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Hypothesis Testing: Non-Parametric

1. Assumptions in Hypothesis Testing

Overview

Assumptions ensure that the results of hypothesis tests are valid and reliable. Violations of these assumptions can lead to incorrect conclusions.

1.1 Randomness

  • Definition: Data must be collected randomly to avoid bias.
  • Reason: Randomness ensures that the sample represents the population.
  • Example: A random sample of voters provides an unbiased estimate of election preferences.

1.2 Independence of Observations

  • Definition: Each observation must be independent, meaning the occurrence of one observation should not influence another.
  • Reason: Dependence introduces bias and invalidates statistical tests.
  • Example: Measuring blood pressure in a single individual multiple times violates independence.

1.3 Large Sample Size

  • The Central Limit Theorem (CLT) ensures that the sampling distribution of the mean becomes approximately normal with a sufficiently large sample size (\(n > 30\)).

t-Tests

  • Large sample size is needed when population variance is unknown.

Formula:

\[ t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \]

Python Example: 

from scipy.stats import ttest_1samp
import numpy as np

# Simulated data
data = np.random.normal(loc=50, scale=5, size=40)
stat, p_value = ttest_1samp(data, popmean=50)
print(f"t-statistic: {stat}, p-value: {p_value}")

Proportion Tests

  • \(np \geq 5\) and \(n(1-p) \geq 5\) ensure the normal approximation of the binomial distribution.

Formula:

\[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1-p_0)}{n}}} \]

Python Example: 

from statsmodels.stats.proportion import proportions_ztest

successes = 48  # Number of successes
n = 100         # Sample size
p0 = 0.5        # Hypothesized proportion

stat, p_value = proportions_ztest(count=successes, nobs=n, value=p0)
print(f"Z-statistic: {stat}, p-value: {p_value}")

Chi-Square Tests

  • Each expected frequency should be \(\geq 5\).

Formula:

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

Python Example: 

import numpy as np
from scipy.stats import chi2_contingency

data = np.array([[50, 30], [20, 100]])
chi2, p_value, dof, expected = chi2_contingency(data)
print(f"Chi2-statistic: {chi2}, p-value: {p_value}")

2. Parametric vs. Non-Parametric Tests

Parametric Tests

  • Definition: Tests that assume the data follows a specific distribution (e.g., normal distribution).
  • Examples: t-tests, ANOVA.
  • Advantages: More powerful when assumptions are met.
  • Disadvantages: Sensitive to violations of assumptions.

Non-Parametric Tests

  • Definition: Tests that do not assume a specific data distribution.
  • Examples: Wilcoxon tests, Kruskal-Wallis test.
  • Advantages: Robust to non-normal data and outliers.
  • Disadvantages: Less powerful than parametric tests for normal data.

3. Non-Parametric Tests

3.1 Wilcoxon Signed-Rank Test

  • Purpose: Tests whether the median of paired data differs from a hypothesized value.
  • Hypotheses:
  • \(H_0\): The medians are equal.
  • \(H_a\): The medians are not equal.

Mathematical Formula:

  1. Compute differences (\(d_i\)) between paired samples.
  2. Rank absolute differences.
  3. Calculate the sum of signed ranks (\(W\)).

Python Example: 

from scipy.stats import wilcoxon

# Paired data
before = [88, 85, 90, 92, 91]
after = [84, 82, 88, 89, 86]

# Perform Wilcoxon signed-rank test
stat, p_value = wilcoxon(before, after)
print(f"Statistic: {stat}, p-value: {p_value}")

3.2 Wilcoxon-Mann-Whitney Test

  • Purpose: Compares medians of two independent samples.

  • Hypotheses:

    • \(H_0\): The distributions are the same.
    • \(H_a\): The distributions are different.

Python Example: 

from scipy.stats import mannwhitneyu

# Two independent samples
group1 = [88, 85, 90, 92, 91]
group2 = [84, 82, 88, 89, 86]

# Perform Mann-Whitney U test
stat, p_value = mannwhitneyu(group1, group2)
print(f"Statistic: {stat}, p-value: {p_value}")

3.3 Kruskal-Wallis Test

  • Purpose: Tests whether medians of three or more groups are equal.
  • Hypotheses:
    • \(H_0\): All group medians are equal.
    • \(H_a\): At least one group median is different.

Mathematical Formula:

\[ H = \frac{12}{n(n+1)} \sum \frac{R_i^2}{n_i} - 3(n+1) \]

Python Example: 

from scipy.stats import kruskal

# Three groups
group1 = [88, 85, 90, 92, 91]
group2 = [84, 82, 88, 89, 86]
group3 = [80, 81, 85, 87, 83]

# Perform Kruskal-Wallis test
stat, p_value = kruskal(group1, group2, group3)
print(f"Statistic: {stat}, p-value: {p_value}")

4. Chi-Square Tests

4.1 Chi-Square Test of Independence

  • Purpose: Determines if two categorical variables are independent.
  • Hypotheses:
    • \(H_0\): Variables are independent.
    • \(H_a\): Variables are not independent.

Python Example: 

data = np.array([[50, 30], [20, 100]])
chi2, p_value, dof, expected = chi2_contingency(data)
print(f"Chi2-statistic: {chi2}, p-value: {p_value}")

4.2 Chi-Square Goodness-of-Fit Test

  • Purpose: Tests if an observed distribution matches an expected distribution.
  • Hypotheses:
    • \(H_0\): Observed distribution matches the expected.
    • \(H_a\): Observed distribution does not match the expected.

Python Example: 

from scipy.stats import chisquare

# Observed and expected frequencies
observed = [18, 22, 20, 25, 15]
expected = [20, 20, 20, 20, 20]

# Perform goodness-of-fit test
chi2, p_value = chisquare(f_obs=observed, f_exp=expected)
print(f"Chi2-statistic: {chi2}, p-value: {p_value}")

Summary

This guide provides a detailed theoretical and practical understanding of hypothesis testing, including parametric and non-parametric methods. Each method includes assumptions, formulas, and Python code examples for real-world applications.