Allometry - Determining If A System Is Scaling With Allometry

Determining If A System Is Scaling With Allometry

To determine whether isometry or allometry is present, an expected relationship between variables needs to be determined to compare data to. This is important in determining if the scaling relationship in a dataset deviates from an expected relationship (such as those that follow isometry). The use of tools such as dimensional analysis is very helpful in determining expected slope. This ‘expected’ slope, as it is known, is essential for detecting allometry because scaling variables are comparisons to other things. Saying that mass scales with a slope of 5 in relation to length doesn’t have much meaning unless you know the isometric slope is 3, meaning in this case, the mass is increasing extremely fast. For example, different sized frogs should be able to jump the same distance according to the geometric similarity model proposed by Hill 1950 and interpreted by Wilson 2000, but in actuality larger frogs do jump longer distances. Dimensional analysis is extremely useful for balancing units in an equation or in our case, determining expected slope. A few dimensional examples below: M-Mass L-Length V-Volume (which is also L cubed because a volume is merely length cubed)

If trying to find the expected slope for the relationship between mass and the characteristic length of an animal (see figure) you would take the units of mass (M=L3, because mass is a volume-volumes are lengths cubed) from the Y axis and divide them by the X axis (in this case L). Your expected slope in this case is 3 (L3/ L1, 3/1=3). This is the slope of a straight line, but most data gathered in science does not automatically fall neatly in a straight line, so data transformations are useful. It is also important to keep in mind what you are comparing in your data. Comparing head length to head width can yield different results than comparing to body length. Sometimes a characteristic length such as head length may scale differently to its width than body length.

A common way to analyze data such as those collected in scaling is to use log-transformation. It is beneficial to transform both axes using logarithms and then perform a linear regression. This will normalize the data set and make it easier to analyze trends using the slope of the line. Before analyzing data though, it is important to have a predicted slope of the line to compare your analysis to. After data are log transformed and linearly regressed. You can then use least squares regression with 95% confidence intervals or reduced major axis analysis. Sometimes the two analyses can yield different results, but often they do not. If the expected slope is outside the confidence intervals then there is allometry present. If mass in our imaginary animal scaled with a slope of 5 and this was a statistically significant value, then mass would scale very fast in this animal versus the expected value. It would scale with positive allometry. If the expected slope is 3 and in reality in a certain organism mass scaled with 1 (assuming this slope is statistically significant), then it would be negatively allometric.

Another example: Force is dependent on the cross-sectional area of muscle (CSA), which is L2. If comparing force to a length, then your expected slope is 2. Alternatively, this analysis may be accomplished with a power regression. Plot the relationship between your data onto a graph. Fit this to a power curve (depending on your stats program, this can be done multiple ways) and it will give you an equation with the form y=Zx^number. That “number” is the relationship between your data points. The downside to this form of analysis is that it makes it a little more difficult to do statistical analyses.