Is there a difference between semipartial correlation and regression coefficient in multiple regression?

Question

I am preparing a presentation about multiple regression. Most of my sources seem to equal unstandardized coefficients in multiple regression with the semipartial correlation of that IV with the DV. But one book says there is a slight difference:

both terms have the same enumerator, but the differ in the denominator: the semipartial correlation coefficient has a quare root in the denominator (sqr(1-r²), but the regression coefficient ß has none (1-r²). the author states that the more the predictors correlate, the more will the two values differ.

I could not find this information anywhere else. is this a fact or what should i think of it?

Answer 1

While thorough and ultimately correct, the comment of @ttnphns given to the question is slightly misleading in the sense that it focuses on the similarities between the standardized regression coefficient and the partial correlation, while the more obvious comparison would be between standardized regression coefficient and the more closely related semipartial correlation [but see the thoughtful answer of @ttnphns in response to my post, clarifying his point about partial correlations].

Indeed, the only difference is that the semipartial takes the square root of the denominator. The result is that the semipartial is bounded between -1 and +1, while Beta is not.

Aside from the algebraic similarities, semipartial correlations are also conceptually closest to regression coefficients. In a regression analysis, we try to measure the unique explanatory power of predictors, i.e. the unique part of the total variance of Y that can be explained by X1, controlled for the other X-variables. That is, we residualize each X on other predictors to get its unique effect, but we do not residualize Y, as in the partial correlation.

For an excellent Powerpoint presentation on this topic, see these slides by Michael Brannick of the University of South Florida.

Answer 2

In their answer @Soporiferous correctly says of the relation between the semipartial correlation and the standardized beta coefficient, and hastes to label my old comment to the question (about the sooner existence of a relation between the beta and partial correlation) "slightly misleading". But in my comment, I implied Beta of another regression (another dependent variable) than @Soporiferous seemingly implied.

Let us have 3 variables X, Y, Z.

Partial correlation between X and Y (Z partialled out from both) is

$r_{xy.z} = \frac{r_{xy} - r_{xz}r_{yz} }{\sqrt{ (1-r_{xz}^2)(1-r_{yz}^2) }}$.

While semipartial or part correlation between X and Y (with Z partialled out from Y) is

$r_{x(y.z)} = \frac{r_{xy} - r_{xz}r_{yz} }{\sqrt{1-r_{yz}^2 }}$.

@Soporiferous correctly notices (by linking to an outer source) that the last formula is very similar to the formula of a beta regression coefficient:

$\beta = \frac{r_{xy} - r_{xz}r_{yz} }{{1-r_{yz}^2 }}$, with the only difference being is that root taken in the denominator.

True observation; however mind that this beta is $\beta_y$ in the regression where X is the dependent while Y and Z are predictors. Semipartial $r_{x(y.z)}$ squared is equal to the rise of R-square in reaction to the inclusion of Y to the model having consisted only of Z.

So, the semipartial is structurally similar to $\beta_y$, - and not to $\beta_x$ (of regression where Y is the dependent which idea would probably come to mind at first place "by default").

But partial $r_{xy.z}$ is related to both the $\beta_y$ (where X is the dependent) and to the $\beta_x$ (where Y is the dependent):

$\beta_x = \frac{r_{xy} - r_{xz}r_{yz} }{{1-r_{xz}^2 }}$.

Saying (in my comment to the question and in my answer linked just now) that regression coefficient beta is directly related to partial correlation I meant that regression where X is a regressor and the partial correlation considered is, too, between the regressor X and the dependent Y. The very title of the question induces such interpretation of it.