A Primer on Allometric Equations and on Curve Fitting

I. Allometry

Using "allometric" equations is a fundamental skill for comparative physiologists. Because the text has little on this topic (p. 192ff), I've prepared the following exercises to help you hone your skills. The exercises are best done on a computer (e.g., with Excel), but you can easily do them "by hand" on graph paper.  The text (Schmidt-Nielsen, Appendix C) gives a very brief introduction to allometry.

Also, I encourage you to download the mammalmass02.xls spreadsheet, which contains data on several species of mammals.  Trying the various manipulations (below) on those data will reinforce the points made here.

These exercises will require some effort on your part. However, if you want to be able to read the literature of comparative physiology, medicine, and related fields, you must learn these skills.

Allometry ("different measure") or scaling refers to how some aspect of the morphology or the physiology of a group of animals changes in proportion with a change in body size. Thus one might study the allometry of oxygen consumption, brain size, limb length, offspring number, home range size, or whatever. For example, one can ask, is blood volume a constant proportion of body mass? In other words, does a shrew have proportionately the same amount of blood as does an elephant? Because body size is the dominant physiological variable, the study of scaling is central to many aspects of physiology.

Start with a simple example -- how brain size changes with body size (hypothetical data!).

  1. Plot (on an arithmetic graph) the following data on brain mass (or brain/body) on the Y axis (vertical) and body mass on the X axis (horizontal). Here are some sample data that you should plot. [Calculate Y/X (thus, Y as a proportion of X) and fill in the table.]

Y (brain)

X (body)

Y/X (brain/body)

0.2

1
  

0.4

4
  

0.6

6
  

0.7

8
  

Note first that big animals have (not surprisingly) bigger brains than do small animals. But note also that the relationship you've graphed isn't linear but curvilinear. Specifically, brain size isn't increasing as fast as is body mass.

 

Convince yourself of this by computing for each species the ratio of brain mass to body mass (Y/X in column 3), and then plot this ratio as a function of body mass.

What can we conclude? Big animals have big brains, but their brains are a relatively smaller proportion of their overall mass. This would lead naturally to a series of questions concerning the physiological reasons for this scaling relationship. Why, for example, aren't brains a constant proportion of body mass?

Now examine another set of data:

Y

X

Y/X

1

1
  

4

2
  

9

3
  

16

4
  

 

(3) Plot and compute these data as above. Note that the ratio of Y/X increases as X increases, just the opposite of what we found with brain size.

 

( 4) Finally, plot these data:

Y

X

Y/X

3

1
  

6

2
  

12

4
  

15

5
  

How does Y scale?

* * *

We have seen three basic ways that dependent variables (Y's) can scale.

 

From Graphs to Equations

Now, in addition to using graphical approaches to "visualize" scaling relationships , one can also use statistical approaches to fit a line to the data and thus compute "power functions" [when used in physiology or morphology, power functions are called "allometric" equations.]

Physiologists regularly estimate such equations because they are a very simple and concise way of summarizing vast amounts of data, and obviously take much less space to report than does a graph. Also, they also are easy to manipulate algebraically, and they can be used predictively (e.g., to predict brain size of some animal from knowledge of its body mass). Scaling relationships are often well fitted by the general formula

Y = a X b

where Y is any physiological or morphological variable, a is a proportionality constant that characterizes that variable for a given group of organisms, X is body mass (usually changed in the physiological literature to M to symbolize body mass), and the exponent b describes the strength and direction of the effect of body size on Y.

The shape of the curve:

Learn a few simple tricks and you can easily look at one of these equations and immediately draw the general shape of the scaling relationship -- no calculators are necessary!

What are the basic tricks?  Let's start with the exponent b, which can take four basic values: b = positive, negative, one, or zero…

If b is positive (thus b > 0), then Y must get bigger as X increases.

Convince yourself of this by calculating (and then plotting) Y for several values of M in the following equations. You don't need a calculator for these computations, simply pick "simple" M values (say 1, 9, 100) :

Y = 0.3 M2

Y = 5.6 M.5

Y = 2.5 M1

 

If b is negative (b < 0), then Y must get smaller as M increases.

 

Convince yourself of this by calculating Y for the above equations but change all the b values to negative ones (e.g., Y = 0.3 M-.5).

 

If b is zero, then Y is a constant (always = a) and does not change as body mass increases. (Recall that anything raised to the zero power is always one).

In the special case when b is 1 (and only if b = 1), then Y is a constant proportion of M (see the example above). In other words the ratio Y/M is independent of M and is always equal to a.

To prove this, simply divide both sides of the equation by M and then cancel:

Y = (a M 1)

Y/M = (a M 1) / M

Y/M = a

 

Self-Test: On arithmetic plots, only two values of b produce straight lines. What are those two values?

The "Height" of the Curve

The "height" of the curve above the X axis depends on the value of a for any given M b. One simple way to see this is to assume that M = 1. In this case, Y = a for any b (recall 1 raised to any power is 1), and thus Y increases directly with a and only with a.

Convince yourself of this by plotting on a single arithmetic graph the following allometric equations for a few values of M:

Y = 1 M 2

Y= 10 M2

Using these simple tricks, you should now be able to look at any allometric equation and draw its general shape without a computer or calculator. Test yourself by making up several equations, drawing them, and then verifying the general shape using a calculator and graph paper (or computer).

Conversely, you should be able to do just the opposite -- that is, to look at a graph and say which of two curves has the higher value of a and also estimate roughly whether b is > 1, 0 < b < 1, b = 0, or b < 1.

