When you have multiple correlation coefficients (possibly from different runs of an experiment) you can perform the following tests:

1. The default test: are all the correlation coefficients the same

For this test H0 is: all correlation coefficients are equal. In the following example two waves contain four correlation coefficients and their respective sample size.

 corWave sizeWave 0.65 31 0.47 42 0.77 53 0.69 23

To run the test execute the following command:

`StatsMultiCorrelationTest/T=1/Q corWave,sizeWave`

The results appear in the "Multi-Correlation Test" table:

 n 4 ChiSquared 5.7686 degreesF 3 Critical 7.81473 zw 0.799829 rw 0.663941 chiSquaredP 5.90534

In this case the Chi-squared value is smaller than the critical value so H0 can't be rejected. The weighted correlation coefficient is rw=0.663941 and zw is its Fisher's z-transform.

2. Testing when the correlation coefficients are unequal:

 corWave1 sizeWave 0.46 31 0.42 42 0.77 53 0.69 23

To run the test execute the command:

`StatsMultiCorrelationTest/T=1/Q corWave1,sizeWave`

The results appear in the Multi-Correlation Test table:

 n 4 ChiSquared 9.46775 degreesF 3 Critical 7.81473 zw 0.733971 rw 0.625489 chiSquaredP 9.7349

In this case the Chi-squared statistic is greater than the critical value so H0 must be rejected. At this point it may be of interest to perform the multi-comparisons between the different correlation coefficients which can take the form of a Tukey test. To run the test use the command:

`StatsMultiCorrelationTest/T=1/Q /TUK corWave1,sizeWave`

The results appear in the Tukey Multi-Correlation Test table:

 Pair Difference SE q qc conclusion R3_vs_R0 0.410334 0.20702 1.9821 3.63316 1 R3_vs_R1 0.459953 0.194475 2.3651 3.63316 1 R3_vs_R2 -0.11268 0.187083 0.60231 3.63316 1 R2_vs_R0 0.523016 0.166905 3.13363 3.63316 1 R2_vs_R1 0.572636 0.15106 3.79067 3.63316 0 R1_vs_R0 -0.04961 0.17515 0.28329 3.63316 1

As one might expect, the q-values indicate greatest variation between corWave1 and corWave1 and so the hypothesis R2=R1 must be rejected.

It is interesting to note the effect of sample size. Using the same corWave1 as above we have increased the number of samples corresponding to the highest correlation coefficient:

 corWave1 sizeWave1 0.46 31 0.42 25 0.77 72 0.72 23

Repeating the last test:

`StatsMultiCorrelationTest/T=1/Q /TUK corWave1,sizeWave1`

The results appear in the "Multi-Correlation Test" table and in the "Tukey Multi-Correlation Test" table:

 n 4 ChiSquared 8.86816 degreesF 3 Critical 7.81473 zw 0.808126 rw 0.668555 chiSquaredP 9.1714

Again the Chi-squared statistic is larger than the critical value, but this time, the Tukey test gives:

 Pair Difference SE q qc conclusion R3_vs_R0 0.410334 0.20702 1.9821 3.63316 1 R3_vs_R1 0.459953 0.218466 2.10538 3.63316 1 R3_vs_R2 -0.112683 0.179573 0.627504 3.63316 1 R2_vs_R0 0.523016 0.158441 3.30102 3.63316 1 R2_vs_R1 0.572636 0.173129 3.30757 3.63316 1 R1_vs_R0 -0.0496193 0.201456 0.246304 3.63316 1

At least at the 0.05 significance level, the Tukey test does not find any combination of correlation coefficients where the hypothesis of Ri=Rj can be rejected.

3. Increasing the significance to 0.1 we have:

`StatsMultiCorrelationTest/T=1/Q /TUK/ALPH=0.1 corWave1,sizeWave1`
 n 4 ChiSquared 8.86816 degreesF 3 Critical 6.25139 zw 0.808126 rw 0.668555 chiSquaredP 9.1714

and

 Pair Difference SE q qc conclusion R3_vs_R0 0.410334 0.20702 1.9821 3.24045 1 R3_vs_R1 0.459953 0.218466 2.10538 3.24045 1 R3_vs_R2 -0.112683 0.179573 0.627504 3.24045 1 R2_vs_R0 0.523016 0.158441 3.30102 3.24045 0 R2_vs_R1 0.572636 0.173129 3.30757 3.24045 0 R1_vs_R0 -0.0496193 0.201456 0.246304 3.24045 1

At the 0.1 significance level we find that the equality of R2 with both R0 and R1 is rejected.

4. Example of testing contrasts

Suppose the hypothesis that we want to test is: r0+r2=r1+r3. The appropriate contrast wave is:

 constrastWave 1 -1 1 -1

To run the test execute the command:

`StatsMultiCorrelationTest/T=1/Q /CONT=contrastWave  corWave1,sizeWave1`

The results appear in the Multi-Correlation Test table:

 n 4 ChiSquared 8.86816 degreesF 3 Critical 7.81473 zw 0.808126 rw 0.668555 chiSquaredP 9.1714 ContrastSE 0.381656 ContrastS 0.425257 Contrast_Critical 2.79548

The first part of the table consists of the results of the standard multi-correlation test as in (1) above. The contrast results consist of the SE value, the contrast statistic S and the critical value. Clearly, ContrastS<<Contrast_Critical and so the hypothesis defined by the contrast equation above is accepted. Forum Support Gallery