1. Data Handling & Analysis Allometry & Log-log Regression Andrew Jackson a.jackson@tcd.ie

2. Linear type data How are two measures related?

3. Length – mass relationships How does the mass of an organism scale with its length? Related to interesting biological and ecological processes – Metabolic costs – Predation or fishing/harvesting – Diet – Ecological scaling laws, fractals and food-web architecture

4. Mass of a cube How does a cube scale with its length? – Mass = Density x Volume – Volume = L 1 X L 2 x L 3 = L 3 – Volume = aL b Where a = 1 and b = 3 – So if the cube remains the same shape (i.e. it stays a cube) How does mass change if length is doubled? 2L 1 X 2L 2 x 2L 3 = Mass x 2 3 – Isometic scaling The object does not change shape as is grows or shrinks

5. Mass of a sphere How does mass of a sphere change with length? Volume = (4/3)πr 3 = (4/3)π(L 3 /2 3 ) Again, mass changes with Length 3 The difference here, compared with the cube, is the coefficient of Length (4/3)π(L 3 /2 3 ) = (4/3)π(1/2 3 ) (L 3 ) So, we have – Volume = (some number)L 3 – Volume = aL b Where b = 3 in this case

6. A general equation Mass = aLength b – Where we might expect b = 3 Take the log of both sides – Log(M) = log(aL b ) – Log(M) = log(a) + log(L b ) – Log(M) = log(a) + b(log(L)) – Y = b 0 + b 1 (X) Log(a) = b 0 So…. a = exp(b 0 ) b 1 = b and is simply the power in the allometric equation

7. What do these coefficients mean? On a log-log scale what does the intercept mean? – The intercept is the coefficient, or the multiplier, of length – Mass = aLength b – Spheres and cubes differ only in their coefficients – So a = exp(b 0 ) tells us how the shapes differ between two species

8. What does the slope mean? If b1, the slope and coefficient of log(Length) is 3, then the fish grows isometrically – Its shape stays the same

9. What do these coefficients mean? If b1, the slope and coefficient of log(Length) is < 3, then the fish becomes thinner as it grows

10. What do these coefficients mean? If b1, the slope and coefficient of Length is > 3, then the fish becomes fatter as it grows

11. Brain and body mass relationships Instead of plotting brain mass against body mass Plot log(brain mass) against log(body mass)

12. Regress brain on body mass Use regression analysis Log(BRAIN) = b 0 + b 1 (Log(BODY)) What value would b 1 take if brains scaled isometrically with body size? A sensible null model would be Brain = aBody 1 i.e. that brain size is a constant proportion of body size In reality, would you expect b 1 to actually be larger or smaller than this? What are the biological reasons that might govern this relationship?

13. Common allometric relationships Length scales with Mass 1/3 Surface area scales with Mass 2/3 Metabolic rate scales with Mass 3/4 Breathing rate or Heart rate with Mass 1/4 Abundance of species scales with (Body Mass) 3/4 – except parasites (Hechinger et al 2011, Science, 333, p445-448)