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Suppose \(R_{1.23}\) is a multiple correlation coefficient between \(X_{1}\) and \((X_{2}, X_{3})\). Then which of the following statements cannot be true ?

  • A.\(R_{1.23}\) is a maximum correlation between \(X_{1}\) and \((aX_{2} + a_{3}X_{3})\) where \(a_{1}\) and \(a_{2}\) are any real numbers.
  • B.\(R_{1.23}\) is a simple correlation between \(X_{1}\) and \(\hat{X}_{1}\), where \(\hat{X}_{1}\) is a line of best fit based on \(X_{2}\) and \(X_{3}\).
  • C.\(R_{1.23} = -0.4\)
  • D.\(R_{1.23} = 0.4\)

Answer: C

The statement that cannot be true is \(R_{1.23} = -0.4\).

The multiple correlation coefficient is the correlation between \(X_{1}\) and its best linear predictor \(\hat{X}_{1}\) from \(X_{2}, X_{3}\), and equivalently the maximum correlation between \(X_{1}\) and any linear combination of \(X_{2}, X_{3}\) (statements A and B, despite the paper's slips in the coefficient names). A maximum of correlations over combinations and their negatives is never negative, so \(0 \le R_{1.23} \le 1\).

So 0.4 is a possible value, and \(-0.4\) is impossible.

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