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Anderson-Darling Normality Test

Source: R/43-NormalityTest.R
ad.test.Rd

Performs the Anderson-Darling test for normality to assess whether a given sample comes from a normal distribution. This implementation follows the methodology from the nortest R package.

Usage

ad.test(x)

Arguments

x

A numeric vector of data values. Missing values will be automatically removed.

Value

A list with class "htest" containing the following components:

  • statistic - The Anderson-Darling test statistic (A)

  • p.value - The p-value for the test

  • method - The name of the method ("Anderson-Darling normality test")

  • data.name - The name of the data used in the test

Details

The Anderson-Darling test is a statistical test of whether a given sample of data is drawn from a normal distribution. The test is a modification of the Kolmogorov-Smirnov (K-S) test and gives more weight to the tails than the K-S test.

This implementation requires a minimum sample size of 8 observations. The test statistic is calculated as: $$A^2 = -n - \frac{1}{n}\sum_{i=1}^n(2i-1)[\ln(p_i) + \ln(1-p_{n+1-i})]$$ where \(p_i = \Phi((x_i - \mu)/\sigma)\) and \(\Phi\) is the cumulative distribution function of the standard normal distribution.

The p-value is calculated using different approximation formulas depending on the value of the adjusted test statistic:

  • For AA < 0.2: \(p = 1 - \exp(-13.436 + 101.14 \times AA - 223.73 \times AA^2)\)

  • For 0.2 ≤ AA < 0.34: \(p = 1 - \exp(-8.318 + 42.796 \times AA - 59.938 \times AA^2)\)

  • For 0.34 ≤ AA < 0.6: \(p = \exp(0.9177 - 4.279 \times AA - 1.38 \times AA^2)\)

  • For 0.6 ≤ AA < 10: \(p = \exp(1.2937 - 5.709 \times AA + 0.0186 \times AA^2)\)

  • For AA ≥ 10: \(p = 3.7 \times 10^{-24}\)

where \(AA = A^2 \times (1 + 0.75/n + 2.25/n^2)\).

References

This implementation is based on the nortest package:

Gross, J. and Ligges, U. (2015). nortest: Tests for Normality. R package version 1.0-4. https://CRAN.R-project.org/package=nortest

The original test is described in:

Anderson, T.W. and Darling, D.A. (1954). A Test of Goodness of Fit. Journal of the American Statistical Association, 49(268), 765-769.

Stephens, M.A. (1974). EDF Statistics for Goodness of Fit and Some Comparisons. Journal of the American Statistical Association, 69(347), 730-737.

See also

ad.test for the original implementation in the nortest package. shapiro.test for the Shapiro-Wilk normality test. ks.test for the Kolmogorov-Smirnov test.

Other normality_test: cvm.test(), dagostino.test(), jb.test.modified(), pearson.test(), sf.test()

Examples

if (FALSE) { # \dontrun{
# Test a sample from normal distribution
set.seed(123)
normal_data <- rnorm(100)
ad.test(normal_data)

# Test a sample from non-normal distribution
exponential_data <- rexp(50)
ad.test(exponential_data)

# Test with real data
if (require("datasets")) {
  ad.test(iris$Sepal.Length)
}
} # }