 # Non-Parametric Test

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.

## 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, H0K 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:

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

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:

$$\begin{array}{l}U_{1}= n_{1}n_{2}+\frac{n_{1}(n_{1}+1)}{2}-R_{1}\end{array}$$

$$\begin{array}{l}U_{2}= n_{1}n_{2}+\frac{n_{2}(n_{2}+1)}{2}-R_{2}\end{array}$$

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.

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.

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## Frequently Asked Questions on Non-Parametric Test

Q1

### What is meant by a non-parametric test?

The non-parametric test is one of the methods of statistical analysis, which does not require any distribution to meet the required assumptions, that has to be analyzed. Hence, the non-parametric test is called a distribution-free test.

Q2

### What is the advantage of a non-parametric test?

The advantage of nonparametric tests over the parametric test is that they do not consider any assumptions about the data.

Q3

### Is Chi-square a non-parametric test?

Yes, the Chi-square test is a non-parametric test in statistics, and it is called a distribution-free test.

Q4

### Mention the different types of non-parametric tests.

The different types of non-parametric test are:
Kruskal Wallis Test
Sign Test
Mann Whitney U test
Wilcoxon signed-rank test

Q5

### When to use the parametric and non-parametric test?

If the mean of the data more accurately represents the centre of the distribution, and the sample size is large enough, we can use the parametric test. Whereas, if the median of the data more accurately represents the centre of the distribution, and the sample size is large, we can use non-parametric distribution.