Anderson darling normality test

Anderson-Darling normality test is a type of statistical test which is used to check whether the range of tested data overlaps with the theoretical range and thus confirm or deny hypotheses made earlier (L. Jäntschi, S.D. Bolboacă 2018, p. 1). This test was created in 1954 by Theodore Wilbur Anderson and Donald Allan Darling as a result of modification of the Cramer-von Mises (CVM) and the Kolmogorov-Smirnov (K-S) tests (N. Mohd Razali, Y. Bee Wah 2011 p. 24).

Advantages and disadvantages

The most important difference in Kolmogorov-Smirnov Test and Anderson-Darling Test is that in the Kolmogorov-Smirnov test critical values are not dependent on the distribution that is being tested (this statement is true only if all parameters tested are known). However, in the Anderson-Darling Test critical values are determined based on the distribution that has been tested. The advantage of this is that the Anderson-Darling Test is much more flexible and better shows how the tested values are arranged, while the Kolmogorow-Smiernov Test gives much stiffer results. The disadvantage of Anderson-Darling Test is that for each distribution that is tested, the critical value must be calculated separately (Anderson-Darling Test 2012, Anderson-Darling Test).

Anderson-Darling Formula

The original Formula of Anderson-Darling test took the form given below (H. Shin, Y. Jung, C. Jeong, J.H. Heo 2012 p. 107):: $A_{n}^{2}=n\int _{-\infty }^{\infty }{\frac {[F_{n}(x)-F(x)]^{2}}{F(x)\;[1-F(x)]}}\,dF(x)$ However, usually is used a simplified version of this formula (H. Shin, Y. Jung, C. Jeong, J.H. Heo 2012 p. 107):: $A_{n}^{2}=-n-{\frac {1}{n}}\sum \limits _{i=1}^{n}\left(2i-1)[\log F_{X_{i}}+\log(1-F_{X_{n+1-i}})\right]$

Test for normality

Research carried out by statisticians proved that the Anderson-Darling Test is much better than other types of tests and "is the most powerful EDF (Empirical Distribution Function) test" (N. Mohd Razali, Y. Bee Wah 2011 p. 24).

To calculate The Anderson-Darling Normality Test, follow steps given below (R.B. D'Agostino 1986, p. 372):

1. Rank the data as follows::

$X_{1}\leq ...\leq X_{n}$

1. Calculate $Y_{i}$ - where $Y_{i}$ stands for normalized values and $i=1,...,n$::

$Y_{i}={\frac {X_{i}-{\overline {X}}}{S}}$

1. Determine the value of $P_{i}$::

$P_{i}={\Phi (Y_{i})}=\int _{-\infty }^{Y_{i}}{\frac {e^{-t_{2}^{2}}}{\sqrt {2\Pi }}}\,dt$

1. Calculate the $A^{2}$ factor::

$A^{2}=-\sum \limits _{i=1}^{n}\left[(2i-1){\frac {\log P_{i}+\log(1-P_{n+1-i})}{n}}-n\right]$

1. Calculate the $A^{*}$ factor, where $A^{*}$ is the modfied statistic::

$A^{*}=A^{2}(1+{\frac {0,75}{n}}+{\frac {2,25}{n^{2}}})$

1. Reject the hypothesis for which the $A^{*}$ factor is higher than a given level of significance for 0.10, 0.05, 0.025, 0.01 and 0.005.

The Anderson-Darling Normality Test can be used only, if $n\geq 8$.

See also Statistical power

Examples of Anderson darling normality test

• Example 1: Anderson-Darling Normality Test is used in many scientific fields, such as in the field of biology. For example, researchers may use this test to measure the size of cells in a sample population, to determine if the distribution of cell sizes is normal.
• Example 2: Anderson-Darling Normality Test is also used in economics. For instance, economists may use this test to measure the income of households in a certain area and to identify if the distribution of income among the households is normal.
• Example 3: Anderson-Darling Normality Test is also used in the field of psychology. For example, psychologists may use this test to measure the responses of a group of participants to a certain stimulus and to determine if the responses follow a normal distribution.

Other approaches related to Anderson darling normality test

One of the other approaches related to Anderson-Darling normality test is the Shapiro-Wilk test which is used to assess the normality of data. This test is often used in a variety of fields, such as psychology, biology, economics, and medicine. It is based on the assumption that the data is normally distributed and is usually used in small sample sizes.

• The Shapiro-Francia test is another approach used to assess the normality of data. This test is mainly used in research studies with small sample sizes, as it is more sensitive than the Shapiro-Wilk test.
• The Kolmogorov-Smirnov test is another popular approach used to assess the normality of data. This test is based on the assumption that the data is normally distributed and is usually used in large sample sizes.
• The Q-Q plot (quantile-quantile plot) is a graphical approach used to assess the normality of data. This test is based on the assumption that the data is normally distributed and is usually used to determine if two sets of data have the same distribution.

Overall, there are a variety of approaches used to assess the normality of data, such as the Anderson-Darling normality test, the Shapiro-Wilk test, the Shapiro-Francia test, the Kolmogorov-Smirnov test, and the Q-Q plot. Each of these tests has its own advantages and disadvantages and should be used depending on the size of the sample and the type of data being tested.

Anderson darling normality test — recommended articles

Statistical significance — Nominal scale — Cronbach Alpha — Three-Way ANOVA — One-tailed test — Systematic sampling techniques — Probability density function — Maximum likelihood method — Quantitative variable

References

• Anderson-Darling Test (2012) Anderson-Darling Test, "e-Handbook of Statistical Methods"
• D'Agostino R.B. (1986) Godness-of-Fit-Techniques, Marcel Dekker Inc., New York, p. 372-373
• Jäntschi L., Bolboacă S.D. (2018) Computation of Probability Associated with Anderson-Darling Statistic, "Mathematics 2018", nr 6(6), p. 1-16
• Mohd Razali N., Bee Wah Y. (2011) Power Comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling Tests, "Journal of Statistical Modeling and Analitics", nr 2(1), p. 21-33
• Shin H., Jung Y., Jeong C., Heo J.H. (2012) Assessment of modified Anderson-Darling test statistics for the generalized extreme value and generalized logistic distributions, " Stochastic Environmental Research and Risk Assessment", nr 26(1), p. 105-114

Author: Patrycja Czerwiec