Question

If r 12 =0.3, r 23 = 0.41, r 31 =0.18, find the partial correlation coefficient
between x 1 and x 2 keeping x 3 as constant. Find the multiple correlation
coefficient R 1.23 , R 2.13

Answer 1

$1. \text{ Partial correlation coefficient between } x_1 \text{ and } x_2\\ \text{keeping }x_3\text{ as const}:\\ r_{12.3}=\frac{r_{12}-r_{13}r_{23}}{\sqrt{1-r_{13}^2}\sqrt{1-r_{23}^2}}=\frac{0.3-(0.18)(0.41)}{\sqrt{1-0.18^2}\sqrt{1-0.41^2}}\approx 0.25.\\ 2. \text{ Multiple correlation coefficient:}\\ R_{1.23}=\sqrt{\frac{r_{12}^2+r_{13}^2-2r_{12}r_{13}r_{23}}{1-r_{23}^2}}=\sqrt{\frac{0.3^2+0.18^2-2(0.3)(0.18)(0.41)}{1-0.41^2}}\approx 0.31.\\ R_{2.13}=\sqrt{\frac{r_{12}^2+r_{23}^2-2r_{12}r_{23}r_{13}}{1-r_{13}^2}}=\sqrt{\frac{0.3^2+0.41^2-2(0.3)(0.41)(0.18)}{1-0.18^2}}\approx 0.47.$