What's the best nonparametric test for your data?

1 Comparing two independent groups

If you want to compare the median or the distribution of two independent groups, such as males and females, treatment and control, or pre and post, you can use the Mann-Whitney U test (also known as the Wilcoxon rank-sum test). This test ranks the data from both groups and compares the sum of ranks for each group. It tests the null hypothesis that the two groups have the same median (or distribution). The Mann-Whitney U test is the nonparametric equivalent of the independent t-test.

2 Comparing two paired groups

If you want to compare the median or the distribution of two paired groups, such as before and after, matched pairs, or repeated measures, you can use the Wilcoxon signed-rank test. This test ranks the absolute differences between the paired observations and compares the sum of positive and negative ranks. It tests the null hypothesis that the median difference between the paired groups is zero (or that the distributions are the same). The Wilcoxon signed-rank test is the nonparametric equivalent of the paired t-test.

3 Comparing more than two independent groups

If you want to compare the median or the distribution of more than two independent groups, such as different levels of a factor, different treatments, or different locations, you can use the Kruskal-Wallis H test. This test ranks the data from all groups and compares the mean rank for each group. It tests the null hypothesis that the medians (or distributions) of all groups are equal. The Kruskal-Wallis H test is the nonparametric equivalent of the one-way ANOVA. If the Kruskal-Wallis H test is significant, you can use post-hoc tests, such as the Dunn's test, to compare the groups pairwise.

4 Comparing more than two paired groups

If you want to compare the median or the distribution of more than two paired groups, such as repeated measures over time, different conditions within subjects, or different blocks within subjects, you can use the Friedman test. This test ranks the data within each block (or subject) and compares the mean rank for each group. It tests the null hypothesis that the medians (or distributions) of all groups are equal. The Friedman test is the nonparametric equivalent of the repeated measures ANOVA. If the Friedman test is significant, you can use post-hoc tests, such as the Wilcoxon signed-rank test or the Nemenyi test, to compare the groups pairwise.

5 Testing the association between two variables

If you want to test the association between two variables, such as the correlation between two continuous variables, the association between two ordinal variables, or the agreement between two nominal variables, you can use different nonparametric tests depending on the type and scale of your data. For example, you can use the Spearman's rank correlation coefficient to measure the strength and direction of the monotonic relationship between two continuous or ordinal variables. You can use the Kendall's tau coefficient to measure the degree of concordance or discordance between two ordinal variables. You can use the Chi-square test of independence to test if there is a significant relationship between two nominal variables. You can use the Cohen's kappa coefficient to measure the agreement between two nominal variables.

6 Testing the goodness of fit of a distribution

If you want to test the goodness of fit of a distribution, such as the normality, uniformity, or exponentiality of your data, you can use different nonparametric tests depending on the distribution you want to test. For example, you can use the Kolmogorov-Smirnov test to compare your data with any theoretical distribution and test if they are significantly different. You can use the Shapiro-Wilk test to test if your data follows a normal distribution. You can use the Chi-square goodness of fit test to test if your data follows a discrete distribution, such as a binomial or a Poisson distribution.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?