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I want to compare the means between groups 1 vs. 2 and also 2 vs. 3. All three groups are nonnormal, as the Shapiro-Wilk test resulted in p values of 0.03, 0.04 and <0.0001 respectively. n is 10, 13 and 38 respectively, and the Levene test for equal variances resulted in p values of 0.22 for 1 vs. 2 and 0.44 for 2 vs. 3 (so they seem to have equal variances). Both, the parametric t-test and the nonparametric Wilcoxon test resulted in insignificant differences with p values larger than 0.5. Which test is more appropriate to this situation? The groups do not follow a normal distribution, have different sample size, but seem to be homoscedastic. Your help will be very much appreciated.

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The Mann-Whitney U (also called Wilcoxon rank sum) test is the nonparametric "equivalent" of the two sample t-test. Sounds like you did the right test(s).

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