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First published online October 31, 2008
Journal of Experimental Biology 211, 3581-3587 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.023317
Traditional allometric analysis fails to provide a valid predictive model for mammalian metabolic rates
1 Department of Biology, Colorado State University, Fort Collins, CO 80523,
USA
2 Department of Environmental Science and Policy, George Mason University,
Fairfax, VA 22030, USA
* Author for correspondence (e-mail: packard{at}lamar.colostate.edu)
Accepted 23 September 2008
 |
Summary |
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 TOP Summary INTRODUCTIONMETHODS AND RESULTSDISCUSSIONReferences |
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Key words: allometry, scaling, logarithms, transformations, metabolic rates
|
INTRODUCTION |
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TOPSummaryINTRODUCTIONMETHODS AND RESULTSDISCUSSIONReferences |
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). The model generated a wave of new interest in the
discipline (Agutter and Wheatley,
2004; Glazier,
2005; da Silva et al.,
2006), and it also re-opened the long-simmering debate about the
`true' value for the scaling factor in the allometric equation
(Hoppeler and Weibel, 2005).
However, this debate about the value for the scaling exponent may be
premature, because investigators on all sides of the issue have for years
unknowingly used an unreliable procedure to estimate parameters in the
equation (see below). This problem has gone largely unrecognized because
allometric equations typically have not been validated in the scale of
measurement.
The relationship between metabolic rate and body mass is usually assumed to
follow a simple, two-parameter power function:
 | (1) |
;
Glazier, 2005;
da Silva et al., 2006). It is
unclear whether this expression emerged from strictly theoretical
considerations; or whether it was adopted in the era prior to digital
computers because of the relative ease in manipulating logarithmic
transformations of empirical data conforming with such an equation; or whether
the answer lies in some combination of these and other factors
(Kleiber, 1961;
Gould, 1966;
Heusner, 1987). Regardless of
the origin of the function, however, values for the predictor and response
variables are seldom examined in the original arithmetic scale
(Smith, 1984) but, instead,
are immediately transformed to their logarithms, at which point the expression
assumes the form:
 | (2) |
;
Agutter and Wheatley, 2004;
Glazier, 2005;
da Silva et al., 2006). A
straight line commonly is fitted to the data by the method of ordinary least
squares (Glazier, 2005), after
which parameters in the allometric equation are estimated by
back-transformation to the arithmetic scale
(Smith, 1984).
The aforementioned (`traditional') approach to allometric analysis has not
changed appreciably since the time of Kleiber, Benedict and Brody
(Kleiber, 1932;
Benedict, 1938;
Brody, 1945). The approach
nonetheless is beset by a variety of problems, most of which result from the
use of logarithmic transformations. First, transformation profoundly alters
the relationship between predictor and response variables
(Emerson and Stoto, 1983;
Jansson, 1985;
Osborne, 2002), so influential
outliers may go undetected, remain in the data set, and bias parameter
estimates in the fitted statistical model
(Packard and Boardman, 2008a).
Second, the two-parameter power function (Eqn
1) underlying the traditional allometric analysis may not provide
a good fit to the data (Zar,
1968; Albrecht and Gelvin,
1987; Albrecht,
1988; Packard and Boardman,
2008a; Packard and Boardman,
2008c), in which case parameter estimates again may be inaccurate
and misleading. Third, the statistical model obtained by back-transformation
from logarithms is one that predicts geometric means for Y instead of
arithmetic means (Miller,
1984; Smith, 1993;
Hayes and Shonkwiler, 2006).
And fourth, a straight line fitted to logged values may undergo distortional
rotation owing to the fact that squared residuals are not equivalent for large
and small values of the original response variable
(Zar, 1968;
Jansson, 1985;
McCuen et al., 1990; Pandy and
Nguyen, 1999; Packard and Boardman,
2008b).
