### log-log transformations

```Data Handling & Analysis
Allometry &
Log-log Regression
Andrew Jackson
[email protected]
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 the
shapes 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
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)
```