assessing scaling relationships: uses, abuses, and ...

DIFFERING PERSPECTIVES

Int. J. Plant Sci. 175(7):754–763. 2014. 䉷 2014 by The University of Chicago. All rights reserved. 1058-5893/2014/17507-0003$15.00 DOI: 10.1086/677238

ASSESSING SCALING RELATIONSHIPS: USES, ABUSES, AND ALTERNATIVES Karl J. Niklas1,* and Sean T. Hammond† *Section of Plant Biology, School of Integrative Plant Science, Cornell University, Ithaca, New York 14853, USA; and †Department of Biology, University of New Mexico, Albuquerque, New Mexico 87131, USA

Editor: William E. Friedman Premise of research. Workers have relied on fitting a straight line to logarithmically transformed data to determine biological scaling relationships without testing the assumption that error is normal and additive on the logarithmic scale. Methodology. We review the history of this practice, the pros and cons of log transformation, and the use of model Type I and II regression protocols. Using standard statistical protocols and the Akaike Information Criterion, we then evaluate linear and nonlinear models applied to a large interspecific data set and a smaller intraspecific data set to reexamine the hypothesis called diminishing returns, which states that the surface areas of mature leaves may fail to increase one-to-one (isometrically) as lamina dry mass increases. Pivotal results. The error structures of both data sets were multiplicative and lognormal and thus complied with a linear model, which obtained log-log linear lines with slopes less than 1; i.e., the data were consistent with the hypothesis of diminishing returns. Conclusions. History shows that log transformation has always been a controversial practice. However, the extent to which linear or nonlinear models comply with a particular data set is generally transparent using standard statistical protocols (e.g., analysis of residuals). Previous scaling analyses using log-transformed data therefore are likely generally valid. Nevertheless, the error structure in every data set should be assessed to determine whether linear or nonlinear regression models are appropriate. Reliable algorithms are available for this purpose. Keywords: allometry, diminishing returns, linear and nonlinear models, scaling, WBE theory.

might be expected, the rapid growth in this field of inquiry has not been easy or without debate concerning allometric theory and practice. The debate over theoretical underpinnings has focused largely on the theory proposed by Geoffrey West, James Brown, and Brian Enquist and its subsequent reincarnations (see Price et al. 2012), which attempt to explain why so many scaling relationships are governed by quarter-power rules or multiples thereof (West et al. 1997, 1999, 2000). This mechanistic explanation offers an overarching perspective on biological scaling phenomena ranging from the relationship between metabolic rates and body size to standing biomass at the ecosystem level. A parallel debate continues to revolve around statistical inference and which among alternative regression models should be used to determine trends in data sets or testing theoretical predictions. This concern should not come as a surprise. History reveals that the selection of how to analyze biological data sets was hotly debated by none other than Julian S. Huxley, who is generally (but perhaps incorrectly) judged to be the father of allometric analysis, and D’Arcy Thompson, who was arguably the best biometrician of his day. This debate continues today and for very legitimate reasons (see, e.g., Packard et al. 2011; Packard 2013, 2014), and it is not likely to cease in the foreseeable future, although much hinges on a consensus, so much so that the editors of this journal took the somewhat

I hope that you will learn not merely results, or formulae applicable to cases that may possibly occur ..., but the principles on which these formulae depend, and without which the formulae are mere mental rubbish. (attributed to James Clerk Maxwell 1831–1879)

Introduction Biologists have long sought methods to predict how physiological, morphological, ecological, and evolutionary processes and patterns scale with respect to organismic size. They have also searched for a theoretical (mechanistic) explanation for empirically observed scaling relationships. These interests have experienced a renaissance by virtue of scaling analyses, which trace their early history to eugenics and the anthropological quest to quantify intelligence. Indeed, the past few years have been particularly robust in the number of publications quantifying (and attempting to explain) sizedependent phenomena ranging from the scaling of respiration and metabolic rates to the scaling of entire ecosystems. As 1

Author for correspondence; e-mail: [email protected].

Manuscript received May 2014; revised manuscript received May 2014; electronically published August 1, 2014.

