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Allometry and Log-Log Regression: Scaling Relationships in Biology and Ecology

This analysis explores the relationship between length and mass in biological organisms using allometry and log-log regression. It addresses how the mass of organisms scales with length, revealing crucial insights into metabolic costs, predation, diet, and ecological scaling laws. The study employs geometric shapes like cubes and spheres to illustrate how mass changes with dimensions, introducing concepts such as isometric scaling and the coefficients in allometric equations. Additionally, it examines brain-body mass relationships through regression analysis, highlighting common allometric scaling patterns in biology.

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Allometry and Log-Log Regression: Scaling Relationships in Biology and Ecology

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  1. Data Handling & AnalysisAllometry &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 = 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

  5. 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

  6. 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

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

  13. 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)

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