What the Mann-Whitney U test is and when to use it

The Mann-Whitney U test is a non-parametric statistical test used in psychology to determine whether there is a significant difference between two independent groups on a continuous or ordinal variable. This test is particularly useful when comparing groups that are not related, such as testing differences in scores between participants from different backgrounds. For example, it could assess whether children from varying socio-economic environments show different levels of gender stereotyping in their responses to stories.

Key features of the Mann-Whitney U test

• Independent groups - The test compares two separate samples, where participants in one group are not matched or related to those in the other.
• Ordinal or continuous data - It works with ranked data or measurements that can be ordered.
• Hypothesis testing - Typically used for a directional (one-tailed) or non-directional (two-tailed) hypothesis, focusing on whether one group tends to have higher or lower ranks than the other.
• U statistic - The test calculates a value called U, which reflects the degree of overlap in ranks between the groups.

Steps for calculating Mann-Whitney U

To perform the Mann-Whitney U test, follow a systematic process that involves ranking the data and applying formulas to find the U value.

The calculation process

1. Combine all scores from both groups and rank them from lowest to highest.
2. If there are ties, assign the average rank to tied scores.
3. Calculate the sum of ranks for group A (R_a) and group B (R_b).
4. Use the formulas to find U_a and U_b, where n_a and n_b are the sample sizes of groups A and B.
5. Select the smaller U value as the test statistic (U).
6. Compare U to critical values from a table to assess significance; if it is equal to or less than the critical value, the results are significant.

Formula for U_a:

$U_a=n_a{n}_b+\frac{n_a(n_a+1)}{2}-\mathrm{\Sigma }{R}_a$

Where:

• U_a = U value for group A
• n_a = Number of participants in group A
• n_b = Number of participants in group B
• ΣR_a = Sum of ranks for group A

Formula for U_b:

$U_b=n_a{n}_b+\frac{n_b(n_b+1)}{2}-\mathrm{\Sigma }{R}_b$

Where:

• U_b = U value for group B
• n_a = Number of participants in group A
• n_b = Number of participants in group B
• ΣR_b = Sum of ranks for group B

These steps convert raw data into ranks, helping to identify if one group systematically outranks the other.

Worked example - Calculating Mann-Whitney U

Participants were tested on their recall of details from psychological case studies, with scores out of 100 (higher scores indicate better recall). Group A (n=6) consisted of participants with prior psychology experience, while group B (n=7) had no prior experience. The scores are as follows:

Step 1: Rank all scores combined

Combined and ranked scores (lowest to highest): 28 (rank 1), 31 (rank 2), 33 (rank 3), 35 (rank 4), 38 (rank 5), 41 (rank 6), 42 (rank 7), 45 (rank 8), 65 (rank 9), 68 (rank 10), 72 (rank 11), 81 (rank 12), 83 (rank 13).

Group A ranks: 8, 5, 4, 7, 1, 9

ΣR_a = 34

Group B ranks: 11, 12, 6, 13, 2, 10, 3

ΣR_b = 57

Step 2: Calculate U_a

$U_a=6\times 7+\frac{6\times (6+1)}{2}-34=42+\frac{42}{2}-34=42+21-34=29$

Step 3: Calculate U_b

$U_b=6\times 7+\frac{7\times (7+1)}{2}-57=42+\frac{56}{2}-57=42+28-57=13$

Step 4: Select the smaller U

U = smaller of 29 and 13, so U = 13.

Determining statistical significance

Once the U value is calculated, compare it to critical values from a Mann-Whitney U table to decide if the results are statistically significant.

Key considerations for significance

• Level of significance (p-value) - Common levels are p ≤ 0.05 (5% chance of error) or p ≤ 0.01 (1% chance of error).
• One-tailed vs. two-tailed - Use a one-tailed test for a directional hypothesis (e.g., group A will score higher); use two-tailed for non-directional.
• Critical value - This depends on sample sizes (n_a and n_b) and the p-level. The observed U must be less than or equal to the critical value for significance.
• Interpretation - If U ≤ critical value, reject the null hypothesis (there is a significant difference). If U > critical value, accept the null hypothesis (no significant difference).