754

NIKLAS & HAMMOND—ASSESSING SCALING RELATIONSHIPS unusual step of inviting this contribution as a companion piece to the article published by Gary C. Packard, entitled “Assessing Allometric Growth by Leaves and the Hypothesis of Diminishing Returns” (Packard 2014). In light of this invitation, the goals of this article are twofold: (1) to provide a historical perspective on how allometric relationships have been (and continue to be) assessed using linear models and (2) to illustrate quantitatively the use (and abuse) of alternative regression protocols. For this purpose, we use a large (n p 3874) interspecific data set and a smaller (n p 164) intraspecific data set for leaf surface area and lamina dry mass to examine a hypothesis known as diminishing returns, which proposes that leaf surface area often fails to increase one-to-one with increasing lamina dry mass. Much of what follows could be rendered in strictly mathematical terms, but this would not achieve our goals because it would obscure a key point: the selection of the mathematical and statistical tools used to assess scaling relationships often rests as much on the objectives of the researcher as on the structure of a data set. Some may find this assertion disagreeably unscientific. However, consider that there are at least two objectives for doing a regression analysis. One reason for using regression analyses is to try to predict the numerical value of a variable of interest based on the measurement of another variable of interest. This is routinely done in the experimental sciences where one variable of interest is under the control of the researcher and the other is the response variable, as, for example, the concentration of salt and its effect on cell turgor pressure. Under these circumstances, a researcher may evaluate any number of regression curves to see which, if any, provides the most robust predictive model. The curve that provides the best fit is the curve of choice. The second reason for using regression analysis is to see whether a data set conforms to a theoretical prediction. This is routinely done in scaling analysis, as, for example, determining whether growth rates increase as a quarter power (or multiple thereof) with respect to body mass. These contrasting modus operandi reveal that the objective of regression analysis can be either post hoc in the context of finding the best predictive regression curve or ad hoc in the context of trying to test the prediction of a specific theory. For this and other reasons, what follows will focus on what might rightly be called philosophical issues regarding irksome questions such as “What model best fits my data?” or “Do my data fit a theoretical prediction?”

A Little History Before examining the raison d’eˆtre for regression analyses, a brief excursion into history is required, since it shows that the debate over how to fit data is not new (and still far from resolved). In the latter part of the nineteenth century, a number of researchers concerned with quantifying human intelligence redirected their research from the study of facial characteristics (a practice very reminiscent of phrenology) to the study of cerebral biometry. By 1897, the Dutch researcher Eugene Dubois worked out a quantitative means to determine how evolved any given organism was by comparing the mass of its

755

brain (e for encephalon) to the mass of its body (s for soma) using a formula of his own invention, i.e., e p cs r,

(1)

where c is the coefficient of cephalization and r is the coefficient of relation (Dubois 1897, p. 368). A year later, the French researcher Lapicque applied Dubois’s formula to study the brains of various varieties of dogs and subsequently published a log-log graph wherein the data were fitted with a straight line (Lapicque 1907) such that the numerical values of c and r defined the y-intercept and the slope of the line, respectively. The Dubois-Lapicque power law equation would come to be known as the allometric equation largely due to the work of Julian S. Huxley in the first half of the twentieth century (Huxley 1924, 1927a, 1927b, 1932). Perhaps the most famous example of Huxley’s examples of heterogonic development (Pe´zard 1918) was the relationship observed between the mass of the large chela and the total body mass (minus the mass of the chela) of the male fiddler crab Uca pugnax. Huxley adopted the notation y p bx k,

(2a)

which when log transformed yields log y p log b ⫹ k log x,

(2b)

where y is the mass of a body part (in this example, the male crab’s chela) and x is total body mass. In the case of male fiddler crabs, Huxley (1932, fig. 3) found that b p 0.0073 and k p 1.62 for x ≤ 1.1 g, whereas b p 0.083 and k p 1.255 for x 1 1.1 g. Notice that x did not include the mass of the chela (i.e., x and y were not autocorrelated and y was not total body mass). Two additional points are important. First, Huxley used log-transformed data, and, second, he observed that the scaling relationship between y and x required two regression curves rather than one to adequately fit the entire data set. The fact that he used two regression curves in this example indicates that he had no theoretical expectation other than that equations (2) provided the best fit. Indeed, Huxley was an advocate of using different regression curves to fit data (e.g., Huxley 1921, 1924). Nevertheless, Huxley (1932) later conceived of equations (2) as a true biological law—one that provided an a priori theory for understanding the effects of size on morphology. Although subsequent workers have argued for a more pluralistic approach to the quantification of relative size—alternatives that do not rest exclusively on a power formula (e.g., Smith 1980, 2009; Harvey 1982; Chappell 1989; Packard 2014)—Huxley’s first detractor was the individual to whom he dedicated his 1932 book, D’Arcy Thompson, who wrote, the formula is mathematical rather than biological; there is a lack of either biological or physical significance in a growth-rate which happens to stand, during part of an animal’s life, at 62 per cent. compound interest. Julian Huxley holds, and many hold with him, that the exponential or logarithmic formula, or compound-interest law, is of gen-

