• Allometry is the study of how physical traits change as overall body size of an individual changes
• Mathematically, it is expressed as:

\[ Y = aX^{b} \]

• \(Y\): the biological trait being measured
• \(X\): body mass
• \(a\): a constant proportionality factor
• \(b\): the scaling exponent, describing how fast \(Y\) changes relative to \(X\)

• For our question, the equation looks more like this and suggests that brain size increases exponentially with body mass:

\[ \text{Brain size} = a\,(\text{Body mass})^{b} \]

• To get a linear regression, we must take the log of each side:

\[ \log (Y) = \log (aX^{b}) \]

\[ \log Y = b \log X + \log a \]

Depending on the value of the exponent \(b\), traits can scale:

• Isometrically (\(b = 1\)): the trait grows at the same rate as the body
• Positively (\(b > 1\)): the trait grows faster than body size
• Negatively (\(b < 1\)): the trait grows slower than body size

For our data, this looks like:

\[ \log (\text{Brain size}) = \log (a \text{Body Size}^{b}) \]

\[ \log \text{Brain size} = b \log \text{Body Size} + \log a \]

We predict that brain size will scale isometrically with body size. Since larger mammals are typically more biologically complex than smaller ones, it is likely that they have bigger, more complex brains.

• Using linear regressions, we can take messy exponential relationships and linearize them by taking the log of the equation to see clear trends.
• In our data, brain size tends increase with body size for the mammals included in the study