722
MICHAEL FOOTE
TABLE 1. Selected occupancy statistics, expressed as the median of all by-stage values, showing the effect of spatial resolution.
Quantity n
k
k=n pest
ðk=nÞ=pest
Points* 454
4.09
0.00982 0.00331 2.60
1.3×104km2† 188
2.93 0.0185
0.00521 3.17
Size of sampling unit 5.1×104km2† 131
2.67 0.0226
0.00606 3.46
2.0×105km2† 92
2.42 0.0285
0.00744 3.51
8.2×105km2† 61
2.14 0.0369
0.00879 4.13
*All collections with the same geographic coordinates within a stage combined into a single sampling unit. †Equal-area cells formed by dividing a Lambert cylindrical equal-area projectionwith the same number of latitudinal and longitudinal divisions: 200, 100, 50, and 25, respectively.
occupancy, based on a specified quota of sites or of coverage (Chao et al. 2014), could potentially serve the purpose. One of the reasons for pursuing the approach of this paper, however, was the realization that with the data analyzed here, standardizing the number of sites, n, does not quite eliminate the dependence between n and k=n evident in the raw data. For example, if for each stage we sample n = 21 equal-area cells at random (the number corresponding to the smallest value for any stage)—replicating the subsampling and averaging the results—the original num- ber of cells is still significantly correlated with the standardized mean occupancy (rs= −0.32, p < 0.01) (see also Fig. 3C). The circumstances under which rarefaction and related approaches eliminate the number-of-sites bias deserve further attention. One obvious potential limitation is that different occupancy distributions with the same value of p but different values of μlog and σlog are expected to yield different mean values of k (k>0), even if n is held fixed. For log-normal distributions typical of those estimated herein, decreasing μlog while increasing σlog, in such a way as to keep p constant, results in higher values of k (eq. A15) and lower values of S (eq. A7). As noted earlier, the joint probability of
occupancy and detection is treated as a property of species, without reference to among-site variation. To explore the possible effects of such variation, I carried out some simple simulations, in which each of S species is assigned an occupancy probability, psp,i, and each of n sites is assigned a detection effect, psite,j, equal to the probability that a species
truly occupying the site will be found. Thus, the probability, pij, that species i will be observed at site j is equal to the product psp,i×psite,j. The overall mean observation probability for species i, pi, is equal to Pn
tion we would like to estimate. Values of psp,i and psite,j were drawn at
j¼1 pij=n; this is the quantity whose distribu-
random from separate log-normal distribu- tions. For simplicity, all distributions had pmax=1.0, which was not treated as a fitted parameter, and all had σlog=1.0. Values of μlog were then assigned to yield five mean prob- abilities for species (0.05 through 0.25 in increments of 0.05) and five mean probabilities for sites (0.1 through 0.5 in increments of 0.1). All 25 combinations of species- and site- specific distributions were simulated, with 100 replicate simulations per combination. The true values of S and n were set to 1000 and 100, respectively, for each simulation; in general, fewer simulated species and sites actually ended up with nonzero occupancy. For each simulation, each value of pij was compared to a uniform random number to determine whether the species i was found at site j. The value of k for each species was then tabulated as the number of sites at which the species was observed. The parameters were then estimated as with the empirical data; only nonzero values of k were included, and n was taken to be the number of sites with at least one species. The simulated probabilities were treated
according to three different regimens. (1) Species-specific occupancy probabilities were kept independent of site-specific detec- tion probabilities, as in Dorazio et al. (2006).
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