756

INTERNATIONAL JOURNAL OF PLANT SCIENCES

eral application to cases of differential growth-rates. I do not find it to be so: any more than we have found organ, organism or population to increase by compound interest or geometrical progression, save under exceptional circumstances and in transient phase. Undoubtedly many of Huxley’s instances shew increase by compound interest, during a phase of rapid and unstinted growth; but I find many others following a simple-interest rather than compound-interest law. (Thompson 1942)

Thompson reevaluated the same data sets originally used by Huxley and showed that simple linear equations often fit some of the untransformed data as well as or better than equations (2) and that linearly correlated pairwise data remain linear when log transformed. Indeed, many early workers failed to provide a sound theoretical basis for accepting a power function relationship, whereas reevaluations of their data indicate that statistical analysis of untransformed data can give results equally as good as those of the transformed data (e.g., Smith 1980). Another obvious difficulty facing early researchers was that the power functions emerging from their studies were strictly empirical. There was no underlying theory as to why or how such relationships come into being. As early as 1932, an otherwise positive reviewer of Huxley’s now famous book pointed out that allometric formulas taking the form of equations (2) are necessarily empirical. Of the causes of differential growth we have little knowledge; their investigation is the problem at issue. A variety of possible relations, in fact, reduce approximately to this formula. But it is not the object of the formula to establish the correctness of a particular hypothesis as the cause of differential growth; it merely expresses the observed facts with considerable accuracy in a simple way, so that many very significant features emerge which would not otherwise do so. (Pantin 1932)

This evaluation was fair and prescient in the sense that it draws attention to the two objectives for employing regression analyses: whether the objective is to empirically assess and describe trends in a data set or to evaluate whether a theory predicts trends observed in a data set. There have been numerous attempts to devise a mechanistic explanation for the persistent observation that many scaling relationships conform to equations (2) and that many of these relationships appear to be governed by exponents taking on the numerical value of 1/4 or multiples thereof. One intriguing theory is that organisms are geometrically four-dimensional rather than three-dimensional objects such that surface area scales as the 3/4 power of volume rather than as the 2/3 power of volume (Blum 1977). The expectation that organisms are non-Euclidean emerges from the fact that every cell has an internal surface (in the form of convoluted membranes) and that the surface area of an n-dimensional object scales as n ⫺ 1, whereas volume scales as n. Accordingly, if internal surfaces comprise a fourth dimension within cells, tissues, and so on, surface area will be proportional to volume raised to the 3/4 power, which is more or less consistent with the available data. However, this feature does not explain why so many other biological phenomena appear to obey quarter-power laws. In contrast, West et al. (1997, 1999, 2000) have proposed a comprehensive and far-reaching theory to explain the quar-

ter-scaling relationships observed across diverse organisms (also see Price et al. 2012). Although criticized on empirical as well as theoretical grounds and challenged by alternative conceptual approaches (see Banavar et al. 1999; Dodds et al. 2001; Darveau et al. 2002; Weibel 2002), the West, Brown, Enquist (WBE) theory currently remains the most successful, so much so that it has been called the allometric “theory of everything.” However, the issue here is not whether the WBE theory (or any of its incarnations and modifications) is correct. The issue is that this theory provides an array of predictions that have encouraged numerous researchers to gather huge data sets to test them. This flurry of research brings us back to the two aforementioned questions: “What equation best fits my data?” and “Do my data fit a theoretical (WBE) prediction?”

