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**Interpreting Anderson-Darling Test Statistics**

**Questions:**

What does it mean for the inverse Gauss distribution to have an A-D test value of 1.67895 and the Loglogistic distribution to have an A-D test value of 6.78744? Does the A-D test have a unique distribution, meaning that it is not a conventional F Test or χ² (chi-squared) test? Is an A-D test value of 1.68 approximately four times better than an A-D test value of 6.79? How can the test values be compared?

**Response:**

The A-D test value is simply the average squared difference between the empirical cumulative function and the fitted cumulative function, with a special weighting designed to accentuate the tails of the distribution. There are many good references for this, including *Simulation Modeling and Analysis* by Law & Kelton. What this means is that in an absolute sense A-D values can be compared from one distribution to another. An A-D test value of 6.78744 versus one of 1.67895 implies that the average squared distance between the empirical and fitted cumulative functions (including the effects of the preferential weighting of the tails) is four times as big in one case versus another.

A potential drawback for the A-D test is that it does not have a convenient, unique test distribution, like the χ² test does. Actually, to be fair to the A-D test, even the χ² statistic only approximately follows the χ² distribution in the case where fit parameters have been estimated (see Law & Kelton). Because the A-D test doesn't have a usable test distribution, we can't calculate p-values and critical values for the test, except in special distributions under special conditions, and even in those cases only approximately. There is a very brief discussion of this in Law & Kelton as well, but most of @RISK's treatment of this is taken from the very specialized book *Goodness-of-Fit Techniques* by D'Agostino & Stephens.

last edited: 2012-08-04

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