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.

Good Nigt mr.charles, i want to ask you about the simbol of ~T(n-3), what does it mean? especially the T simbol. Thank you.

from Indonesia 🙂

x ~ T(n-3) means that the random variable x has a t distribution with n-3 degrees of freedom. It can also mean that x has approxomately a t distribution with n-3 degrees of freedom.

Charles

If I understand it correct this is versus the alternative that their IQ:s are different.

Now if I want to make a one sided test of this, should go about the same way as in previous examples?

Gustaf,

This is a one sided test.

Charles

Dear Professor Zaiontz,

I am a medical student persueing master degree from China. I am really greatful for your sharing this wonderful formula with us on the web, for I want to take this statistic into my writing article. But I wonder where this formula comes from. Would you please tell me the references? Thanks a lot.

My best wishes.

Yoyo Gong

The reference is Howell, D. C. (2010). Statistical methods for psychology (7th ed.). Wadsworth, Cengage Learning.

Charles

Thanks a lot. This is quite useful for me.

Hi Charles,

Thank you very much for providing the well illustrated example. I noticed one typo error in the formular t (r12-r13)^2/4*(1-r23)^3 should be (r12+r13)^2/4*(1-r23)^3

Hi Baogui,

Good catch. This is indeed a typo. I have now corrected the formula. Thanks for you help.

Charles

Great. Do you have any formular or procedure to test if there is significant difference between more than two independent samples? I get one in the note of Alan Pickering http://homepages.gold.ac.uk/aphome/correlnotes.doc, but I am not sure it is correct.