Business Mathematics and Statistics · Ch 9 — Correlation and Regression Analysis

Regression Coefficients — byx and bxy

Correlation tells us how strongly two variables are linearly related, but not how to predict one from the other. Regression fits a straight line through the bivariate data for exactly this purpose — estimating one variable's value given the other's.

Because either variable could be the one being predicted, there are genuinely two regression relationships, each with its own regression coefficient:

Note

Regression Coefficients

$b_{yx} = \dfrac{\sum(x_i-\bar x)(y_i-\bar y)}{\sum(x_i-\bar x)^2} \qquad\qquad b_{xy} = \dfrac{\sum(x_i-\bar x)(y_i-\bar y)}{\sum(y_i-\bar y)^2}$

$b_{yx}$ (' $y$ on $x$ ') is the slope used to estimate $y$ from a given $x$ ; $b_{xy}$ (' $x$ on $y$ ') is the slope used to estimate $x$ from a given $y$ .

Both coefficients always carry the same sign as $r$ (all three share the same numerator, $\sum(x-\bar x)(y-\bar y)$ ), and are linked to $r$ by a useful built-in check:

Note

Relation Between r, byx and bxy

$r^2 = b_{yx}\cdot b_{xy}$

Once both regression coefficients are found, squaring the previously-computed $r$ for the same data and comparing it against $b_{yx}\cdot b_{xy}$ is a reliable, easy cross-check on the whole calculation. …

Definition 5Regression Coefficient of y on x (byx)

The slope of the line used to estimate $y$ from a given $x$ : $b_{yx} = \sum(x-\bar x)(y-\bar y)/ …

Definition 6Regression Coefficient of x on y (bxy)

The slope of the line used to estimate $x$ from a given $y$ : $b_{xy} = \sum(x-\bar x)(y-\bar y)/ …

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