ARTICLE IN PRESS
Journal of Theoretical Biology 228 (2004) 149–153
Testing the allometric scaling laws
Pekka Kaitaniemi*
Hyytial
124, FIN-35500 Korkeakoski, Finland
. a. Forestry Field Station, University of Helsinki, Hyytial
. antie
.
Received 2 October 2003; received in revised form 8 December 2003; accepted 12 December 2003
Abstract
Allometric scaling laws have received increasing attention due to the recent theoretical advancements. However, existing evidence
suggests that the scaling relationships may vary a lot without much consistency, which poses a challenge to the applicability of
general theories. In this report, I demonstrate that much of the discrepancy may be an artefact caused by the limited use of methods
for estimating the parameters in the allometric scaling equations. I suggest alternative procedures that can be utilized to avoid biased
interpretations. The comments are largely applicable to any research that involves parameterization of equations.
r 2003 Elsevier Ltd. All rights reserved.
Keywords: Allometry; Model; Parameter estimation; Regression
1. Introduction
Various size measures of organisms have been long
known to exhibit specific allometric scaling relationships. Recently, the theoretical studies by West et al.
(1999b) suggested a new mechanistic explanation to
these scaling laws, often empirically detected in organisms. The theory predicts numerous scaling relationships
at multiple levels of biological organization, and the
authors claim it can provide a general framework for
linking the structure and function of organisms from
cells to ecosystems (West et al., 1999b; Enquist, 2002).
Therefore, it is likely that there will be an increasing
number of studies designed to test the validity of the
theoretical predictions for different organisms and traits.
However, previous evidence suggests that the scaling
relationships may, in fact, vary a lot without much
consistency. For plants, Niklas (1994, p. 314) states that
‘each scaling relation is functional only within a narrow
domain of size and form’, and for animals, Dodds et al.
(2001) claim ‘after a systematic review of the available
empirical data and theoretical arguments, we find
evidence that there may not be a simple scaling law
for metabolic rate’. It thus appears that the applicability
of a single general theory might become easily challenged. However, as I will demonstrate in this paper,
*Tel.: +358-3-3355217; fax: +358-3-3355555.
E-mail address: pekka.j.kaitaniemi@helsinki.fi (P. Kaitaniemi).
0022-5193/$ - see front matter r 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jtbi.2003.12.007
much of the discrepancy in the previous results may be
an artefact caused by the methods used to determine the
parameters in the allometric scaling equations.
2. Allometric scaling laws
Allometric scaling laws have the form Y ¼ bX a ; and
they represent the dependence of the variable Y on the
variable X as a function that involves the normalization
constant b and the scaling exponent a: The constant b
may be specific for a particular system whereas the
scaling exponent a is expected to follow the theoretical
predictions (West et al., 1999b).
A ‘traditional’ method for determining the scaling
exponent is log-transforming the original variables and
then fitting a linear reduced major axis (RMA)
regression to the data. Although better alternatives exist
(Isobe et al., 1990), this method is acceptable, if it is not
clear which variable should be treated as the independent and which as the dependent, and if both variables
are subject to measurement error. An ordinary leastsquares (OLS) regression, in turn, is recommended for
predictive purposes where the aim is to illustrate the
relationship (Niklas, 1994, p. 333). If a model predicts
that one variable clearly depends on the other (e.g. West
et al., 1999a), then OLS regression should be preferred
even if there exists measurement error in the values of
the variables (Isobe et al., 1990).
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P. Kaitaniemi / Journal of Theoretical Biology 228 (2004) 149–153
However, besides linear regression, other methods for
parameter estimation do also exist, and it is not obvious
which of them produces the best fit (Haefner, 1996). It is
also a known fact that the number of possible solutions
increases as the number of parameters in an equation
increases. As I will show, even the two parameters in the
allometric equation leave scope for alternative values
that can all be accepted using standard selection criteria.
3. Methods
As an example, I use the analysis of data described in
Kaitaniemi and Ruohom.aki (2003). It consists of
allometric scaling relationships between branch and
stem diameter, shoot number (comparable to leaf
number), tree height, and branch and tree biomass in
the mountain birch, Betula pubescens ssp. czerepanovii
(Orlova) H.amet-Ahti. Figs. 9 and 10 in Kaitaniemi and
Ruohom.aki (2003) show the data and the allometric
relationships I investigated.
In the present paper, I used four different methods for
determining the values of a and b: First, I calculated
both the OLS regression and the RMA regression for
data that were first log-transformed. These calculations
were made with the procedures REG and CORR in the
SAS statistical analysis software (SAS Institute Inc.,
Cary, NC, USA). Second, I determined both parameters
with untransformed data using the SAS procedure
NLIN and selecting the Marquardt method for iteration. Finally, I set the value of the scaling exponent
according to the theoretical value given by West et al.
(1999a), and then used the procedure NLIN for
determining the value of the coefficient b alone.
To compare the resulting four parameterizations, I
used the multiple working hypotheses approach (e.g.