Data Transformation, Regression Protocols, and Models These questions are surprisingly difficult to answer, less so for technological reasons and more so for philosophical ones. For example, it is easy and common to log transform data for allometric analyses. However, it is not always obvious whether this is a good thing to do. As noted, D’Arcy Thompson’s statistical analysis of Huxley’s data gave equally good fits using untransformed data. Log transformation can also give one false impressions. For example, log-log regression curves, especially in the case of broad interspecific comparisons, can obtain high coefficients of correlation that are often interpreted to indicate that y values can be accurately predicted based on their corresponding x values. This problem arises when the variation in y is large, as is often the case in biology. Under these circumstances the predictive capacity of log-log regression curves can be remarkably low. It is therefore always a good idea to study regression residuals and percent prediction errors. Yet another problem with using log-transformed data is that standard regression techniques fit a line to the mean value of the y variable, but the mean of log-transformed variables is the median of the lognormal distribution (Gould 1966; Sokal and Rohlf 1981). Therefore, without correction, values reported for the antilog of the y-intercept of the regression line are consistently biased (Prothero 1986; Niklas 1994). The correction factor, however, is simple; it can be estimated from the standard error of log b. Unfortunately, many workers to fail to report this important statistical parameter. Nevertheless, workers continue to use transformed data generally for three reasons. First, many forms of transformed data reduce the problem of working with outliers. Second, logtransformed data typically comply with the statistical assumptions of normality and homoscedacity (Kermack and Haldane 1950; Sokal and Rohlf 1981). Third, it provides a convenient means of examining proportionality that is unaffected by the unit of measurement, since the slope of the loglog regression line (log y2 ⫺ log y1)/(log x 2 ⫺ log x1) becomes (y2/ y1)/(x2/x1) when converted to the arithmetic scale. For these reasons, log transformation has been advocated (Mascaro et al. 2014) when dealing with biological data and has become a standard in most scaling analyses (e.g., Gould 1966; Peters 1983; Calder 1984; Schmidt-Nielson 1984; see, however, Smith 2009).

NIKLAS & HAMMOND—ASSESSING SCALING RELATIONSHIPS

757

Fig. 1 a, Bivariate plot of 3874 pairwise measurements of laminae surface area and laminae dry mass (original units, cm2 and g) fitted with a linear and nonlinear model. b, Bivariate plot of nonlinear model residuals versus dry mass. The frequency distribution of the nonlinear model residuals (see inset) indicates that the error structure in the data set does not conform to the assumptions of the nonlinear model.

Another controversial issue is the type of regression analysis used in scaling analyses, because the type of analysis can profoundly influence the numerical values of scaling exponents and thus the extent to which observed exponents accord statistically with those predicted by a particular theory when the data are not exceptionally well correlated (Sokal and Rohlf 1981; Seim 1983; Niklas 1994). Based on standard statistical inference, ordinary least squares regression analysis (denoted here as OLS) can be used provided that four conditions are met: (1) the error term ␧ is normally distributed with a mean of 0 and constant variance, (2) the distribution of log y is normal at each value of log x, (3) the variance of log y is constant across the range of log x, and (4) log x is an independent variable whose values are known without error (Sokal and Rohlf 1981); i.e., OLS is asym-

metric such that the slope and resulting interpretation of the data are changed when the variables assigned to y and x are reversed. Unfortunately, these four assumptions are rarely if ever true for many biological data sets. Certainly, there is no independent variable when we regress the mass of a body part against the mass of the remaining organism (as in the case of the chela of a fiddler crab). For these reasons, many workers have turned to model Type II regression analyses, e.g., major axis (MA) or reduced major axis (RMA, also known as standard major axis regression analyses; see Niklas 1994; Warton et al. 2006). Model Type II analyses typically identify the variables of interest as y2 and y1 rather than y and x to indicate that both variables of interest are interdependent; i.e., model Type II regression protocols are symmetric such that a single line defines the bivariate rela-

758


Fig. 2 a, Bivariate plot of log10-transformed 3874 pairwise measurements of laminae surface area and laminae dry mass fitted against a linear model. b, Bivariate plot of linear model residuals versus dry mass. The frequency distribution of the linear model residuals (see inset) indicates that the error structure in the data set conforms to the assumptions of the linear model.

tionship between the two variables of interest, regardless of which variable is used as y2 and y1. The choice of which kind of model Type II regression analysis to use still remains a matter of debate, even among statisticians. Each model Type II regression protocol involves different assumptions about the error structure and the variance relations between y2 and y1 to estimate the slope and y-intercept of regression curves (Niklas 1994). In this regard, it is useful to note that MA is sensitive to the absolute measurement scales used to quantify y2 and y1 and it is not especially robust when switching the coordinate axes, whereas

RMA is insensitive to both of these concerns. RMA is also less sensitive to assumptions about the error structure in a data set. For these and other reasons, RMA has become the standard regression protocol in allometric analyses (e.g., Deng et al. 2012; Price et al. 2012; Dillon and Frazier 2013; DongMei et al. 2013). Nevertheless, as recently pointed out by Xiao et al. (2011), the manner in which the data are modeled and expressed affects the error term in the analysis. Specifically, using Huxley’s notation (see eqq. [2]), in the case of a linear regression on log10-transformed data, the error ␧ is assumed

