MEASURING OCCUPANCY


sum is divided among taxa and among how many taxa. There are 960 sampled genera. The coverage estimate implies that there are at least 490 unsampled genera, accounting for a summed occupancy probability of 6.18, and that the sampled genera account for a summed occupancy probability of 31.28. The estimated mean occupancy probability for sampled genera is therefore 0.0326 (31.28/960). The lower bound on the number of unsampled genera gives upper bounds for the mean occupancy probability equal to 0.0126 (6.18/490) for unsampled genera and 0.0258 (37.46/[960 + 490]) for all genera. The truncated log-normal fit yields estimates of mean occupancy probability equal to 0.0329 for sampled genera (eq.A14); 0.0073 for unsampled genera (eq. A12); and 0.0210 for all genera (eq. A10). Thus, the estimates of the summed occupancy probabilities accounted for by the sampled versus unsampled genera are quite similarwith the twomethods;where they differ is in the estimated or implied number of unsampled genera and therefore the mean occupancy probability of unsampled genera and of all genera. Carrying out the same calculations for all


stages, the coverage-based estimate of mean occupancy probability for all taxa is consis- tently higher than the estimate based on the method of this paper (Fig. 13A), whereas the two estimates are nearly identical for sampled taxa (Fig. 13B). For these data, the standard error of the coverage-based estimates is gen- erally smaller. Presumably this result reflects the fact that the method is distribution free. If for each stage we focus exclusively on


genera that are known both immediately before and immediately after the stage (as in Fig. 8), we can use the sampled genera to produce occupancy estimates for all genera and compare these estimates to direct tabula- tions, because we know how many of the through-ranging genera have k=0. Figure 14 compares the two methods with respect to estimated mean occupancy for all genera (Fig. 14A) and the proportion of genera sampled at least once (Fig. 14B). The coverage-based estimates tend to overpredict these values, again an expected consequence of the estimation of a lower bound on the number of unsampled species.


(A)


0.000 0.005 0.010 0.015 0.020 0.025 0.030


0.000 0.005 0.010 0.015 0.020 0.025 0.030 (B) 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.010 0.015 0.020 0.025 0.030 0.035 0.040


Estimate from fitting log-normal distribution of occupancy probabilities


FIGURE 13. Estimated mean occupancy probabilities, comparing log-normal fit (abscissa) to coverage- based estimate (ordinate). Each point is a stage. Error bars are ±1 SE, based on bootstrap resampling of genera within stages. Consistent with the results of Fig. 12, the two methods yield comparable occupancy probabilities for sampled taxa, whereas the log-normal fit yields lower estimates for unsampled taxa and for all taxa combined.


p (sampled taxa) p (all taxa)


719


These last results suggest that the method


developed herein may allow improved estimates of the mean occupancy probability of all taxa, sampled and unsampled, but it would be premature to generalize without further empirical and simulation studies and other approaches to estimating the


Estimate from coverage theory


Estimate from coverage theory (upper bound)


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