We re-examined data for the basal metabolic rate (BMR) of 626 species of
mammals (Savage et al., 2004)
to illustrate how applying the traditional method for allometric analysis can
result in biased and misleading estimates for parameters in a two-parameter
allometric equation. Species represented in this comprehensive sample varied
in size from a 2.4 g shrew to a 3672 kg elephant. We focused on a modified
data set created by binning logged values for body mass [Appendix 2 in the
study by Savage and colleagues (Savage et
al., 2004)], because the resulting estimate of 0.737 for the
allometric exponent is regarded by many workers as providing strong support
for the concept of 3/4-power scaling (Brown
et al., 2004; Farrell-Gray and
Gotelli, 2005; West and Brown,
2005). The dimension for each of the bins was 0.1 in logged units
for body mass, and each bin yielded one representative (`average') value
irrespective of the number of species assigned to it. Binning was undertaken
by Savage and colleagues (Savage et al.,
2004) to prevent the preponderance of small species in the full
sample from exerting undue influence on estimates for parameters in the
allometric equation; but the use of binned values also facilitates graphical
analysis by avoiding the visual clutter that would accompany the display of
more than 600 values in a single plot. Preliminary examination of values for
the full 626 species, coupled with the results of an independent study of the
same data set (Hui and Jackson,
2007), indicates that none of our conclusions was affected by
using binned values.
We do not address in our study subsidiary issues like phylogenetic
independence of measurements for different species
(Garland et al., 2005) or
assumptions of least-squares regression
(Warton et al., 2006).
Instead, our treatment follows the same general approach that was used by
Savage and colleagues (Savage et al.,
2004), thereby enabling us to make detailed comparisons of their
findings with our own. We carried all calculations to six decimal places
before rounding to three.
|
METHODS AND RESULTS |
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TOPSummaryINTRODUCTIONMETHODS AND RESULTSDISCUSSIONReferences |
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). This
step was followed by preliminary statistical analyses
(Packard and Boardman, 2008a)
wherein a straight line was fitted to logarithmic transformations by ordinary
least squares and a two-parameter power function was fitted to values in the
arithmetic scale by non-linear regression
(Motulsky and Christopoulos,
2004). Computations were performed in SigmaPlot (version 10.0 from
Systat Software, Inc., San Jose, CA, USA), which uses the
Marquardt–Levenberg algorithm to fit non-linear functions by an
iterative process that minimizes the sum of squares for residuals
(Marquardt, 1963). Allometric
exponents estimated by the two procedures were quite different: 0.737 from the
slope of the straight line fitted to logarithms (R2=0.986)
and 0.909 from the two-parameter power function fitted to values in the
arithmetic scale (R2=0.997). However, logarithmic
transformation failed to linearize the data (e.g. a plot of residuals for
logBMR against predictions yielded a pattern in the form of an inverted
parabola) (see also Kozlowski and
Konarzewski, 2005), and the equation resulting from
back-transformation seriously underestimated metabolic rate for the elephant,
the largest species in the sample. In addition, neither of the analyses passed
the test for constancy of variances (Spearman rank correlation between
absolute values for residuals and observed values for the predictor variable)
(Kutner et al., 2004).
The unbalanced distribution of arithmetic values for body mass was also of
concern. The elephant (which was the only species represented in that bin) was
nearly an order of magnitude heavier than the next species in the sample. This
observation, coupled with the decidedly different estimates for allometric
exponents, raised the possibility that the elephant was an unduly influential
outlier in the non-linear regression
(Anscombe, 1973;
Stevens, 1984;
Osborne and Overbay, 2004). We
subsequently discovered that Cook's Distance, which is a sensitive measure of
the influence of a data point on parameters in the fitted model
(Kutner et al., 2004), was an
extraordinary 4600 for the elephant. Any data point for which Cook's Distance
exceeds 4 is likely to exert undue influence, so we treated the elephant as a
statistical outlier and removed it from the data set.
Fitting the allometric equation
A straight line fitted to the remaining 51 logged values yielded a
statistically significant equation (Fig.
1A), even though a plot of residuals against predicted values
indicated once again that a straight line was not an appropriate model
(Fig. 1B). Nonetheless,
R2 was extraordinarily high
(Fig. 1A), and the estimate of
0.728 for the allometric exponent is similar to the estimate from examination
of the full data set (i.e. exclusion of the elephant had little effect on the
outcome). The analysis passed the test for normality (P=0.288 by
Kolmogorov–Smirnov test) (Kutner et
al., 2004) but it failed the one for constancy of variances
(P=0.002). Consequently, confidence limits for the slope and
intercept are unlikely to be reliable
(Myers, 1986;
Finney, 1989).
|
;
Finney, 1989). However, the
parameter estimates themselves may be better than those obtained by
back-transformation (Myers,
1986), and the predictive equation is efficient in terms of
minimizing residual variance (Asselman,
2000).
