Nonparametric tests like Kruskal-Wallis and Friedman are lifesavers when your data doesn't play nice with normal distributions. They use ranks instead of actual values, making them more robust against outliers and weird data shapes.

These tests help you compare multiple groups or repeated measures without assuming anything about your data's distribution. They're perfect for ordinal data or when ANOVA's assumptions are too strict for your biological samples.

Nonparametric Tests for Multiple Samples

Principles and Assumptions

• Nonparametric tests are statistical methods that do not rely on assumptions about the underlying distribution of the data (normality or equal variances)
• Nonparametric tests are useful when the assumptions of parametric tests (ANOVA or t-tests) are violated or when the data is measured on an ordinal scale
• The null hypothesis in nonparametric tests for comparing multiple samples states that the samples come from populations with the same distribution
• The alternative hypothesis suggests that at least one sample comes from a population with a different distribution

Rank-Based Approach

• Nonparametric tests for comparing multiple samples are based on the ranks of the observations rather than their actual values
• Ranking observations makes nonparametric tests less sensitive to outliers and extreme values
• The test statistics for nonparametric tests (Kruskal-Wallis and Friedman) are calculated based on the ranks of the observations
• Degrees of freedom for nonparametric tests are equal to the number of samples or paired samples minus one (k - 1)

Kruskal-Wallis Test for Independent Samples

Test Overview and Purpose

• The Kruskal-Wallis test is a nonparametric alternative to the one-way ANOVA for comparing multiple independent samples
• The test is used to determine if there are statistically significant differences between three or more independent samples when the assumptions of one-way ANOVA are not met
• The Kruskal-Wallis test is suitable for data measured on an ordinal scale or when the distribution of the data is not normal

Test Procedure and Interpretation

• To conduct the Kruskal-Wallis test, rank all observations from smallest to largest, ignoring the sample membership
• Calculate the sum of ranks for each sample
• The test statistic for the Kruskal-Wallis test, H, is calculated based on the ranks of the observations in each sample and follows a chi-square distribution under the null hypothesis
• If the Kruskal-Wallis test results in a statistically significant difference, post-hoc pairwise comparisons using Dunn's test or the Mann-Whitney U test with a Bonferroni correction can be used to determine which samples differ significantly from each other

Friedman Test for Paired Samples

Test Overview and Purpose

• The Friedman test is a nonparametric alternative to the repeated measures ANOVA for comparing multiple paired samples
• The test is used to determine if there are statistically significant differences between three or more paired samples (repeated measurements on the same subjects) when the assumptions of repeated measures ANOVA are not met
• The Friedman test is suitable for data measured on an ordinal scale or when the distribution of the data is not normal

Test Procedure and Interpretation

• To conduct the Friedman test, rank the observations within each subject from smallest to largest
• Calculate the sum of ranks for each paired sample
• The test statistic for the Friedman test, Q, is calculated based on the ranks of the observations within each subject and follows a chi-square distribution under the null hypothesis
• If the Friedman test results in a statistically significant difference, post-hoc pairwise comparisons using the Wilcoxon signed-rank test with a Bonferroni correction can be used to determine which paired samples differ significantly from each other

Interpreting Nonparametric Test Results

Statistical Significance and Effect Size

• A statistically significant result in the Kruskal-Wallis test indicates that at least one of the independent samples comes from a population with a different distribution, suggesting that the factor of interest has an effect on the response variable
• A statistically significant result in the Friedman test indicates that at least one of the paired samples comes from a population with a different distribution, suggesting that the factor of interest has an effect on the response variable
• The effect size for the Kruskal-Wallis and Friedman tests can be calculated using the eta-squared (η²) statistic, which represents the proportion of variance in the response variable explained by the factor of interest

Biological Significance and Reporting Results

• When interpreting the results of nonparametric tests in biological data analysis, it is essential to consider the biological significance of the findings in addition to the statistical significance
• Consider the limitations of the study design and potential confounding factors when interpreting results
• The results of Kruskal-Wallis and Friedman tests should be reported in a clear and concise manner, including the test statistic, degrees of freedom, p-value, effect size, and a brief interpretation of the findings in the context of the research question and biological system under study