There are a variety of technical points pertaining to B1 tests of the ZSM SAD prediction. We include them here in the supplemental materials in a bullet point format.
1. Note that in the published significance test by Volkov et al. (2003) they incorrectly calculated degrees of freedom. Volkov et al. used a three-parameter lognormal and claimed to only have one parameter for the ZSM; we use here a two-parameter lognormal (plus a lost degree of freedom to make the bins sum up) and use the correct count of three parameters for the anZSM (m,J,S with no lost degree of freedom). The parameter S can be replaced by  , still leaving three parameters. |
2. It is known that different binning schemes favor goodness-of-fit in whatever region is broken into the most bins (Magurran 1988, McGill 2003b). This is especially problematic when using Preston log2 binning which places high emphasis on species with abundances in the region of 1–8 individuals, the same region where the ZSM is highly flexible in fitting data (Fig. 1, main text). This has the effect (presumably undesired but never made explicit) of de-emphasizing fit for species with >8 individuals, where the lognormal often performs better than the ZSM (McGill 2003a). To address this, McGill (2003a) evaluated eight different measures of goodness-of-fit. Volkov et al.(2003) used the  statistic on Preston log2 binned data. The remaining authors (Alonso and McKane 2004, Etienne and Olff 2004, Olszewski and Erwin 2004, He 2005) used a single criterion for goodness-of-fit, in each case based on some form of likelihood. |
| 3. Other goodness-of-fit measures also suffer from biases. Anything based on the cumulative density function (CDF) is biased toward a good fit in the region where the CDF rises most steeply (i.e., the mode of the SAD). The likelihood gives excess weight to outliers (Hilborn and Mangel 1997) and hence favors a good fit in both tails. |
| 4. At a minimum, if one uses the same measure of goodness of fit to choose parameters and to generate a significance value for goodness-of-fit, then an adjustment to P values is needed to compensate for this circularity (Lilliefors 1967). This has not been implemented in the context of neutral theory. |
| 5. Some authors generated confidence intervals on their parameter estimates (Etienne and Olff 2004, Olszewski and Erwin 2004) while the remainder did not. This is important because it turns out that the estimated confidence intervals are quite large. On the BCI data, m has high support from about 0.05 to 0.5 and theta varies from about 35–55 (Etienne and Olff 2004), while on a different data set the 95% confidence interval of m varies from about 0.1 to 1.0 and theta shows similarly large ranges (Olszewski and Erwin 2004). It would appear that the likelihood surfaces of the ZSM are extremely flat (see also McGill 2003a). |
6. Likelihood can be used as a measure of fit by using p(n|,m,J)=< (n|,m,J)>/<S> with <> given by Eq. 7 in Volkov et al. and <S> given by the equation on p 171 in Hubbell (2001). We caution that likelihood for the anZSM depends very sensitively on the formula for <S>. |
| 7. Two authors merely explore whether the ZSM performs the least bit better than the lognormal (Volkov et al. 2003, Olszewski and Erwin 2004). But McGill (2003a,b) suggested that this was a hypotheticodeductive test and that neutral theory must be shown to fit not just better but statistically significantly better than the null (lognormal) hypothesis. He (2005) and Alonso (2004) adopt this approach. Etienne and Olff (2004) used the Bayesian equivalent of this idea. |
| 8. Note that Olszewski and Erwin’s paper explores fossil data time-averaged across many 100,000’s of years (and generations) which is known to distort the shape of the SAD (McGill 2003b, Volkov et al. 2003), making for a bad test of neutral theory however important their work may be for paleontology. Thus we do not consider it a test of neutral theory and it is therefore not included in our list of empirical tests, but we do discuss their methodological approaches for level B1 tests of the ZSM SAD prediction. |
| 9. Volkov et al. and He used Preston’s discrete approximation to the lognormal (Preston 1948), but there is probably no reason to use such an approximation when modern computing makes it trivial to use the true (continuous) lognormal used by the other authors. Etienne used the Poisson lognormal, which on the one hand is desirable because it gives discrete abundances, but it does not fit as well as the lognormal (B. J. McGill, unpublished manuscript). |
LITERATURE CITED
Alonso, D., and A. J. McKane. 2004. Sampling Hubbell's neutral theory of biodiversity. Ecology Letters 7:901–910.
Etienne, R. S. 2005. A new sampling formula for neutral biodiversity. Ecology Letters 8:253–260.
Etienne, R. S., and H. Olff. 2004. A novel genealogical approach to neutral biodiversity theory. Ecology Letters 7:170–175.
He, F. 2005. Deriving a neutral model of species abundance from fundamental mechanisms of population dynamics. Functional Ecology 19:187–193
Hilborn, R., and M. Mangel. 1997. The ecological detective: confronting models with data. Princeton University Press, Princeton, New Jersey, USA.
Lilliefors, H. W. 1967. On the Kolmogorov-Smirnov test for normality with mean and variance unknown. Journal of the American Statistical Association 64:399–402.
Magurran, A. E. 1988. Ecological diversity and its measurement. Princeton University Press, Princeton, New Jersey, USA.
McGill, B. J. 2003a. A test of the unified neutral theory of biodiversity. Nature 422:881–885.
McGill, B. J. 2003b. Strong and weak tests of macroecological theory. Oikos 102:679–685.
Olszewski, T. D., and D. H. Erwin. 2004. Dynamic response of Permian brachiopod communities to long-term environmental change. Nature 428:738–741.
Preston, F. W. 1948. The commonness and rarity of species. Ecology 29:254–283.
Volkov, I., J. R. Banavar, S. P. Hubbell, and A. Maritan. 2003. Neutral theory and relative species abundance in ecology. Nature 424:1035–1037.