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# Non-Parametric Tests

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Non-parametric tests are also known as distribution-free tests. These are statistical tests that do not require normally-distributed data.

Non-parametric tests are used as an alternative when Parametric Tests cannot be carried out. Non-parametric tests include the Kruskal-Wallis and the Spearman correlation. These are used when the alternative parametric tests (e.g. one-way ANOVA and Pearson correlation) cannot be carried out because the data doesn't meet the required assumptions.

## The framework of non-parametric tests

Non-parametric tests determine the value of data points by assigning + or - signs, based upon the ranking of data. The analysis process involves numerically ordering data and identifying its ranking number. Data is assigned a ‘+’ if it is greater than the reference value (where the value is expected/hypothesised to fall) and a ‘-’ if it is lower than the reference value. This ranked data is then used as data points for a non-parametric statistical analysis.

This example data set illustrates how non-parametric tests are ranked:

Data set: 25, 16, 6, 16, 30. The reference value was calculated as 20.

 X₁ X₂ X₃ X₄ X₅ -6 -16 -16 +25 +30

The data is ranked in numerical order from the lowest (6) to the highest (30). As there are two instances of the value of 16, both are assigned with a ranking of 2.5. The reference value was predicted as 20; therefore, two 25 and 30 have positive values and the rest have negative values.

### When to use non-parametric tests

Non-parametric tests are tests with fewer restrictions than parametric tests. It is appropriate to use non-parametric tests in research in different cases. For example:

• When data is nominal. Data is nominal when it is assigned to groups, these groups are distinct and have limited similarities (e.g. responses to ‘What is your ethnicity?’).

• When data is ordinal. That is when data has a set order or scale (e.g. ‘Rate your anger from 1-10’.)

• When there have been outliers identified in the data set.

• When data was collected from a small sample.

However, it is important to note that non-parametric tests are also used when the following criteria can be assumed:

• At least one violation of parametric tests assumptions. E.g., data should have similar homoscedasticity of variance: the amount of ‘noise’ (potential experimental errors) should be similar in each variable and between groups.

• Non-normal distribution of data. In other words, data is likely skewed.

• Randomness: data should be taken from a random sample from the target population.

• Independence: the data from each participant in each variable should not be correlated. This means that measurements from a participant should not be influenced or associated with other participants.

## Non-parametric statistical tests

The table below shows examples of non-parametric tests. It includes their parametric test equivalent, the method of data analysis the test uses, and example research that is appropriate for each statistical test.

 Non-parametric test Equivalent parametric test Purpose of statistical test Example Wilcoxon rank-sum test Paired t-test Compares the mean value of two variables obtained from the same participants. The difference in depression scores before and after treatment. Mann-Whitney U test Unpaired t-test Compares the mean value of a variable measured from two independent groups. The difference between depression symptom severity in a placebo and drug therapy group. Spearman correlation Pearson correlation Measures the relationship (strength/direction) between two variables. The relationship between fitness test scores and the number of hours spent exercising. Kruskal Wallis test One-way analysis of variance (ANOVA) Compares the mean of two or more independent groups (uses a between-subject design and the independent variable needs to have three or more levels.) The difference in average fitness test scores of individuals who exercise frequently, moderately, or do not exercise. Friedman's ANOVA One-way repeated measures ANOVA Compares the mean of two or more dependent groups (uses a within-subject design and the independent variable needs to have three or more levels.) The difference in average fitness test scores during the morning, afternoon, and evening.

Research using non-parametric tests has many advantages:

• Statistical analysis uses computations based on signs or ranks. Thus, outliers in the data set are unlikely to affect the analysis.

• They are appropriate to use even when the research sample size is small.

• They are less restrictive than parametric tests as they don't have to meet as many criteria or assumptions. Therefore, they can be applied to data in various situations.

• They have more statistical power than parametric tests when the assumptions of parametric tests have been violated. This is because they use the median to measure the central tendency rather than the mean. Outliers are less likely to affect the median.

• Many non-parametric tests have been a standard in psychology research for many years: the chi-square test, the Fisher exact probability test, and the Spearman’s correlation test.

Non-parametric tests also have disadvantages that we should consider:

• The mean is considered the best and a standard measure of central tendency because it uses all the data points within the data set for analysis. If data values change, then the mean calculated will also change. However, this is not always the case when calculating the median.

• As these tests don’t tend to be vastly affected by outliers, there is an increased likelihood of research carrying out a Type 1 error (essentially a ‘false positive', rejecting the null hypothesis when it should be accepted). This reduces the validity of the findings.

