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Shapiro-Wilk Test

Overview

The Shapiro-Wilk test is a popular method for assessing the normality of a dataset. It evaluates whether the sample data comes from a normally distributed population by calculating a test statistic and corresponding \(p\)-value.

Hypotheses

  • Null Hypothesis (\(H_0\)): The data is normally distributed.
  • Alternative Hypothesis (\(H_1\)): The data is not normally distributed.

Computation of the Test Statistic W

We compute the Shapiro-Wilk test statistic \(W\) using the following steps:

  1. Order the Data: Let \(X_1, X_2, \dots, X_n\) be the sample data sorted in ascending order such that \(X_{(1)} \leq X_{(2)} \leq \dots \leq X_{(n)}\).

  2. Expected Values: Calculate the expected values \(m_i\) for a sample of size \(n\) from a standard normal distribution. These values represent the means of the order statistics. Let \(\mathbf{m}^T = [m_1, m_2, \dots, m_n]\) be the vector of these expected values.

  3. Covariance Matrix: We use a covariance matrix \(\Sigma\) of the order statistics from the normal distribution to generate weights \(a_i\). These weights are computed to optimize the sensitivity of the test to departures from normality. The vector of weights is denoted by \(\mathbf{a}^T = [a_1, a_2, \dots, a_n]\):

    \[ [a_1, a_2, \dots, a_n] = \frac{[m_1, m_2, \dots, m_n]\Sigma^{-1}}{\sqrt{[m_1, m_2, \dots, m_n]\Sigma^{-1}\Sigma^{-1}[m_1, m_2, \dots, m_n]^T}} \]

  4. Test Statistic \(W\): Then, we compute the test statistic \(W\) as:

    \[ W = \frac{\left( \sum_{i=1}^{n} a_i X_{(i)} \right)^2}{\sum_{i=1}^{n} (X_i - \bar{X})^2} \]

    where

    • \(X_{(i)}\) is the \(i\)-th ordered data point,
    • \(a_i\) is the corresponding weight from the vector \(\mathbf{a}\),
    • \(\bar{X}\) is the sample mean.

    The numerator represents the squared linear combination of the ordered sample, and the denominator is the total variance of the sample data.

Deriving the p-Value

Once we compute the test statistic \(W\), the \(p\)-value is obtained by comparing \(W\) to the distribution of \(W\) under the null hypothesis of normality. The \(p\)-value represents the probability of observing a test statistic as extreme as \(W\) under the assumption that the null hypothesis is true.

  • We reject the null hypothesis if the \(p\)-value is small (e.g., less than \(\alpha = 0.05\)). This suggests the data does not follow a normal distribution.
  • If the \(p\)-value is large (greater than or equal to \(\alpha\)), we do not reject the null hypothesis.

Decision Rule

  • If \(p\)-value \(\leq \alpha\), reject \(H_0\) (the data is not normally distributed).
  • If \(p\)-value \(> \alpha\), fail to reject \(H_0\) (the data is normally distributed).

Python Implementation

import numpy as np
from scipy import stats

# Generate a sample dataset
# data = np.random.normal(0, 1, 1000)
data = np.random.normal(1, 10, 1000)

# Perform Shapiro-Wilk test
stat, p_value = stats.shapiro(data)
print(f"Shapiro-Wilk Test: Statistic={stat:.4f}, p-value={p_value:.4f}")

# Interpretation
alpha = 0.05
if p_value <= alpha:
    print("Reject H_0: The data is not normally distributed.")
else:
    print("Fail to reject H_0: The data is normally distributed.")

In summary, the Shapiro-Wilk test uses the ordered sample data and precomputed weights to compute the test statistic \(W\) to determine whether the data is likely to have come from a normal distribution. It is generally considered one of the most powerful normality tests, particularly for small to moderate sample sizes.