Non-Parametric Test

Non-parametric test is a statistical analysis method that does not assume the population data belongs to some prescribed distribution which is determined by some parameters. Due to this, a non-parametric test is also known as a distribution-free test. These tests are usually based on distributions that have unspecified parameters.

A non-parametric test acts as an alternative to a parametric test for mathematical models where the nature of parameters is flexible. Usually, when the assumptions of parametric tests are violated then non-parametric tests are used. In this article, we will learn more about a non-parametric test, the types, examples, advantages, and disadvantages.

A non-parametric test in statistics does not assume that the data has been taken from a normal distribution. A normal distribution belongs to a parametrized family of probability distributions and includes parameters such as mean, variance, standard deviation, etc. Thus, a non-parametric test does not make assumptions about the probability distribution's parameters.

Non-Parametric Test Definition

A non-parametric test can be defined as a test that is used in statistical analysis when the data under consideration does not belong to a parametrized family of distributions. When the data does not meet the requirements to perform a parametric test, a non-parametric test is used to analyze it.

Reasons to Use Non-Parametric Tests

It is important to access when to apply parametric and non-parametric tests in order to arrive at the correct statistical inference. The reasons to use a non-parametric test are given below:

• When the distribution is skewed, a non-parametric test is used. For skewed distributions, the mean is not the best measure of central tendency, hence, parametric tests cannot be used.
• If the size of the data is too small then validating the distribution of the data becomes difficult. Thus, in such cases, a non-parametric test is used to analyze the data.
• If the data is nominal or ordinal, a non-parametric test is used. This is because a parametric test can only be used for continuous data.

Parametric tests are those that assume that the data follows a normal distribution. Examples include ANOVA and t-tests. There are many different methods available to perform a non-parametric test. These tests can also be used in hypothesis testing. Some common non-parametric tests are given as follows:

Mann-Whitney U Test

This non-parametric test is analogous to t-tests for independent samples. To conduct such a test the distribution must contain ordinal data. It is also known as the Wilcoxon rank sum test.

Null Hypothesis: \(H_{0}\): The two populations under consideration must be equal.

Test Statistic: U should be smaller of

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

where, \(R_{1}\) is the sum of ranks in group 1 and \(R_{2}\) is the sum of ranks in group 2.

Decision Criteria: Reject the null hypothesis if U < critical value.

Wilcoxon Signed Rank Test

This is the non-parametric test whose counterpart is the parametric paired t-test. It is used to compare two samples that contain ordinal data and are dependent. The Wilcoxon signed rank test assumes that the data comes from a symmetric distribution.

Null Hypothesis: \(H_{0}\): The difference in the median is 0.

Test Statistic: W. W is defined as the smaller of the sums of the negative and positive ranks.

Decision Criteria: Reject the null hypothesis if W < critical value.

Sign Test

This non-parametric test is the parametric counterpart to the paired samples t-test. The sign test is similar to the Wilcoxon sign test.

Null Hypothesis: \(H_{0}\): The difference in the median is 0.

Test Statistic: The smaller value among the number of positive and negative signs.

Decision Criteria: Reject the null hypothesis if the test statistic < critical value.

Kruskal Wallis Test

The parametric one-way ANOVA test is analogous to the non-parametric Kruskal Wallis test. It is used for comparing more than two groups of data that are independent and ordinal.

Null Hypothesis: \(H_{0}\): m population medians are equal

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

where, N = total sample size, \(n_{j}\) and \(R_{j}\) are the sample size and the sum of ranks of the j^th group

Decision Criteria: Reject the null hypothesis if H > critical value

The best way to understand how to set up and solve a hypothesis involving a non-parametric test is by taking an example.

Suppose patients are suffering from cancer. They are divided into three groups and different drugs were administered. The platelet count for the patients is given in the table below. It needs to be checked if the population medians are equal. The significance level is 0.05.

