Eric G. Lamb, James F. Cahill, and Mark R. T. Dale. 2006. A nonlinear regression approach to test for size-dependence of competitive ability. Ecology 87:1452–1457.

Appendix B. Three tables and a figure showing the full results of the analysis of the Linum experiment and a figure showing the results of the analysis of the Lymnaea data set.

FIG. B1. Results of a traditional analysis of the Linum experiment using the log response ratio (Hedges et al. 1999). Bars in the figure are log response ratios for each of the five fertilizer treatments (gm-2 NPK) with error bars are of standard deviation. There were significant fertilization effects on the competitive response of Linum when the log response ratio data were analyzed using ANOVA (SPSS for Windows v.9) with fertilization as a fixed factor and block as a random factor (F4,27 = 3.345, P = 0.024). A subsequent Tukey test only identified significant differences between the 0 and 8 gm-2 fertilization treatments. The conclusions from this analysis of the response ratios analysis are very different from the power curve analysis. This figure suggests that competitive ability in the zero fertilizer treatment was higher than in the 16 and 32 gm-2 while competitive ability was clearly highest in the high-fertilizer treatments in the power curve analysis (Main text Fig. 1).

TABLE B1. Summaries of the models generated during the analysis of the Linum data set. Global models have a single curve for the entire data set, while separate models have a curve for each treatment. The number of parameters is the number of coefficient values that were estimated from the data for that model. The residual sums of squares and degrees of freedom values are the sum of the residuals from the individual curves for each treatment.

TABLE B2. Results of extra-sums of squares tests to compare models from the Linum experiment. Global models have a single curve for all treatments while separate models have individual curves for each treatment. The first two tests demonstrate that models with a k₃ parameter, indicating nonlinearity, are better than linear models. The third test indicates that a model with separate k₂ and k₃ parameters for each treatment is not significantly better than single model, but the fourth and fifth tests indicate that models with one parameter in common but the other different between treatments are significantly better than the global model. Since one of the separate models with one common parameter is not a nested subset of the other model, they cannot be compared with the F test, and instead should be compared using AIC (Table B1).

TABLE B3. Pairwise comparisons of models from the Linum experiment. Each test compared a global curve for two fertilization treatments with the k₃ parameter value set to 0.2531 (the k₃ parameter value from the individual model with a common k₃) to individual models with a separate k₂ parameter for each treatment. With Bonferroni correction only P values less than 0.005 should be considered significant.

FIG. B2. Power curve for Lymnaea fecundity with raw data points for the two species and drying treatments overlain. The gray line is the 1:1 reference line. Any position above that line indicates higher fecundity at higher density, while any position below the line indicates higher fecundity at lower density. Only one line is shown on this figure as these data were best described by a single curve with k3=1.

LITERATURE CITED

Hedges, L.V., J. Gurevitch, and P.S. Curtis. 1999. The meta-analysis of response ratios in experimental ecology. Ecology 80:1150–1156.