Appendix A. Detailed instructions for implementing and interpreting a power curve analysis.
A power curve analysis of competition data involves nonlinear regression using a power function. The y-variable measures should be taken on an individual or population experiencing competition and the x-variable measured on a paired individual or population experiencing low or no competition. A power function takes the form:
y = k1+k2xk3
The resulting relationship is linear if k3=1, a saturating curve if k3<1, and an accelerating curve if k3>1 (Fig. A1).
| |
| FIG. A1. Examples of a range of line behaviours possible with a power function of the form: y = k1+k2xk3. Only an isometric relationship like the solid line, where the y-intercept (k1) is 0 and k3=1, is suitable for analysis using a ratio-based index of competitive ability. Any linear relationship where k3=1 is suitable for ANCOVA. |
General procedure for a power curve analysis
A power curve analysis involves multiple stages where candidate models are developed and compared. Competing models are distinguished using the extra-sums of squares test and AIC, tests that determine whether a simple model or a more complex one best describes a data set (Bates and Watts 1988; Sokal and Rohlf 1995; Motulsky and Christopoulos 2004). Details on parameter constraints for the coefficients, interpretation of coefficient values, and use of the extra-sums of squares test follow.
1. Fit a single (global) curve with common k1 (if necessary), k2, and k3 coefficients for all treatments. 2. Determine whether the k1 parameter (y-intercept) is necessary. a. Fit a global curve with the k1 parameter set to zero. b. Compare the model with the k1 parameter set to zero to the full global model from Step 1 using the extra-sums of squares test and AIC to test whether the intercept is significantly different from zero. In this extra-sums of squares test the model with k1=0 is the simpler model and the global model is the more complex one. c. A significant F test indicates that the more complex global model explains significantly more variation in these data and the k1 parameter should be retained. If the models are not significantly different then the k1 parameter is unnecessary and further analyses can proceed with using only the k2 and k3 parameters. 3. Determine whether the overall relationship is significantly different from linear. a. Fit a global curve with the k3 parameter set to one b. Compare this model to the model retained in Step 2 using the extra-sums of squares test and AIC. If the models are not significantly different then the interaction is size-independent and the k3 parameter can be removed from the model. c. If the relationship is linear then analysis should proceed using the more powerful and flexible ANCOVA with the y-variable as the response variable and the x-variable as a covariate. 4. Determine whether different curves best describe each treatment level. a. Fit individual curves to each treatment with all coefficients independent between treatments. b. Fit individual curves with one coefficient in common for each treatment but the other coefficient allowed to vary. This step allows the effects of experimental treatments that only impact one of the parameters to be detected. c. If the power function includes all three parameters then six candidate models should be developed (three models with one parameter in common, and two varying between treatments, and three models with two parameters in common and one allowed to vary). d. Each the models from Steps 4a and 4b should be compared to the global model using the extra-sums of squares test and AIC. AIC must be used in cases where two models with different common parameters are to be compared; an extra sums of squares test is precluded because neither model is a nested subset of the other. 5. If the separate curves from step 4 are found to be significantly better than the global curve and if more than one treatment is to be compared, make pairwise comparisons to determine which treatments are significantly different from each other. a. Develop both a global curve for the two treatments to be compared, and individual models for each treatment. If common parameters between treatments were identified in Step 4, then the values of those parameters from Step 4 should be entered directly into the model. b. Compare the global and individual pairwise models with extra-sums of squares tests. Because numerous pairwise tests may be required at this stage, Bonferroni adjustments should be used to correct for experiment-wise error. 6. Interpretation of the final curves should follow Table A1.TABLE A1. Interpretations of power curve coefficients. Curves with k1>0 and k2<0 are also possible but biologically unlikely. See Fig. A1 for example curves for different parameter combinations.
