# Wilcoxon

Wilcoxon signed ranks test is a non-parametric test for a difference in central location (median) between two paired samples.

The requirements of the test are:

- Two paired samples measured on an ordinal or continuous scale.
- Samples have similar shape distributions, although the distributions need not be normal.

## Arranging the dataset

Data in existing Excel worksheets can be used and should be arranged in a List dataset layout. The dataset must contain two ordinal or continuous scale variables.

When entering new data we recommend using New Dataset to create a new **2 variables** dataset ready for data entry.

**Using the test**

**Using the test**

To start the test:

- Excel 2007:

Select any cell in the range containing the dataset to analyse, then click**Compare Pairs**on the**Analyse-it**tab, then click**Wilcoxon**. - Click
**Variable X**and**Variable Y**and select the variables to compare. - Click
**Alternative hypothesis**and select the alternative hypothesis to test. - Click
**OK**to run the test.

Excel 97, 2000, 2002 & 2003:

Select any cell in the range containing the dataset to analyse, then click **Analyse **on the **Analyse-it **toolbar, click **Compare Pairs** then click **Wilcoxon**.

X ≠ Y to test if the median(X) is not equal to median(Y). |

X > Y to test if the median(X) is greater than median (Y). |

X < Y to test if the median(X) is less than median (Y). |

The report shows the number of observations analysed, and, if applicable, how many missing values were excluded. Since the test uses the ranked difference between paired observations, rank summary statistics for positive, negative and zero differences are shown.

The Wilcoxon statistic and hypothesis test are shown. The *p*-value is the probability of rejecting the null hypothesis, that the samples have the same median, when it is in fact true. A significant p-value implies that the two samples have different medians.

**METHOD ** When the number of cases is ≤25 an exact p-value is calculated, based on the assumption of no ties (see [2]). If a few ties are present (the number of ties is shown next to the p-value) the p-value will be conservative. For > 25 cases a normal approximation, with correction for ties, is used (see [2]).

## Median difference point estimate

If both samples are continuous the median difference and confidence interval can be calculated to quantify the difference between the samples in terms that can be practically evaluated.

To calculate the median difference and confidence interval:

- If the Summary statistics dialog box is not visible click
**Edit**on the**Analyse-it**tab/toolbar. - Enter
**Confidence interval**to calculate for the median difference. The level should be entered as a percentage between 50 and 100, without the % sign.**NOTE**The Hodges-Lehman method used to calculate the confidence interval can be extremely time-consuming for large sample sizes. If a confidence interval is not required, or the sample sizes are large, leave the confidence level blank and the calculation will be skipped. - Click
**OK**.

The median differences and confidence interval are shown.

** METHOD ** The median difference and confidence interval are calculated using the Hodges-Lehman method (see [2]).

## References to further reading

- Handbook of Parametric and Nonparametric Statistical Procedures (3rd edition)

David J. Sheskin, ISBN 1-58488-440-1 2003; 609. - Practical Non-parametric Statistics (3rd edition)

Conover W.J. ISBN 0-471-16068-7 1999; 352-362.

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