Discussion
Suppose \(r_{12.3}\) is a partial correlation coefficient between \((X_{1}, X_{2})\) and \(X_{3}\). Then which of the following statements is not correct ?
Answer: C
The statement that is not correct is \(r_{12.3} \ge 0\) always.
A partial correlation is an ordinary correlation, between the residuals of \(X_{1}\) and \(X_{2}\) after each is regressed on \(X_{3}\) (A), so it lies between -1 and 1 (B) and can be negative. For example, with \(r_{12} = 0\), \(r_{13} = r_{23} = 0.5\), the formula gives
\[ \begin{aligned} r_{12.3} &= \frac{0 - 0.25}{\sqrt{0.75 \times 0.75}} \\ &= -\frac{1}{3} \end{aligned} \]
Option D is the standard formula for the partial correlation in terms of the simple correlations, so it is correct.
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