• Non-parametric tests are considered as appropriate for hypothesis testing only, as they do not calculate or estimate effect sizes (a quantitative value that tells you how much two variables are related) or confidence intervals. This means that researchers cannot identify how much the independent variable affects the dependent variable and how significant these findings are. Therefore, the utility of the findings is limited and its validity is also difficult to establish.

## Non-Parametric Tests - Key takeaways

• Non-parametric tests are also known as distribution-free tests. These are statistical tests that do not require normally distributed data.
• Non-parametric tests determine the value of data points via assigning + or - signs, based upon the ranking of data. The analysis process involves numerically ordering data and identifying their rank number. This ranked data is used as data points for non-parametric statistical analysis.
• Examples of non-parametric tests are Wilcoxon Rank sum test, Mann-Whitney U test, Spearman correlation, Kruskal Wallis test, and Friedman’s ANOVA test. All of these tests have alternative parametric tests.

• The advantages of non-parametric tests are:

• The shape of the distribution does not matter because these tests use the median rather than the mean as the measure of central tendency.

• The analysis isn't vastly affected by outliers.

• These tests have more statistical power than parametric tests when the assumptions of parametric tests have been violated.

• The disadvantages of non-parametric tests are:

• These tests are less powerful because the computations do not consider all of the data within the data set, as data is not vastly affected by outliers.

• There is an increased likelihood of having a Type 1 error.

• These tests are limited in utility as they do not give statistical analysis findings concerning effect size and confidence intervals.

Non-parametric tests are also known as distribution-free tests. These are statistical tests that do not require normally-distributed data for the analysis.

• Non-parametric tests should be used when data is not normally distributed.

• At least one of the assumptions of the parametric test has been violated.

• Data is nominal or ordinal.

• There are outliers in the data set.

• The sample size is small.

The difference between the two types of tests is that non-parametric tests use the median to measure the central tendency value for statistical analysis whereas parametric tests measure the mean.

Non-parametric tests can also be very sensitive tests as analysis also accounts for outliers. The presence of these increases the likelihood of having a Type 1 error, reducing the validity of findings.

The Kruskal-Wallis test. It is used to compare the mean of two or more independent groups (uses a between-subject design). An example of research that could use the Kruskal-Wallis test is to measure the difference in average fitness scores of individuals who exercise frequently, moderately, or do not exercise.

## Final Non-Parametric Tests Quiz

Question

What is the definition of a non-parametric test?

Non-parametric tests are also known as distribution-free tests, these are statistical tests that do not require normally-distributed data for the analysis tests to be employed.

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Question

When is it appropriate to use non-parametric tests?

• Data is nominal (data assigned to groups, these groups are distinct and have limited similarities eg responses to "What is your ethnicity?")
• Ordinal (data with a set order / scale eg “rate your anger from 1-10”), there are outliers within the data,
• If data has been collected from a small sample.

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Question

What is the criterion of non-parametric tests?

The following criterion is required for non-parametric tests:

• At least one violation of parametric tests assumptions,
• Non-normally distributed data
• Data is random (taken from random sample)
• Data values ​​are independent from one another (no correlation between data collected from each participant)

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Question

What is the definition of nominal and ordinal data?

Nominal data is when data is assigned to groups that are distinct from each other. An example of nominal data is the response from “What is your ethnicity?”. Whereas, ordinal data is defined as data with a set scale / order. For example the response from "Rate your anger from a scale of 1-10".

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Question

Why does data need to be ranked prior to carrying out non-parametric data analysis?

Data needs to be ranked prior to statistical analysis as these ranked values ​​are used as data points for the analysis rather than the raw values ​​obtained from the experiment / observation.

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Question

What is the 'reference value'?

The reference value is where the researchers predict / hypothesise where the median value is expected to fall.

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Question

What do '+' and '-' ranked values ​​indicate?

Data is assigned as '+' if it is greater than the reference value and data that is '-' is lower than the reference value.

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Question

Rank the following data values and assign them with the correct sign.

Researchers hypothesised that the reference value would be 13. The dataset is: 3, 5, 3, 19, 16, 21, 14.

-3, -3, -5, +14, +16, +19, +21

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Question

What do '+' and '-' signs mean in terms of ranking data for non-parametric analysis?

Data is assigned as '+' if it is greater than the reference value and ‘-’ if it is lower than the reference value.

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Question

What are the most common non-parametric tests?

Examples of common non-parametric tests are: Wilcoxon Rank sum Test, Mann-Whitney U test, Spearman correlation, Kruskal Wallis test and Friedman's ANOVA test.