* * *

From Arithmetic to Logarithmic Plots

The above exercises generate mainly curvilinear graphs (i.e., the lines are curved, not straight). For statistical reasons, curvilinear relationships are difficult to manipulate, so statisticians attempt to convert curved lines into straight ones. For allometric relationships, this transformation is easy. Specifically, all allometric relationships yield straight lines when plotted on logarithmic coordinates rather than on arithmetic ones! Let's start with the general allometric equation:

Y = a Mb

Plotting on log-log paper is equivalent to a log transform of the data points. To see this take the logs of both sides of the above equation:

log Y = log a + b log M

This equation is now in the same general form as the familiar equation for a straight line (a "linear regression")

Y = a + b M

Convince yourself that this trick straightens curvilinear relationships by re-plotting some of the above data on log-log graph paper (equivalently, you can simply plot the logs of X and of Y on arithmetic paper). Allometric relationships that were curvilinear on arithmetic plots become linear on log-log plots.  [For fun (ok, I have a warped sense of humor), try log 10 and log e transformations and see if both work.]

Þ On the first exam, you will be expected to be able to look at an allometric equation and then to draw its general shape on logarithmic coordinates and on arithmetic coordinates

It should now be obvious one reason why scaling data in physiology (and other fields) are routinely plotted on logarithmic coordinates -- these generate straight lines. Log-log plots have additional advantages.

One important one is that a very large range of body sizes can be plotted on the same graph ("mouse to elephant" curves) without having all of the small species scrunched onto the left hand side of the graph.

They have a disadvantage, however. Log-log plots make data appear "less noisy" (i.e., less variable) than they actually are -- so they can be visually deceptive.

*         *          *

Applying What We Have Learned:

Next, let's use allometric equations to investigate the metabolic "intensity" of a single gram of animal tissue. Suppose we wanted to know whether the metabolic rate of one gram of tissue of a small animal is the same as that of a large animal. Similarly, is the metabolic rate of one cell of a small animal the same as that of a large animal?

To answer that question, we could of course measure metabolic rates of one gram of tissue obtained from "biopsies" taken from a variety of different sized animals. Alternatively, we could simply measure the "whole animal" metabolic rate of different sized animals, and then calculate the metabolic rate of a gram of different sized animals.

So we gather the data, and then (using standard statistical methods) estimate the allometric equation for metabolic rate as a function of (whole) body mass:

                         E = a M 0.75

where is a standard physiological symbol for rate of metabolism (there should be a dot "." above E, to signify that E is a  rate).

To compute the metabolic rate of one gram of tissue (the so called "mass specific" metabolic rate = E/M), simply divide both sides of the equation by M, thus

                  E/M = rate/gram

                                       =  a M.75 / M

                                                        = a M (.75 - 1)

                                                                         = a M (-.25)

This simple manipulation tells us how mass-specific metabolic rate changes with body mass. Because b is negative, we know immediately that the metabolic rate of a gram of tissue of a small animal is higher than that of a large animal.

You should be able to draw the general shape of the above relationship on both arithmetic and logarithmic coordinates.

 

*         *           *

Comparing Different Sized Animals:

Physiologists frequently want to compare physiological capacities of sets of animals that differ in some way (habitat, age, etc.). Imagine that you measured heart mass for several carnivores, and you wanted to test the hypothesis that "marathon" type predators such as dogs have larger hearts (hence better oxygen delivery capacities) than do "sprint" type (wimply) predators such as cats. If you were lucky (or smart) enough to have measured heart sizes in dogs and cats that were exactly the same body size, then you could just compare values directly.

However, very often, the dogs and cats you are able to study won't be all of the same body size. So any direct comparison of their heart sizes would be confounded by differences in body size as well as difference in mode of predation.

Can you "correct" for size differences so that you can still test for any associations between "relative" heart mass with predation mode? Easily. To "correct" for size, physiologists first calculate an overall allometric relationship for heart mass vs. body mass, and then look at the deviations (= "residuals", the vertical lines shown below) of the heart mass for a particular species from the expected heart size for an animal of its body size. As is obvious, (dogs red) have relatively bigger hearts than cats (blue). This of course explains why dogs make so much better pets

Note that in the past (and, alas, sometimes still in the present!) physiologists have often tried to correct for body size by computing "mass-specific" heart rates, metabolic rates, or whatever. These measures do not in fact correct for body size unless the dependent variable scales as M1 (isometry). Why? Recall from above that

Y/M = a M (b-1)

 

It should be obvious that Y/M is still dependent on M unless b = 1. In this case (b-1) = (1-1), so

Y/M = a M (b-1) = a M0 = a.

Hence Y/M is always constant (always equals a) and is thus independent of M. But for no other value of b is this the case.

Consequently, mass-specific rates can legitimately be used to gain insight into the physiological "intensity" of a gram of tissue, but such rates should not be used to correct for size -- use residual analyses instead. [Partial correlation and analysis of covariance are alternative statistical methods.] One can also report "mass-independent" units. If for example metabolic rate scales with b = 0.738, then one can calculate the mass-independent metabolic rate for a given animal using the following relationship:

ml 02 g-0.738 h-1.

A final note on terminology. The expression "allometry" means different measures or different proportions. It refers to the fact that Y is not a constant proportion of X for all b > or < 1. However, in the case that b = 1, then Y is a constant proportion of X.  [Prove this to yourself, algebraically.]  This is called, not surprisingly, "isometry" ("same" metric).