# Chi-square test to see if set of dependent correlations are equal

I've got four dependent correlations (i.e., from the same sample) that involve predictor variables $(A,B,C,D)$ and an outcome variable $(E)$. $N=172$.

$$\begin{array}{c|cccc}{\rm Statistic}&AE&BE&CE&DE\\\hline r&-.480&.276&.395&.327\\p&< .01&< .01&< .05&< .01\end{array}$$

I am trying to run three chi-square tests on this set of four dependent correlations, where $H_0: r_{AE} = r_{BE} = r_{CE} = r_{DE} = 0$; and $H_1$ is simply "NOT $H_0$". Assuming $H_0$ is rejected, I will then be testing: $H_0: r_{AE} = r_{BE} = r_{CE} = r_{DE}$; with $H_1$ being "NOT $H_0$" again. Assuming that $H_0$ is rejected again, I will then be testing $H_0: r_{BE} = r_{CE} = r_{DE}$.

SPSS cannot help with me comparing dependent correlations and Field's "Discovering Statistics..." (2013) only discusses comparing two dependent correlations, not an entire set. So, I have been directed to several places so far, with no luck:

1. "Multicorr" by Steiger (http://www.statpower.net/Software.html). I simply cannot figure out how to use it.
2. A paper by Steiger ("Tests for Comparing Elements of a Correlation Matrix", Psychological Bulletin, 1980, 87(2)). But I cannot figure out how/where my numbers fit with the formulas he has provided there.

Ideally, I'd just like the mathematical formula(s) I can use to get my chi-square observed value (I have the critical value tables already). Any guidance that could be offered would be most appreciated.

Okay, so with some help I managed to figure out how to use "Multicorr". Here's the input files that I created (the formatting of these files is discussed in the Multicorr documentation):

First Test

CORRELATION MATRIX -- IDENTITY
5  172   4 9876
(5F7.2)
1.00
-0.480 1.00
-0.158 0.276  1.00
-0.246 0.395  0.234  1.00
-0.257 0.327  0.122  0.362  1.00
2    1    0.0
3    2    0.0
4    2    0.0
5    2    0.0


The first block of numbers is the correlation matrix for all of the variables involved (i.e., A, B, C, D, E). The second block of numbers is the indices of the correlations that I want to test, and what I want to test them against (e.g., "2 1" is 0.276)

Second Test

CORRELATION MATRIX -- EQUALITY
5  172   4 9876
(5F7.2)
1.00
-0.480 1.00
-0.158 0.276  1.00
-0.246 0.395  0.234  1.00
-0.257 0.327  0.122  0.362  1.00
2    1    1
3    2    1
4    2    1
5    2    1


By putting "1" as the value to test against, Multicorr assumes that you are testing correlations AGAINST EACH OTHER. So all the indices that are tested against "1" are, in fact, tested against each other (e.g., if you put "1" for half of them and "2" for the other half, they would only be tested against the ones with the same number)

Third Test

PAPER -- CORRELATION MATRIX -- LIMITED EQUALITY
5  172   3 9876
(5F7.2)
1.00
-0.480 1.00
-0.158 0.276  1.00
-0.246 0.395  0.234  1.00
-0.257 0.327  0.122  0.362  1.00
3    2    1
4    2    1
5    2    1


The output of Multicorr, on the last line, gives you the chi-square value, the degrees of freedom, and the p-value.

Note: You are going to need DOS Box to run Multicorr.