NIKLAS & HAMMOND—ASSESSING SCALING RELATIONSHIPS

759

Fig. 3 a, Bivariate plot of 164 pairwise measurements taken from a single Sassafras albidum tree of laminae surface area and laminae dry mass (original units, cm2 and g) fitted with a linear and nonlinear model. b, Bivariate plot of nonlinear model residuals versus dry mass. The frequency distribution of the nonlinear model residuals (see inset) indicates that the error structure in the data set does conform to the assumptions of the nonlinear model (see also fig. 4).

to be normally distributed and additive on the logarithmic scale log 10 y p log 10 b ⫹ k log 10 x ⫹ e,

(3a)

which corresponds to a lognormally distributed multiplicative error on the arithmetic scale y p bx ke.

In the case of nonlinear regression on untransformed data, the error term is normally distributed and additive on the arithmetic scale,

(3b)

y p bx k ⫹ e.

(4)

Thus, linear regression is required when the error is multiplicative and lognormal, whereas nonlinear regression is required

760


Fig. 4 a, Bivariate plot of log10-transformed 164 pairwise measurements taken from a single Sassafras albidum tree of laminae surface area and laminae dry mass fitted against a linear model. b, Bivariate plot of linear model residuals versus dry mass. The frequency distribution of the linear model residuals (see inset) indicates that the error structure in the data set conforms to the assumptions of the linear model (see also fig. 3).

when the error is additive and normal. Since both multiplicative and additive errors occur in nature, the error structure in a particular data set dictates whether linear or nonlinear regression should be used, since no single data set can manifest both forms of error simultaneously.

A Test Case: Diminishing Returns To illustrate the use and abuse of equations (3) and (4), we turn to a data set and test the hypothesis called diminishing returns, which emerges from the observation that the surface

area (S) of the laminae of mature leaves often fails to increase one-to-one (isometrically) with increasing lamina dry mass (M; see Niinemets et al. 2006, 2007; Milla and Reich 2007; Niklas et al. 2007, 2009; Price and Enquist 2007; Li et al. 2008; Niklas and Cobb 2008, 2010); i.e., the slopes of the linear regression curves that fit bivariate plots of log S versus log M are often less than unity (S ∝ Mk!1.0). This phenomenology, which has been reported for leaves collected from the same plant or species, or across many different species, is not universal since some comparisons obtain log-log linear relationships with slopes that are statistically indistinguishable from

NIKLAS & HAMMOND—ASSESSING SCALING RELATIONSHIPS Table 1 Numerical Values for the y-Intercept (b) and Slope (k), 95% Confidence Intervals (CIs), and Akaike Information Criterion (AIC) Values for Linear and Nonlinear Regression Models Applied to 3874 Pairwise Data for Laminae Surface Area (S; cm2) and Laminae Dry Mass (M; g)

b (95% CI) k (95% CI) AIC Note.

Linear model (log10S p log10b ⫹ klog10M ⫹ ␧)

Nonlinear model (S p bMk ⫹ ␧)

82.40 (78.75–86.20) .932 (.921–.943) 2188.8

95.20 (91.06–99.35) .975 (.957–.992) 40,895

Analyses are based on the protocols of Xiao et al. (2011).

1 (e.g., Niklas et al. 2007), nor does this imply an upper theoretical limit to leaf size as is sometimes claimed. We selected the hypothesis to illustrate the use of linear and nonlinear models for two reasons. First, it has been subjected to recent inquiry using data provided by Price and Enquist (2007) and challenged on the basis that the “allometric equations that purportedly support the hypothesis of diminishing returns may be inaccurate and misleading” (Packard 2014, p. 742), and, second, because we have access to a recently updated data set consisting of 3874 measurements of S and M spanning more than 2000 species representative of six species groupings (i.e., ferns, graminoids, forbs, shrubs, trees, and vines; for a previous version, see Niklas et al. 2007). We approach the untransformed data first and apply linear and nonlinear regression curves to determine which provides the best fit. In both cases, we evaluate the error distribution graphically as a frequency distribution of regression residuals and by using the Akaike Information Criterion (AIC; Akaike 1974; Bozdogan 1987), using the software provided by Xiao et al. (2011), which is exemplary of model Type I regression protocols. Analysis reveals that the linear regression model for log10transformed S and M data fits the data better than does the nonlinear regression model; i.e., the error structure is multiplicative, heteroscedactic, and lognormal. This is graphically apparent when the frequency distributions of the model residuals are inspected (figs. 1, 2). Analysis of the data also reveals that the slope of the linear regression model is statistically significantly less than 1 (i.e., k ! 1.0); i.e., the regression model supports the hypothesis of diminishing returns (table 1). Comparable analyses using loge-transformed data reveal no meaningful differences in the parameters reported in table 1. Finally, a linear regression model was found to provide the best fit to each of the six species groupings represented in the data. These models identified slopes within a numerical range of 0.904 ≤ k ≤ 0.989. With the exception of two species groupings (forb and tree species), the upper 95% confidence interval of each slope was less than 1; i.e., the hypothesis of diminishing returns was supported by the analyses of four out of the six data subsets (data not shown but consistent with those reported by Niklas et al. 2007; table 1).

