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Example 4.3: Comparing Goodness-of-Fit Tests

See CAPGOF in the SAS/QC Sample Library

A weakness of the chi-square goodness-of-fit test is its dependence on the choice of histogram midpoints. An advantage of the EDF tests is that they give the same results regardless of the midpoints, as illustrated in this example.

In Example 4.2, the option MIDPOINTS=0.2 TO 1.8 BY 0.2 was used to specify the histogram midpoints for GAP. The following statements refit the lognormal distribution using default midpoints (0.3 to 1.8 by 0.3).

   title1 'Distribution of Plate Gaps';
   legend1 frame cframe=ligr cborder=black position=center;
   proc capability data=plates noprint;
      var gap;
      specs  lsl = 0.3 llsl = 2  clsl=black
             usl = 0.8 lusl = 20 cusl=black;
      histogram /
         lognormal (l=1  color=yellow w=3)
         vaxis  = axis1
         legend = legend1
         cfill  = purple
         cframe = ligr;
      inset n mean(5.3) std='Std Dev'(5.3) skewness(5.3) /
         header = 'Summary Statistics' cfill = blank
         pos    = ne
         cfill  = blank;
      axis1  label=(a=90 r=0);

The histogram is shown in Output 4.3.1.

Output 4.3.1: Lognormal Curve Fit with Default Midpoints
caphex3a.gif (7936 bytes)

A summary of the lognormal fit is shown in Output 4.3.2. The p-value for the chi-square goodness-of-fit test is 0.0822. Since this value is less than 0.10 (a typical cutoff level), the conclusion is that the lognormal distribution is not an appropriate model for the data. This is the opposite conclusion drawn from the chi-square test in Example 4.2, which is based on a different set of midpoints and has a p-value of 0.2756 (see Output 4.2.2). Moreover, the results of the EDF goodness-of-fit tests are the same since these tests do not depend on the midpoints. When available, the EDF tests provide more powerful alternatives to the chi-square test. For a thorough discussion of EDF tests, refer to D'Agostino and Stephens (1986).

Output 4.3.2: Printed Output for the Lognormal Curve
Distribution of Plate Gaps

The CAPABILITY Procedure
Fitted Lognormal Distribution for gap

Parameters for Lognormal Distribution
Parameter Symbol Estimate
Threshold Theta 0
Scale Zeta -0.58375
Shape Sigma 0.499546
Mean   0.631932
Std Dev   0.336436
Goodness-of-Fit Tests for Lognormal Distribution
Test Statistic DF p Value
Kolmogorov-Smirnov D 0.06441431   Pr > D >0.150
Cramer-von Mises W-Sq 0.02823022   Pr > W-Sq >0.500
Anderson-Darling A-Sq 0.24308402   Pr > A-Sq >0.500
Chi-Square Chi-Sq 6.69789360 3 Pr > Chi-Sq 0.082

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Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.