Circular Two Sample Tests

This example covers four tests that are available in the StatsCircularTwoSampleTest operation. All tests are used to compare two samples of circular data which represent either raw data or means of data (second order analysis).

For the first two examples we use the waves Sample1, Sample2 and Sample3:

Sample1

AngleRadius
2.071091.05535
2.204791.11541
1.66371.12639
2.148611.01445
2.37850.990459
2.04210.985494
2.148611.02701
1.984040.944105
1.874460.986208
1.971710.888526
1.813220.976384

Sample2

AngleRadius
2.097450.925728
1.868661.03347
1.959951.00445
2.017050.932245
2.045591.06179
2.045060.999727
2.426681.0344
2.036051.0751
2.336021.0398
2.188390.935943
1.922621.03721

Sample3

AngleRadius
1.815040.925728
1.932311.03347
1.840011.00445
1.83530.932245
1.856391.06179
1.814320.999727
1.936071.0344
1.899471.0751
1.579291.0398
1.847270.935943
1.866961.03721

Non-Parametric Paired-Sample Test

To test if the pairs represented by corresponding rows in Sample1 and Sample2 are the same (H0) execute the command:

StatsCircularTwoSampleTest/Q/T=1/NPR Sample1,Sample2

The reusults of the test are summarized in the table:

numPairs11
Rp0.350962
Critical1.04402
P-Value0.739203

Since the statistic Rp is smaller than the critical value we can't reject H0 that the two paired distributions are the same.

Parametric Paired-Sample Test

To test if the pairs represented by corresponding rows in Sample1 and Sample2 are the same (H0) execute the command:

StatsCircularTwoSampleTest/Q/T=1/PPR Sample1,Sample2

The reusults of the test are summarized in the table:

numPairs11
xBar-0.0520224
yBar-0.0247836
F0.246806
Critical4.25649
P-Value0.786411

As expected the parameteric test results in the same conclusion that H0 can't be rejected. To repeat the test and compare Sample1 to Sample3 execute the command:

StatsCircularTwoSampleTest/Q/T=1/PPR Sample1,Sample3

The reusults of the test are summarized in the table:

numPairs11
xBar0.170992
yBar0.0770423
F4.62463
Critical4.25649
P-Value0.0415419

In this case the pair-wise equality of the samples is rejected.

Two-Sample Parametric Second Order Test

Suppose you have two samples of means of circular data contained in Sample4 and Sample5 as shown below.

Sample4

AngleRadius
1.951560.998453
2.024990.966636
1.9230.900775
1.932130.952201
2.119020.980929
1.907440.993908
2.098110.999118
2.033580.901898

Sample5

AngleRadius
1.860720.994223
1.742220.930861
1.984120.90284
1.660551.0654
1.946681.00991
2.034871.02034
1.830050.957592
1.790430.956647
1.685440.973875
1.624410.944371
2.059691.02666

To test if the means of the populations from which the two samples were taken are equal (H0), execute the following command:

StatsCircularTwoSampleTest/Q/T=1/PSOA Sample4,Sample5

The reusults of the test are summarized in the table:

Samples18
xBar1-0.398434
yBar10.872187
r10.958885
a11.99931
Samples211
xBar2-0.25697
yBar20.935076
r20.969743
a21.83899
F4.03807
Critical3.63372
P-Value0.0380423

Since the F statistic is greater than the critical value we reject H0.

Two-Sample Non-Parametric Second Order Test

To test the same data using a non-parametric test, execute the command:

StatsCircularTwoSampleTest/Q/T=1/NSOA Sample4,Sample5

The reusults of the test are summarized in the table:

Total_Points19
Watson_U20.202153
Critical_Tiku0.184557
Approx_P0.0345285
Critical0.184103

Clearly the Watson_U2 statistic is greater than the critical value and H0 (equality of the means) must be rejected.