Hodges-Ajne Test

The Hodges-Ajne test provides a non-parametric measure for the angular uniformity of the data.

Uniform angular distribution

We create the distribution using the following command:

Make/O/N=30 data1=mod(2*pi*(1+enoise(1))/2,2*pi)

The data are displayed in the polar graph below.

Picture0

To run the test execute the command:

StatsHodgesAjneTest/T=1/Q data1

The results are shown in the Hodges-Anje Stats table:

Input_Points30
m10
Critical6
alpha0.05
P0.559632

This means that the least number of data that can be found on one side of any diameter of this circle is 10. Since the critical value is 6, H0 of uniformity is accepted with a P-value 0f 0.559632. Note that in this case m must exceed the critical value to accept H0.

Working with a non-uniform distribution

We create the non-uniform distribution by executing the command:

Make/O/N=30 data2=mod(pi*(1+enoise(1))/3,2*pi)

StatsHodgesAjneTest/T=1/Q data2

Input_Points30
m0
Critical6
alpha0.05
P5.5e-08

In this case the operation found a diameter where there were no data on one side. This is a clear violation of uniformity and H0 must be rejected.

Testing uniformity against a specific direction alternative

Using data2 from above we first test against an alternative with mean direction pi/8. To execute the test select the blue line below and type Ctrl-Enter:

StatsHodgesAjneTest/T=1/Q/SA=(pi/8) data2

The results are shown in the Hodges-Anje Stats table:

Input_Points30
90_degree_Points28
C_statistic2
Critical9
P_value8.6e-07

In this case the test for uniformity indicates that we must reject H0 in favor of the alternative. If we execute:

StatsHodgesAjneTest/T=1/Q/SA=(pi/3) data2

Input_Points30
90_degree_Points30
C_statistic0
Critical9
P_value0

We observe that there is an even stronger rejection of H0 in favor of the alternative concentration about pi/3.