A Little Philosophy The forgoing comparison of contending models can be criticized because it does not address the problem of heterosce-

761

dasticity directly. That is, our comparison of linear and nonlinear models did not resolve the problem that emerges when subpopulations of randomly collected variables evince different variances. Clearly, the data from each species in our interspecific data set represent a subpopulation, and there are very good reasons to suppose that they may not have the same variance. This is a concern in interspecific analyses because heteroscedasticity can invalidate critical assumptions about modeling errors. We cannot canonically resolve this concern other than to say that (1) scaling analyses seek to quantify interspecific size-dependent trends that typically employ heteroscedastic data, (2) there are excellent protocols to evaluate the problem of heteroscedasticity (Xiao et al. 2011), and (3) these protocols were used in the previous analyses. At this juncture, we return to a point made earlier, i.e., the choice of which model to use is often a matter of a researcher’s philosophy about modeling. To illustrate this point, we turn to an unpublished data set consisting of 164 leaves collected from a single tree of Sassafras albidum and perform the same analyses as those used previously to examine the efficacy of linear and nonlinear models in the context of interspecific data. In this intraspecific comparison, both models give statistically comparable results as gauged by the frequency distributions of regression residuals (figs. 3, 4), the 95% confidence intervals of b and k, and the AIC values, although the latter indicate that the linear model is superior (table 2). Likewise, both models support the hypothesis of diminishing returns, since the slope obtained by each model is numerically less than 1. However, the validation of this hypothesis is not the point. The real issue is how each model might be used given that the two models yield very similar results. For example, the linear model provides a direct test of the hypothesis of diminishing returns even if it is not the most statistically robust of the two models. Likewise, the nonlinear model could be used to predict S based on values of M even if it was not the most statistically robust of the two. Indeed, the ways in which the two models could be used are not mutually exclusive and thus are a matter of choice. Finally, since alternative mechanistic explanations for diminishing returns predict alternative S versus M scaling exponents (e.g., Niklas et al. 2009), no claim can be made regarding a general regression model or the relative frequency of the occurrence of diminishing returns. Many more studies are required to evaluate whether it occurs in a few, many, or

Table 2 Numerical Values for the y-Intercept (b) and Slope (k), 95% Confidence Intervals (CIs), and Akaike Information Criterion (AIC) Values for Linear and Nonlinear Regression Models Applied to 164 Pairwise Data for Laminae Surface Area (S; cm2) and Laminae Dry Mass (M; g)

b (95% CI) k (95% CI) AIC Note.

Linear model (log10 S p log10 b ⫹ klog10 M ⫹ ␧)

Nonlinear model (S p bMk ⫹ ␧)

315.92 (302.08–330.39) .747 (.704–.790) ⫺475.1

307.59 (295.26–320.253) .704 (.653–.756) 1484.7

Analyses are based the protocols of Xiao et al. (2011).

762


the majority of species. Indeed, as of now, both of the models in table 2 can be said to work only for this one tree.

Concluding Remarks It is clear from history as well as contemporary studies that workers continue to debate the types of statistical models and the types of regression protocols that should be applied when studying scaling relationships. However, there is evidence that a consensus is being achieved among workers interested in scaling analyses, one that favors the use of linear regression when the error structure of a data set is multiplicative, heteroscedactic, and lognormal and the use of nonlinear models when the error is additive, homoscedastic, and normal. The choice of which model to use therefore is not arbitrary because (1) the error structure in a particular data set dictates whether a linear or nonlinear model is required and because (2) no data set can manifest both error structures simultaneously. Nevertheless, there are two different philosophical approaches to inspecting data statistically: one that seeks simply to find the best fit to the data and another that seeks to test a theoretical prediction. These approaches are not mutually exclusive as is illustrated by our reexamination of the hypothesis of diminishing returns. However, fitting data with regression models that have no apparent biological meaning is an unprofitable endeavor. Given sufficient effort, a mathematical model can be found to fit any nonrandom data set. Yet, to be useful, a model must provide some mechanistic insight. In this context, power functions have been shown to illuminate scaling relationships mechanistically, albeit not to the exclusion of other models. Finally, although the numerical values of the scaling exponents governing these power functions have been emphasized, their