Validating the allometric equation
Next, we back-transformed the equation for the line fitted to logarithms
and displayed the resulting function on bivariate graphs together with the
function obtained by non-linear regression
(Fig. 2A,B). The line from
back-transformation is a good descriptor for values in the logarithmic scale
(Fig. 2A), but it fails to
predict values for large animals in the arithmetic scale
(Fig. 2B). In contrast, the
non-linear regression predicts consistently higher values for the response
variable in the logarithmic domain (Fig.
2A) while performing much better than the alternative model for
large animals in the arithmetic scale (Fig.
2B).
|
). The equation
from back-transformation is the better predictor of BMR for animals between
102 and 104 g in body mass
(Fig. 2C, inset), but clearly
is the inferior function for predicting BMR of larger mammals
(Fig. 2C). Thus, the equation
from back-transformation is useful for predicting metabolic rates of animals
weighing less than 104 g but the function is not valid over the
full range in body size. The function fitted by non-linear regression is a
better descriptor over the entire range in body size, even though it is more
biased than the function from back-transformation in the mid-range for body
size. (Part of the bias in the function fitted by non-linear regression is the
result of forcing the line to pass through the origin and could be eliminated
by fitting a three-parameter function instead. However, the scaling exponent
in the three-parameter function differs only slightly from that for the
two-parameter function, i.e. 0.667 vs 0.686, respectively.)
Curvilinearity in the allometric relationship
The scaling exponent for small species is predicted to be smaller than that
for large species (Savage et al.,
2004) – a prediction that seems to be confirmed by the
observed curvilinearity (concave upward) in the relationship between logBMR
and log body mass (Fig. 1A,B).
Such a curvilinear pattern of variation in log-transformed data should be
cause for concern, because it calls into question the underlying allometric
model (Eqn 1). Nevertheless, we
examined data for large and small mammals separately to see whether the
aforementioned prediction was realized. Binned data for mammals weighing less
than 260 g were taken to represent small species whereas those for mammals
weighing more than 260 g were taken to represent large ones
(Savage et al., 2004). The
elephant was omitted from the analyses because of the likelihood that it is an
outlier.
Straight lines fitted to transformed values for both small and large
species yielded statistically significant equations with high values for
R2 (Fig.
3A,B). The analysis of values for small species passed tests for
normality (P=0.065) and homogeneity of variances (P=0.059).
The analysis for large species, however, passed the test for normality
(P=0.298) but not that for homoscedasticity (P=0.016).
Scaling exponents for small and large species were 0.678 and 0.797,
respectively, which seemingly confirmed expectation
(Savage et al., 2004).
|
The alternative methods for fitting the allometric equation to data for small mammals yielded functions that are reasonably good visual fits to values in the original scale (Fig. 3E). The non-linear function fitted to values for large mammals is also a good fit graphically (Fig. 3F) but the equation obtained by back-transformation seriously underestimates BMR for species with masses between 75 and 150kg (Fig. 3F).
Scaling exponents estimated for large and small species by the traditional
method are quite different (Fig.
3A,B) whereas exponents estimated by non-linear regression are
quite similar (Fig. 3C,D).
Indeed, the 95% confidence interval for the exponent estimated by non-linear
regression for small mammals (i.e. the group for which such limits can be
reliably computed) is 0.615–0.697, which includes the exponent estimated
for large mammals (Fig. 3D) as
well as the one for all species exclusive of the elephant
(Fig. 1C). Thus, analyses of
different subsets of the data by non-linear regression lead to a common
estimate for a scaling exponent (in the range 0.656–0.686) – not
to the different exponents predicted by the aforementioned theoretical model
(Savage et al., 2004).
|
DISCUSSION |
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TOPSummaryINTRODUCTIONMETHODS AND RESULTSDISCUSSIONReferences |
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;
Jansson, 1985;
Osborne, 2002). Logarithmic
transformation results in an overall compression of distributions for the
variables, but the compression is greater at the high ends of the scales than
at the low ends (Emerson and Stoto,
1983; Jansson,
1985; Osborne,
2002). This disproportionate compression causes small values for
the variables to have a large effect on parameters in the fitted equation and
large values to have a small effect
(Glass, 1969;
Jansson, 1985;
Packard and Boardman,
2008b).
By way of example (Jansson,
1985), consider a straight line fitted to logarithms of 0.9 and
1.1 at one level for X and to 1.9 and 2.1 at a higher level for
X. Predictions for logs of the response variable Y are 1.0
and 2.0, respectively, with all residuals having absolute values of 0.1. Such
a balanced distribution of residuals indicates that logged values for
Y were weighted equally in fitting the line by ordinary least
squares.
