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# Wilcoxon Signed-Rank Test

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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.

Essentially, they allow us to make conclusions based on our experiments.

You’re are conducting an experiment where you ask participants to solve math problems after 8 hours of sleep, and then you repeat the experiment. Still, this time participants have to solve math problems after no sleep.

You find that in the no-sleep condition, participants scored ten points lower. The results look promising, BUT this is not enough to conclude that the lack of sleep made a difference. To conclude that the difference you found was not just due to chance, we need to conduct a statistical analysis.

The accepted level of statistical significance in psychology is <0.05. We reject the null hypothesis if there is less than a 5% chance our results are due to chance.

## 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.

Within-participants design involves testing the same group of participants twice, and they experience every condition. The data used in a Wilcoxon signed-rank test is ordinal data, and it is a repeated measures or matched design.

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.

 Participant Mood before school Mood after school 1 3 7 2 2 7 3 6 5 4 2 4 5 8 9 6 2 7 7 10 4 8 5 5 9 6 5 10 4 3

#### 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.

 Participant Mood before school Mood after school Difference scores 1 3 7 -4 2 7 7 0 3 6 5 -1 4 2 4 -2 5 8 9 1 6 2 7 -5 7 10 4 6 8 5 5 0 9 6 5 1 10 8 4 4

#### 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.

 Participant Mood before school Mood after school Difference scores Ranks Signed ranks 1 3 7 -4 5.5 -5.5 2 7 7 0 - - 3 6 5 -1 2 -2 4 2 4 -2 4 -4 5 8 9 1 2 2 6 2 7 -5 7 -7 7 10 4 6 8 8 8 5 5 0 - - 9 6 5 1 2 2 10 8 4 4 5.5 5.5

#### 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.

Critical W values can be found in statistical tables. They depend on your sample and the level of significance.

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.

## Wilcoxon Signed-Rank Test - Key takeaways

• Parametric tests require your data to meet assumptions before conducting the test, e.g., 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.
• Wilcoxon’s signed-rank test is equivalent to the paired t-test. We use it to test the statistical significance of data from research using within-participants designs.
• Wilcoxon signed-rank test can be conducted by first calculating the difference scores, ranking the difference scores, calculating the sum of positive and the sum of negative ranks and determining the observed Wilcoxon test statistic W.
• The Wilcoxon signed-rank test is interpreted by comparing the observed W value to the critical W value. If the observed W value is smaller or equal to the critical W value, you can reject your null hypothesis.
• Non-parametric tests like the Wilcoxon signed-rank test are less powerful than their parametric equivalents. This means they are less likely to find a difference if one is in the data.

## Frequently Asked Questions about Wilcoxon Signed-Rank Test

The Wilcoxon signed-rank test is used for calculating the statistical significance of results from research that used within-participants design, but the data obtained did not meet assumptions of the paired t-test. It uses ordinal data.

The test statistic for the Wilcoxon signed-rank test is W.

The Wilcoxon signed-rank test is a non-parametric statistical test, used to analyse data from within-participants research designs.

The Wilcoxon test doesn’t make assumptions about the population. The test is only appropriate for within-participants designs.

First, calculate difference scores for each participant. Next, rank the difference between these scores. Separately calculate the sum of positive and negative ranks. The smaller sum is the observed Wilcoxon value of W. If your observed W value is smaller or equal to the critical W value you can reject your null hypothesis.

## Final Wilcoxon Signed-Rank Test Quiz

Question

What are the assumptions of a Wilcoxon test?

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Answer

The Wilcoxon test doesn’t make assumptions about the population.

Show question

Question

What research design is the Wilcoxon signed-rank test appropriate for?

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Answer

Within-participants

Show question

Question

What is a within-participants design?

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Answer

Within-participants design involves testing the same group of participants twice, under two different conditions.

Show question

Question

What is the parametric equivalent of the Wilcoxon signed-rank test?

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Answer

Independent t-test

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Question

When do we reject the null hypothesis?

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Answer

We reject the null hypothesis if there is less than a 5% chance that our results are due to chance.

Show question

Question

What is the Wilcoxon signed-rank test?

Show answer

Answer

The Wilcoxon signed-rank test is a non-parametric statistical test used to analyse data from within-participants research designs.

Show question

Question

What is the test statistic for the Wilcoxon signed-rank test?

Show answer

Answer

The test statistic for the Wilcoxon signed-rank test is W.

Show question

Question

How to conduct the Wilcoxon signed-rank test?

Show answer

Answer

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

Step 1: calculating difference scores

Step 2: ranking difference scores

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

Step 4: determining the Wilcoxon test statistic W.

Show question

Question

How are difference scores calculated?

Show answer

Answer

To calculate difference scores, we need to subtract the second measurement value from the first one for each participant.

Show question

Question

How are difference scores ranked?

Show answer

Answer

• Difference scores are ranked from the smallest to the greatest difference. For this part, we ignore the signs, e.g. we treat -5 as a 5.
• We ignore 0 values in our ranking.
• We have to take ties into account, meaning if we get repeating values, we have to calculate the mean rank for them.
• Signs are added back to appropriate ranks.

Show question

Question

How to determine the Wilcoxon test statistic W?

Show answer

Answer

Wilcoxon test statistic W is either the sum of all positive or negative ranks, depending on which value is the smallest.

Show question

Question

How do we know if our results are statistically significant after calculating the Wilcoxon test statistic W?

Show answer

Answer

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 is equal to or less than the critical W value.

Show question

Question

What does the critical W value depend on?

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Answer

The critical W value depends on the sample and the level of statistical significance (usually 0.05).

Show question

Question

What is the limitations of the Wilcoxon signed-rank test?

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Answer

As a non-parametric test it is less powerful than its parametic equivalent.

Show question

Question

What does it mean that a non-parametric test is less powerful?

Show answer

Answer

Less powerful means that it’s less likely to find a difference if there is one in our data.

Show question

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