Discussion
For the three random variables X1, X2, X3 let \(\hat{X}_{1}\) and \(\hat{X}_{2}\) be the fitted values of X1 and X2 respectively after regressing on the linear function of X3. The partial correlation coefficient between X1 and X2 after removing the effect of X3 is the simple correlation coefficient between :
Answer: D
The partial correlation is the correlation between \(X_{1} - \hat{X}_{1}\) and \(X_{2} - \hat{X}_{2}\).
Removing the effect of \(X_{3}\) means regressing \(X_{1}\) and \(X_{2}\) on \(X_{3}\) and keeping the residuals \(X_{1} - \hat{X}_{1}\) and \(X_{2} - \hat{X}_{2}\), the parts not explained by \(X_{3}\). The partial correlation \(r_{12.3}\) is the simple correlation between these residuals.
Correlating \(X_{1}\) with \(\hat{X}_{2}\), or \(\hat{X}_{1}\) with \(X_{2}\), mixes in the part explained by \(X_{3}\). Subtracting \(X_{3}\) itself, as in \(X_{1} - X_{3}\), removes \(X_{3}\) with coefficient 1 instead of the fitted regression coefficient.
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