Back-transformation of the predicted values yields a geometric (not
arithmetic) mean of 10 at the first level for X and 100 at the second
level for X. Observed values corresponding to the prediction of 10
are 7.9 and 12.6 whereas those corresponding to the prediction of 100 are 79.4
and 125.9. Thus, absolute values for residuals expressed on the scale of
measurement are 2.1, 2.6, 20.6 and 25.9, despite the fact that all residuals
were ±0.1 in the log domain. This difference between the scales is
important because the fitted line minimizes the sum of the squared residuals
regardless of the scale in which the data are expressed
(Zar, 1968). Whereas the
squares for the residuals in the log domain are identical (i.e. 0.01), the
square for the largest of the values in the arithmetic domain (i.e. 670.8) is
more than two orders of magnitude larger than that for the smallest value
(i.e. 4.4).
At each level for X, the smaller of the two measurements lies below the fitted line and the larger one lies above it. Consequently, the smaller values in the arithmetic scale have a disproportionate influence on the elevation of the line fitted to logarithms because residuals for large and small values are identical in the logarithmic domain. Additionally, the smaller of the two measurements lying above (or below) the fitted line is associated with the lower level for X and the larger with the higher level. Thus, the smaller value in the arithmetic scale has a disproportionate influence on the slope of the line fitted to logarithms (again, because residuals are identical in the log domain).
Depending on the distributions of the variables, both the slope and
intercept of the straight line may be affected in unexpected ways
(Glass, 1969;
Jansson, 1985;
McCuen et al., 1990; Pandy and
Nguyen, 1999; Packard and Boardman,
2008b), and these effects later are transmitted by
back-transformation to the two-parameter allometric equation. This disparate
influence of small and large values is apparent in the current study in graphs
of the alternative equations in both logarithmic and arithmetic scales
(Fig. 2). The linear regression
on transformed values was rotated in a counter-clockwise direction
(Fig. 2A), and the result was a
poor fit of the back-transformation to data for large animals
(Fig. 2B). The general problem
outlined here probably occurs commonly in data sets that include animals
spanning large ranges in size (Glass,
1969; Packard and Boardman,
2008b): the traditional procedure provides good predictions for
small animals but poor predictions for large ones.
Savage and colleagues used the binning procedure in an attempt to reduce
the disproportionate influence of the many small species in the sample and
thereby obtain a more reliable estimate for the scaling exponent
(Savage et al., 2004). On the
other hand, Glazier argued that binning actually caused values for large
species to have too great an influence on parameters in the allometric
equation owing to an increase in proportional representation for large species
(Glazier, 2008). Both these
suggestions, however, are based on a misunderstanding of the logarithmic
transformation. First, binning by logs for body mass had the effect of
expanding the scale at the lower end of the distribution and compressing it at
the upper end, thereby maintaining a skew in the distribution of masses
expressed in grams and causing small species to be over-represented in the 51
bins exclusive of the elephant. For example, the arithmetic mean for 51
back-transformed values for body mass is 31,364 g. A total of 42 bins (of
which one was empty) were available to accommodate species with masses below
the average, and 12 (of which two were empty) were available to accommodate
species with masses above the average. Small species consequently continued to
be `over-represented' in the data set. Second, the line fitted to logarithms
was `transformation biased' by the undue influence of the small species
(Jansson, 1985;
Packard and Boardman, 2008b),
leading to rotation of the line, to underestimation of the allometric
coefficient from back-transformation, and to overestimation of the allometric
exponent (Fig. 2B). The minor
influence of large species is why deletion of the elephant from the data set
had little effect on the allometric exponent estimated by the traditional
method.
Logarithmic transformations
Logarithmic transformations have a long history of use in allometric
analyses, so it is useful here to consider briefly the reasons for performing
such transformations and to ask whether the transformations continue to have
application.
;
Smith, 1993). This application
of log transformations was based on the implicit assumption that the data
conformed with a two-parameter power function
(Smith, 1984). The assumption
was seldom addressed, and inappropriate equations were sometimes fitted
(Albrecht and Gelvin, 1987;
Albrecht, 1988;
Packard and Boardman, 2008a;
Packard and Boardman, 2008c).
However, the form of the underlying equation now is moot because the advent of
sophisticated PCs and software for both graphical and statistical analysis has
rendered linearization unnecessary
(Motulsky and Christopoulos,
2004). ). The
main problem with unbalanced distributions is that the largest animals in the
sample may exert undue influence on parameters that are estimated by methods
based on least squares. Indeed, the problem of imbalance was apparent in data
examined here (Fig. 1C).
