We now consider the case where the two sample pairs are not drawn independently because the two correlations have one variable in common.
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 (e.g. the correlation between Mother and Son is calculated by the function =CORREL(B4:B23,C4:C23) and is shown in cells H5 and G6).
The 95% confidence interval is calculated in the usual way using the fact that the standard error is the reciprocal of the square root term in the definition of t.
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.
Real Statistics Functions: The following array functions are provided in the Real Statistics Resource Pack.
Correl2OverlapTTest(r12, r13, r23, n, alpha, lab): array function which outputs the difference between the correlation coefficients r12 and r13, t statistic, p-value (two-tailed) and the lower and upper bound of the 1 – alpha confidence interval, where r12 is the correlation coefficient between the first and second samples, r13 is the correlation coefficient between the first and third samples, r23 is the correlation coefficient between the second and third samples and n is the size of each of the three samples. If lab = TRUE then the output takes the form of a 5 × 2 range with the first column consisting of labels, while if lab = FALSE (default) then the output takes the form of a 5 × 1 range without labels; if alpha is omitted it defaults to .05.
Corr2OverlapTTest(R1, R2, R3, alpha, lab) performs the two sample correlation test for samples R1, R2 and R3 where R2 is the overlapping sample. This array function return the array from =Correl2OverlapTTest(r12, r13, r23, n, alpha, lab) where r12 = CORREL(R1, R2), r13 = CORREL(R1, R3), r23 = CORREL(R2, R3) and n = the common sample size for R1, R2 and R3.
Figure 3 – Test using Real Statistics function
The same output is produced by the function
Observation: We can perform another version of this two sample correlation test using the Fisher transformation, as shown in Figure 4.
Figure 4 – Fisher analysis for Example 1
As you can see from range F15:F16, the 95% confidence interval calculated this time (taking absolute values) is (02202, .87714), which is not so different from the interval calculated in Figure 2, namely (016408, .834219).
Real Statistics Functions: The following array functions are provided in the Real Statistics Resource Pack to implement the Fisher test described above.
Correl2OverlapTest(r12, r13, r23, n, alpha, lab)
Corr2OverlapTest(R1, R2, R3, alpha, lab)
These functions are identical to Correl2OverlapTTest(r12, r13, r23, n, alpha, lab) and Corr2OverlapTTest(R1, R3, R2, alpha, lab), except that the Fisher transformation is used as described above and the output only has three elements: difference between the correlations and the end points of the confidence interval.
For Example 1, the output from =Correl2OverlapTest(F6,G6,G4,F8,,TRUE) is as shown in range F14:F16 of Figure 3. The same output is produced by the array function