linear regression (stats)

A technique to predict unknown Y scores based on correlated X scores. The resulting Y that we find for a given X is known as $Y^{\prime }$ or “Y prime”.

The linear regression equation is $Y^{\prime }=bX+a$ where $b$ is the slope of the line, X is the score on the X axis and $a$ is the Y intercept. The accuracy of this prediction depends on how good of an $r$ value we end up with. We could also look this up through the standard error of the estimate, but it’s a more advanced concept.

The formula for slope is $b=\frac{N(\Sigma XY)-(\Sigma X)(\Sigma Y)}{N(\Sigma X^2)-(\Sigma X{)}^2}$. Formula for the y-intercept is $a=\bar{Y}-(b)(\bar{X})$ where the bar is the mean of that variable.

Definitions

predictor variable : aka X. This is the value that someone did research about.

criterion variable : aka Y. This is the effect-side of predictor variable.

linear regression equation : the equation that produces the value of $Y^{\prime }$ at each X. It defines the straight line that summarizes a relationship.

proportion of variance accounted for : aka $r^2$. It’s like effect size aka magnitude of the difference, but b/c we can’t say that changing X = a change to Y.. it’s named differently. The general idea here is that we could have data like (x=1, y=2), (x=1, y=3). Same X, but with different variance in y. If you have (x=1, y=2)and (x=2,y=6).. we can ascribe some of that difference to a change in X. It’s scale is 0 (no change in Y as X changes) or $\pm 1$ (Y only chnages when X changes).