For each of the four examples below, the slope of a power function is given comp

Question

For each of the four examples below, the slope of a power function is given comparing two measurements of a group of organisms. For each, give the slope you would expect if the relationship was isometric, and state whether the observed relationship is isometric or allometric. The log of the first variable would be plotted on the y-axis and the log of the second on the x-axis. If needed, use this link for reference https://goo.gl/H0NYUR

arm length vs. body height for infant humans: b=1.2

brain mass vs. body mass in humans: b=0.66

mass of skeleton vs. body length in whales: b=3.07

mass of skeleton vs. body length in terrestrial mammals: b=3.25

Explanation / Answer

            Y = axb (a power function)

or alternatively log y=log a+b (log x)

–       Y = body part being measured in relationship to the size of the organism

–       X = measure of size used for basis of comparison

–       a = Y intercept

–       b = scaling exponent or Slope

Now if the relation is isometric then in each case the value of b (slope) must be 1 as there will no exponential growth.

We know that

•         If b < 1 ------------negative allometry

•         If b >1   ---------- positive allometry

•         If b=1 ------------isometry

Now

1.    Arm length vs. body height for infant humans: b=1.2

Here b > 1 thus it indicates positive allometry

2.    Brain mass vs. body mass in humans: b=0.66

     Here b < 1 thus it indicates negative allometry

3.    Mass of skeleton vs. body length in whales: b=3.07

Here b > 1 thus it indicates positive allometry

4.    Mass of skeleton vs. body length in terrestrial mammals: b=3.25

        Here b > 1 thus it indicates positive allometry

So, the observed relationship is allometric

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