1. Non – Parametric Test
Dr. Anshul Singh Thapa

2. An Introduction
Most of the statistical tests require an important assumption to be met if they are to be correctly applied. This assumption is that data of the population from which a sample or samples are drawn is normally distributed. These statistical tests allow considerable latitude or deviation from normality. If the distribution from which a sample is drawn is non-normal, then these statistical tests will not yield meaningful results.
A second assumption upon which most of the tests rest is that meaningful sample statistics, such as mean and standard deviation, can be derived from the sample(s) and used to estimate the corresponding population parameters. Data which are nominal or ordinal in nature do not yield meaningful results.

3. Statisticians have devised alternate procedures which can be used to test hypothesis about data which are non – normal or for which meaningful sample statistics cannot be calculated.
Since these tests do not depends on the shape of the distribution, they are called distribution free tests.
These tests do not depends upon the population parameters, such as the mean and variance therefore they are also called Non – Parametric Tests

4. Uses and application of Non – Parametric Test
It is apparent that there are number of factors involved in choosing whether or not to use non parametric test, these are:
Level of measurement
Sample size and
Sample distribution
It must be remembered that for each of the main parametric test there is non parametric test available. In general these test falls into following categories:
Tests of differences between groups (independent sample).
Test of differences between variables (dependent test).
Tests of relationship between variables.

5. Tests of differences between groups (independent sample).
Usually when we have two samples that we want to compare concerning their mean value for some variable of interest, we would use the t-test for independent sample. The non parametric alternatives for this test are the Wald-Wolfowitz runs test, the Mann – Whitney U test.
If we have multiple groups, we would use analysis of variance i.e., ANOVA/MANOVA. The non-parametric equivalents to this method are the Kruskal Wallis analysis of ranks and the Median test.

6. Test of differences between variables (dependent test).
If we want to compare two variables measured in the same sample we would customarily use the t-test for dependent sample (paired t-test). Non – parametric alternatives to this test are the sign test and Wilcoxon’s matched pairs test.
If there are more than two variables that were measured in the same sample, then we would customarily use repeated measure ANOVA. Non – parametric alternative to this methods are Friedman’s two – way analysis of variance.

7. Tests of relationship between variables.
To express a relationship between two variables one usually computes the correlation coefficient. The Non – parametric equivalents to the standard correlation coefficient of Pearson ‘r’ are Spearman and Kendall rank correlations.
The appropriate non – parametric statistics for testing the relationship between two variables are chi-square.

8. Alternative for parametric and non parametric tests
Correlation test
Pearson
Spearman
Independent Sample
2 Groups
Independent sample t – test
Mann – Whitney test
(U – test)
More than 2 Groups
One way independent measures ANOVA
Kruskal – Wallis Test
Repeated Measure,
2 conditions
Paired t – test
Wilcoxon test
More than 2 conditions
One way repeated measures ANOVA
Friedman’s Test