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?


Purpose: Test for distributional adequacy  The AndersonDarling Test
The AndersonDarling test (Stephens, 1974) is used to test if a sample of data comes from a specific distribution. It is a modification of the KolmogorovSmirnov (KS) test and gives more weight to the tails of the distribution than does the KS test. The KS 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 AndersonDarling 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 AndersonDarling test for the following
distributions:


AndersonDarling procedure  Details on the construction and interpretation of the AndersonDarling test statistic, \(A^2\), and examples for several distributions are outlined in Chapter 1.  
ShapiroWilk test for normality  The ShapiroWilk Test For Normality
The ShapiroWilk 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 ShapiroWilk test the reader is referred
to the original
Shapiro and Wilk (1965) paper and the tables in
Pearson and Hartley (1972).
