Wilcoxon Signed-Rank Test

Parametric statistical tests require your data to meet certain assumptions. For example, they may require your data to be normally distributed. When your data doesn’t fall into these parameters or meet the assumptions for a statistical test, you can use a non-parametric test. The Wilcoxon sighed-rank test is the non-parametric equivalent of a paired t-test.

What are statistical tests?

Statistical tests tell us about the statistical significance of our results in hypothesis testing. They help us identify if the variables we were testing (for instance, in experimental manipulation) have a statistically significant relationship and, if so, how far that relationship extends. It helps us identify that the results are not a product of pure chance, and we can then confidently reject the null hypothesis.

When to use non-parametric statistical tests?

Parametric tests require your data to meet assumptions before you can conduct the test, e.g., that data within a population should be normally distributed and shouldn’t include any outliers. Non-parametric statistical tests don’t make any assumptions, which means you can use them if your data violates the assumptions of parametric tests.

Example of normally distributed data, Pixabay

Wilcoxon signed-rank test assumptions

Wilcoxon’s signed-rank test is equivalent to the paired t-test, an inferential statistic. We use paired-tests to test the statistical significance of data from research using within-participants research designs.

To use the parametric test (paired t-test), the difference scores (difference of the scores a participant got in both conditions) would normally be distributed. Wilcoxon’s signed-rank test doesn’t make that assumption, so we can use it if our difference scores are not normally distributed or contain outliers (extreme scores).

The Wilcoxon signed-rank test formula

W = test statistic

Nr = sample difference scores, excluding pairs

sgn = sign

difference between corresponding scores

R = rank

How to conduct the Wilcoxon signed-rank test?

Wilcoxon signed-rank test can be conducted in main four steps:

1. Calculating difference scores

2. Ranking these difference scores

3. Calculating the sum of positive and sum of negative ranks

4. Determining the Wilcoxon test statistic W.

Now let’s examine how to conduct the four steps using a worked example.

Examples of the Wilcoxon signed-rank test

The experimenter wants to investigate whether students’ mood changes after school. She recruits ten students and asks them to rate their mood in the morning before school starts and then again at the end of the school day.

Step 1: calculating difference scores

We need to subtract the second measurement value (mood before school) from the first (mood after school) to calculate difference scores.

Step 2: ranking difference scores

Here, we rank scores from the smallest to the greatest difference. For this part, we ignore the signs, e.g., we treat -5 as a 5.

1. Ignore 0 values.

2. Take ties into account:

   • If you get repeating values, you have to calculate the mean rank for them, e.g. we have three ‘1s’, which are the three smallest values in our ranking. Instead of assigning them with ranks 1, 2 and 3, we will assign the mean rank 2 to all of them. (1+2+3)/3 = 2

   • The next value following our ‘1s’ is ‘2’; it’s the fourth smallest difference, therefore, it will be assigned rank 4.

   • The next value is ‘4’, we have two ‘4s’, which are the 5th and 6th smallest differences in our data set. Their mean rank will be 5.5 because (5+6)/2=5.5.

   • The next smallest difference is 5, it is our 7th smallest value, so its rank will be 7.

3. The last thing at this stage is to add the signs back to the ranks. Add a minus sign to all ranks of negative difference scores.

Step 3: calculating the sum of positive and sum of negative ranks

Sum of positive ranks:

• w+ = 2+8+2+5.5 = 17.5

Sum of negative ranks:

• w- = 5.5+2+4+7=18.5

Step 4: determining Wilcoxon test statistic W

Wilcoxon test statistic W is either the sum of all positive or negative ranks, depending on which value is the smallest. In our case, the smallest value was the sum of the positive ranks (17.5). Therefore our Wilcoxon test statistic W=17.5.

Wilcoxon signed-rank test interpretation and Wilcoxon signed-rank test significance

Our null hypothesis is that there will be no difference in mood ratings before and after school.

To know if our results are statistically significant we need to compare our observed value of W to a critical value of W. We can reject the null hypothesis if our observed W value (17.5) is equal to or less than the critical W value.

For our sample (n=10) and level of significance (α <0.05), the critical W value is 8.

Since our observed W value is larger than the critical W value (17.5>8) we fail to reject our null hypothesis. The experimenter can conclude that school did not affect students’ moods.

Limitations of the Wilcoxon singed-rank test

There’s a reason why we should use the parametrical, paired t-test if we can. It’s important to remember that non-parametrical tests should only be used as your second option because they are less powerful. This means they are less likely to find a difference if there is one in our data. Our manipulation might have been effective, but because the Wilcoxon test is less sensitive, it didn’t detect our results to be significant.