 7. Product and Process Comparisons
7.2. Comparisons based on data from one process
7.2.1. Do the observations come from a particular distribution?

Anderson-Darling and Shapiro-Wilk tests

Purpose: Test for distributional adequacy The Anderson-Darling Test

The Anderson-Darling test (Stephens, 1974) is used to test if a sample of data comes from a specific distribution. It is a modification of the Kolmogorov-Smirnov (K-S) test and gives more weight to the tails of the distribution than does the K-S test. The K-S test is distribution free in the sense that the critical values do not depend on the specific distribution being tested.

Requires critical values for each distribution The Anderson-Darling test makes use of the specific distribution in calculating critical values. This has the advantage of allowing a more sensitive test and the disadvantage that critical values must be calculated for each distribution. Tables of critical values are not given in this handbook (see Stephens 1974, 1976, 1977, and 1979) because this test is usually applied with a statistical software program that produces the relevant critical values. Currently, Dataplot computes critical values for the Anderson-Darling test for the following distributions:
• normal
• lognormal
• Weibull
• extreme value type I.
Anderson-Darling procedure Details on the construction and interpretation of the Anderson-Darling test statistic, $$A^2$$, and examples for several distributions are outlined in Chapter 1.
Shapiro-Wilk test for normality The Shapiro-Wilk Test For Normality

The Shapiro-Wilk test, proposed in 1965, calculates a $$W$$ statistic that tests whether a random sample, $$x_1, \, x_2, \, \ldots, \, x_n$$ comes from (specifically) a normal distribution . Small values of $$W$$ are evidence of departure from normality and percentage points for the $$W$$ statistic, obtained via Monte Carlo simulations, were reproduced by Pearson and Hartley (1972, Table 16). This test has done very well in comparison studies with other goodness of fit tests.

The $$W$$ statistic is calculated as follows: $$W = \frac{\left( \sum_{i=1}^n a_i x_{(i)} \right)^2} {\sum_{i=1}^n (x_i - \bar{x})^2} \, ,$$ where the $$x_{(i)}$$ are the ordered sample values ($$x_{(1)}$$ is the smallest) and the $$a_i$$ are constants generated from the means, variances and covariances of the order statistics of a sample of size $$n$$ from a normal distribution (see Pearson and Hartley (1972, Table 15).

For more information about the Shapiro-Wilk test the reader is referred to the original Shapiro and Wilk (1965) paper and the tables in Pearson and Hartley (1972). 