SL 4.10—Regression x on y

Regression direction matters when predicting one variable from another.

The regression of $x$ on $y$ predicts $x$ from a known value of $y$ and minimises squared residuals measured in the $x$-direction. It is not interchangeable with the regression of $y$ on $x$.

Example

If $x$ is fuel used and $y$ is distance, an $x$-on-$y$ line may predict fuel from a given distance. It should not be rearranged to predict distance from fuel; use the separately fitted $y$-on-$x$ line for that direction.

State which variable is being predicted, choose that variable 'on' the known variable, and keep prediction within a defensible data range.

A high correlation does not make the two regression lines identical.