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First published online October 31, 2008
Journal of Experimental Biology 211, i-a (2008)
Copyright © 2008 The Company of Biologists Limited
doi: 10.1242/jeb.026211
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Inside JEB |
IS ALLOMETRIC SCALING A MATHEMATICAL ARTEFACT?
kathryn{at}biologists.com
Every zoology undergraduate knows Max Kleiber's `elephant to mouse' curve.
In the early 1930s, Kleiber plotted the body masses and metabolic rates of
animals ranging in size from ring doves up to steers on a log graph and found
a rather simple relationship; the metabolic rates scaled as the 
power
of the animals' body masses. This was later refined by F. G. Benedict, who
restricted the curve to mammals, and the method is now known as allometric
scaling. However, the reliability of this scaling factor has always been
questioned, and never more so than since a theoretical model, published in
1997 by Geoffrey West and colleagues, claimed to explain the pleasing
relationship. But Gary Packard and Geoffrey Birchard were suspicious. Could
everyone have been missing the point for more than 70 years? What if the data
simply didn't fit the assumptions that underpin Kleiber's classic curve and
the power relationship was just an artefact of mathematical
manipulation (p.
3581)?
Turning to a data set of body masses and metabolic rates for 626 species
ranging from 2.4 g shrews up to a 3672 kg elephant assembled by Van Savage and
colleagues in 2004, the duo tested whether all of the data points were equally
valid and whether there were any statistical outliers that should be ignored.
They found that the elephant was so far out there was no way it could be
included in the calculation. Having ruled out the elephant, the pair plotted
the data on a logarithmic scale to get a metabolic scaling factor that was
close to , before replotting the data from the logarithmic plot on an
arithmetic scale to see how well it predicted the animals' metabolic rates.
Packard and Birchard explain that although the graph predicted the smaller
animals' metabolic rates well, it failed for larger animals. However, when
they recalculated the scaling coefficient using a different method (non-linear
regression), the value was between 0.656 and 0.686 and predicted all of the
animals' metabolic rates well.
So why have scientists been using log transformations to derive the
allometric scaling factor when it could well be overestimating the
relationship? Packard and Birchard explain that scientists traditionally
replotted their data on log graphs to `linearize' complex data sets over
several orders of magnitude. But they explain that this assumption was only
true if the `data conformed with a two paramater power function', and the
relationship between animals' body masses and their metabolic rates does not.
No one had tested this assumption, and consequently the log transformation
introduced a new relationship between metabolic rate and body mass that over
estimated the metabolic scaling factor. On top of that, no one had checked for
outliers, such as the elephant, in the data set and having derived the scaling
factor, no one went back to check that it correctly predicted a mammal's
metabolic rate from its body mass.
Packard adds, `Our work certainly calls into question the validity of
"Kleiber's Law", but points to a larger and more general problem
with the standard method for allometric analysis.' Doubtless this is not the
final word in the allometric scaling debate, but it could be another nail in
the power coffin.
References
Packard, G. C. and Birchard, G. F. (2008).
Traditional allometric analysis fails to provide a valid predictive model for
mammalian metabolic rates. J. Exp. Biol.
211,3581
-3587.
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