Regression Lines, Prediction and Strength

Use the regression of \(y\) on \(x\) to predict \(y\) from a given \(x\), and the regression of \(x\) on \(y\) to predict \(x\) from a given \(y\). Substitution into the appropriate fitted equation is exactly how a fitted response is estimated.

When both equations are drawn with \(x\) horizontal and \(y\) vertical, the \(x\)-on-\(y\) line has the greater gradient magnitude for \(0<|r|<1\), assuming non-zero variances. The lines coincide exactly when \(|r|=1\). At \(r=0\), they reduce to \(y=\overline{y}\) and \(x=\overline{x}\).

The regression of \(y\) on \(x\) minimises the sum of squared vertical residuals; the regression of \(x\) on \(y\) minimises the sum of squared horizontal residuals. Choose the line by the variable being predicted, including when neither variable has an established causal role.