ANDERSON DARLING K SAMPLE TEST

ANDERSON DARLING K SAMPLE TEST

Analysis Command

The k-sample Anderson-Darling test is a nonparametric statistical procedure that tests the hypothesis that the populations from which two or more groups of data were drawn are identical. Each group should be an independent random sample from a population.

This test is part of the MIL-HDBK-17 standard. In the terminology of MIL-HDBK-17, data can be either "structured" (i. e., groups) or "unstructured" (i.e., ungrouped). Unstructured data can often be simpler to analyze. Therefore the Anderson-Darling k-sample test is used to determine if a structured data set can in fact be treated as an unstructured data set. Dataplot supports most of the techniques in chapter 8 of the MIL-HDBK-17 as support for the RECIPE analysis.

The k-sample Anderson-Darling statistic is

where

• h_j = the number of values in the combined samples equal to z_j
• H_j = the number of values in the combined samples less than z_j plus one half the number of of values in the combined samples equal to z_j
• F_ij = the number of values in the ith group which are less than z_j plus one half the number of values in this group which are equal to z_j

where k is the number of samples (groups), n_i is the number of observations in group i, x_ij is the jth observation in the ith group, and z₁, z₂ ..., z_L are the distinct values in the combined data set ordered from smallest to largest (L is less than n if there are tied observations).

Chapter 8 of the MIL-HDBK-17 derives the formulas for the critical values of the Anderson-Darling test statistic. These formulas are rather involved and not given here.

Dataplot uses the ANDYK routine from the MIL-HDBK-17 to compute the Anderson-Darling k sample test.

ANDERSON DARLING K SAMPLE TEST <y> <groupid> <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable;
          <groupid> is group (sample) identifier variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

ANDERSON DARLING K SAMPLE TEST Y1 GROUP
ANDERSON DARLING K SAMPLE TEST Y1 GROUP SUBSET GROUP > 2

None

ANDERSON DARLING K SAMPLE is a synonym for ANDERSON DARLING K SAMPLE TEST.

LEVENE TEST	= Compute Levene's test.
BOX PLOT	= Generate a box plot.
RECIPE FIT	= Perform a RECIPE analysis.
GRUBBS TEST	= Compute a Grubbs test for outliers.
ANDERSON DARLING TEST	= Anderson-Darling one sample goodness of fit test.

"MIL-HDBK-17 Volume 1: Guidelines for Characterization of Structural Materials", Depeartment of Defense, chapter 8. The URL for MIL-HDBK-17 is http://mil-17.udel.edu/.

Tolerance Studies in Reliability of Materials

1998/6

SKIP 25
READ VANGEL32.DAT Y X B
ANDERSON DARLING K SAMPLE TEST Y X


The following output is generated:

  
           **********************************************
           **      ANDERSON DARLING K SAMPLE TEST Y X  **
           **********************************************
      
      
                   ANDERSON-DARLING K-SAMPLE TEST FOR COMMON GROUPS
      
     1. STATISTICS
           NUMBER OF OBSERVATIONS                     =       45
           NUMBER OF GROUPS                           =        3
           ANDERSON-DARLING K-SAMPLE TEST STATISTIC   =    155.3625
      
     2. FOR ANDERSON-DARLING K-SAMPLE TEST STATISTIC
           95         % POINT                         =    1.992743
      
     3. CONCLUSION (AT THE 5% LEVEL):
           THE GROUPS ARE SIGNIFICANTLY DIFFERENT.
           THUS: GROUPS SHOULD BE TREATED AS STRUCTURED.