# Independent variables formulas in multiple regression

I have a variable X1 = (a - b) / (a + b). This variable shows a higher correlation to Y that any of (a, Y) and (b, Y).

In a multiple regression model like Y ~ X1, X2, does it make sense to use the X1 formula, or should I always use the base variables a and b?

In this post somebody pointed out that

Intuitively you'd be a lot more confident about inferences from an observed ratio of 1 (boys to girls) if it came from seeing 100 boys and 100 girls than > from seeing 2 and 2. Consequently, if you have covariates you'll have more information about their effects and potentially a better predictive model.

Fine, but can the multiple linear model rebuild the same (X1, Y) predictive relationship just by a and b least square analysis?

• As @jbowman gets at below, I think your theory about how the world is likely to work is important for choosing how you specify your model... do you have a good theoretical reason to believe a variable Z = (a-b)/(a+b) is related to Y? – Michael Bishop Dec 5 '11 at 0:19

The short answer is no. The relationship $Y = \beta_1 (a-b)/(a+b) + \beta_2 X_2 + e$ is not the same as $Y = \beta_1 a + \beta_2 b + \beta_3 X_2 + e$, and you can't get from one to the other.