Angular Distance Test Example

In this example we have three groups of data contained in the numeric waves data1, data2 and data3. The values represent angles in radians:

data1data2data3
2.1732.342692.55387
2.154952.427432.52051
2.249842.359772.60009
2.160182.323472.60166
2.125172.347092.57084
2.176032.370792.47116
2.196942.270662.58406
2.170192.390492.4725
2.204842.364722.54683
2.241692.439492.46343
2.184062.269812.47491
2.245352.403762.53199
2.190742.407252.51363
2.187962.429272.54139
2.14962.298482.59919

The data are shown here in a polar plot:

Picture0

Multi-sample testing for angular dispersion

The three input waves were created using uniformly distributed random numbers centered on 125, 135 and 145 degrees respectively. We therefore expect the angular distance test to indicate low dispersion. To run the test execute the command:

  StatsAngularDistanceTest/T=1/Q/ANGM data1,data2,data3

The results of the test are shown in the "Kruskal-Wallis Test" table:

Num_Groups3
Num_Valid_Data45
Alpha0.05
H3.83614
Hc_Chi_square_approx5.99146
P_Chi_square_approx0.146891
Hc_Wallace_approx6.04367
P_Wallace_approx0.15194

The H statistic is clearly below both the Chi-square and Wallace approximate critical values so the hypothesis H0: no effective dispersion can't be rejected at the default 0.05 significance level. Note that in this case we have used the /ANGM flag to compute the relative dispersion of the data about each sample's mean instead of providing sample means using the /ANGW flag. Here the operation first computes the mean of each sample and then performs the Kruskal-Wallis test on the difference of each datum from its corresponding mean value.

Two-sample testing for angular dispersion

Here we run a different test on the first two waves that we tested above. The operation computes the mean direction of the two samples and then performs the nonparametric Wilcoxon test on the differences between each datum and the mean direction. To run the test just execute the command:

  StatsAngularDistanceTest/T=1/Q/ANGM data1,data2

The results of the test are shown in the "Mann-Whitney Wilcoxon Test" table:

m15
n15
TotalPoints30
U_statistic74
Up_statistic151
P_Value0.946777

As one would expect, we get a P-value which is much greater than alpha (0.05) so H0: no dispersion can't be rejected.

Two-sample testing for angular dispersion with user-specified mean direction

In this example we test the same samples (which were generated as data centered at 125 and 135 degrees respectively) relative to mean direction of 135 degrees for both. To run the test execute the command:

  StatsAngularDistanceTest/T=1/Q/ANG={2.35619,2.35619} data1,data2

The results of the test are shown in the "Mann-Whitney Wilcoxon Test" table:

m15
n15
totalPoints30
U_statistic225
Up_statistic0
P_Value6.38465e-09

Clearly, the P-value is less than alpha and so we reject H0.