Wilcoxon Signed-Rank Test

The Wilcoxon Signed Rank Test is a non-parametric statistical test used to compare two related samples, matched samples, or repeated measurements on a single sample. It is often used as an alternative to the paired t-test when the data cannot be assumed to be normally distributed.  The test evaluates the differences between pairs of observations, ranking the absolute differences while ignoring zeros. These ranks are then assigned a positive or negative sign, based on the direction of the difference. The test statistic is computed by summing the ranks of positive and negative differences. 

The Wilcoxon Signed Rank Test is similar to the Mann-Whitney U Test in that both test the difference in variance between groups rather than compare the groups means as you would in a t-test.  The Mann-Whitney U Test is only applied to compare the variances of two independent groups while the Wilcoxon Signed Rank Test is used to compare the variances of two dependent groups.  As such, the Wilcoxon Signed Rank Test is suitable for exploring the relationship between pre and post test (or pre and post treatment) type data where the same individuals (or subjects) are for both groups.  

The W statistics is calculated as the minimum of the absolute difference between the positive and negative differences. This value is typically written as W = min(T+, T-) where T+ is the sum of the positive ranks and T- is the sum of the negative ranks.  This calculated W value (or W test) is compared to a W critical values obtained from a W table which cross references the number of values in the analysis (N) with the level of significance (ɑ).  If W-test is less than or equal to W-critical, then the null hypothesis is rejected in favor of the alternative hypothesis.  If W-test is greater than W-critical, then you will fail to reject the null hypothesis meaning that there is not sufficient evidence to support the alternative hypothesis.  

The following video provides an overview of the Wilcoxon Signed-Rank Test as well as several examples of its application. 

Here are the notes for the presentation.