As the size of the 3 groups is not same the Kruskal Wallis test is used.

\(H_{0}\): Population medians are same

\(H_{1}\): Population medians are different

\(n_{1}\) = 5, \(n_{2}\) = 3, \(n_{3}\) = 4

N = 5 + 3 + 4 = 12

Now ordering the groups and assigning ranks

\(R_{1}\) = 18.5, \(R_{2}\) = 21, \(R_{3}\) = 38.5,

Substituting these values in the test statistic formula, \(\left ( \frac{12}{N(N+1)}\sum_{1}^{m} \frac{R_{j}^{2}}{n_{j}}\right ) - 3(N+1)\)

H = 6.0778.

Using the critical value table, the critical value will be 5.656.

As H < critical value, the null hypothesis is rejected and it is concluded that there is no significant evidence to show that the population medians are equal.

Depending upon the type of distribution that the data has been obtained from both, a parametric test and a non-parametric test can be used in hypothesis testing. The table given below outlines the main difference between parametric and non-parametric tests.

Non-parametric tests are used when the conditions for a parametric test are not satisfied. In some cases when the data does not match the required assumptions but has a large sample size then a parametric test can still be used. Some of the advantages and disadvantages of a non-parametric test are listed as follows:

Advantages of Non-Parametric Test

The advantages of a non-parametric test are listed as follows:

• Knowledge of the population distribution is not required.
• The calculations involved in such a test are shorter.
• A non-parametric test is easy to understand.
• These tests are applicable to all data types.

Disadvantages of Non-Parametric Test

The disadvantages of a non-parametric test are given below:

• They are not as efficient as their parametric counterparts.
• As these are distribution-free tests the level of accuracy is reduced.

Related Articles:

• Summary Statistics
• Probability and Statistics
• T-Distribution
• P-value

Important Notes on Non-Parametric Test

• A non-parametric test is a statistical test that is performed on data belonging to a distribution whose parameters are unknown.
• It is used on skewed distributions and the measure of central tendency used is the median.
• Kruskal Wallis test, sign test, Wilcoxon signed test and the Mann Whitney u test are some important non-parametric tests used in hypothesis testing.

What is a Non-Parametric Test?

A non-parametric test in statistics is a test that is performed on data belonging to a distribution that has flexible parameters. Thus, they are also known as distribution-free tests.

When Should a Non-Parametric Test be Used?

A non-parametric test should be used under the following conditions.

• The distribution is skewed.
• The size of the distribution is small.
• The data is nominal or ordinal.

What is the Test Statistic Used for the Mann-Whitney U Non-Parametric Test?

The Mann Whitney U non-parametric test is the non parametric version of the sample t-test. The test statistic used for hypothesis testing is U . U should be smaller of \(U_{1} = n_{1}n_{2}+\frac{n_{1}(n_{1}+1)}{2}-R_{1}\) or \(U_{2} = n_{1}n_{2}+\frac{n_{2}(n_{2}+1)}{2}-R_{2}\)

What is the Test Statistic Used for the Kruskal Wallis Non-Parametric Test?

The parametric counterpart of the Kruskal Wallis non parametric test is the one way ANOVA test. The test statistic used is H = \(\left ( \frac{12}{N(N+1)}\sum_{1}^{m} \frac{R_{j}^{2}}{n_{j}}\right ) - 3(N+1)\).

What is the Test Statistic Used for the Sign Non-Parametric Test?

The smaller value among the number of positive and negative signs is the test statistic that is used for the sign non-parametric test.

What is the Difference Between a Parametric and Non-Parametric Test?

A parametric test is conducted on data that is obtained from a parameterized distribution such as a normal distribution. On the other hand, a non-parametric test is conducted on a skewed distribution or when the parameters of the population distribution are not known.

What are the Advantages of a Non-Parametric Test?

A non-parametric test does not rely on the assumed parameters of a distribution and is applicable to all data types. Furthermore, they are easy to understand.