Whereas transformation may produce a better distribution of values
(Fig. 1A), it also introduces a
bias that may be more serious than the problem that the transformation was
intended to correct (Jansson,
1985). The bias results from the change in error structure of the
statistical model on back-transformation to the arithmetic scale
(Glass, 1969;
Manaster and Manaster, 1975;
Miller, 1984;
Jansson, 1985;
Smith, 1993;
Hayes and Shonkwiler, 2006)
coupled with the distortion that is introduced by the non-linear relationship
between logs and values in the original scale
(Jansson, 1985;
McCuen et al., 1990;
Pandey and Nguyen, 1999;
Hui and Jackson, 2007;
Packard and Boardman, 2008b).
A better way than transformation for addressing the problem of unbalanced
distributions might be to make greater use of regression diagnostics and model
validation (Anscombe, 1973;
Snee, 1977;
Kutner et al., 2004). ;
Smith, 1984). Unfortunately,
few authors examine the transformed variables for heteroscedasticity, so the
resulting analysis may fail to correct the problem that led to transformation
in the first place; the logged data examined here provide but one example of
the failure of transformation to achieve its intended goal
(Fig. 1B). Even when the
original data are heteroscedastic and transformation results in a
homoscedastic distribution, the potential exists for introducing a
transformation bias that is evident only on back-transformation to the
arithmetic scale. However, the problem of heteroscedasticity probably will
become moot as more investigators discover statistical procedures that
accommodate different assumptions concerning distributions of data in the
original scale (Lane, 2002;
Cox et al., 2008).
Validating the model
Regardless of the means by which an allometric equation is fitted to data,
it is essential that the model be validated
(Snee, 1977;
Emerson and Stoto, 1983;
Myers, 1986;
Finney, 1989;
Kutner et al., 2004). A
graphical display is the most effective way to verify that the fitted model
actually describes the data on which it is based
(Anscombe, 1973).
Unfortunately, validation in the traditional allometric analysis typically is
limited to a display of values in the logarithmic scale, which often has
little bearing on the relationship between metabolic rate and body mass
(Fig. 2A,B). A good fit of a
linear function to logarithms does not imply a good fit of the re-expressed
equation to data in the arithmetic scale
(McCuen et al., 1990;
Pattyn and Van Huele, 1998).
Thus, proper validation requires that the allometric equation be shown against
the backdrop of data in the scale of measurement
(Emerson and Stoto, 1983;
Myers, 1986;
Finney, 1989).
Implications for theoretical models
The data set compiled by Savage and colleagues is widely regarded to be one
of the very best (Savage et al.,
2004). It comes as no surprise, therefore, that the statistical
analysis performed on those data by Savage and colleagues
(Savage et al., 2004) is
viewed by many in the scientific community as offering strong evidence in
support of the concept of 3/4-power scaling for metabolic rate on body mass in
mammals (Brown et al., 2004;
Farrell-Gray and Gotelli,
2005; West and Brown,
2005). However, Savage and colleagues
(Savage et al., 2004) seem to
have omitted three critical steps from their investigation: they apparently
did not (1) examine their data for potential outliers, (2) test assumptions
underlying their statistical analysis, or (3) validate the allometric model in
the original scale. Our re-analysis of data exclusive of the elephant, which
was an apparent outlier, revealed that a linear equation fitted to
log–log transformations failed tests for both linearity and constancy of
variances, and that the two-parameter power function estimated by
back-transformation did not predict metabolic rates of large animals in the
sample. Consequently, the earlier estimate of 3/4-power scaling is not well
supported, thereby calling into question the validity of theoretical models
that purport to explain such a scaling factor in mammals (e.g.
West et al., 1997;
Banavar et al., 1999;
Darveau et al., 2002).
The traditional approach to allometric research is to fit a straight line
to logarithmic transformations and then back-transform the resulting equation
to the arithmetic scale (Smith,
1984; Agutter and Wheatley,
2004; Glazier,
2005; da Silva et al.,
2006). Consequently, biases of the kinds shown in the current
study and elsewhere (Glass,
1969; Hui and Jackson,
2007; Packard and Boardman,
2008a; Packard and Boardman,
2008b; Packard and Boardman,
2008c) are likely to occur commonly in published research on
allometry. For this reason, scaling coefficients and exponents reported in the
literature should be interpreted with a healthy dose of skepticism.
|
Acknowledgments |
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