## The Wilcoxon Test

This test is the non-parametric equivalent of the paired samples t-test. The null and alternative hypotheses are the same as those for the paired samples t-test although they are often stated in terms of the median, thus:

H_{0}: In the population the median difference is zero

H_{1}: In the population the median difference is not zero

However, an assumption behind this test is that the population distributions are symmetrical, in which case the mean and medians are identical.

**Worked Example**

Blood pressures (mmHg) |
|||
---|---|---|---|

Subject |
Standing |
Supine |
Difference |

1 | 132 | 136 | 4 |

2 | 146 | 145 | 1 |

3 | 135 | 140 | 5 |

4 | 141 | 147 | 6 |

5 | 139 | 142 | 3 |

6 | 162 | 160 | -2 |

7 | 128 | 137 | 9 |

8 | 137 | 136 | -1 |

9 | 145 | 149 | 4 |

10 | 151 | 158 | 7 |

11 | 131 | 120 | -11 |

12 | 143 | 150 | 7 |

Mean |
140.83 | 143.33 | 2.50 |

SD |
9.49 | 10.83 | 5.50 |

The frugal Minitab output simply shows the estimated median difference and the 94.5% confidence interval (with non-parametric tests it is not always possible to achieve exact probability levels). Since this confidence interval contains zero, we fail to reject the null hypothesis that the population median difference is zero.

The SPSS output provides, among other things, the p-value for a two-tailed test (0.099).

The conclusion is that there is insufficient evidence to suggest a difference in standing and supine systolic blood pressures

**Computer output from Minitab and SPSS**

**The Wilcoxon Test using Minitab**

To perform this test use **
Stat > Nonparametrics > 1-sample Wilcoxon...** after creating a column of differences between the pairs in the sample.

Wilcoxon Signed Rank Confidence Interval

Estimated |
Achieved |
|||
---|---|---|---|---|

N | Median | Confidence | Confidence Interval | |

Difference |
12 | 3.00 | 94.5 | (-1.00, 6.00) |

**The Wilcoxon Test using SPSS**

Use**
Analyze > Nonparametric Tests > 2 Related Samples...** to obtain the following output

Wilcoxon Signed Ranks Test