Business Mathematics and Basic Statistics · Ch 16 — Bivariate Statistics — Correlation and Regression

Regression Coefficients — byx and bxy

Correlation tells us how strongly two variables are linearly related, but it does not by itself give a way to predict one variable's value from the other's. Regression is the technique for doing exactly that: fitting a straight line through the bivariate data so that, given a value of one variable, the corresponding value of the other can be estimated.

Because either variable could be the one we want to predict, there are genuinely two regression relationships to consider, not one — predicting $y$ from a given $x$ , and predicting $x$ from a given $y$ — and each has its own regression coefficient, a slope-like quantity built from the same covariance and variances already used for Pearson's $r$ .

Note

Regression Coefficients

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

$b_{yx}$ (read 'regression coefficient of $y$ on $x$ ') is the slope used to predict $y$ from $x$ ; $b_{xy}$ ('of $x$ on $y$ ') is the slope used to predict $x$ from $y$ .

Both coefficients always carry the same sign as $r$ (all three come from the same numerator, the sum of cross-products $\sum(x-\bar{x})(y-\bar{y})$ ), and they are linked to Pearson's coefficient by a useful identity worth remembering as a built-in check on any calculation:

Note

Relation Between $r$ , $b_{yx}$ and $b_{xy}$

$r^2 = b_{yx}\cdot b_{xy}$ …

Definition 1Regression Coefficient of y on x (byx)

The slope of the line used to predict $y$ from a given $x$ : $b_{yx} = \text{Cov}(x,y) …

Definition 2Regression Coefficient of x on y (bxy)

The slope of the line used to predict $x$ from a given $y$ : $b_{xy} = \text{Cov}(x,y) …

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