Data Handling & Analysis Allometry & Log-log Regression

Data Handling & AnalysisAllometry &Log-log Regression Andrew Jackson a.jackson@tcd.ie

Linear type data • How are two measures related?

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

Mass of a cube • How does a cube scale with its length? • Mass = Density x Volume • Volume = L1 X L2 x L3= L3 • Volume = aLb • 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? • 2L1 X 2L2 x 2L3 = Mass x 23 • Isometic scaling • The object does not change shape as is grows or shrinks

Mass of a sphere • How does mass of a sphere change with length? • Volume = (4/3)πr3 = (4/3)π(L3/23) • Again, mass changes with Length3 • The difference here, compared with the cube, is the coefficient of Length • (4/3)π(L3/23) = (4/3)π(1/23) (L3) • So, we have • Volume = (some number)L3 • Volume = aLb • Where b = 3 in this case

A general equation • Mass = aLengthb • Where we might expect b = 3 • Take the log of both sides • Log(M) = log(aLb) • Log(M) = log(a) + log(Lb) • Log(M) = log(a) + b(log(L)) • Y = b0 + b1(X) • Log(a) = b0 • So…. a = exp(b0) • b1 = b and is simply the power in the allometric equation

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 = aLengthb • Spheres and cubes differ only in their coefficients • So a = exp(b0) tells us how theshapes differ between two species

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

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

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

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

Regress brain on body mass • Use regression analysis • Log(BRAIN) = b0+ b1(Log(BODY)) • What value would b1 take if brains scaled isometrically with body size? • A sensible null model would be • Brain = aBody1 • i.e. that brain size is a constant proportion of body size • In reality, would you expect b1 to actually be larger or smaller than this? • What are the biological reasons that might govern this relationship?

Common allometric relationships • Length scales with Mass1/3 • Surface area scales with Mass2/3 • Metabolic rate scales with Mass3/4 • Breathing rate or Heart rate with Mass1/4 • Abundance of species scales with (Body Mass)3/4 • except parasites (Hechinger et al 2011, Science, 333, p445-448)

Data Handling & Analysis Allometry & Log-log Regression