1.7. Non-Parametric test for Hypothesis Testing: Mann-Whitney U Test

The Mann-Whitney U Test, also referred to as the Wilcoxon Rank Sum Test, is a non-parametric statistical method used to compare two samples or groups.

This test evaluates whether the two sampled groups are likely to come from the same population, essentially questioning if these two populations have the same data distribution. In other words, it seeks evidence to determine whether the groups originate from populations with different levels of a variable of interest. Consequently, the hypotheses in a Mann-Whitney U Test are as follows [1]:

• The null hypothesis (H0) posits that the two populations are equal.

• The alternative hypothesis (H1) suggests that the two populations are not equal.

Some researchers view this as a comparison of the medians between the two populations, while parametric tests compare the means between two independent groups. In specific cases, where the data have similar shapes (as per the assumptions), this interpretation is valid. However, it's important to note that medians are not directly involved in the calculation of the Mann-Whitney U test statistic. Two groups could have the same median and still show significant differences according to the Mann-Whitney U test.

To illustrate the application of the test, it was generated two sets of sample data (old_drug and new_drug) using a normal distribution with different means. This creates datasets suitable for illustrating the Mann-Whitney U test.

Some key assumptions for the Mann-Whitney U Test are detailed below [1, 2]: 

• The variable being compared between the two groups must be continuous (able to take any number in a range – for example, age, weight, height, or heart rate). This is because the test is based on ranking the observations in each group. 

• The data are assumed to take a non-normal, or skewed, distribution. If your data are normally distributed, the unpaired Student’s t-test should be used to compare the two groups instead.  The next figure illustrates this point.

• While the data in both groups are not assumed to be Normal, the data are assumed to be similar in shape across the two groups. 

• The data should be two randomly selected independent samples, meaning the groups have no relationship to each other. If samples are paired (for example, two measurements from the same group of participants), then a paired samples t-test should be used instead. 

• A sufficient sample size is needed for a valid test, usually more than 5 observations in each group.

The lowest value here is assigned the rank 1 and the second lowest value is assigned the rank 2 and so on.

We assign the mean rank when there are ties in a sample to make sure that the sum of ranks in each sample of size n is same. Therefore, the sum of ranks will always be equal to n(n+1)/2.

import numpy as np

import pandas as pd

import matplotlib.pyplot as plt

from scipy.stats import mannwhitneyu

# Data for old and new drugs

old_drug = [7, 8, 4, 9, 8]

new_drug = [3, 4, 2, 1, 1]

# Step 1: Plot frequency bars for each drug

def plot_frequency_bars(old_drug, new_drug):

    labels, counts_old = np.unique(old_drug, return_counts=True)

    _, counts_new = np.unique(new_drug, return_counts=True)

    fig, ax = plt.subplots(1, 2, figsize=(12, 6))

    ax[0].bar(labels, counts_old, color='blue', alpha=0.7)

    ax[0].set_title('Old Drug')

    ax[0].set_xlabel('Number of Sleepwalking Cases')

    ax[0].set_ylabel('Frequency')

    ax[1].bar(labels, counts_new, color='green', alpha=0.7)

    ax[1].set_title('New Drug')

    ax[1].set_xlabel('Number of Sleepwalking Cases')

    ax[1].set_ylabel('Frequency')

    plt.tight_layout()

    plt.show()

plot_frequency_bars(old_drug, new_drug)

# Combine the samples and create a DataFrame

data = pd.DataFrame({

    'Sleepwalking Cases': old_drug + new_drug,

    'Group': ['OD']*len(old_drug) + ['ND']*len(new_drug)

})

# Step 2: Rank the combined data

data['Rank'] = data['Sleepwalking Cases'].rank()

# Separate the ranks by group

rank_sum_old = data[data['Group'] == 'OD']['Rank'].sum()

rank_sum_new = data[data['Group'] == 'ND']['Rank'].sum()

# Step 3: Compute U1 and U2 using the ranks

n1 = len(old_drug)

n2 = len(new_drug)

R1 = rank_sum_old

R2 = rank_sum_new

U1 = n1 * n2 + (n1 * (n1 + 1)) / 2 - R1

U2 = n1 * n2 + (n2 * (n2 + 1)) / 2 - R2

print(f"Rank sum for old drug (R1): {R1}")

print(f"Rank sum for new drug (R2): {R2}")

print(f"U1: {U1}")

print(f"U2: {U2}")

# Step 4: Use scipy to compute the p-value

u_statistic, p_value = mannwhitneyu(old_drug, new_drug, alternative='two-sided')

print(f"Mann-Whitney U statistic: {u_statistic}")

print(f"P-value: {p_value}")

# Display the ranks table

print("\nData with ranks:")

print(data)

# Conclusion based on p-value

alpha = 0.05

if p_value < alpha:

    print("Reject the null hypothesis: There is a significant difference between the two drugs.")

