Interactions - using ratio of variables

Question

I have 3 variables, colony size, colony age and growth rate (colony size/age).

I am interested to predict various other properties ($y$) of a colony using these 3 variables;

$y = a_1 \text{ size} + a_2\text{ age} + a_3\text{ growth rate}$

since growth rate is really size/age, I was thinking it is effectively a kind of interaction between size and age.

$y = a_1 \text{ size} + a_2\text{ age} + a_3\text{ (size/age)}$

Normally I see one models interactions as the product of the variables, not as their ratio. I was wondering about the consequences of using the ratio, not the product?

Answer 1

Given the basic arithmetic:

$$\frac{x}{y} = x \times \frac{1}{y}$$

this does not really matter. So growth rate in is: $\text{size} \times (1 / \text{age}) $ and you already have this effect in your interaction. If it makes interpretation easier you can always work with reverse age rather than age as independent variable.

Answer 2

As size = age * growth rate, you could also see size as interaction between age and growth rate.