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1. Metabolic Scaling Theory (MST) predicts a ‘universal scaling law’ of tree growth. Proponents claim that MST has strong empirical support: the size-dependent growth curves of 40 out of 45 species in a Costa Rican forest have scaling exponents indistinguishable from the MST prediction.
2. Here, we show that the Costa Rican study has been misinterpreted. Using Standardized Major Axis (SMA) line-fitting to estimate scaling exponents, we find that four out of five species represented by more than 100 stems have scaling exponents that deviate significantly from the MST prediction. On the other hand, sample sizes were too small to make strong inferences in the cases of 33 species represented by fewer than 50 stems.
3. Recently, it has been argued that MST is useful for predicting average scaling exponents, even if individual species do not conform to the theory. We find that the mean scaling exponent of the Costa Rican trees is greater than predicted (across-species mean = 0.44), and hypothesize that scaling exponents in natural forests will generally be greater than predicted, because the theory fails to model asymmetric competition for light.
4.Synthesis. We highlight shortcomings in the interpretation of data used in support of a key MST prediction. We recommend that future research into biological scaling should compare the merits of alternative models rather than focusing attention on tests of a single theory.
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Metabolic Scaling Theory (MST) has proven immensely stimulating to plant ecologists because it provides ‘a quantitative, predictive framework for understanding the structure and dynamics of an average idealized forest’ (Enquist, West & Brown 2009). Proponents of this ‘zeroth-order’ theory maintain that it enables useful generalizations to be made about tree growth by focusing attention on two key processes – the optimization of water transport through the xylem and the biomechanics of stem wood (West, Enquist & Brown 2009). Yet its sweeping generalizations and bold assumptions have also generated debate and drawn much criticism (Muller-Landau et al. 2006; Reich et al. 2006). The counterargument is that other processes that contribute to resource capture, transportation and metabolism have a significant influence on tree growth, and that their omission, or superficial treatment, gives rise to mismatches between MST predictions and empirical realities (e.g. Reich et al. 2006; Muller-Landau et al. 2006; Coomes and Allen 2007a,b).
Here, we consider whether one of MST’s central claims is supported by data. Enquist et al. (1999) argued that tree stem diameter growth dD/dt scales with stem diameter (D) according to a power function:
( eqn 1)
They emphasized that the scaling exponent, α, was equal to 1/3, arguing that differences in the life history of species affected the scaling coefficient, β, but had little influence on α. In the face of much criticism, Enquist et al. have recently conceded that scaling exponents can vary greatly among species, and have shifted their position to arguing that MST ‘predicts a basin of attraction, around which actual biological networks cluster’ (Price, Enquist & Savage 2007). We interpret this phrase as meaning that MST is useful in predicting the average value of α taken across species.
Enquist et al. (1999) demonstrated support for their theory by modelling the growth of 45 tree species found in a Costa Rican forest, for which stem diameters had been measured and re-measured 20 years later. To estimate α and β, they integrated the growth function given in eqn 1 to get:
( eqn 2)
where D0 and Dt are stem diameter measurements made on a tree during an initial census and after time interval t. If the MST prediction that α = 1/3 is correct, then it follows that:
( eqn 3)
Enquist et al. (1999) used Reduced Major Axis (RMA) regression to fit straight lines through scatterplots of D02/3 vs. Dt2/3 for each species. Note that RMA regression is identical in all but name to Standardized Major Axis (SMA) line-fitting, and we use the latter terminology for the reasons set out in Warton et al. (2006). SMA line-fitting is a least-squares method that differs from linear regression only in the direction in which errors from the line are measured and is regarded as the most appropriate approach when the purpose of a study is to describe how two size variables are related (Warton et al. 2006). Enquist et al. (1999) reasoned that the slope of the fitted line (m) would only be equal to 1.000 if α was equal to 1/3. Testing whether slopes differ significantly from 1.000 provides a reasonable way of testing the hypothesis that α = 1/3, but it is important to note that it does not provide an estimate of α. In other words, it identifies species whose growth curves deviate from the MST prediction, but is unable to quantify the degree of deviation.
