Tests of difference: Mann- whitney and wilcoxon

Question 1

Overview of Tests of Difference

Answer 1

• Purpose: To determine if there are significant differences between two sets of data.
• Types: Mann-Whitney U test (for independent samples) and Wilcoxon Signed-Rank test (for related samples).

Question 2

Mann-Whitney U Test

Answer 2

• Definition: A non-parametric test used to compare differences between two independent groups when the data does not meet the assumptions of normality.
• When to Use:
  o Two independent groups (e.g., treatment vs. control).
  o Ordinal data or continuous data that is not normally distributed.

Question 3

Steps for Conducting Mann-Whitney U Test

Answer 3

1. Rank all the data: Combine both groups and rank the scores from lowest to highest.
2. Calculate U: Use the ranks to calculate the U statistic for each group:
   o U1=R1−n1(n1+1)2U_1 = R_1 - \frac{n_1(n_1 + 1)}{2}U1=R1−2n1(n1+1) (for group 1)
   o U2=R2−n2(n2+1)2U_2 = R_2 - \frac{n_2(n_2 + 1)}{2}U2=R2−2n2(n2+1) (for group 2)
   o Choose the smaller U value for the final analysis.
3. Determine significance: Compare the U value against critical values from the Mann-Whitney distribution table or calculate a p-value.

Question 4

Example of Mann-Whitney U Test

Answer 4

• Scenario: Comparing stress levels between two independent groups of students (Group A: Meditation, Group B: No meditation).
• Data Collection: Collect stress scores from both groups.
• Analysis: Use the Mann-Whitney U test to assess if there is a significant difference in stress levels.

Question 5

Wilcoxon Signed-Rank Test

Answer 5

• Definition: A non-parametric test used to compare two related samples or matched pairs to assess whether their population mean ranks differ.
• When to Use:
  o Related samples (e.g., pre-test vs. post-test scores).
  o Ordinal data or continuous data that is not normally distributed.

Question 6

Steps for Conducting Wilcoxon Signed-Rank Test

Answer 6

1. Calculate the differences: For each pair, subtract one score from the other.
2. Rank the absolute differences: Assign ranks to the absolute differences, ignoring the sign.
3. Assign signs to the ranks: Apply the original signs (positive or negative) to the ranks based on the direction of the difference.
4. Calculate the W statistic: Sum the ranks for positive differences (T+) and negative differences (T-):
   o W=min⁡(T+,T−)W = \min(T+, T-)W=min(T+,T−)
5. Determine significance: Compare the W value against critical values from the Wilcoxon signed-rank table or calculate a p-value.

Question 7

Example of Wilcoxon Signed-Rank Test

Answer 7

• Scenario: Evaluating the effectiveness of a new study program by comparing students’ test scores before and after the program.
• Data Collection: Collect pre-test and post-test scores from the same students.
• Analysis: Use the Wilcoxon signed-rank test to assess if there is a significant difference in scores before and after the program.

Question 8

Reporting Results

Answer 8

• Format: When reporting results, include the test statistic, sample sizes, and p-value.
  o Example: “The Mann-Whitney U test revealed a significant difference in stress levels between Group A (Mdn = 25) and Group B (Mdn = 30), U = 45, p < 0.05.”
  o Example: “The Wilcoxon signed-rank test indicated a significant increase in test scores after the program (T+ = 15, p < 0.01).”

Question 9

Advantages and Limitations

Answer 9

• Advantages:
  o Non-parametric tests do not require normality, making them suitable for ordinal or non-normally distributed data.
  o Can be used with small sample sizes.
• Limitations:
  o Less powerful than parametric tests if the assumptions of the parametric tests are met.
  o May not provide as much information as parametric tests regarding effect sizes.