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

\( ADK = \frac{n-1}{n^2(k-1)} \sum_{i=1}^{k}{[\frac{1}{n_i} \sum_{j=1}^{L}{h_j \frac{(nF_{ij} - n_{i}H_{j})^2}{H_j(n-H_j) - nh_{j}/4}}]} \)

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 values in the combined samples equal to z_j
• F_ij = the number of values in the i-th 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 j-th observation in the i-th 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

The following statistics are also supported:
LET A = ANDERSON DARLING K SAMPLE TEST Y X


LET ALPHA = <value>
LET A = ANDERSON DARLING K SAMPLE TEST CRITICAL VALUE Y X


with Y denoting the response variable, X denoting the group-id variable, and ALPHA denoting the significance level for the critical value.

In addition to the above LET command, built-in statistics are supported for about 20+ different commands (enter HELP STATISTICS for details).

None

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

LEVENE TEST	= Compute Levene's test for homogeneity of variances.
BOX PLOT	= Generate a box plot.
RECIPE FIT	= Perform a RECIPE analysis.
GRUBBS TEST	= Compute a Grubbs test for outliers.
GOODNESS OF FIT	= Assess univariate distributional goodness-of-fit using Anderson-Darling, Kolmogorov-Smirnov, chi-square, or PPCC method.

"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
SET WRITE DECIMALS 4
ANDERSON DARLING K SAMPLE TEST Y X
    
    The following output is generated:
    
    
            Anderson-Darling K-Sample Test for Common Groups
 
Response Variable: Y
Group-ID Variable: X
 
H0: The Groups Are Homogeneous
Ha: The Groups Are Not Homogeneous
 
Summary Statistics:
Total Number of Observations:                        45
Number of Groups:                                     3
Minimum Batch Size:                                  15
Maximum Batch Size:                                  15
 
Test Statistic Value:                          155.3624
Test Statistic Standard Error:                   0.5108
 
 
            Conclusions (Upper 1-Tailed Test)
 
------------------------------------------------------------------------
                                                                    Null
        Null   Significance           Test       Critical     Hypothesis
  Hypothesis          Level      Statistic    Region (>=)     Conclusion
------------------------------------------------------------------------
 Homogeneous          50.0%       155.3624         1.1524         REJECT
 Homogeneous          75.0%       155.3624         1.4969         REJECT
 Homogeneous          90.0%       155.3624         1.8070         REJECT
 Homogeneous          95.0%       155.3624         1.9926         REJECT
 Homogeneous          97.5%       155.3624         2.1536         REJECT
 Homogeneous          99.0%       155.3624         2.3407         REJECT
 Homogeneous          99.9%       155.3624         2.7310         REJECT