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 ¾ 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 ¾ 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 ¾, 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 the¾ 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 ¾ power coffin.
