Key Nonparametric Statistical Tests

Nonparametric statistical tests are essential tools in Advanced Quantitative Methods, especially when data doesn't meet normality assumptions. These tests, like the Mann-Whitney U and Wilcoxon signed-rank tests, help analyze ordinal and non-normally distributed continuous data effectively.

1. Mann-Whitney U test

   • Compares differences between two independent groups when the dependent variable is ordinal or continuous but not normally distributed.
   • Ranks all data points from both groups together, then calculates the U statistic based on these ranks.
   • Useful for small sample sizes and when assumptions of parametric tests (like normality) are violated.

2. Wilcoxon signed-rank test

   • Tests for differences between two related samples or matched observations, focusing on the ranks of the differences.
   • Suitable for ordinal data or continuous data that do not meet normality assumptions.
   • Provides a nonparametric alternative to the paired t-test.

3. Kruskal-Wallis test

   • Extends the Mann-Whitney U test to compare three or more independent groups.
   • Ranks all data points across groups and assesses whether the rank distributions differ significantly.
   • Ideal for ordinal data or continuous data that is not normally distributed.

4. Friedman test

   • A nonparametric alternative to the repeated measures ANOVA, used for comparing three or more related groups.
   • Analyzes the ranks of the data across different conditions or time points.
   • Useful for ordinal data or continuous data that does not meet the assumptions of parametric tests.

5. Spearman's rank correlation coefficient

   • Measures the strength and direction of association between two ranked variables.
   • Suitable for ordinal data or continuous data that is not normally distributed.
   • Provides insight into monotonic relationships, where one variable tends to increase as the other does.

6. Chi-square test of independence

   • Assesses whether there is a significant association between two categorical variables in a contingency table.
   • Compares observed frequencies with expected frequencies under the assumption of independence.
   • Requires a sufficient sample size to ensure validity of results.

7. Sign test

   • A simple nonparametric test used to determine if there is a median difference between paired observations.
   • Focuses on the direction of differences (positive or negative) rather than their magnitude.
   • Useful for small sample sizes and when data does not meet normality assumptions.

8. Kolmogorov-Smirnov test

   • Compares the distribution of a sample with a reference probability distribution or compares two samples.
   • Tests for differences in the shape of distributions, making it useful for assessing normality.
   • Nonparametric and applicable to continuous data.

9. Kendall's tau

   • Measures the strength and direction of association between two variables using the ranks of the data.
   • More robust to ties than Spearman's rank correlation and provides a measure of ordinal association.
   • Suitable for small sample sizes and non-normally distributed data.

10. McNemar's test

    • A nonparametric test used for paired nominal data to determine if there are differences in proportions.
    • Commonly applied in before-and-after studies or matched case-control studies.
    • Focuses on changes in responses rather than the magnitude of changes.