Haefner, 1996, pp. 22–24). In the present study, it meant
that alternative values for model parameters were
compared in parallel with each other to find the values
that statistically best fit the data at hand. The conceptual
basis of the model itself remained the same. However,
the same approach would have been appropriate, if I
had compared alternative structural formulations of a
model instead of different parameter values. In that
case, I would have judged the fit of data with different
mechanistic models that are based on different hypotheses about the causal relationships, and that way predict
different parameter values for the allometric scaling
equations, or perhaps predict completely different
equations.
Two different indices of model fit were used in the
comparisons, the coefficient of determination r2 ; and the
Akaike’s Information Criterion (AIC, Akaike, 1973).
The former simply states the proportion of variation in
the response variable that can be explained by the
regression equation. The latter is a more abstract
measure which describes the likelihood that the data
fits a specific model. It is defined as AICi ¼ 2 log Li þ
2Vi ; where Li is the maximum likelihood for the
candidate model i; and Vi is the number of parameters
estimated from the data for the model i:
AIC belongs to a larger group of information criteria
that can all be used as criteria for selecting the best from
among the set of alternative models (Zucchini, 2000).
The model that exhibits the lowest absolute value of
AIC is judged the best, but if the difference in the
AIC values is low for the alternative models then the
alternatives also deserve serious consideration. A rough
rule of thumb is that models with a difference less than
two have substantial support, and models with a
difference larger than ten might be omitted from further
consideration (Burnham and Anderson, 2002, p. 446).
The requirement is that the alternative models are
compared with an identical set of data. The number of
parameters in the different models may vary, but
increasing their number adds a penalty to the value of
AIC. In the present study, the number of parameters
was constant for all models.
To obtain r2 ; the values predicted by different
parameterizations were correlated with the observed
values. To obtain AIC for the fit, the predicted values
were regressed against the observed values with no
intercept term in the procedure REG. This procedure
checked the deviation of the match between the
variables from the perfect 1:1 relationship.
In addition to the above criteria, I calculated the 95%
confidence limits for the scaling exponent a to get an
indication on what the conclusions would have been, if
a statistical criterion had been used to compare the
exponents in different models.
4. Results and discussion
The four parameterization methods produced quite
variable parameter values (Table 1). When the results of
the purely empirical methods were compared with each
other, the values of the scaling exponent, which are of
interest here, showed up to 1.5-fold differences. There
were also noticeable differences in the values of r2 ;
although they generally indicated a reasonably good fit
in all cases. What is most important, is that none of the
methods was able to consistently produce the best fit.
The largest values of r2 and the smallest values of AIC
were not associated with each other, or with a particular
parameterization method. This shows that the final
conclusions may depend on the choice of method, if the
researcher decides to use just one of the alternatives.
The confidence limits of the scaling exponents, in
turn, indicated that in more than half of the cases the
theoretical value was not included within the limits, even
when it provided an acceptable fit. For example, as
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P. Kaitaniemi / Journal of Theoretical Biology 228 (2004) 149–153
151
Table 1
Parameters and fit statistics of allometric scaling equations fitted to data by Kaitaniemi and Ruohom.aki (2003; Figs. 9 and 10). Four different linear
or nonlinear regression methods were used to fit the parameters of equations for different branching orders. D=diameter, M=dry mass, N=shoot
number, H=tree height. OLS refers to ordinary least-squares regression and RMA to reduced major axis regression. Value of a is represented with
95% confidence limits. r2 is coefficient of determination and AIC is Akaike’s Information Criterion
a
95% CL for a
r2 a
3.13
2.89
2.74
2.45
0.38
0.39
0.39
0.41
—
0.34–0.44
0.36–0.42
0.38–0.45
0.87
0.87
0.87
0.87
235.0
233.1
225.8
229.7
Theoretical, nonlinear
Empirical, nonlinear
Empirical, linear, OLS
Empirical, linear, RMA
1.84
1.90
2.24
1.85
0.38
0.37
0.30
0.38
—
0.35–0.39
0.28–0.33
0.35–0.40
0.72
0.72
0.71
0.72
800.0
785.9
685.3
804.9
0
0
0
0
Theoretical, nonlinear
Empirical, nonlinear
Empirical, linear, OLS
Empirical, linear, RMA
0.33
4.73
1.47
0.99
2.00
1.31
1.61
1.73
—
1.05–1.57
1.45–1.77
1.56–1.90
0.67
0.73
0.71
0.70
721.6
701.9
703.0
716.4
(d) N ¼ bDa
>0
>0
>0
>0
