Two-Sample Non-Parametric Tests

1. Mann-Whitney U Test (Wilcoxon Rank-Sum Test)

The Mann-Whitney U test, also known as the Wilcoxon rank-sum test, is a non-parametric statistical test used to compare the distributions of two independent groups. It is particularly useful when the assumptions of a parametric test, such as the independent two-sample t-test, are not met (e.g., non-normality or ordinal data).

Key Features

• Purpose: Tests whether the distributions of two independent groups are the same, or whether one group tends to have larger (or smaller) values than the other.
• Assumptions:
  • The two groups are independent of each other.
  • The observations are ordinal, interval, or ratio (but do not require normality).
  • The two samples are randomly drawn.
• Null Hypothesis (\(H_0\)): The two groups have the same distribution (no difference in medians or ranks).
• Alternative Hypothesis (\(H_1\)): The distributions are different, or one group tends to have higher values than the other.

How the Test Works

Step 1: Combine all data from both groups and assign ranks to the values. The smallest value gets rank 1, the second smallest rank 2, and so on. If there are ties, assign the average of the tied ranks.

Step 2: Calculate the sum of ranks for each group:

• \(R_1\): Sum of ranks for group 1
• \(R_2\): Sum of ranks for group 2

Step 3: Compute the U-statistic for each group:

\[U_1 = n_1 n_2 + \frac{n_1 (n_1 + 1)}{2} - R_1\]

\[U_2 = n_1 n_2 + \frac{n_2 (n_2 + 1)}{2} - R_2\]

where \(n_1\) and \(n_2\) are the sample sizes for groups 1 and 2.

Step 4: The Mann-Whitney U statistic is:

\[U = \min(U_1, U_2)\]

Step 5: Determine significance:

• For large sample sizes (\(n_1, n_2 > 20\)), use a normal approximation with a Z-score.
• For small sample sizes, use the exact distribution of \(U\).

Interpretation

• If the p-value is small (e.g., \(p < 0.05\)): Reject \(H_0\); the two groups have significantly different distributions.
• If the p-value is large: Fail to reject \(H_0\); no evidence of a difference.

Which Group is Larger if H_0 is Rejected?

Compare the mean rank of each group. A higher mean rank indicates that the values in that group tend to be larger.

Python Implementation

import numpy as np
from scipy.stats import mannwhitneyu

# Example: comparing test scores of two teaching methods
group_a = np.array([85, 78, 92, 88, 76, 95, 89, 82, 91, 87])
group_b = np.array([72, 68, 81, 75, 70, 77, 74, 69, 73, 71])

# Perform Mann-Whitney U test
stat, p_value = mannwhitneyu(group_a, group_b, alternative='two-sided')

print(f"U statistic: {stat}")
print(f"P-value: {p_value:.4f}")

# Mean ranks
combined = np.concatenate([group_a, group_b])
ranks = np.argsort(np.argsort(combined)) + 1
mean_rank_a = ranks[:len(group_a)].mean()
mean_rank_b = ranks[len(group_a):].mean()
print(f"Mean rank Group A: {mean_rank_a:.1f}")
print(f"Mean rank Group B: {mean_rank_b:.1f}")

alpha = 0.05
if p_value < alpha:
    print("Reject H0: Significant difference between groups.")
    if mean_rank_a > mean_rank_b:
        print("Group A tends to have larger values.")
    else:
        print("Group B tends to have larger values.")
else:
    print("Fail to reject H0.")

Equivalence Note

The Mann-Whitney U test and Wilcoxon rank-sum test are statistically equivalent. The choice of terminology often depends on the software:

• SPSS: "Mann-Whitney U test"
• R: "Wilcoxon rank-sum test" (wilcox.test)
• Python scipy: mannwhitneyu or ranksums

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2. Mood's Median Test

Mood's Median Test is a non-parametric hypothesis test used to compare the medians of two or more groups. It is particularly useful when data is non-normal, ordinal, or contains outliers, as it focuses solely on the median rather than the mean or variance.

Key Features

• Non-parametric: Does not require assumptions about the distribution.
• Robust to Outliers: Focuses on the median.
• Applicable for Two or More Groups.
• Null Hypothesis (\(H_0\)): All groups have the same median.
• Alternative Hypothesis (\(H_1\)): At least one group has a different median.

Assumptions

1. The samples are independent.
2. The data is continuous or ordinal.
3. The groups are random samples from the population.

How It Works

Step 1: Calculate the overall median of the combined dataset.

Step 2: For each group, classify data points as "above" or "below" the overall median.

Step 3: Construct a contingency table showing the counts of values above and below the overall median for each group.

