Goodness of Fit > Anderson-Darling Goodness of Fit
What is the Anderson-Darling Test?
The Anderson-Darling Goodness of Fit Test (AD-Test) is a measure of how well your data fits a specified distribution. It’s commonly used as a test for normality.
Performing the AD-Test by Hand
The hypotheses for the AD-test are:
H0: The data comes from a specified distribution.
H1: The data does not come from a specified distribution.
The formula is:
n = the sample size,
F(x) = CDF for the specified distribution,
i = the ith sample, calculated when the data is sorted in ascending order.
As you can probably see, the test statistic is cumbersome to calculate by hand. The general steps are:
Step 1: Calculate the AD Statistic for each distribution, using the formula above.
Step 2: Find the statistic’s p-value (probability value). The formula for the p-value depends on the value for the AD statistic from Step 1. The following formulas are taken from Agostino and Stephen’s Goodness of Fit Techniques.
|AD statistic||P-Value Formula|
|AD ≥ 0.60||p = exp(1.2937 – 5.709(AD)+ 0.0186(AD)2|
|0.34 < AD* < .60||p = exp(0.9177 – 4.279(AD) – 1.38(AD)2|
|0.20 < AD* < .34||p = 1 – exp(-8.318 + 42.796(AD)- 59.938(AD)2|
|AD≤ 0.20 < AD* < .34||p = 1 – exp(-13.436 + 101.14(AD)- 223.73(AD)2|
If you are comparing several distributions, choose the one that gives the largest p-value; this is the closest match to your data.
The steps are basically the same, except that software will do the legwork for you and calculate the AD statistic and the p-value. All you have to do is Step 2 above: compare your AD-test p-values to your alpha levels.
- SPSS: The test is not available in SPSS at the time of writing.
- R: use the ‘nortest’ package outlined here.
D’Agostino, R. & Stephens, M. (1986). Goodness-of-fit-techniques (Statistics: a Series of Textbooks and Monographs, Vol. 68) 1st Edition. Dekker.
Stephanie Glen. "Anderson-Darling Test & Statistic: Definition, Examples" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/anderson-darling-test/
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