Appendix B. Details of reanalysis of results published in Meehan (2006) and Enquist and Niklas (2001).
General Approach
In Fig. 4 of our paper we report reanalyses of data originally reported by Meehan (2006) and Enquist and Niklas (2001). We first analyzed the data following the protocols reported in the original paper. We replicated the original analysis as closely as possible including bin widths, positions of bin edges, and exclusion of data points. Results are reported as kernel density estimates (basically histograms) of the fitted exponents.
Reanalysis of Meehan (2006)
Our reanalysis of Meehan (2006) focuses on the "local scale" size distribution results reported in his Fig. 3. Meehan’s original analysis used simple logarithmic binning to fit size distributions in litter and soil community data originally published by Petersen and Luxton (1982). Meehan’s original analysis included additional variables in a multivariate framework (trophic level and temperature). We ignore these here for simplicity and our estimates of the size distribution exponents appear to be unaffected by excluding these variables (i.e., his simple logarithmic binning estimates including the over variables are very similar to our estimates excluding them). While Meehan’s final model fit all sites and trophic levels using a single model, the exponent derived using all of the data is similar to those fit to each site and trophic level separately (Meehan 2006). Therefore, we fit each distribution separately yielding a total of 12 distributions.
We used the bin edges reported in the original paper, but changed the analysis slightly to make it consistent with the standard description of logarithmic binning given in our paper. Meehan (2006) calculated the average mass of an individual occurring in a body size bin based on the reported body size data and used this as the value of x instead of simply using the bin center. In our reanalysis we used the mass at the center of the bin (i.e., the geometric mean of the linearly scaled bin edges) for both our replication of Meehan’s analysis and our reanalysis. Results using Meehan’s original definition of average mass were qualitatively similar.
The raw data in Meehan (2006) consists of an average mass and a total abundance for each species at each site. Not having the actual masses of each individual will affect maximum likelihood estimation in unknown ways. Therefore we reanalyzed this data by following all of the protocols reported in Meehan (2006), but normalized the counts by the linear width of the logarithmic bins. This allowed us to illustrate the dangers of simple logarithmic binning despite the limitations of the data.
Data for this reanalysis was kindly provided by Timothy Meehan, who may be contacted regarding access to the data by email at tdmeehan@entomology.wisc.edu.
Reanalysis of Enquist and Niklas (2001)
Our reanalysis of Enquist and Niklas (2001) focuses on size distribution results reported in their Figs. 2 and 3. The original analyses consisted of fitting diameter distributions for Alwyn Gentry’s tree communities (Phillips and Miller 2002) using linear binning. We used the same data used in Enquist and Niklas 2001, provided by BJE. When binning the data we follow all of their original protocols including the exclusion of diameter bins with less than 5 individuals.
We reanalyzed this data using maximum likelihood estimation based on the Pareto distribution. The minimum diameter recorded in the Gentry data is 2.5cm, so a = 2.5. The results are qualitatively similar using the Truncated Pareto distribution with the maximum size set to the maximum size of an individual tree observed at the site.
Caveat: These results are not intended to comment directly on the form of the tree-size distribution for two reasons. First, tree size distributions may deviate from simple power laws (Coomes et al. 2003, Muller-Landau et al. 2006, Coomes and Allen 2007), and the analyses to establish whether the power-law is the best model for the distribution are beyond the scope of this paper. Second, the Gentry data (Phillips and Miller 2002) on which these results are based are actually are not ideal for evaluating the form of the tree size distribution. Gentry’s data include single trees that branch below breast height and therefore have multiple stem measurements recorded for an individual. Unfortunately, the Gentry data does not identify the stems with the individual that they came from making it impossible to back calculate the basal stem diameter for an individual. As a result these data have typically been treated as if every stem is it’s own individual (e.g., this is how Enquist and Niklas analyze the data in their original paper, and therefore how we do so here). As a result, there is likely a bias towards an overrepresentation of smaller individuals as small multiple stems from a single tree are counted as separate individuals. Thus, caution should be taken when interpreting these results in terms of actual individual tree diameter distributions. Regardless, these results clearly demonstrate the importance of using unbiased methods for estimating power law exponents.
The Gentry data is
available from
the
LITERATURE CITED
Coomes, D. A., and R. B. Allen. 2007. Mortality and tree-size distributions in natural mixed-age forests. Journal of Ecology 95:27–40.
Coomes, D. A., R. P. Duncan, R. B. Allen, and J. Truscott. 2003. Disturbances prevent stem size-density distributions in natural forests from following scaling relationships. Ecology Letters 6:980–989.
Enquist, B. J., and K. J. Niklas. 2001. Invariant scaling relations across tree-dominated communities. Nature 410:655–660.
Meehan, T. D. 2006. Energy use and animal abundance in litter and soil communities. Ecology 87:1650–1658.
Muller-Landau, H. C., R. S. Condit, K. E. Harms, C. O. Marks, S. C. Thomas, S. Bunyavejchewin, G. Chuyong, L. Co, S. Davies, R. Foster, et al. 2006. Comparing tropical forest tree size distributions with the predictions of metabolic ecology and equilibrium models. Ecology Letters 9:589–602.
Petersen, H., and M. Luxton. 1982. A Comparative-Analysis Of Soil Fauna Populations And Their Role In Decomposition Processes. Oikos 39:287–388.
Phillips, O., and J. S. Miller. 2002. Global Patterns of Plant Diversity: Alwyn H. Gentry's Forest Transect Data Set. Missouri Botanical Garden Press, St. Louis, Missouri.