Analysis of the Costa Rican data set using this method produced apparently compelling evidence in support of MST (Enquist et al. 1999). For 40 of the 45 species examined, the 95% confidence intervals (CIs) of slope estimates overlapped with 1.0, suggesting that α = 1/3 was consistent with data for the vast majority of species. Lines fitted to D02/3 vs. Dt2/3 using linear regression had high r2 values – 0.95 on average – conveying an impression that these inferences carried statistical weight. In addition, the cross-species mean of the slope estimates was 1.038, with a CI of 1.01–1.08. Although the confidence intervals indicated a departure from the null hypothesis, Enquist et al. (1999) passed this off as ‘essentially not different from the predicted value of 1.0’ and who could deprive them of such an indulgence when the mean slope was so remarkably close to 1.00? The conclusions drawn from these analyses seem compelling and have been repeated in several subsequent papers (Enquist, West & Brown 2009). However, we take issue with the way the results were interpreted and illustrate our concerns by revisiting the Costa Rican data set.
Our first criticism is that sample sizes were generally too small to detect departures from the theoretical predictions. For each of the 45 species, we fitted straight lines to the D202/3 vs. D02/3 relationship using SMA line-fitting (implemented using line.cis in the smatr library of R; Warton et al. 2006) and plotted the slope estimates from these analyses against sample sizes (Fig. 1). We used our slope estimates because line.cis provides exact CIs, whereas the previous approach is only approximate, and also because several errors and curious anomalies were identified in Table S1 of Enquist et al. (1999) (see Supporting Information section). A narrowing of the CIs with sample size is apparent in Fig. 1a, in accordance with the basic tenets of sampling theory (e.g. Cochran 1953). Of the five species represented by more than 100 stems, four had slope values that differed significantly from 1.0 (P ≤ 0.05; Muller-Landau et al. 2006). On the other hand, 23 of the 24 species represented by fewer than 30 samples had wide CIs that overlapped with 1.0 (Fig. 1a). Hence the MST hypothesis was rejected in the majority of cases when species were well sampled, but the tests had insufficient power to reject the null hypothesis in the case of the poorly sampled species. Even though many species had been poorly sampled, we nevertheless detected highly significant differences in the slopes of the 45 species (tested using slope.com in the smatr library, likelihood ratio test = 80.8, P =0.0006).
A better idea of how many samples are needed to reject the MST hypothesis can be ascertained by analysing a data set from New Zealand. The data set consists of stem diameters of 3334 trees of Nothofagus solandri var. cliffortioides growing naturally in the Southern Alps, and measured over a 19-year period (Coomes and Allen 2007a,b). When all 3334 trees were analysed using the extended approach, we found the slope of the SMA line was 1.058 (CI = 1.051–1.065), which is significantly different from 1.00 (t =31, P < 0.00001). Next, the data set was randomly divided into subsets to obtain samples ranging in size from 5 to 3000 stems, and regression lines were fitted to each subset. The relationship between slope estimates and sample size are shown in Fig. 1b: each data point and CI is the average slope obtained from 100 random subsets. We were unable to reject the null hypothesis that α = 1/3 unless the sample size exceeded about 50 stems, because for smaller samples the 95% CI of m overlapped with 1.0. Thus, the null hypothesis would have been incorrectly accepted if inference had been made from a data set consisting of fewer than about 50 trees. Assuming that the amount of unexplained variation in growth rates observed in the New Zealand data set is typical of that found in other data sets, then our analyses suggest a sample size of at least 50 trees per species is desirable for testing MST theory.
