In this section we will consider the case where the two sample pairs are not drawn independently, usually either because the two correlations have one variable in common or because the two variables are correlated at one moment in time and again at another moment in time.
Example 1: IQ tests are given to 20 couples. The oldest son of each couple is also given the IQ test with the scores displayed in Figure 1. We would like to know whether the correlation between son and mother is the significantly different from the correlation between son and father.
Figure 1 – Data for Example 1
We will use the following test statistic
where S is the 3 × 3 sample correlation matrix and
For this problem the results are displayed in Figure 2, where the upper part of the figure contains the correlation matrix S.
Figure 2 – Analysis for Example 1
Since p-value = .042 < .05 = α (or t < t-crit) we reject the null hypothesis, and conclude that the correlation between mother and son is significantly different from the correlation between father and son.