Anderson-Darling test

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Overview

The Anderson-Darling test, named after Theodore Wilbur Anderson, Jr. (1918–?) and Donald A. Darling (1915–?), who invented it in 1952^[1], is one of the most powerful statistics for detecting most departures from normality. It may be used with small sample sizes n ≤ 25. Very large sample sizes may reject the assumption of normality with only slight imperfections, but industrial data with sample sizes of 200 and more have passed the Anderson-Darling test. ^{[citation needed]}

The Anderson-Darling test assesses whether a sample comes from a specified distribution. The formula for the test statistic A to assess if data (note that the data must be put in order) comes from a distribution with cumulative distribution function (CDF) F is

A² = − N − S

where

The test statistic can then be compared against the critical values of the theoretical distribution (dependent on which F is used) to determine the P-value.

The Anderson-Darling test for normality is a distance or empirical distribution function (EDF) test. It is based upon the concept that when given a hypothesized underlying distribution, the data can be transformed to a uniform distribution. The transformed sample data can be then tested for uniformity with a distance test (Shapiro 1980).

In comparisons of power, Stephens (1974) found A² to be one of the best EDF statistics for detecting most departures from normality.^[2] The only statistic close was the W² (Cramér von-Mises test) statistic.

Procedure

(If testing for normal distribution of the variable X)

1) The data of the variable X that should be tested is sorted from low to high.

2) The mean, , and standard deviation, s, are calculated from the sample of X.

3) The values of X are standardized as follows:

4) With the standard normal CDF Φ, A² is calculated using:

5) A^{2 *} , an approximate adjustment for sample size, is calculated using:

6) If A^{2 *} exceeds 0.752 then the hypothesis of normality is rejected for a 5% level test.

Note:

1. If s = 0 or any P_i = (0 or 1) then A² cannot be calculated and is undefined.

2. Above, it was assumed that the variable X_i was being tested for normal distribution. Any other theoretical distribution can be assumed by using its CDF. Each theoretical distribution has its own critical values, and some examples are: lognormal, exponential, Weibull, extreme value type I and logistic distribution.

3. Null hypothesis follows the true distribution (in this case, N(0, 1)).

See also

• Kolmogorov-Smirnov test
• Shapiro-Wilk test
• Smirnov-Cramér-von-Mises test

External links

• US NIST Handbook of Statistics

References

1. ↑ Anderson, T. W.; Darling, D. A. (1952). "Asymptotic theory of certain "goodness-of-fit" criteria based on stochastic processes". Annals of Mathematical Statistics 23: 193-212.
2. ↑ Stephens, M. A. (1974). "EDF Statistics for Goodness of Fit and Some Comparisons". Journal of the American Statistical Association 69: 730-737.