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Wilcoxon Signed-Rank Test

Parametric statistical tests require your data to meet certain assumptions. For example, they may need your data to be normally distributed.

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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 need 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.

• We will start with a quick summary of statistical tests and then get stuck into the Wilcoxon signed-rank test assumptions.
• Moving on, we will look at the Wilcoxon signed-rank test formula and go through the steps of calculating it.
• To help with your learning, we will look at and go through an example of the Wilcoxon signed-rank test. The example will explain the Wilcoxon signed-rank interpretation and Wilcoxon signed-rank test significance.

## 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.

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

You conduct an experiment asking participants to solve math problems after 8 hours of sleep and then 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.

## 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.

Within-participant 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 measure 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 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. $W=\underset{i=1}{\overset{Nr}{\sum \left[}}sgn\left({x}_{2,i}-{x}_{1,i}\right)·{R}_{i}\right]$

W = test statistic

Nr = sample difference scores, excluding pairs

sgn = sign

X1,i, X2,i, X3,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.

 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

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

 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 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.

 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 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.

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.

### 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.

Remember, in experimental research; the alternative hypothesis can only be accepted if the findings are statistically significant.

If the findings are not statistically significant, the researcher should reject the alternative hypothesis and accept the null hypothesis, even if a trend is observed.

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.

## Wilcoxon Signed-Rank Test - Key takeaways

• Non-parametric statistical tests have limited and less restrictive 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 interpretation compares the observed W value to the critical W value.
• You can reject your null hypothesis if the observed W value is smaller or equal to the critical W value.

The Wilcoxon signed-rank test is used for calculating the statistical significance of results from research that used a within-participants design, but the data obtained does not meet the assumptions of the paired t-test, e.g., if it collects 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 the 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.

## Test your knowledge with multiple choice flashcards

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

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

A Wilcoxon-signed rank test is a form of inferential test, true or false?

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