Step 4: Apply the chi-square test of independence to the contingency table:

\[\chi^2 = \sum \frac{(O - E)^2}{E}\]

where \(O\) is the observed frequency and \(E\) is the expected frequency under the null hypothesis.

Step 5: Evaluate p-value to determine significance.

Worked Example

Scenario: Compare median test scores of two student groups.

• Group A: \([50, 55, 60, 65, 70]\)
• Group B: \([45, 50, 55, 60, 65]\)

Python Implementation

import numpy as np
from scipy.stats import chi2_contingency

def moods_median_test(*groups):
    """
    Perform Mood's Median Test to compare medians of multiple groups.

    Parameters:
    - groups: Variable number of arrays representing the groups.

    Returns:
    - chi2_stat: The chi-square statistic.
    - p_value: The p-value for the test.
    - contingency_table: The contingency table used.
    """
    combined_data = np.concatenate(groups)
    overall_median = np.median(combined_data)

    contingency_table = []
    for group in groups:
        above_median = np.sum(group > overall_median)
        below_median = np.sum(group < overall_median)
        contingency_table.append([above_median, below_median])

    contingency_table = np.array(contingency_table).T
    chi2_stat, p_value, _, _ = chi2_contingency(contingency_table)

    return chi2_stat, p_value, contingency_table

# Example Data
group_a = np.array([50, 55, 60, 65, 70])
group_b = np.array([45, 50, 55, 60, 65])

chi2_stat, p_value, contingency_table = moods_median_test(group_a, group_b)

print(f"Chi-Square Statistic: {chi2_stat:.4f}")
print(f"P-value: {p_value:.4f}")
print(f"Contingency Table:\n{contingency_table}")

Output:

Chi-Square Statistic: 0.4000
P-value: 0.5271
Contingency Table:
[[3 2]
 [2 3]]

Interpretation: The p-value (0.5271) is greater than 0.05, so we fail to reject the null hypothesis. No significant difference in medians.

Advantages and Limitations

Advantages	Limitations
Robust to outliers	Ignores variability within groups
Non-parametric	Low power compared to parametric tests
Simple to implement	Requires independence
Works with multiple groups

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3. Kruskal-Wallis Test

The Kruskal-Wallis test is the non-parametric extension of one-way ANOVA, used to compare the distributions of three or more independent groups.

Key Features

• Purpose: Test whether the medians (or distributions) of three or more groups differ.
• Null Hypothesis: All groups have the same distribution.
• Alternative: At least one group differs.
• Relation: When there are only two groups, the Kruskal-Wallis test is equivalent to the Mann-Whitney U test.

Test Statistic

\[H = \frac{12}{N(N+1)} \sum_{i=1}^{k} \frac{R_i^2}{n_i} - 3(N+1)\]

where \(N\) is the total number of observations, \(k\) is the number of groups, \(n_i\) is the size of group \(i\), and \(R_i\) is the sum of ranks for group \(i\).

Under \(H_0\), \(H\) follows approximately a \(\chi^2\) distribution with \(k - 1\) degrees of freedom.

Python Implementation

from scipy.stats import kruskal

group_a = [85, 78, 92, 88, 76]
group_b = [72, 68, 81, 75, 70]
group_c = [90, 95, 88, 92, 87]

stat, p_value = kruskal(group_a, group_b, group_c)

print(f"H statistic: {stat:.4f}")
print(f"P-value: {p_value:.4f}")

alpha = 0.05
if p_value < alpha:
    print("Reject H0: At least one group differs significantly.")
else:
    print("Fail to reject H0.")

Post-Hoc Testing

If the Kruskal-Wallis test is significant, use pairwise Mann-Whitney U tests with a correction for multiple comparisons (e.g., Bonferroni correction) to determine which specific groups differ.

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Comparison: Two-Sample Parametric vs Non-Parametric Tests

Feature	Two-Sample t-Test	Mann-Whitney U	Mood's Median
Assumption	Normality, equal variances	None (ordinal+)	None
Tests	Means	Distributions/ranks	Medians
Power	Highest (when normal)	Moderate	Low
Robustness	Sensitive to outliers	Robust	Very robust
Multiple groups	Use ANOVA	Use Kruskal-Wallis	Can handle

Guideline for Choosing

• Normal data, equal variances: Use the two-sample t-test (or Welch's t-test for unequal variances).
• Non-normal data, continuous: Use the Mann-Whitney U test.
• Data with extreme outliers: Use Mood's Median Test.
• Three or more groups: Use Kruskal-Wallis (non-parametric) or ANOVA (parametric).