corresponding normalization constants (i.e., the y-intercepts of regression curves on log transformations) have not. While the scaling exponents show proportional relationships on the log scale, the normalization constants dictate absolute size. Though vastly different species exhibit similar exponents for any given relationship, the normalization values can differ a great deal and may reflect quantifiable differences in evolutionary and individual life histories. Perhaps unsurprisingly, the disagreement about the importance of the normalization constant dates back to the seminal publications of Huxley and Teissier (1936). The two differed in opinion enough that their simultaneously published articles—in English and French, respectively—differ by only a single sentence, with Teissier endorsing the biological significance of normalization constants. The absence of a wider interest in the importance of normalization constants and the lack of an underlying theory explaining their significance are strikingly similar to scaling theory before the emergence of the WBE theory. The lack of attention can be seen as both a detriment to scaling theory and an immense opportunity for future research.

Acknowledgments We thank Drs. Patrick Herendeen (Chicago Botanic Garden) and William E. Friedman (Department of Organismic and Evolutionary Biology and the Arnold Arboretum, Harvard University), who invited us to provide an opportunity to stimulate a discussion around modeling biological scaling relationships. We also thank Dr. Gary C. Packard (Colorado State University) for comments and suggestions during the review process.

Literature Cited Akaike H 1974 A new look at the statistical model identification. IEEE Trans Autom Control 19:716–723. Banavar JR, A Maritan, A Rinaldo 1999 Size and form in efficient transportation networks. Nature 399:130–132. Blum JJ 1977 On the geometry of four dimensions and the relationship between metabolism and body mass. J Theor Biol 64:599–601. Bozdogan H 1987 Model selection and Akaike’s information criterion (AIC): the general theory and its analytical extensions. Psychometrika 52:345–370. Calder WA III 1984 Size, unction, and life history. Harvard University Press, Cambridge, MA. Chappell R 1989 Fitting bent lines to data, with applications to allometry. J Theor Biol 138:235–256. Darveau CA, RK Suarez, RD Andrews, PW Hochachka 2002 Allometric cascade as a unifying principle of body mass effects on metabolism. Nature 417:166–170. Deng J, W Zuo, Z Wang, Z Fan, M Ji, J Ran, G Wang, et al 2012 Insights into plant size-density relationships from models and agricultural crops. Proc Natl Acad Sci USA 109:8600–8605. Dillon ME, MR Frazier 2013 Thermodynamics constraints allometric scaling of optimal developmental time in insects. PLoS ONE 8: e84308. doi:10.1371/journal.pone.0084308. Dodds PS, DH Rothman, JS Weitz 2001 Re-examination of the “3/ 4-law” of metabolism. J Theor Biol 209:9–27. DongMei J, C XueCui, M KePing 2013 Leaf functional traits vary with the adult height of plant species in forest communities. J Plant Ecol 7:68–76.

Dubois E 1897 Sur le rapport de l’ence´phale avec la grandeur du corps chez les Mammife`res. Bull Soc Anthropol Paris 5:337–374. Gould SJ 1966 Allometry and size in ontogeny and phylogeny. Biol Rev 41:587–640. Harvey PH 1982 On rethinking allometry. J Theor Biol 95:37–41. Huxley JS 1921 Studies in dedifferentiation. II. Dedifferentiation and resorption in Pterophora. Q J Microsc Sci 65:643–692. ——— 1924 Constant differential growth ratios and their significance. Nature 114:895–896. ——— 1927a Further work on heterogonic growth. Biol Zent 47: 151–163. ——— 1927b On the relation between egg-weight and body-weight in birds. J Linn Soc Zool 36:457–466. ——— 1932 Problems of relative growth. Methuen, London. Huxley JS, G Teissier 1936 Terminology of relative growth. Nature 137:780–781. Kermack KA, JBS Haldane 1950 Organic correlation and allometry. Biometrika 37:30–41. Lapicque L 1907 Tableau general des poids somatiques et ence´phaliques dans les espe`ces animales. Bull Soc Anthropol Paris 9:248– 269. Li G, D Yang, S Sun 2008 Allometric relationships between lamina area, lamina mass and petiole mass of 93 temperate woody species vary with leaf habit, leaf form and altitude. Funct Ecol 22:557–564. Mascaro J, CM Litton, RF Hughes, A Uowolo, SA Schnitzer 2014 Is logarithmic transformation necessary in allometry? ten, one-hundred, one-thousand-times yes. Biol J Linn Soc 111:230–233.

