14.2 Mann-Whitney U Test

The Mann-Whitney U test is a powerful tool for comparing two independent samples when data isn't normally distributed or is ordinal. It's like a nonparametric version of the t-test, helping us spot differences between groups without assuming normal distributions.

This test combines and ranks data from both samples, calculates rank sums, and uses the U statistic to determine if there's a significant difference. For larger samples, we can use z-scores, while smaller samples rely on critical value tables.

Nonparametric Tests for Two Independent Samples

Appropriateness of Mann-Whitney U Test

• Compares two independent samples not related or paired in any way (treatment vs control group)
• Serves as a nonparametric alternative to the independent samples t-test when data is ordinal or assumptions of normality are violated (non-normal distribution, presence of outliers)
• Tests the null hypothesis that the two populations have the same distribution against the alternative hypothesis that they have different distributions (location, shape, or variability)

Execution of Mann-Whitney U Test

• Combines observations from both samples into a single ordered array and assigns ranks from lowest to highest (1, 2, 3...)
  • Averages the ranks for tied observations (2.5 for two observations tied for 2nd and 3rd rank)
• Calculates the rank sum for each sample ($R_1$ for sample 1 and $R_2$ for sample 2)
• Determines the sample sizes of the two groups ($n_1$ for sample 1 and $n_2$ for sample 2)
• Calculates the U statistic for each sample using the formulas:
  1. $U_1 = n_1n_2 + \frac{n_1(n_1+1)}{2} - R_1$
  2. $U_2 = n_1n_2 + \frac{n_2(n_2+1)}{2} - R_2$
• Selects the smaller U value ($U_{min}$) for further analysis

Calculation of U statistic

• Follows a known distribution for sample sizes greater than 20, allowing for z-score calculation
  • Relies on tables to find critical values for smaller sample sizes (n < 20)
• Calculates the mean of the U statistic: $\mu_U = \frac{n_1n_2}{2}$
• Calculates the standard deviation of the U statistic: $\sigma_U = \sqrt{\frac{n_1n_2(n_1+n_2+1)}{12}}$
• For large sample sizes, calculates the z-score: $z = \frac{U_{min} - \mu_U}{\sigma_U}$
• Compares the calculated z-score or $U_{min}$ to the critical value at the desired significance level (0.05, 0.01)

Interpretation of Mann-Whitney results

• Rejects the null hypothesis if the calculated z-score or $U_{min}$ is less than the critical value, concluding a significant difference between the two populations (treatment effect)
• Fails to reject the null hypothesis if the calculated z-score or $U_{min}$ is greater than the critical value, concluding insufficient evidence to suggest a significant difference (no treatment effect)
• Reports results using appropriate language and statistical terminology, including the U statistic, sample sizes, significance level, and conclusion drawn from the test
• Discusses the implications of the findings in the context of the research question or problem being addressed (practical significance, limitations, future research)