Anderson Darling Test

The Anderson-Darling test or Anderson-Darling adaptation test is a statistical test that can be used to determine whether the frequency distribution of the data in a sample deviates from a given hypothetical probability distribution . The best known and most common application of this goodness-of-fit test is as a normality test to examine a sample for normal distribution . It is named after the American mathematicians Theodore Wilbur Anderson and Donald Allan Darling , who first described it in 1952. Further detailed research on this test is from Michael A. Stephens.

Test description

The Anderson-Darling test is based on a transformation of the values sorted by size in the sample into a uniform distribution based on the distribution function of the given hypothetical probability distribution. The distance between the transformed sample data and the distribution function of the uniform distribution acts as the test variable. Various corrections are available for greater weighting of the marginal areas and for use in the case of unknown expected values and variances . Stephens also described a method for the direct estimation of the p-value from the test variable . This is calculated using the formulas as a distribution function of the given hypothetical probability distribution ${\ displaystyle A}$ ${\ displaystyle F}$

${\ displaystyle A ^ {2} = - nS \ ,,}$

With

${\ displaystyle S = \ sum _ {k = 1} ^ {n} {\ frac {2k-1} {n}} \ left [\ ln F (Y_ {k}) + \ ln \ left (1-F (Y_ {n + 1-k}) \ right) \ right].}$

The test's null hypothesis is the assumption that the frequency distribution of the data in the sample matches the given hypothetical probability distribution. A p-value less than 0.05 as the result of the Anderson-Darling test is therefore to be interpreted as a significant deviation from the given distribution. In contrast, a p-value greater than 0.05 does not necessarily mean that the frequency distribution of the data corresponds to the specified distribution.

The Anderson-Darling test can be used from a sample size of n≥8. Its most common use is as a normality test to compare the distribution of a sample with the normal distribution . The decision as to whether the values of a sample are normally distributed is essential for the choice of statistical tests for further analyzes. While certain procedures such as the t-test and the analysis of variance assume normally distributed samples, non-parametric tests such as the Mann-Whitney U-test , the Wilcoxon signed rank test , the Kruskal-Wallis test or the Use Friedman test as an alternative.

Alternative procedures

An alternative to the Anderson-Darling test for general use as a test of adaptation is the Kolmogorow-Smirnow test , which can also be used to compare a sample with a hypothetical probability distribution. Compared to this, the Anderson-Darling test takes into account certain critical values, which makes it more sensitive than the Kolmogorow-Smirnow test. However, these critical values are dependent on the given probability distribution; corresponding tables are currently available for the normal distribution , the logarithmic normal distribution , the exponential distribution , the Weibull distribution , the type I extreme value distribution and the logistic distribution . The Kolmogorow-Smirnow test has the advantage over the Anderson-Darling test that it can also be used to compare the distribution of two samples. The same applies to the Cramér von Mises test .

For special use as a normality test, the Anderson-Darling test is one of the most selective statistical methods. Alternatives for this application are the Shapiro-Wilk test , which is in most cases comparable in terms of test strength , the Jarque-Bera test and also the Kolmogorow-Smirnow test, which, however, as a test for normal distribution only has a low test strength and in comparison to other normality tests are not recommended. The Lilliefors test , which is a special adaptation of the Kolmogorow-Smirnow test for the test for normal distribution, is also inferior to the Anderson-Darling test in terms of test strength.

literature

Theodore Wilbur Anderson, Donald Allan Darling: Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes. In: Annals of Mathematical Statistics. 23 (2) / 1952, pp. 193-212, JSTOR 2236446
Michael A. Stephens: EDF Statistics for Goodness of Fit and Some Comparisons. In: Journal of the American Statistical Association. 69/1974, pp. 730-737, doi : 10.1080 / 01621459.1974.10480196 JSTOR 2286009
Michael A. Stephens: Goodness of Fit, Anderson – Darling Test of. In: Encyclopedia of Statistical Sciences. John Wiley & Sons, 2006, doi : 10.1002 / 0471667196.ess0041.pub2

Web links

NIST / SEMATECH e-Handbook of Statistical Methods: Anderson-Darling Test (English)
Free Excel Template for Anderson-Darling Test (German)