MEASURING OCCUPANCY


mean value of k is 2.69, and the mean value of k/n is 0.039. Also shown are the predicted values of k corresponding to the maximum- likelihood estimate of the distribution of p, which itself is given in Figure 5B. The mean of this distribution is 0.021, distinctly lower than k=n. This mean tacitly incorporates the occu- pancy probability of all genera, including unsampled genera, assuming they are all drawn from the same underlying distribution. In addition to estimating the occupancy probability distribution of all taxa, we can estimate the part of the distribution corre- sponding to subsets of taxa with specified values of k. Those with k>0 preferentially sample higher values of p than those with k=0 (Fig. 5C,D), and those with progressively higher values of k reflect ever higher values of p (Fig. 5D). Even though the taxa with k>0 represent a biased subset of the entire prob- ability distribution (Fig. 5C), fitting that dis- tribution allows this bias to be circumvented. Truncating the log-normal at pmax is admittedly ad hoc and inelegant. John Alroy (personal communication 2016) has kindly suggested an alternative approach: to model the probabilities as a logit-normal distribution, in which logit(p), equal to log[p/(1 − p)], is normally distributed (Coull and Agresti 1999; Dorazio et al. 2006: p. 847). With p bounded between 0 and 1, logit(p) is unbounded from –∞to∞, no ad hoc truncation is necessary, and the distribution can be modeled with only two parameters, μlogit (the mean of the logit of p) and σlogit (the standard deviation of the logit of p). With the low values of p characteristic of the data analyzed here, the log-normal and logit- normal are rather similar. Moreover, by the time the truncation point is reached, the probability density has generally trailed off nearly to zero, so the fitted log-normal prob- ability curve does not “drop off a cliff” at pmax. The case in Figure 5 is typical. If for each stage we compare the probability density at pmax to the maximum value of the density, the ratio of the two has a median ~1×10−4, and it is never higher than ~1×10−3. The median tail area to the right of pmax is ~0.3%, and the maximum is ~2%. Figure 6 shows results of fitting three different distributions to each stage: (1) constant p;


713


(2) truncated log-normal p; and (3) logit- normal p. Figure 6A compares the predicted and observed standard deviation of k.Ifwe assume p is constant for each stage and use the Haldane–Fisher approach to estimate it, the predicted variance in k is substantially lower than what is observed. If we fit a logit-normal, the predicted variance is too high. This is because most genera have observed occupancy values, k, far to the left of the theoretical maximum value of n, that is, their observed proportional occupancies, k/n, are far below 1.0. However, the logit-normal integrates all the way to p= 1.0 (i.e., logit[p]=∞) and there- fore predicts some high values of k that increase the expected variance. (Similar results [not shown] obtain if we fit a log-normal truncated at p=1.0.) If instead we fita truncated log-normal, we accurately capture the observed variation in k. Calculating the sample size–corrected AIC for each model for each stage (Fig. 6B), we see that the Akaike weights generally favor the truncated log- normal (median weight 0.80 vs. 0.20 for the logit-normal and 0.0 for the constant-p model) and that the median truncation point is around 0.25 (Fig. 6C). The agreement between predicted and


observed variance in k is one sign that the truncated log-normal gives a reasonable fitto the data. Another sign is that the dependence between the number of cells, n, and the estimate of mean occupancy probability, p, essentially vanishes with this model (Spearman rank-order correlation coefficient: rs= −0.13, p > 0.25), whereas some bias, albeit weak, persists if we fit the logit-normal (rs= −0.35, p<0.01) (Fig. 7). The lack of correlation between n and p is to be expected if the data conform to the modeled form of the distribution, because n is explicitly taken into account in the likelihood function (Appendix). Whether to prefer the logit-normal or the


truncated log-normal is somewhat a question of truth versus beauty. For aesthetics the logit- normal takes the prize, but given the data considered here, the truncated log-normal generally fits better, enough so to justify the extra parameter. I will therefore stick with the truncated log-normal for the rest of this paper. However, there is no obvious reason to


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