y- y=r(s_y)/(s_x)(x- x), x- x=r(s_x)/(s_y)(y- y) 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. A prediction retains residual uncertainty. Choose the correct regression direction and avoid unjustified extrapolation beyond the observed data range. Perfect positive: r=1 Perfect negative: r=-1 Strong positive: r 0.8 Strong negative: r -0.8 Weak positive: r 0.4 Weak negative: r -0.4 No linear correlation: r=0 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= y and x= x.

\[y-\bar y=r\frac{s_y}{s_x}(x-\bar x),\qquad x-\bar x=r\frac{s_x}{s_y}(y-\bar y)\]

Regression equations

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.

A prediction retains residual uncertainty. Choose the correct regression direction and avoid unjustified extrapolation beyond the observed data range.

Prediction cautions

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 \(y=\bar y\) and \(x=\bar x\).

Regression Lines, Prediction and Strength

Regression Lines, Prediction and Strength