Non-parametric tests are experiments that do not require the underlying population for assumptions. It does not rely on any data referring to any particular parametric group of probability distributions. Non-parametric methods are also called distribution-free tests since they do not have any underlying population. In this article, we will discuss what a non-parametric test is, different methods, merits, demerits and examples of non-parametric testing methods.

Table of Contents:

• What is a Non-parametric Test?
  • Non-parametric T Test
  • Non-parametric Paired T-Test
• Non-parametric Test Methods
  • Mann Whitney U Test
  • Sign Test
  • Wilcoxon Signed-Rank Test
  • Kruskal Wallis Test
• Advantages and Disadvantages
• Applications
• FAQs

What is a Non-parametric Test?

Non-parametric tests are the mathematical methods used in statistical hypothesis testing, which do not make assumptions about the frequency distribution of variables that are to be evaluated. The non-parametric experiment is used when there are skewed data, and it comprises techniques that do not depend on data pertaining to any particular distribution.

The word non-parametric does not mean that these models do not have any parameters. The fact is, the characteristics and number of parameters are pretty flexible and not predefined. Therefore, these models are called distribution-free models.

Non-Parametric T-Test

Whenever a few assumptions in the given population are uncertain, we use non-parametric tests, which are also considered parametric counterparts. When data are not distributed normally or when they are on an ordinal level of measurement, we have to use non-parametric tests for analysis. The basic rule is to use a parametric t-test for normally distributed data and a non-parametric test for skewed data.

Non-Parametric Paired T-Test

The paired sample t-test is used to match two means scores, and these scores come from the same group. Pair samples t-test is used when variables are independent and have two levels, and those levels are repeated measures.

Non-parametric Test Methods

The four different techniques of parametric tests, such as Mann Whitney U test, the sign test, the Wilcoxon signed-rank test, and the Kruskal Wallis test are discussed here in detail. We know that the non-parametric tests are completely based on the ranks, which are assigned to the ordered data. The four different types of non-parametric test are summarized below with their uses, null hypothesis, test statistic, and the decision rule. 

Kruskal Wallis Test

Kruskal Wallis test is used to compare the continuous outcome in greater than two independent samples.

Null hypothesis, H₀:  K Population medians are equal.

Test statistic:

If N is the total sample size, k is the number of comparison groups, Rj is the sum of the ranks in the jth group and nj is the sample size in the jth group, then the test statistic, H is given by:

\(H = \left ( \frac{12}{N(N+1)}\sum_{j=1}^{k} \frac{R_{j}^{2}}{n_{j}}\right )-3(N+1)\)

Decision Rule: Reject the null hypothesis H₀ if H ≥ critical value

Sign Test

The sign test is used to compare the continuous outcome in the paired samples or the two matches samples.

Null hypothesis, H₀: Median difference should be zero 

Test statistic: The test statistic of the sign test is the smaller of the number of positive or negative signs.

Decision Rule: Reject the null hypothesis if the smaller of number of the positive or the negative signs are less than or equal to the critical value from the table.

Mann Whitney U Test

Mann Whitney U test is used to compare the continuous outcomes in the two independent samples. 

Null hypothesis, H₀: The two populations should be equal.

Test statistic:

If R₁ and R₂ are the sum of the ranks in group 1 and group 2 respectively, then the test statistic “U” is the smaller of:

\(U_{1}= n_{1}n_{2}+\frac{n_{1}(n_{1}+1)}{2}-R_{1}\)

\(U_{2}= n_{1}n_{2}+\frac{n_{2}(n_{2}+1)}{2}-R_{2}\)

Decision Rule: Reject the null hypothesis if the test statistic, U is less than or equal to critical value from the table.

Wilcoxon Signed-Rank Test

Wilcoxon signed-rank test is used to compare the continuous outcome in the two matched samples or the paired samples.

Null hypothesis, H₀: Median difference should be zero.

Test statistic: The test statistic W, is defined as the smaller of W+ or W- .

Where W+ and W- are the sums of the positive and the negative ranks of the different scores.

Decision Rule: Reject the null hypothesis if the test statistic, W is less than or equal to the critical value from the table.

Advantages and Disadvantages of Non-Parametric Test

The advantages of the non-parametric test are:

• Easily understandable
• Short calculations
• Assumption of distribution is not required
• Applicable to all types of data

The disadvantages of the non-parametric test are:

• Less efficient as compared to parametric test
• The results may or may not provide an accurate answer because they are distribution free

Applications of Non-Parametric Test

The conditions when non-parametric tests are used are listed below:

• When parametric tests are not satisfied.
• When testing the hypothesis, it does not have any distribution.
• For quick data analysis.
• When unscaled data is available.

For more Maths-related articles, visit BYJU’S –  The Learning App to learn with ease by exploring more videos.