Estimating values of α for species and communities
Enquist et al. (1999) tested whether α values differed significantly from the MST prediction, but did not provide an estimate of the α values themselves. Given that the slopes of the 45 SMA lines hover closely around 1.00, it might seem reasonable to suppose that α values cluster closely around 1/3, but is that the case? Here, we estimate α values by extending the Enquist et al. (1999) approach. We chose to build on their method because we recognized its elegance in dealing with continuous variation in growth, and in using SMA line-fitting to estimate how one variable scaled against another (Warton et al. 2006). Instead of assuming that α = 1/3, we fitted SMA lines to the relationship between D201−σ and D01−σ for 250 values of σ ranging from −2.0 to 2.5. The scaling exponent (α) was taken to be the value of σ that produced a slope of 1.000. The CIs were calculated by examining the range of σ values which produced SMA slopes that were statistically indistinguishable from 1.00 at P =0.05 (R code and New Zealand data are provided in section ‘Supporting Information’).
We illustrate the adapted method by estimating α for the best-sampled species in the Costa Rican data set: Luehea speciosa is a small tree in the Tiliaceae for which 381 trees were measured. The relationship between D202/3 and D02/3 is plotted in Fig. 2a: the slope of the SMA line is 1.102 and the CI excludes 1.00 (CI = 1.066–1.140). Using our approach, we predict that α = 0.815 (CI = 0.660–0.963). When D201−0.815 is plotted against D01−0.815 (Fig. 2b), we find the regression slope is exactly 1.00 and not significantly different from that value (CI = 0.965–1.035). So the best-supported value of α is quite different from the theoretical value predicted by MST. However, one could be forgiven for believing that the theoretical model fits the data excellently if the assessment was based on visual comparison of the regression lines in Fig. 2a and b, a point to which we shall return later.
Applying the adapted method to all 45 species, we found that scaling exponents ranged from −0.88 to 1.34, with only half of the estimates lying between 0.25 and 0.75 (Fig. 3a). Of the five species represented by over 100 samples, four α values differed significantly from 1/3 (α = −0.16, 0.12, 0.62 and 0.82) and one was indistinguishable from the MST prediction (α = 0.18). In contrast, 23 of the 24 species represented by fewer than 30 individuals had such wide confidence intervals that the estimates were not significantly different either from 1/3 or from zero (i.e. no relationship between growth and size), further illustrating the inadequacy of such data for testing MST. Growth curves for the 45 species are shown in Fig. 3b; these curves were estimated by entering our α and β estimates into the following function derived from eqn 2:
( eqn 4)
where the initial diameter (D0) was set at 10 cm (Fig. 3b). Equation 4 was used to predict the diameter of trees after 100 years of growth (with D0 = 10 cm) when α and β were estimated by (a) the variable-α model versus (b) the MST model (Fig. 3c). The analysis demonstrates that interspecific differences in α are biologically significant as well as statistically significant: the estimates of D100 obtained by the two approaches are only weakly correlated (Pearson’s product-moment correlation, r =0.51).
Our analysis demonstrates that α differs among species, but could it be that MST correctly predicts the ‘basin of attraction’ of α values? For the Costa Rican data set, we estimated an across-species mean value for α of 0.44 (CI = 0.31–0.57). These summary statistics could be interpreted as providing support for MST, because the confidence interval overlaps with 1/3, but it is also hard to reject the null hypothesis when species differ so greatly in α values. Furthermore, the mean species-level estimate takes no account of the degree of confidence associated with each estimate. We used a meta-analysis approach which weights each α value by the inverse of its standard deviation (using meta.summaries in the rmeta library of R, assuming fixed effects) and obtained an across-species mean of 0.47 (CI = 0.40–0.54); the CIs should be interpreted cautiously as they depend upon the assumption made in the meta-analysis model. Further evidence that α exceeds 1/3 comes from analysis of the entire data set, disregarding information about species identity: the scaling exponent was 0.66 (CI = 0.60–0.72). These analyses do not support the ‘basin of attraction’ idea.
