## 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 H_{0} 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 H_{0} 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 H_{0} 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[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:

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 R_{i}=R_{j} 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.