714 (A) 8


constant p logit-normal p truncated log-normal p


6


MICHAEL FOOTE


suppose that it is preferable in general; the logit-normal, as well as other distributions, should certainly be considered in future work. Another test of the truncated log-normal


4 2


2468 Standard deviation of k (predicted)


(B) 20


model is to use it to predict observations that were not involved in estimating the distribution. The distribution of p predicts the proportion of taxa with k=0, but—bearing in mind that what we mean by k=0 is that we know the taxon existed but it was not found at any sites—this prediction is not generally testable directly. One exception involves the subset of genera present in the stages immedi- ately before and immediately after the stage in question, the three-timers and part-timers of Alroy (Alroy 2008; Alroy et al. 2008). We know these were extant during the stage and therefore know which ones have k=0 (the part-timers) and which have k>0 (the three- timers). For each stage, I fitted a distribution only to these through-ranging genera,


15 10 5


including only those with k>0. I then used the fitted distribution to predict the mean value of k, including k=0; this mean correlates well with the observed value (Fig. 8A). The pre- dicted proportion of genera with k>0 also provides a reasonable estimate (Fig. 8B). Although the truncated log-normal may not


0 0.0 (C) 8 6 0.2 0.4 0.6 0.8 Akaike weight in favor of truncated log-normal 1.0


be perfect, it is credible enough to consider applying it to some real occupancy problems. I will present a few examples showing how occupancy patterns are modified by the bias- correction approach. I will not interpret these biologically in any great detail.


A Few Applications Permo-Triassic Distributions.—Some genera 4 2


and species are observed to be exceptionally widespread in the aftermaths of mass extinctions, particularly the end-Permian event (Schubert and Bottjer 1995; Brayard et al. 2006: Figs. 7–8), but does this phenomenon reflect just a few genera or clades (Chen et al. 2005), or is it


0 0.1 0.2 0.3 0.4 0.5 Fitted truncation point for log-normal distribution


FIGURE 6. Alternative models for fitting Phanerozoic occupancy data. A, Observed standard deviation of k compared with that predicted by model of: constant p (crosses); logit-normal p (open squares); and truncated log-normal p (closed circles). B, Akaike weights of truncated log-normal. C, Fitted truncation points. Truncated log- normal distribution is overall the best model of these three.


Number of stages


Number of stages


Standard deviation of k (observed)


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