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CSS Statistics QUESTION #1982
Question 1
The partial correlation coefficient \(r_{12.3}\) between \(X_1\) and \(X_2\) after eliminating the linear effect of \(X_3\) is computed as the simple correlation between:
  • \(X_1\) and \(X_2\) directly
  • The residuals \(X_{1.3}\) and \(X_{2.3}\) from their respective regressions on \(X_3\)✔️
  • The standardized values of \(X_1\) and \(X_2\)
  • The ranked values of \(X_1\) and \(X_2\)
Correct Answer Explanation
The partial correlation \(r_{12.3}\) is defined as the Pearson correlation between the residuals \(X_{1.3} = X_1 - \hat{X}_{1|X_3}\) and \(X_{2.3} = X_2 - \hat{X}_{2|X_3}\). This removes the linear influence of \(X_3\) from both \(X_1\) and \(X_2\) before computing their correlation.