HUTCHINSONIAN RATIOS AND LOG-NORMAL DISTRIBUTIONS.

It has been proposed that competition among members of a species assemblage leads to constant size ratios among the species ranked in order of average individual size, or to large minimum size ratios (for references, see Simberloff and Boecklen, 1981). Simberloff and Boecklen (1981) present statistical tests for constancy of size ratios and for large minimum size ratios in species assemblages (sensu Hutchinson, 1959) and apply these tests to appropriate data from the literature. The tests for ratio constancy are based on those developed by Barton and David (1956) to determine whether points along a line are unlikely to be the result of independent uniform-random placement. The test for minimum size ratios is based on Irwin's (1955) probability distribution function for the minimum segment produced by independent uniform-random placement of points along a unit line. To determine whether sets of species manifest constant size ratios or large minimum ratios, Simberloff and Boecklen use these tests to determine whether species are constantly spaced or exhibit large minimum distances along a log-scaled line. Asking whether the logarithms of sizes are constantly spaced or have a large minimum distance along a line is equivalent to asking whether size ratios are constant or exhibit large minima along an arithmetic line since log(a/ b) = log a log b, log(b/c) = log b log c, etc. Simberloff and Boecklen find, in a compilation of species assemblages adduced to demonstrate the existence of constant ratios and large minimum ratios, only seven of 21 claims of ratio constancy and seven of 18 claims of large minima supported at the 5% level, and only about half the claims supported at the 30% level. Schoener (1984), Colwell and Winkler (1984), and Case et al. (1983) challenge these results, claiming that the tests used by Simberloff and Boecklen assume a biologically unrealistic uniform distribution of sizes. Colwell and Winkler suggest that a modal distribution is more appropriate; Schoener and Case et al. recommend a log-normal distribution. Furthermore, they claim that the assumption of a uniform distribution biases the tests toward failing to reject the null hypothesis. The tests used by Simberloff and Boecklen assume a log-uniform distribution of sizes-not a uniform distribution. Nevertheless, a log-normal distribution of sizes may be more realistic and the performance of the BartonDavid and Irwin test statistics, as used by Simberloff and Boecklen, should be investigated under this assumption, Critics of null models contend that particular assumptions of null models may invalidate them as appropriate reference points for evaluating evolutionary processes that are assumed to structure communities and species interactions (Hendrickson, 1981; Diamond and Gilpin, 1982) or that historical effects (Colwell and Winkler, 1984) or multiple causality (Quinn and Dunham, 1983) may insulate community theory from direct refutation by null models. Criticisms of the assumptions of null models have the advantage that they may be addressed in a rigorous and quantitative manner. Here we present a sensitivity analysis that assesses the performance of the Barton-David and Irwin test statistics when data are drawn from a log-normal distribution. We compare the cumulative distributions of the test statistics generated from the log-normal distribution to those generated by the Barton-David and Irwin probability distribution functions. The cumulative distributions for each test statistic are based upon the average of five sets of 5,000 replicates for species assemblages of five, six, and ten. Averaging the five sets of 5,000 replicates produces precise estimates with minimal sampling variation.