Anderson-Darling Test¶

Overview¶

The Anderson-Darling test is an enhancement of the Kolmogorov-Smirnov test, designed to assess whether a sample comes from a specific distribution, such as the normal distribution. It is particularly sensitive to deviations in the tails, making it especially useful for detecting departures from normality in smaller samples.

Hypotheses¶

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

Computation of the Anderson-Darling Test Statistic¶

To compute the Anderson-Darling test statistic \(A^2\), follow these steps:

Order the Data: Start with a sample dataset \(X_1, X_2, \dots, X_n\) and sort it in ascending order, giving \(X_{(1)} \leq X_{(2)} \leq \dots \leq X_{(n)}\).
Standardize the Data: Standardize each data point to have a mean of 0 and a variance of 1. Note that this transformation does not impose normality on the data. For each data point \(X_i\), calculate the standardized value \(Z_i\):

\[ Z_i = \frac{X_i - \mu}{\sigma} \]

where \(\mu\) is the sample mean and \(\sigma\) is the sample standard deviation.
Calculate the Empirical Distribution Function (EDF): For each ordered standardized value \(Z_{(i)}\), compute the CDF of the normal distribution, \(F(Z_{(i)})\).
Compute Test Statistic \(A^2\):

\[ A^2 = -n - \frac{1}{n} \sum_{i=1}^{n} \left[ (2i-1) \left( \ln(F(Z_{(i)})) + \ln(1 - F(Z_{(n+1-i)})) \right) \right] \]

where \(n\) is the sample size and \(F(Z_{(i)})\) is the CDF of the normal distribution at each ordered standardized data point \(Z_{(i)}\).

This formula adjusts for the lower and upper tails, giving the test higher sensitivity to deviations in the distribution tails.

Deriving the p-Value¶

After calculating the test statistic \(A^2\), it is compared to critical values specific to the Anderson-Darling distribution for the desired distribution type and sample size. Critical values are selected based on the significance level \(\alpha\) (e.g., 0.01, 0.05, or 0.10).

If \(A^2\) is greater than the critical value for a given \(\alpha\), the \(p\)-value will be below \(\alpha\), leading to rejection of the null hypothesis.
If \(A^2\) is less than the critical value, the \(p\)-value will be above \(\alpha\), indicating insufficient evidence to reject the null hypothesis.

Decision Rule¶

If \(A^2 > \text{critical value}\) at significance level \(\alpha\), reject \(H_0\) (the data is not normally distributed).
If \(A^2 < \text{critical value}\), fail to reject \(H_0\) (the data is normally distributed).

Python Implementation with `stats.anderson`¶

import numpy as np
from scipy import stats

np.random.seed(0)

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

# Perform Anderson-Darling test
result = stats.anderson(data)
statistic = result.statistic
print(f"Anderson-Darling Test: Statistic={statistic}")

# Display critical values
for significance_level, critical_value in zip(result.significance_level, result.critical_values):
    if statistic >= critical_value:
        print(f"At {significance_level}% significance level: Reject H_0. The data is not normally distributed.")
    else:
        print(f"At {significance_level}% significance level: Fail to reject H_0. The data is normally distributed.")

Can I Get a p-Value from `stats.anderson`?¶

The stats.anderson() function in SciPy does not directly provide a \(p\)-value; it only returns the test statistic and critical values at specific significance levels.

Why No Direct p-Value?¶

The Anderson-Darling test has predefined critical values based on simulation or theoretical distribution tables for each significance level (e.g., 15%, 10%, 5%, 2.5%, and 1% for the normal distribution). Since the Anderson-Darling statistic distribution varies depending on the sample size and the specific distribution being tested, calculating an exact \(p\)-value is complex.

Approximate p-Values (for Normality Testing)¶

If you need an approximate \(p\)-value for normality testing, there are two options:

Option 1: Use statsmodels

The statsmodels library provides an implementation with an approximate \(p\)-value:

import numpy as np
from statsmodels.stats.diagnostic import normal_ad

np.random.seed(0)
data = np.random.normal(0, 1, 1000)

statistic, p_value = normal_ad(data)
print(f"Anderson-Darling Test: Statistic={statistic}, p-value={p_value}")

Option 2: Interpret Based on Critical Values

If you want a rough approximation without additional libraries, you can interpret the \(p\)-value range based on how the test statistic compares to the provided critical values:

If the test statistic is below the critical value for a certain significance level, the \(p\)-value is higher than that significance level.
If the test statistic is above the critical value for a certain significance level, the \(p\)-value is lower than that significance level.

Python Implementation with `normal_ad`¶

import numpy as np
from statsmodels.stats.diagnostic import normal_ad

np.random.seed(0)

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

# Perform Anderson-Darling test for normality with p-value
statistic, p_value = normal_ad(data)
print(f"Anderson-Darling Test: Statistic={statistic}, p-value={p_value}")

# 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.")

`stats.anderson` vs `normal_ad`: Which to Choose?¶

If you need the Anderson-Darling test for general distribution testing (not limited to normality) and do not need a \(p\)-value, then stats.anderson is the better choice. It allows testing for normality, exponential, Weibull, logistic, and extreme value distributions and provides critical values for each.
If your goal is specifically normality testing and you need a \(p\)-value, then normal_ad from statsmodels is ideal. This implementation is designed specifically for normality testing and includes an approximate \(p\)-value, making it easier to interpret results in a traditional hypothesis testing framework.

Recommendation¶

If your focus is normality testing with a clear \(p\)-value interpretation, use normal_ad. For flexibility to test against multiple distributions, use stats.anderson and interpret results based on the provided critical values.