera with a given value of k varies with n.
The Permo-Triassic example (Fig. 9) shows
that some details of the temporal pattern of occupancy vary with the spatial resolution of a site or sampling unit. In the data analyzed herein, sampling is at a fine spatial scale, with the equal-area cells defined after the fact, so the number of sampled and unsampled genera is not affected by the resolution imposed. What about bias in the proportion of occupied cells? As resolution increases, that is, as the cell size decreases, both n and k increase, but n increases more rapidly so that k=n decreases, as does estimated mean occupancy probability, p (Table 1). The increase in n would lead us to expect k=n to be less biased with higher resolution, whereas the decline in occupancy probability would lead us to expect the opposite. The former effect is evidently stronger, as the bias, that is, the ratio of k=n to p, decreases with increasing resolution. The aim of this paper has been to estimate an
accurately scaled distribution of occupancy probabilities, from which one can derive means or other statistics, as well as point estimates for individual species. If the goal instead is simply to compare different data sets in terms of the relativemagnitude of theirmean occupancies, then sampling standardization of
50 100
p for subset of species with given value of k
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