Mann-Whitney U Test

 The Mann-Whitney U test is a non-parametric statistical test used to compare the differences between two independent groups on a continuous or ordinal dependent variable. It is often used when the assumptions of the independent samples t-test (such as normality of the data) cannot be met. The Mann-Whitney U test is based on ranks rather than the actual values, making it suitable for comparing distributions that may not be normally distributed.

Key Features of the Mann-Whitney U Test

  1. Non-parametric: It does not assume that the data follows a normal distribution.
  2. Independent Samples: It compares two independent groups or samples.
  3. Ordinal or Continuous Data: The test works with both ordinal and continuous data, as long as the data can be ranked.
  4. Tests the Null Hypothesis: It tests whether the two populations from which the samples were drawn have the same distribution.

Null and Alternative Hypotheses

  • Null Hypothesis (H₀): The two groups come from the same population or have identical distributions (i.e., the distribution of the variable is the same in both groups).
  • Alternative Hypothesis (H₁): The two groups come from different populations or have different distributions (i.e., the distribution of the variable differs between the two groups).

How It Works

The Mann-Whitney U test compares the ranks of values in the two groups. Here's how the test is performed:

  1. Combine the Two Samples: Rank all the data from both groups together, from the smallest to the largest, assigning ranks to each value.

  2. Calculate U Statistic: For each group, calculate the sum of the ranks and use the formula to compute the U statistic. There are two U statistics calculated—one for each group:

    U1=R1−n1(n1+1)2U_1 = R_1 - \frac{n_1(n_1+1)}{2} U2=R2−n2(n2+1)2U_2 = R_2 - \frac{n_2(n_2+1)}{2}

    where:

    • R1R_1 and R2R_2 are the sum of the ranks in each group,
    • n1n_1 and n2n_2 are the number of observations in each group.

  3. Determine the U Value: The test statistic is the smaller of the two U values calculated (either U1U_1 or U2U_2).

  4. Compare the U Statistic to a Critical Value: Compare the calculated U statistic to the critical value from the Mann-Whitney U distribution table (or use a p-value approach). If the U value is smaller than the critical value, the null hypothesis is rejected.

Assumptions

  • Independence: The two groups must be independent of each other.
  • Ordinal/Continuous Data: The dependent variable should be ordinal or continuous, but the data does not need to be normally distributed.
  • Similar Shape of Distribution: The Mann-Whitney U test assumes that the distributions of the two groups are similar in shape, though it does not assume they are identical.

Example

Let’s say you're comparing the test scores between two different teaching methods (Group 1: traditional teaching and Group 2: online teaching).

  1. Rank all the scores from both groups together.
  2. Compute the sum of ranks for each group.
  3. Use the Mann-Whitney formula to calculate U statistics for both groups.
  4. Compare the U statistic to the critical value or p-value to decide whether the teaching methods have different impacts on test scores.

Advantages of the Mann-Whitney U Test

  • Does not assume normality: Ideal for non-normally distributed data.
  • Works with ordinal data: Useful when the data are not measured on an interval or ratio scale.
  • Works with unequal sample sizes: The test can handle different group sizes.

Limitations

  • Less powerful: It can be less powerful than parametric tests (like the t-test) when the assumptions of the t-test are met.
  • Requires independent samples: It cannot be used for paired or dependent samples.

Mann-Whitney U Test vs. t-Test

  • Mann-Whitney U Test: Non-parametric, uses ranks, no assumption of normality, compares distributions.
  • Independent t-Test: Parametric, assumes normality and homogeneity of variance, compares means.

Conclusion

The Mann-Whitney U test is a versatile and widely used test in cases where the assumptions of parametric tests are not met, especially for comparing two independent groups when data are ordinal or continuous but not normally distributed. It offers a robust alternative when dealing with non-parametric data or small sample sizes.

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