Wilcoxon Signed-Rank Test

Statistical Tests: Definition

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.

Wilcoxon Signed-Rank Test Assumptions

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.

Figure 1: Example of normally distributed data.

Wilcoxon’s signed-rank test is equivalent to the parametric paired t-test. 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 if there are outliers (extreme scores).

The Wilcoxon Signed-Rank Test Formula

These are the elements of the Wilcoxon signed-rank test formula you should keep in mind. It may look like gibberish at the moment, but it will make more sense as you learn about the steps and go through an example of the Wilcoxon signed-rank test. $W=\underset{i=1}{\overset{Nr}{\sum [}}sgn(x_{2,i}-x_{1,i})·R_i]$

W = test statistic

Nr = sample difference scores, excluding pairs

sgn = sign

X_1,i, X_2,i, X_3,i ... = the difference between corresponding scores

R = rank

Wilcoxon Signed-Rank Test: Steps

Wilcoxon signed-rank test can be conducted in four main 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

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

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 their mean rank, e.g. we have three ‘1s’, the three smallest values in our ranking. Instead of assigning them 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’, 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, 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 is W = 17.5.

Wilcoxon Signed-Rank Test Significance

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) equals or is 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.

Wilcoxon Signed-Rank Test Interpretation

The null hypothesis in the hypothetical study was there would be no difference in mood ratings before and after school. And the alternative hypothesis is there will be a difference in mood ratings before and after school.

In the study, as the observed W is larger than the critical W, the researcher should accept the null hypothesis and reject the alternative hypothesis.

Therefore, from the research, it can be concluded that school doesn’t 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 one is in our data.

The experimental effect might have been effective, but because the Wilcoxon test is less sensitive, it may not detect the results as significant.