## 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.

To run the test execute the command:

StatsHodgesAjneTest/T=1/Q data1

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

Input_Points | 30 |

m | 10 |

Critical | 6 |

alpha | 0.05 |

P | 0.559632 |

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, H_{0} of uniformity is accepted with a P-value 0f 0.559632. Note
that in this case m must exceed the critical value to accept H_{0}.

### 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_Points | 30 |

m | 0 |

Critical | 6 |

alpha | 0.05 |

P | 5.5e-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 H_{0} 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_Points | 30 |

90_degree_Points | 28 |

C_statistic | 2 |

Critical | 9 |

P_value | 8.6e-07 |

In this case the test for uniformity indicates that we must reject H_{0} in favor of the alternative.
If we execute:

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

Input_Points | 30 |

90_degree_Points | 30 |

C_statistic | 0 |

Critical | 9 |

P_value | 0 |

We observe that there is an even stronger rejection of H_{0} in favor of the alternative concentration about pi/3.