Re: partial correlation and partial correlation coefficient

From: Reef Fish (Large_Nassau_Grouper_at_Yahoo.com)
Date: 02/25/05 

Date: 25 Feb 2005 10:12:01 -0800

Charlie Yuan wrote:
> >
> >Partial correlation:
> > Rho(X1,X2 |X3) =Cov(X1,X2 |X3)/Sqrt(Var(X1 |X3)Var(X2 |X3)),
> >where Cov(X1,X2 |X3) = Cov(X1,X2)-Cov(X1,
> >X3)(Var(X3))^(-1)Cov(X3,X2),
>
> The formula was:
> Rho(X1,X2 |X3) =Cov(X1,X2 |X3)/Sqrt(Var(X1 |X3)Var(X2 |X3)),
> where
> Cov(X1,X2 |X3) = Cov(X1,X2) - Cov(X1,X3)(Var(X3))^( -
> 1)Cov(X3,X2),
>
>
> >
> >and partial Correlation coefficient:
> > r(xy) - r(xz) * r(yz)
> > r(xy|z) = ---------------------------------------- .
> > sqrt( (1 - r(xz)^2) * (1 - r(yz)^2) )
> >
> >
> >Could anybody tell me what are the differences between these two
> >measures? Which one do you think make more sense for my particular
> >problem? I shall appreciate any advice very much.

They are one and the same, apart from the fact that rho generally
denotes the population parameter and r the sample estimate.

The simplest way to think of the partial correlation between x and
y given a set of other variables z (one or more) is that it is the
SIMPLE correlation between two sets of residuals in the regressions
of x on z, and y on z.

This simple concept replaces a set of cumbersome formulas such as
your r(xy|z) above, as well as those partial correlation coefficients
expressed in terms of lower-order partial correlation coefficients.

-- Bob.