Nonparametric Tests to Know for Statistical Inference

Nonparametric tests are essential tools in statistical inference, especially when data doesn't meet normality assumptions. They provide flexible methods for comparing groups, assessing relationships, and estimating distributions, making them invaluable in data science for robust analysis.

1. Wilcoxon Signed-Rank Test

   • Used to compare two related samples or repeated measurements on a single sample to assess whether their population mean ranks differ.
   • Nonparametric alternative to the paired t-test, suitable when the data does not meet normality assumptions.
   • Involves ranking the absolute differences between paired observations and considering the signs of these differences.

2. Mann-Whitney U Test

   • Compares differences between two independent groups to determine if they come from the same distribution.
   • Nonparametric alternative to the independent t-test, useful when data is not normally distributed.
   • Based on ranking all observations from both groups and calculating the U statistic to assess differences.

3. Kruskal-Wallis Test

   • Extends the Mann-Whitney U Test to compare three or more independent groups.
   • Nonparametric alternative to one-way ANOVA, applicable when the assumptions of ANOVA are violated.
   • Evaluates whether the median ranks of the groups differ, using rank sums to compute the test statistic.

4. Friedman Test

   • Used for comparing three or more related groups, assessing differences in treatments across multiple test attempts.
   • Nonparametric alternative to repeated measures ANOVA, suitable for non-normally distributed data.
   • Involves ranking the data within each block and analyzing the ranks to determine if there are significant differences.

5. Sign Test

   • A simple nonparametric test used to determine if there is a median difference between paired observations.
   • Focuses on the direction of differences rather than their magnitude, making it robust to outliers.
   • Suitable for small sample sizes and when data does not meet the assumptions of other tests.

6. Spearman's Rank Correlation

   • Measures the strength and direction of association between two ranked variables.
   • Nonparametric alternative to Pearson's correlation, applicable when data is ordinal or not normally distributed.
   • Assesses how well the relationship between two variables can be described using a monotonic function.

7. Kendall's Tau

   • Another measure of correlation that assesses the strength and direction of association between two variables.
   • Nonparametric alternative to Pearson's correlation, particularly useful for small sample sizes or data with ties.
   • Focuses on the ordinal ranking of data and calculates the degree of concordance and discordance between pairs.

8. Kolmogorov-Smirnov Test

   • Compares a sample distribution with a reference probability distribution or compares two sample distributions.
   • Nonparametric test that assesses the goodness of fit and can be used for one-sample or two-sample scenarios.
   • Evaluates the maximum distance between the empirical distribution function and the theoretical distribution.

9. Permutation Tests

   • A class of nonparametric tests that assess the significance of observed data by comparing it to a distribution generated by rearranging the data.
   • Useful for hypothesis testing without relying on traditional assumptions about the data distribution.
   • Provides exact p-values and can be applied to various statistical tests, enhancing flexibility in analysis.

10. Bootstrap Methods

    • Resampling techniques used to estimate the distribution of a statistic by repeatedly sampling with replacement from the data.
    • Nonparametric approach that allows for the estimation of confidence intervals and standard errors without relying on normality assumptions.
    • Particularly useful for small sample sizes or complex estimators where traditional methods may fail.