Partial correlation between x1 and x2, adjusting for x3, ..., xp

1) Fit regression models to x1 and x2:

   x1hat = b0 +b1x3 +b2x4 + ... + b(p-2)xp      and
   x2hat = b0* +b1*x3 +b2*x4 + ... + b(p-2)*xp

2) Calculate residuals for each:

   resx1 = x1 -x1hat   and   resx2 = x2 -x2hat

3) Now compute the correlation between these residuals:
   corr(resx1, resx2)
   This is the partial correlation between x1 and x2, adjusting
   for x3, ..., xp.

 This idea is useful in factor analysis when we look at the partial correlation
 between the x's after adjusting for the common factor f's.  If the factor 
 analysis model was successful, then after we adjust for the factors f, there
 should not be much correlation among the original x's.