NIKLAS & HAMMOND—ASSESSING SCALING RELATIONSHIPS Milla R, PB Reich 2007 The scaling of leaf area and mass: the cost of light interception increases with leaf size. Proc R Soc B 274:2109– 2114. Niinemets U¨, A Portsmuth, D Tena, M Tobias, S Matesanz, F Valladares 2007 Do we underestimate the importance of leaf size in plant economics? disproportional scaling of support costs within the spectrum of leaf physiognomy. Ann Bot 100:283–303. Niinemets U¨, A Portsmuth, M Tobias 2006 Leaf size modifies support biomass distribution among stems, petioles and mid-ribs in temperate plants. New Phytol 171:91–104. Niklas KJ 1994 Plant allometry. University of Chicago Press, Chicago. Niklas KJ, ED Cobb 2008 Evidence for “diminishing returns” from the scaling of stem diameter and specific leaf area. Am J Bot 95: 549–557. ——— 2010 Ontogenetic changes in the numbers of short vs. long shoots account for decreasing specific leaf area in Acer rubrum L. as trees increase in size. Am J Bot 97:27–37. Niklas KJ, ED Cobb, U¨ Niinemets, PB Reich, A Sellin, B Shipley, IJ Wright 2007 “Diminishing returns” in the scaling of functional leaf traits across and within species groups. Proc Natl Acad Sci USA 104:8891–8896. Niklas KJ, ED Cobb, H-C Spatz 2009 Predicting the allometry of leaf surface area and dry mass. Am J Bot 96:531–536. Packard GC 2013 Is logarithmic transformation necessary in allometry? Biol J Linn Soc 109:476–486. ——— 2014 Assessing allometric growth by leaves and the hypothesis of diminishing returns. Int J Plant Sci 175:742–753. Packard GC, GF Birchard, TJ Boardman 2011 Fitting statistical models in bivariate allometry. Biol Rev 86:549–563. Pantin CFA 1932 Form and size. Nature 129:775–777. Peters RH 1983 The ecological implications of body size. Cambridge University Press, Cambridge.

763

Pe´zard A 1918 Le conditionnement physiologique des caracte`res sexuels secondaires chez les oiseaux. Bull Biol Fr Belg 52:1–176. Price CA, BJ Enquist 2007 Scaling mass and morphology in leaves: an extension of the WBE model. Ecology 88:1132–1141. Price CA, JF Gilooly, AP Allen, JS Weitz, KJ Niklas 2012 Testing the metabolic theory of ecology. Ecol Let 15:1465–1474. Prothero J 1986 Methodological aspects of scaling in biology. J Theor Biol 118:259–286. Schmidt-Nielson K 1984 Scaling: why is animal size so important? Cambridge University Press, Cambridge. Seim E 1983 On rethinking allometry: which regression model to use? J Theor Biol 104:161–168. Smith RJ 1980 Rethinking allometry. J Theor Biol 87:97–111. ——— 2009 Use and misuse of the reduced major axis for line-fitting. Am J Phys Anthropol 140:476–486. Sokal RR, FJ Rohlf 1981 Biometry. 2nd ed. WH Freedman, New York. Thompson DAW 1942 On growth and form. 2nd ed. Cambridge University Press, Cambridge. Warton DI, IJ Wright, DS Falster, M Westoby 2006 Bivariate linefitting methods for allometry. Biol Rev Camb Philos Soc 81:259– 291. Weibel ER 2002 Physiology: the pitfalls of power laws. Nature 417: 131–132. West GB, JH Brown, JB Enquist 1997 A general model for the origin of allometry scaling laws in biology. Science 276:122–126. ——— 1999 The fourth dimension of life: fractal geometry and allometric scaling of organisms. Science 284:167–169. ——— 2000 The origin of universal scaling laws in biology. Pages 87–112 in JH Brown, GW West, eds. Scaling in biology. Oxford University Press, New York. Xiao X, EP White, MB Hooten, S Durham 2011 On the use of logtransformation vs. nonlinear regression for analyzing biological power laws. Ecology 92:1887–1894.