else:

    print("Fail to reject the null hypothesis: There is no significant difference between the two drugs.")

import numpy as np

from scipy.stats import mannwhitneyu, ttest_ind

# Generate normally distributed sample data

np.random.seed(42)

group1 = np.random.normal(loc=10, scale=2, size=30)

group2 = np.random.normal(loc=12, scale=2, size=30)

# Mann-Whitney U Test

u_statistic, p_value_u = mannwhitneyu(group1, group2, alternative='two-sided')

print(f"Mann-Whitney U test: U statistic = {u_statistic}, p-value = {p_value_u}")

# Independent Samples t-Test

t_statistic, p_value_t = ttest_ind(group1, group2)

print(f"Independent samples t-test: t statistic = {t_statistic}, p-value = {p_value_t}")

import numpy as np

import pandas as pd

import matplotlib.pyplot as plt

import seaborn as sns

from scipy.stats import mannwhitneyu

# Seaborn style for visualization

sns.set(style="whitegrid")

# Generate sample data

np.random.seed(42)

normal_data = np.random.normal(loc=10, scale=2, size=30)  # Normal distribution data

skewed_data = np.random.exponential(scale=2, size=30) + 5  # Skewed distribution data

# Combine the samples and create a DataFrame

data_normal = pd.DataFrame({

    'Value': normal_data,

    'Group': ['Normal']*len(normal_data)

})

data_skewed = pd.DataFrame({

    'Value': skewed_data,

    'Group': ['Skewed']*len(skewed_data)

})

# Plot frequency bars and continuous curves side-by-side

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(16, 6))

# Frequency bars for normal data

sns.histplot(normal_data, bins=10, color='blue', alpha=0.6, label='Normal', kde=False, ax=ax1)

# Secondary y-axis for KDE plots for normal data

ax1_2 = ax1.twinx()

sns.kdeplot(normal_data, color='blue', label='Normal KDE', ax=ax1_2)

ax1_2.set_ylim(0, ax1.get_ylim()[1] / len(normal_data))

ax1.set_title('Normal Distribution')

ax1.set_xlabel('Value')

ax1.set_ylabel('Frequency')

ax1_2.set_ylabel('Density')

# Frequency bars for skewed data

sns.histplot(skewed_data, bins=10, color='green', alpha=0.6, label='Skewed', kde=False, ax=ax2)

# Secondary y-axis for KDE plots for skewed data

ax2_2 = ax2.twinx()

sns.kdeplot(skewed_data, color='green', label='Skewed KDE', ax=ax2_2)

ax2_2.set_ylim(0, ax2.get_ylim()[1] / len(skewed_data))

ax2.set_title('Skewed Distribution')

ax2.set_xlabel('Value')

ax2.set_ylabel('Frequency')

ax2_2.set_ylabel('Density')

# Combine legends from both axes

handles1, labels1 = ax1.get_legend_handles_labels()

handles1_2, labels1_2 = ax1_2.get_legend_handles_labels()

ax1.legend(handles1 + handles1_2, labels1 + labels1_2, loc='upper right')

handles2, labels2 = ax2.get_legend_handles_labels()

handles2_2, labels2_2 = ax2_2.get_legend_handles_labels()

ax2.legend(handles2 + handles2_2, labels2 + labels2_2, loc='upper right')

fig.suptitle('Normal vs. Skewed Distribution')

plt.tight_layout(rect=[0, 0, 1, 0.96])

plt.show()

# Combine the samples and rank them

combined_data = pd.concat([data_normal, data_skewed])

combined_data['Rank'] = combined_data['Value'].rank()

# Separate the ranks by group

rank_sum_normal = combined_data[combined_data['Group'] == 'Normal']['Rank'].sum()

rank_sum_skewed = combined_data[combined_data['Group'] == 'Skewed']['Rank'].sum()

n1 = len(normal_data)

n2 = len(skewed_data)

R1 = rank_sum_normal

R2 = rank_sum_skewed

U1 = n1 * n2 + (n1 * (n1 + 1)) / 2 - R1

U2 = n1 * n2 + (n2 * (n2 + 1)) / 2 - R2

print(f"Rank sum for normal data (R1): {R1}")

print(f"Rank sum for skewed data (R2): {R2}")

print(f"U1: {U1}")

print(f"U2: {U2}")

# Display the ranks table

print("\nData with ranks:")

print(combined_data)

# Mann-Whitney U Test

u_statistic, p_value = mannwhitneyu(normal_data, skewed_data, alternative='two-sided')

print(f"Mann-Whitney U statistic: {u_statistic}")

print(f"P-value: {p_value}")

# Conclusion based on p-value

alpha = 0.05

if p_value < alpha:

    print("Reject the null hypothesis: There is a significant difference between the two drugs.")

else:

    print("Fail to reject the null hypothesis: There is no significant difference between the two drugs.")