Theoretical, nonlinear
Empirical, nonlinear
Empirical, linear, OLS
Empirical, linear, RMA
1.58
8.09
3.51
1.42
2.00
1.43
1.68
2.23
—
1.31–1.55
1.54–1.81
2.05–2.41
0.60
0.61
0.61
0.59
4128.0
4100.5
3986.0
4663.5
(e) N ¼ bM a
0
0
0
0
Theoretical, nonlinear
Empirical, nonlinear
Empirical, linear, OLS
Empirical, linear, RMA
3.79
2.54
4.96
4.66
0.75
0.81
0.70
0.72
—
0.75–0.86
0.67–0.74
0.68–0.75
0.95
0.96
0.95
0.95
584.2
583.2
582.3
588.7
(f) N ¼ bM a
>0
>0
>0
>0
Theoretical, nonlinear
Empirical, nonlinear
Empirical, linear, OLS
Empirical, linear, RMA
7.69
9.00
5.80
5.58
0.75
0.72
0.82
0.84
—
0.71–0.74
0.81–0.84
0.82–0.85
0.96
0.96
0.95
0.95
3227.9
3207.0
3429.8
3536.7
(g) H ¼ bM a
0
0
0
0
Theoretical, nonlinear
Empirical, nonlinear
Empirical, linear, OLS
Empirical, linear, RMA
69.00
40.58
33.27
29.15
0.25
0.22
0.25
0.27
—
0.19–0.25
0.22–0.28
0.24–0.30
0.81
0.81
0.84
0.80
522.2
418.5
421.5
427.2
(h) H ¼ bDa
0
0
0
0
Theoretical, nonlinear
Empirical, nonlinear
Empirical, linear, OLS
Empirical, linear, RMA
33.48
33.24
22.71
17.07
0.25
0.46
0.56
0.65
—
0.43–0.49
0.54–0.59
0.61–0.68
0.72
0.70
0.69
0.67
2632.4
3206.0
3228.6
3363.1
Equation
Branching order
Method
0
0
0
0
Theoretical, nonlinear
Empirical, nonlinear
Empirical, linear, OLS
Empirical, linear, RMA
(b) D ¼ bM a
>0
>0
>0
>0
(c) N ¼ bDa
(a) D ¼ bM
a
a
b
AIC
This table gives r2 based on correlation coefficient whereas Kaitaniemi and Ruohom.aki (2003) give the values of adjusted R2 :
regards the relationship between the main stem diameter
and tree height (Table 1(h)), the theoretical value 0.25
was outside the 95% confidence limits for all the
empirically fitted scaling exponents, yet it provided the
best fit on the basis of AIC and r2 : As regards the
relationship between branch diameter and dry mass for
higher order branches (Table 1(b)), the theoretical value
would have been accepted if r2 was the criterion, but
rejected if AIC was the criterion.
In addition to differences in parameter values, the
values predicted by the corresponding scaling equations
showed considerable deviation from each other depending on the value range of variables (Fig. 1). This
indicates the potential size of error as regards the
predictive use of equations.
On the basis of this exercise alone, it is not possible to
make any generalizations regarding the choice of best
parameterization method, or the best criteria for
evaluating models with alternative parameterizations.
In any case, since the scaling laws are essentially models
that produce predictions (West et al., 1999a), I would
prefer the use of AIC or other model validation
techniques for selecting the best parameter values
(Haefner, 1996; Zucchini, 2000), instead of making
simple statistical comparisons of a against the theoretical value. For the same reason, I would also reject the
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P. Kaitaniemi / Journal of Theoretical Biology 228 (2004) 149–153
152
(a)
100
100
10
10
1
1
0.1
0.1
1
10
100
1000
(b)
0.1
0.1
1000000
10000000
100000
1000000
10000
100000
1000
10000
1
10
100
1000
0.1
1
10
100
1000
0.1
1
10
100
1000
1000
100
100
10
10
1
1
0.1
0.1
0.01
0.01
0.001
0.1
(c)
0.001
1
10
100
1000
(d)
1000
10000
1000
100
100
10
10
1
(e)
1
0.1
0.1
0.1
1
10
100
1000
(f)
10000
1000
1000
100
100
10
10
1
1
(g)
0.1
1
10
100
1000
(h)
0.1
0.1
Theoretical, nonlinear
Empirical, nonlinear
Empirical, linear, OLS
Empirical, linear, RMA
1
Fig. 1. Allometric scaling equations of Table 1 plotted on a logarithmic scale.
10
100
1000
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P. Kaitaniemi / Journal of Theoretical Biology 228 (2004) 149–153
use of RMA regression, which is better suited to the
examination of the scaling exponent a alone (Sokal and
Rohlf, 1981; Niklas, 1994), and only when there is no
information about the predictive relationship between
the variables (Isobe et al., 1990).
To search the values of a and b that produce a model
with the best fit, a safe procedure is probably a thorough
experimentation with a sufficiently wide range of
parameter values during model fitting. The use of
efficient parameter estimation methods, such as evolutionary algorithms, would also be preferable. As long as
these are not readily available, I would recommend, as
a precaution, the simultaneous use of the linear and
nonlinear methods presented in this paper, and perhaps
using additional, biological criteria for selecting the
range of data where the quality of predictions is
particularly important. A technique called model averaging can also be applied to base the inferences on the
entire set of models instead of a single best model
(Burnham and Anderson, 2002). It was clear that, if we
stick without good justification to one biased estimate
only, it is likely that the conclusions will be misleading,
and do not help in developing the theories further.
Acknowledgements
The study was funded by the Academy of Finland
grants 44141 and 201997. I thank K.J. Niklas for helpful
comments on the previous version of the manuscript.
153
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