## Wilcoxon Tests Demo

The WilcoxonRankTest operation supports the non-parametric Wilcoxon-Mann-Whitney two sample rank test and the Wilcoxon Signed-Rank test.

**1. Wilcoxon-Mann-Whitney two sample rank test**

In this test the ranks of the data are used to determine if there is any difference between the samples. The data we for the first test are given in the waves data1 and data2:

data1 | data2 |

60.83 | 59.22 |

63.24 | 65.09 |

72.33 | 71.45 |

62.21 | 72.33 |

66.59 | 74.71 |

61.64 | 68.42 |

73.68 | |

70.19 |

To run the rank test execute:

StatsWilcoxonRankTest/T=1/Q/TAIL=4 data1,data2

The results are displayed in the Mann-Whitney Wilcoxon Test table:

m | 8 |

n | 6 |

totalPoints | 14 |

U_statistic | 18.5 |

Up_statistic | 29.5 |

P_Value | 0.512821 |

In this case the P-value is greater than the significance (0.05 by default) so H_{0}:
there is no difference between the results in data1 and data2 must be accepted. The P-value in this
case is computed by:

P(U'<=18.5|8,6)+P(U>=29.5|8,6)= 0.512821

Comparing instead data1 to data3 where data3 is given by:

data3 |

77.81 |

67.08 |

71.55 |

75.46 |

74.46 |

74.61 |

we execute execute the command:

StatsWilcoxonRankTest/T=1/Q/TAIL=4 data1,data3

The results are displayed in the Mann-Whitney Wilcoxon Test table:

m | 8 |

n | 6 |

totalPoints | 14 |

U_statistic | 5 |

Up_statistic | 43 |

P_Value | 0.012654 |

Here the P-value is smaller than the significance and H_{0} must be rejected.

Note that in these examples the size of the waves was relatively small so there was no need to use the approximations for the P-values.

**2. Wilcoxon Signed-Rank test**

This non-parametric test is used for paired samples. The test is based on the ranks of the absolute difference in the values of each pair. The following two waves represent paired measurements:

data4 | data5 |

63.65 | 64.86 |

58.99 | 50.22 |

57.45 | 59.33 |

61.18 | 59.86 |

58.79 | 60.35 |

63.49 | 59.71 |

56.52 | 59.22 |

59.44 | 59.39 |

60.77 | 64.95 |

64.34 | 56.42 |

59.55 | 59.56 |

63.76 | 54.82 |

63.62 | 60.74 |

51.23 | 60.23 |

61.69 | 60.95 |

64.41 | 61.6 |

54.59 | 64.9 |

57.84 | 59.73 |

57.33 | 60.36 |

57.14 | 60.12 |

To run the Signed-Rank test on the waves data4 and data5 execute the command:

StatsWilcoxonRankTest/T=1/Q/WSRT data4,data5

The results are displayed in the Wilcoxon Signed Rank Test table:

n | 20 |

Tp | 96 |

Tm | 114 |

P_lower_tail | 0.378083 |

P_upper_tail | 0.635747 |

P_two_tail | 0.756166 |

Since the P-value for the two-tail hypothesis is much larger than the significance, we conclude that the two members of the pairs in data4 and data5 are the same.