Multi-Correlation Test

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.

corWavesizeWave
0.6531
0.4742
0.7753
0.6923

To run the test execute the following command:

StatsMultiCorrelationTest/T=1/Q corWave,sizeWave

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

n4
ChiSquared5.7686
degreesF3
Critical7.81473
zw0.799829
rw0.663941
chiSquaredP5.90534

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:

corWave1sizeWave
0.4631
0.4242
0.7753
0.6923

To run the test execute the command:

StatsMultiCorrelationTest/T=1/Q corWave1,sizeWave

The results appear in the Multi-Correlation Test table:

n4
ChiSquared9.46775
degreesF3
Critical7.81473
zw0.733971
rw0.625489
chiSquaredP9.7349

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:

PairDifferenceSEqqcconclusion
R3_vs_R00.4103340.207021.98213.633161
R3_vs_R10.4599530.1944752.36513.633161
R3_vs_R2-0.112680.1870830.602313.633161
R2_vs_R00.5230160.1669053.133633.633161
R2_vs_R10.5726360.151063.790673.633160
R1_vs_R0-0.049610.175150.283293.633161

As one might expect, the q-values indicate greatest variation between corWave1[2] and corWave1[1] 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:

corWave1sizeWave1
0.4631
0.4225
0.7772
0.7223

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:

n4
ChiSquared8.86816
degreesF3
Critical7.81473
zw0.808126
rw0.668555
chiSquaredP9.1714

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

PairDifferenceSEqqcconclusion
R3_vs_R00.4103340.207021.98213.633161
R3_vs_R10.4599530.2184662.105383.633161
R3_vs_R2-0.1126830.1795730.6275043.633161
R2_vs_R00.5230160.1584413.301023.633161
R2_vs_R10.5726360.1731293.307573.633161
R1_vs_R0-0.04961930.2014560.2463043.633161

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
n4
ChiSquared8.86816
degreesF3
Critical6.25139
zw0.808126
rw0.668555
chiSquaredP9.1714

and

PairDifferenceSEqqcconclusion
R3_vs_R00.4103340.207021.98213.240451
R3_vs_R10.4599530.2184662.105383.240451
R3_vs_R2-0.1126830.1795730.6275043.240451
R2_vs_R00.5230160.1584413.301023.240450
R2_vs_R10.5726360.1731293.307573.240450
R1_vs_R0-0.04961930.2014560.2463043.240451

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:

n4
ChiSquared8.86816
degreesF3
Critical7.81473
zw0.808126
rw0.668555
chiSquaredP9.1714
ContrastSE0.381656
ContrastS0.425257
Contrast_Critical2.79548

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.