k1 |
k2 |
k3 |
Line Shape |
Biological explanation |
0 |
>0 |
1 |
Linear |
Competition if 0 <k2 <1; facilitation if k2>1, no variation in competitive ability with size. |
0 |
>0 |
0<k3<1 |
Saturating |
Decreasing competitive ability at larger potential size, possible switch from facilitation to competition |
0 |
>0 |
>1 |
Accelerating |
Increasing competitive ability at larger potential sizes, possible switch from competition to facilitation |
<0 |
>0 |
1 |
Linear |
Increasing competitive ability with potential size, possible switch from competition to facilitation |
>0 |
>0 |
1 |
Linear |
Decreasing competitive ability with potential size, possible switch from facilitation to competition |
<0 |
>0 |
0<k3<1 |
Saturating |
Increasing competitive ability with potential size, possible switch from competition to facilitation |
>0 |
>0 |
0<k3<1 |
Saturating |
Decreasing competitive ability with potential size, possible switch from facilitation to competition |
<0 |
>0 |
>1 |
Accelerating |
Increasing competitive ability with potential size, possible switch from competition to facilitation |
>0 |
>0 |
>1 |
Accelerating |
Decreasing competitive ability with potential size, possible switch from facilitation to competition |
Starting values and constraints on power curve coefficients
Because nonlinear regression is implemented as an iterative procedure, starting values for the coefficients need to be specified, and constraints on those coefficients can be included. We recommend that the following starting values be used in all cases: k1 (if needed) =0, k2=1, and k3=1 because a line with these coefficients represents the simplest possible case: a linear relationship without size dependence where there is neither suppression nor facilitation. We recommend that, in all analyses of competition data, k3 be restricted to be positive, because the shape of the relationship does not differ in a biologically significant manner from that when k3 is negative.
Further restrictions on the coefficients will depend on the type of data to be analyzed. For example, when the data are biomass or another measure of the vegetative characteristics of plants started from seed we can assume that all plants started from the same size, and thus we can force the relationship to pass through the origin. With this assumption the y-intercept, k1, can be omitted, increasing the power of the model and simplifying the interpretation of the other coefficients. Also, since any line with a negative slope and zero intercept will yield negative biomass values, in this situation k2 can be restricted to be positive. If the response variable is any measure where the assumption that the relationship extends through the origin cannot be made, then the k1 coefficient is necessary. Examples of this class of data could include measures of fecundity, since it is possible for one individual in a pair to have a non-zero value while the other is at zero. With the coefficient k1 included, no restrictions on the value of k2 are necessary because if k1 is positive curves with negative slopes can yield biologically meaningful values.
Implementing the extra-sums of squares test
The extra-sums of squares test evaluates whether a more complex model is significantly better (more variance explained) than a simpler model given that more parameters need to be estimated in the more complex model. Four values from the regression output are needed to carry out an extra-sums of squares test: the residual sums of squares and degrees of freedom from the global (less complex) model (SSglob, dfglob), and the sum of the residual sums of squares and degrees of freedom from each of the individual curves (SSind, dfind). An F ratio is calculated from these values using the following formula:
F= (SSglob − SSind)/(dfglob − dfind)
(SSind / dfind)
This F ratio has numerator degrees of freedom= dfglob−dfind and denominator degrees of freedom=dfind (Bates and Watts 1988; Motulsky and Christopoulos 2004). A significant F ratio indicates that the individual curves explain significantly more variation in the data than the global curve, and thus the treatment applied had a significant effect on the interaction under study.
If the experimental design includes more than two treatment levels, then more steps are needed. First, the above tests should be done to compare a global curve including all treatments to the sum of the individual curves for each treatment. Second, pairwise comparisons should be done between individual treatments where the global curve is based on only those two treatments of interest. The only restriction on the extra-sums of squares test is that it can only be used to compare nested models. For example, models with separate curves for each treatment are nested within the global model for all treatments, but models of the same data using different equations are not nested. If the models are not nested then Akaike’s information criterion (AIC) can be used to compare models (Motulsky and Christopoulos 2004).
The extra-sums of squares test can also be used to test whether the differences between a series of treatments are due to differences in all parameters, or whether only a single parameter is responsible for the differences. For example, a common k3 parameter may provide a good fit, but the treatments may be best described by separate k2 values. In such a situation the experimental treatments could be interpreted to have affected only one parameter, and hence only one aspect of competitive ability. A similar test can be used to determine if the intercept (k1) is zero or whether the relationship is nonlinear (k3≠ 1). Bonferroni correction should be used to protect experiment-wise error rates when multiple extra-sums of squares tests must be performed (Sokal and Rohlf 1995).
LITERATURE CITED
Bates, D. M., and D. G. Watts. 1988. Nonlinear regression analysis and its applications. John Wiley and Sons, New York, New York, USA.
Motulsky, H., and A. Christopoulos. 2004. Fitting models to biological data using linear and nonlinear regression. Oxford University Press, Oxford, UK.
Sokal, R. R., and J. F. Rohlf. 1995. Biometry. Third edition. W. H. Freeman, New York, New York, USA.