MEASURING OCCUPANCY Occupancy of cohorts as they age


0.05 0.1


0.005 0.01 0.02


Trem 480 Floi Dapi 470 Darr 460 Geologic time (Ma)


FIGURE 10. Raw mean occupancy and estimated mean occupancy probability for Early Ordovician cohorts. Estimates of p as in Fig. 9. Raw occupancy data suggest a peak in the Dapingian and stable values through the Late Ordovician. Bias correction suggests the Dapingian peak is an artifact of a low number of sites and an actual Hirnantian contraction is masked in raw proportional occupancy by a low site count in that stage as well.


probabilities are represented by the median and interquartile range of a distribution of bootstrap estimates of p. The raw occupancy measures suggest expansion of the cohorts; peak occupancy during the Dapingian stage; and relatively stable occupancy during the Late Ordovician (Sandbian–Hirnantian). However, the low cell counts in the Dapingian and Hirnantian, indicated by the peaks in 1/n, suggest that some aspects of these patterns may be suspect. The bias-corrected occupancy estimates indicate an expansion, consistent with the age-and-area model; absence of a peak in the Dapingian; and a decline in occupancy from the Sandbian through Hirnantian. This last feature contrasts with a Hirnantian expansion in geographic range documented by Darroch and Wagner (2015) using an area- rather than occupancy-based measure of genus extent. Waxing and Waning.—Several recent studies


have sought to interpret aggregate occupancy histories of species and genera, irrespective of when they actually lived (e.g., Foote 2007; Foote et al. 2007; Liowand Stenseth 2007). This approach would be expected to average out the biases caused by temporal variation in the number of sites. However, the fact that taxa early and late in their durations tend to be


Sand


Reference l ine: 1 n Kati


450 Hi


First stage


Intermediate stages


Last stage


FIGURE 11. Aggregated occupancy data for all genera with durations of three to six stages, comparing the first stage, the last stage, and all intermediate stages combined. Observed occupancy expressed as mean of k/n ± 1 binomial SE; estimates of p are the mean of 100 bootstrap replicates ± 1 SE. Expansion and contraction of genera are much more pronounced in the bias-corrected occupancy estimate than in raw data.


k n pest


0.04 Genera with durations of 3 6 stages


717


0.02


0.01 0.005


k n pest


more restricted, as demonstrated by these studies, suggests that their lower occupancy probabilities would cause the raw value of k/n to exaggerate occupancy probability more during the intervals of first and last appearance than during intervening intervals (Fig. 1). To explore this possibility, I have aggregated data for all generawith stratigraphic ranges between three and six stages, inclusive; longer ranges that were omitted account for about 15% of the genera. I then estimated occupancy probability separately for the stage of first appearance, the stage of last appearance, and all intermediate stages combined, including only those genera actually sampled. (Because n varies among stages, these estimates use the likelihood function of eq. A8.) As expected, the pattern in bias-corrected estimates of occupancy agrees with that in the raw values of k=n insofar as both measures are lower during first and last stages (Fig. 11). However, the bias correction implies a much more dramatic pattern of expansion and contraction, with occupancy in intermediate stages about four times higher than in first and last stages, versus an 80% difference in k=n. The stronger bias in the first and last stages reflects both the lower values of k=n and, presumably, the higher frequency of genera with k=0, although this last point cannot be tested directly.


Estimated occupancy


Estimated occupancy


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