Multiple and Partial Correlation

Multiple and Partial Correlation

1. With only two predictors
1. The beta weights can be computed as follows:
 (1) (2)

1. Multiple R can be computed several ways. From the simple correlations, as
 (3)
or from the beta weights and validities as
 (4)
1. Semipartial correlations in general equal the square root of complete minus reduced. These are called semipartial correlations because th e variance of the other controlled variable(s) is removed from the predictor, but not from the criterion. Therefore, in the two predictor case, they are equal
 (5)
Using Equation 3 above and some algebra
 (6) (7) (8)
So the relationship between the semipartial correlation and the beta weight from Equations 1 and 7 is
 (9)
1. Partial correlations differ from semipartial correlations in that the partialled (or covaried) variance is removed from both the criterion and the predictor. The squared partial correlation is equal to complete minus reduced divided by 1 minus reduced. In the two variable case the equation is
 (10)
Again using Equation 3 and some more algebra
 (11) (12) (13)
The relation between partial correlations and beta weights for the two predictor problem turns out to be
 (14)
So semipartial correlations are directional but partial correlations are nondirectional.

1. Following Cohen and Cohen (1975, p. 80), we can think of all these in terms of what they call Ballentines (we can call Mickeys)

Here, the total Y variance is a+b+c+e = 1.

The semipartial correlations are:

And the partial correlations are:

1. With more than two predictors
1. First the relation between a multiple R and various partial r's.
 (15)
This should remind the reader of stepwise multiple regression where each new variable is entered while controlling the variance explained by earlier entered variables. Therefore, if we could compute the higher order partial correlations, we could do multiple regression by hand. A recurrence relationship allows us to do just that, which is
 (16)
Unfortunately, the work involved in solving all the necessary partial correlations is about the same as the work required to solve the normal equations in the first place, but at least each step is interpretable. Again in the general case the rel ation between partial correlations and beta weights is
 (17)