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Question

Researchers are trying to identify what would be an appropriate statistical analysis to run to identify the difference in average fitness test scores of participants during the morning, afternoon, and evening. The researchers identified that their data was skewed and there were a few extreme outliers.

Which test should they run?

The appropriate analysis test to use would be the Friedman's ANOVA test, as the data can be assumed to be non-normally distributed. The study used a within-subjects design and the analysis can help identify the difference in average scores between the morning, afternoon, and evening by comparing the ranked median values.

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Question

Is Pearson correlations a parametric or non-parametric test? What is its alternative test?

The Pearson correlation is an example of a parametric test and its non-parametric alternative is the Spearman's rank correlation.

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Question

What is the purpose of using Pearson's and Spearman's rank correlation?

The purpose of these statistical tests is to identify the association (strength and direction) between two variables.

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Question

What are the advantages of non-parametric tests?

The advantages of non-parametric tests are:

• The shape of the distribution does not matter as these tests measure the median rather than the mean as the measure of central tendency.
• Analysis is not vastly affected by outliers.
• These tests have more statistical power than parametric tests when the assumptions of parametric tests have been violated.

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Question

What are the limitations of non-parametric tests?

The limitations of non-parametric tests are:

• These tests are less powerful because the analysis does not take into account the entire data set (identifies the median value of the sample and compares this to the reference value).
• Data is not vastly affected by outliers, so there is an increased likelihood of having a Type 1 error.
• These tests can mostly be used for ‘hypothesis testing’ as they do not give statistical analysis findings concerning effect size and confidence intervals.

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Question

What are the assumptions of a Wilcoxon test?

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

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Question

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

Within-participants

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Question

What is a within-participants design?

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

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Question

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

Independent t-test

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Question

When do we reject the null hypothesis?

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

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Question

What is the Wilcoxon signed-rank test?

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

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Question

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

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

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Question

How to conduct the Wilcoxon signed-rank test?

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.

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Question

How are difference scores calculated?

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

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How are difference scores ranked?

• 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.

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Question

How to determine the 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.

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Question

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

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.

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Question

What does the critical W value depend on?

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

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Question

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

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

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

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

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Question

What type of test is the binomial sign test?

Parametric test

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Question

What is an advantage of the binomial sign test?

• When researchers collect data, it is not always possible to collect data from a normally-distributed sample.
• Researchers can statistically calculate whether the null or alternative hypothesis should be accepted.

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Question

What is the disadvantage of using a binomial sign test?

The sign test is a non-parametric test. Non-parametric tests are known to be less powerful than their parametric alternatives because non-parametric tests use less information in their calculations, such as distributional information, which makes them less sensitive.

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Question

How many steps are there when calculating the binomial sign test?

2

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Question

What is the purpose of the binomial sign test?

The binomial sign test is a statistical test that is used to test the probability of an occurrence happening.

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Question

Which of the following statements is accurate?

The binomial sign test may be used to identify the likelihood of people’s success or failure in planned diet intervention.

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Question

What are the binomial sign test assumptions?

The binomial sign test assumptions are as follows:

• It should be used when testing a difference between values.
• The experiment should use a related design (repeated measures or matched-pairs design)
• This test relies on comparisons, which can be from the same or different participants as long as it is acceptable to compare them, such as being tested after being identified to share a similar characteristic (this is a matched-pairs design)
• Non-normal data – the data of participants should not be equally distributed.

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Question

What would the N be in the following research scenario when calculating the binomial sign test values, ‘the researcher recruited nine participants, but the two showed no difference’?

9

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Should the researcher accept the research findings as significant if the S value is calculated to be higher than the critical value?

No

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If the S value is calculated to be lower than the critical value, then which hypothesis should the researcher accept?

Alternative hypothesis

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Question

Which values are included in the binomial sign test significance table?

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What does a p-value of .05 indicate?

A p-value of .05 means the researcher can say with 95% the results observed/calculated are not due to chance.

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What does N stand for in statistical analyses?

N is the number of participants that are included in the analysis.

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What is the S value in the binomial sign test?

The S value is the sign that is the least frequent when the difference (sign) is calculated before and after the intervention.

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Are participants who show no difference included in the analysis of the binomial sign test?

Yes

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Question

What is an observed value?

An observed value is the result we get when we run a statistical test.

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What is a critical value?

The critical value is a set value that we look at to see if what we have found is due to the variables we are investigating or due to chance.

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Question

A one tailed hypothesis is:

A very specific direction of findings

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What does N stand for?

Number of participants

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What does df stand for?

Degrees of freedom

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