Are standardized betas in multiple linear regression partial correlations?

Question

Since standardized betas are correlation coefficients in bivariate regression, is it the case that standardized betas in multiple regression are partial correlations?

Answer 1

Longer answer.

If I have this right --

Partial correlation:

$$ r_{y1.2} = \frac{r_{y1}-r_{y2}r_{12}}{\sqrt{(1-r^2_{y2})(1-r^2_{12})}} $$

equivalent standardized beta:

$$ \beta_1 = \frac{r_{y1}-r_{y2}r_{12}}{\sqrt{(1-r^2_{12})}} $$

As you see, the denominator is different. Indeed, since the additional term in the denominator of the first thing is between 0 and 1 (inclusive), it looks like $\beta_1$ will almost always be smaller than $r_{y1.2}$ though they could be equal if the two "independent variables" (and here's why I dislike that term) are ... well, independent - or at least uncorrelated.

Answer 2

I've in another question the following covariance matrix C for the three variables X,Y,Z given:
$$ \text{ C =} \small \left[ \begin{array} {rrr} 1&-0.286122&-0.448535\\ -0.286122&1&0.928251\\ -0.448535&0.928251&1 \end{array} \right] $$ From its cholesky-decomposition L $$ \text{ L =} \small \left[ \begin{array} {rrr} X\\Y\\Z \end{array} \right] = \left[ \begin{array} {rrr} 1&.&.\\ -0.286122&0.958193&.\\ -0.448535&0.834816&0.319215 \end{array} \right] $$ we can directly retrieve the partial correlation between Y,Z wrt. X as $ \small corr(Y,Z)_{\cdot X} = 0.958193 \cdot 0.834816 $ Now if we have the variables ordered such that the dependent variable is Z then the betas are computed by inverting the square-submatrix of the range in L which is populated by the independent variables X,Y: $$ L_{X,Y} = \small \left[ \begin{array} {rrr} 1&.\\ -0.286122&0.958193 \end{array} \right] $$ and its inverse, which is inserted into a 3x3 identity-matrix to form the matrix $M$: $$ M = \small \left[ \begin{array} {rrr} 1&.&.\\ 0.298605&1.043631&.\\ .&.&1 \end{array} \right] $$ Then the betas occur by the matrix-multiplication $ \beta = L \cdot M $

$$ \beta =\small \left[ \begin{array} {rrr} X\\Y\\Z \end{array} \right] = \small \left[ \begin{array} {rrr} 1&.&.\\ .&1&.\\ -0.199254&0.871240&0.319215 \end{array} \right] $$

which indicates, that the $\beta_X$ contribution for $Z$ is $\small \beta_X=-0.199254$ and the $\beta_Y$ contribution for $Z$ is $\small \beta_Y=0.871240$ . The unexplained variance in Z is the bottom-right entry squared: $\small resid^2= (0.319215)^2$
We see in $M$ that -being an inverse of a partial cholesky-matrix- it can contain values bigger than $1$ - and as well the Beta-matrix can then have entries bigger than 1.

So - to come back to your question- the partial correlation between $Y$ and $Z$ were the product of the entries in the second column of the L-matrix. The $ \small \beta_Y$ however is the product of the entry in the second column of the Z-row with the inverse of that in the Y-row and the relation between the concepts of partial correlation and $\small \beta$ can be described by this observation.

Additional comment: I find it a nice feature, that we get by this also the compositions of $X$ and $Y$ in terms of $X$ and $Y$ - which of course are trivially 1. It is also obvious, how we would proceed, if we had a second dependent variable, say $W$, and even that scheme can smoothly be extended to compute/show the coefficients of the generalization to the canonical correlation - but that's another story....