Allometry is the study of the relative change in proportion of an attribute compared to another one during organismal growth. These attributes may be morphological, physiological, or otherwise. A well known example of an allometric relationship is skeletal mass and body mass. Specifically, the skeleton of a larger organism will be relatively heavier than that of a smaller organism. Of course it seems obvious that heavier organisms require heavier skeletons. But is it equally clear that heavier organisms requires disproportionately heavier skeletons? So how does the relationship work? Consider the following data:

• a 10 kg organism may need a 0.75 kg skeleton,


• a 60 kg organism may need a 5.3 kg skeleton, and yet


• a 110 kg organism may need a 10.2 kg skeleton.

As you can see by inspecting these numbers, heavier bodies need relatively beefier skeletons to support them. There is not a constant increase in skeletal mass for each 50 kg increase in body mass; skeletal mass increases out of proportion to body mass [2].

Allometric scaling laws are derived from empirical data. Scientists interested in uncovering these laws measure a common attribute, such as body mass and brain size of adult mammals, across many taxa . The data are then mined for relationships from which equations are written.

Allometric Growth

Allometric scaling relationships can be described using an allometric equation of the form,
	f (s)  =  c s d,		(1)
where c and d are constants. The variables s and f (s) represent the two different attributes that we are comparing (e.g., body mass and skeletal mass).

This equation can be used to understand the relationship between two attributes. Specifically, the constant d in this model determines the relative growth rates of the two attributes represented by s and f (s). For simplicity, let's consider the case d > 0 only.

• If d > 1, the attribute given by f (s) increases out of proportion to the attribute given by s. For example, if s represents body size, then f (s) is relatively larger for larger bodies than for smaller bodies.
  
• If 0 < d < 1, the attribute f (s) increases with attribute s, but does so at a slower rate than that of proportionality.
  
• If d = 1, then attribute f (s) changes as a constant proportion of attribute s. This special case is called isometry, rather than allometry.

Using Allometric Equations

Notice that (1) is a power function not an exponential equation (the constant d is in the exponent position instead of the variable s). Unlike other applications where we need logarithms to help us solve the equation, here we use logarithms to simplify the allometric equation into a linear equation. Here's how it works We rewrite (1) as a logarithmic equation of the form,
	log (f (s))  =  log (c s d).		(2)
Then, using the properties of logarithms, we can rearrange (2) as follows,
	log (f)	=  log c + log (s d),
		=  log c + d log s.	(3)
When we change variables by letting,
	y	=  log f,
	b	=  log c,
	m	=  d,
	x	=  log s.
you can see that (3) is in fact the linear equation
	y	=  mx + b.	(4)
Therefore, transforming an allometric equation into its logarithmic equivalent gives rise to a linear equation. Why Bother? By rewriting the allometric equation into a logarithmic equation, we can easily calculate the values of the constants c and d from a set of experimental data. If we plot log s on the x-axis and log f on the y-axis, we should see a line with slope equal to d and y-intercept equal to log c. Remember, the variables x and y are really on a logarithmic scale (since x = log s and y = log f). We call such a plot a log-log plot. Because allometric equations are derived from empirical data, one should be cautious about data scattered around a line of best fit in the xy-plane of a log-log plot. Small deviations from a line of best fit are actually larger than they may appear. Remember, since the x and y variables are on the logarithmic scale, linear changes in the output variables (x and y) correspond to exponential changes in the input variables (f (s) and s). Since we are ultimately interested in a relationship between f and s, we need to be concerned with even small deviations from a line of best fit.

Now let's go back to our fiddler crab as a concrete example.

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