Goodness of fit
The high r2 values reported by Enquist et al. (1999) give a misleading impression of the strength of growth relationships. Plotting Dt2/3 against D02/3 creates the illusion of a close relationship because diameter growth over 20 years is only a fraction of the initial diameter, so D02/3 is being plotted against itself with small numbers added; this is a guaranteed recipe for creating spuriously strong r2 values! Our point is similar to the general concern raised by Nee et al. (2005) in their paper entitled ‘The illusion of invariant quantities in life histories’. Plotting the growth rate of Luehea speciosa, calculated as (D20 − D0)/20, against the initial size of the trees reveals how weak the relationship really is (Fig. 2c).
Metabolic Scaling Theory originally predicted that interspecific differences in life-history characteristics should influence β, but should have little influence on α in the scaling law (eqn 1). We have shown that species in a Costa Rican forest varied considerably in α values, and similar conclusions were reached by Russo, Wiser & Coomes (2007, 2008), who examined the growth of 56 New Zealand tree species. Also the idea that MST predicts a ‘basin of attraction’ around which species cluster (Price, Enquist & Savage 2007) was not upheld by our analyses: the cross-species mean exceeded 1/3 in the Costa Rican analyses. Even if MST made accurate predictions for trees growing without competitors, we hypothesize that the mean α would generally exceed 1/3 in natural forests: in dense populations the growth of small trees is greatly reduced by asymmetric competition for light while the growth of large trees is relatively unaffected, and this systematically alters the mean scaling relationship (Muller-Landau et al. 2006; Coomes & Allen 2007a). Neighbourhood analyses of competition support this hypothesis in the case of Nothofagus solandri var cliffortioides trees growing in the Southern Alps of New Zealand (Coomes & Allen 2007a): α was estimated to be 0.52 when all trees were considered, but was estimated to be 0.20 for the subset of trees that were unfettered by asymmetric competition. More neighbourhood analyses are needed to test whether asymmetric competition for light has a consistent effect on the scaling of growth in forests, or whether MST theory is better supported in some instances.
Metabolic Scaling Theory authors have done ecology a great service by establishing a novel theoretical framework for understanding biological scaling. Like all useful theories, it is transparent in its assumptions and predictions, allowing researchers like us to conduct empirical tests, challenge assumptions and attempt modifications. Models are abstractions of complex biological realities and all are ‘wrong’ in at least some respect (Price, Enquist & Savage 2007): the iterative process of model improvement requires researchers to identify the ways in which model are ‘importantly wrong’ without over-elaborating on minor details (Box 1976). Our contention is that MST is ‘importantly wrong’ in omitting to consider the effects of asymmetric competition, and that this bias translates into systematic misrepresentation of other forest attributes, such as forest size–structure (Coomes et al. 2003; Muller-Landau et al. 2006; Coomes & Allen 2007b).
In this paper, we have adapted the SMA line-fitting to identify problems with the way that Enquist et al. (1999) interpreted statistical results. The SMA line-fitting approach is an elegant way to fit power functions to growth data, but cannot be applied when multiple factors are thought to affect growth, or when growth curves depart from power laws (as is often observed to be the case; e.g. Muller-Landau et al. 2006; Coomes & Allen 2007a; Russo, Wiser & Coomes 2008). Fitting nonlinear models by maximum likelihood estimation is a powerful approach for more complex models, especially when Akaike Information Criteria (or similar indices) are used to compare models (e.g. Uriarte et al. 2004; Canham et al. 2006; Coomes & Allen 2007a; Russo, Wiser & Coomes 2007, 2008). We recommend that more thought is given to the relative merits of SMA line-fitting versus flexible nonlinear models in the context of growth modelling, as the two approaches give different estimates of α, especially when growth is only weakly related to size (see Supporting Information section). We can have an informed debate over the generality and utility of MST theory only after alternative models have been compared objectively.
We thank Dr B.J. Enquist for providing access to the Costa Rican data set. Funding was provided to RBA by the New Zealand Foundation for Research, Science and Technology and to DAC by the British Natural Environmental Research Council.