720 (A) 0.08 p (all taxa)


MICHAEL FOOTE Discussion and Conclusions The treatment so far has been confined to 0.06 0.04 0.02 0.02 (B) 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.3 0.4 0.5 0.6 0.7 0.8 Direct tabulation (from gap analysis)


FIGURE 14. Directly tabulated vs. estimated values ofmean occupancy (A) and proportion of genera sampled (B), comparing the estimates from the log-normal fit(open squares) and the coverage method (closed squares). Abscissa shows direct tabulations of k=n (A), including cases where k=0, and the proportion of genera with k>0, i.e., the three-timer gap statistic (Alroy 2008; Alroy et al. 2008) (B). Estimated mean occupancy probabilities (A) and proportion of genera sampled (B) are based on analysis of genera sampled in the stages immediately before and immediately after the stage in question, and having k>0 during that stage, as in Fig. 8. The coverage-based estimates are generally higher than the direct tabulations and the estimates from the log-normal fit, suggesting that, for these data, the log-normal fit performs better at estimating the occupancy of unsampled taxa.


number of unsampled taxa (Alroy 2010a,b; Chao and Jost 2012; Colwell et al. 2012; Chao et al. 2014).


0.9 1.0 0.04 0.06 Proportion of genera sampled


Coverage theory Log-normal fit


0.08


analyzing occupancy of distributions of taxa, generally coeval ones. Aggregate occupancy histories of noncoeval taxa were originally developed by superimposing point estimates for individual taxa in specific time periods (Foote 2007; Foote et al. 2007; Liow and Stenseth 2007). As seen in Figure 11, such point values of k and n can be combined across taxa to develop a single, bias-corrected estimate of the distribution of occupancy probabilities. For some problems, however, we may still want a point estimate of the occupancy probability of a single taxon during one time interval. For example, Liow et al. (2010) followed occupancy histories of individual species of planktonic microfossils. Although the results of Figure 11 suggest that the rise-and-fall pattern documen- ted by Liow et al. (e.g., their Fig. 3) may be even steeper than in their raw proportional occu- pancy (k/n) data, it would take an unusual set of circumstances—with the number of sites falling and rising systematically through a species’ duration—for the basic occupancy pattern to be spurious. To take another exam- ple, I previously showed that when individual genera expand or contract in geographic range, these changes tend to coincide with expansions and contractions in the areal extent of their preferred habitat (Foote 2014). In that particu- lar case, the conclusions are probably robust, because compatible results were found using both k/n and k to measure geographic range, even though these measures respond in opposite ways to variation in n (Foote 2014: Table A1). Nonetheless, it is worth considering how we might obtain point estimates that are free of the biases discussed here. There are two obvious possibilities. First, we


could assume that all species with the same value of k in a given time interval have the same, constant occupancy probability and apply the Haldane–Fisher method to each one, as in Figure 1B. But this approach seems undesirable, because the assumption of con- stant occupancy probability, even for just a subset of species, leads to inaccurate estimates of the mean probability, if in fact it varies, and because we cannot obtain an estimate for


Estimate


Estimate


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