Date of this Version
Ecology Letters 11 (2008), pp. 311-312; doi: 10.1111/j.1461-0248.2008.01156.x
Russo et al. (2007) tested two predictions of the Metabolic Ecology Model (Enquist et al. 1999, 2000) using a data set of 56 tree species in New Zealand: (i) the rate of growth in tree diameter (dD/dt) should be related to tree diameter (D) as dD/dt = βDα and (ii) tree height (H) should scale with tree diameter as H(D) = γDδ, where t is time, β and γ are scaling coefficients that may vary between species, and α and δ are invariant scaling exponents predicted to equal 1/3 and 2/3, respectively (Enquist et al. 1999, 2000). To this end, Russo et al. (2007) used maximum likelihood methods to estimate α and δ and their two-unit likelihood support intervals. As noted in our original manuscript, the growth–diameter scaling exponent and coefficient covary, complicating the estimation of confidence intervals. We now recognize that the method we used to estimate support intervals (using marginal support intervals with the nuisance parameters fixed) underestimates the breadth of the interval and that the support intervals, properly estimated, should account for the variability in all parameters (Hilborn & Mangel 1997). This can be done in several ways. For example, the Hessian matrix can be used to estimate the standard deviation for each parameter, assuming asymptotic normality. Alternatively, one can systematically vary the parameter for which the interval is being estimated, re-estimate the Maximum likelihood estimates (MLEs) for the other parameters, and take the support interval to be the values of the target parameter that result in log likelihoods that are two units away from the maximum (Edwards 1992; Hilborn & Mangel 1997). A third and more direct approach to comparing data with prediction is to use the likelihood ratio test (LRT), which explicitly tests if a model with a greater number of parameters provides a significantly better fit to the data than a simpler model in which some parameters are fixed at predicted values (Hilborn & Mangel 1997; Bolker in press).
Here, we re-analyze our data using LRTs, present a table revising Tables 1 and 2 from Russo et al. (2007), and reevaluate whether there is statistical support for the predictions of the Metabolic Ecology Model that we tested in Russo et al. (2007). We used LRTs to test, respectively, whether a model in which a,or d, was estimated at its MLE had a significantly greater likelihood than did a model with α = 1/3, or δ = 2/3, for the growth–diameter and height–diameter scaling relationships.