Statistics · Ch 3 — Linear Regression

Regression Lines: Y on X and X on Y

There are two regression lines for any set of bivariate data:

1. Regression line of $Y$ on $X$ — used to estimate $Y$ for a given value of $X$ . It is obtained by minimizing the sum of squares of the vertical deviations of the actual points from the line (errors measured in the $Y$ -direction). In deviation-from-mean form:

(Y - \bar{Y}) = b_{yx}(X - \bar{X})

2. Regression line of $X$ on $Y$ — used to estimate $X$ for a given value of $Y$ . It is obtained by minimizing the sum of squares of the horizontal deviations (errors measured in the $X$ -direction):

(X - \bar{X}) = b_{xy}(Y - \bar{Y})

Here $b_{yx}$ and $b_{xy}$ are the two regression coefficients — the slopes of the two lines (covered in the next section).

Note

Both regression lines always pass through the point $(\bar{X}, \bar{Y})$ — the point formed by the two means. This is a useful check: if a computed line does not pass through the mean point, an arithmetic error has been made somewhere. …

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