Correlation, Estimation and Finding the Means

This section ties the regression coefficients to the correlation coefficient and shows the two standard uses of the regression equations: estimating values and recovering the means.

Relationship with the correlation coefficient. From Property 2,

with the sign taken as the common sign of $b_{yx}$ and $b_{xy}$. Conversely, if $r$ and the standard deviations are known, the coefficients follow directly:

Estimating a value. To estimate $y$ for a given $x$, substitute that $x$ in the line of $Y$ on $X$; to estimate $x$ for a given $y$, substitute that $y$ in the line of $X$ on $Y$. The estimate is only as reliable as the correlation is strong — a value of $r$ close to $\pm 1$ makes the two lines nearly coincide and the estimate dependable, while $r$ near $0$ makes estimation almost worthless.

Finding the means from the two regression lines. Because both lines pass through $(\bar x,\bar y)$, the means are simply the point of intersection of the two regression equations. Solve the two equations simultaneously; the solution $(x,y)$ is $(\bar x,\bar y)$.

Which line is which? When you are handed two lines but not told which is $Y$ on $X$, use Property 3 as a test. Assume one line is $Y$ on $X$ (so its slope, solved as $y$ in terms of $x$, is $b_{yx}$) and the other is $X$ on $Y$ (solve it as $x$ in terms of $y$ to read $b_{xy}$). If the product $b_{yx}\cdot b_{xy}\le 1$, the assumption is correct; if it exceeds $1$, swap the roles. …