- Tests of differences between groups (independent samples);
- Tests of differences between variables (dependent samples);
- Tests of relationships between variables.
- 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 samples; nonparametric alternatives for this test are the Wald-Wolfowitz runs test, the Mann-Whitney U test, and the Kolmogorov-Smir
nov two-sample test . - If we have multiple groups, we would use analysis of variance see ANOVA/MANOVA; the nonparametric equivalents to this method are the Kruskal-Wallis analysis of ranks and the Median test.
Differences between dependent groups.
- If we want to compare two variables measured in the same sample we would customarily use the t-test for dependent samples ( for example, if we wanted to compare students' math skills at the beginning of the semester with their skills at the end of the semester). Nonparametric alternatives to this test are the Sign test and Wilcoxon's matched pairs test.
- If the variables of interest are dichotomous in nature (i.e., "pass" vs. "no pass") then McNemar's Chi-square test is appropriate.
- If there are more than two variables that were measured in the same sample, then we would customarily use repeated measures ANOVA. Nonparametric alternatives to this method are Friedman's two-way analysis of variance and Cochran Q test (if the variable was measured in terms of categories, e.g., "passed" vs. "failed"). Cochran Q is particularly useful for measuring changes in frequencies (proportions) across time.
Relationships between variables. To express a relationship between two variables one usually computes the correlation coefficient. Nonparametric equivalents to the standard correlation coefficient are Spearman R, Kendall Tau, and coefficient Gamma (see Nonparametric correlations).
- If the two variables of interest are categorical in nature (e.g., "passed" vs. "failed" by "male" vs. "female") appropriate nonparametric statistics for testing the relationship between the two variables are the Chi-square test, the Phi coefficient, and the Fisher exact test.
- In addition, a simultaneous test for relationships between multiple cases is available: Kendall coefficient of concordance. This test is often used for expressing inter-rater agreement among independent judges who are rating (ranking) the same stimuli.
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