The following data represent 4 groups of measurements of a single factor. Note that not all waves have the same length. The Kruskal-Wallis test is a non-parametric test that uses ranks of the data to test the null hypothesis that there is no difference among groups.

 wave1 wave2 wave3 wave4 5.1 6.3 5.5 5.2 6.7 5 5.6 5.8 6 3.9 6.7 6.2 6.2 6.2 5.3 6 5.6 6.1 5.3 7 6.3 4.3 6.7 6.3 5.5 7.6 8.3

To run the test execute the following command:

StatsKWTest /T=1/Q wave1,wave2,wave3,wave4

The results are displayed in the Kruskal-Wallis Test table:

 Num_Groups 4 Num_Valid_Data 27 Alpha 0.05 H 2.93763 Hc_Chi_square_approx 7.81473 P_Chi_square_approx 0.401342 Hc_Wallace_approx 7.81984 P_Wallace_approx 0.411107

The H statistic is clearly below the critical value so H0 (data in the input waves are the same) is accepted.

The following wave represents a 5th group of data for the same factor:

 wave5 6.7 7.3 7 6.7 7.2 7.1 7.2

To run the test execute the following command:

StatsKWTest /T=1/Q wave1,wave2,wave3,wave5

The results are displayed in the Kruskal-Wallis Test table:

 Num_Groups 4 Num_Valid_Data 27 Alpha 0.05 H 12.677 Hc_Chi_square_approx 7.81473 P_Chi_square_approx 0.00538994 Hc_Wallace_approx 7.81984 P_Wallace_approx 0.00283628

In this case the statistic H clearly indicates that H0 should be rejected but there is no indication that the variation is mostly due to wave5. To perform a multi-comparison test execute the following command:

StatsNPMCTest/T=1/DHW/Q wave1,wave2,wave3,wave5

The results are shown in the NP Multiple Comparison (DHW) table:

 Diff SE Q Qc Conclusion wave5_vs_wave2 14.7381 4.39832 3.35084 2.63826 0 wave5_vs_wave1 11.1429 4.22577 2.63688 2.63826 1 wave5_vs_wave3 9.28571 4.22577 2.1974 2.63826 1 wave3_vs_wave2 5.45238 4.39832 1.23965 2.63826 1 wave3_vs_wave1 1.85714 4.22577 0.43948 2.63826 1 wave1_vs_wave2 3.59524 4.39832 0.817411 2.63826 1

Checking the Q statistic we see that most of the variation is between wave5 and the other waves with maximum difference between wave5 and wave2. Note that in the last operation we used the /DHW flag because the input waves were not of the same length.

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