Example 4.2: Fitting Lognormal, Weibull, and Gamma Curves

HISTOGRAM Statement

Example 4.2: Fitting Lognormal, Weibull, and Gamma Curves

See CAPCURV in the SAS/QC Sample Library

To find an appropriate model for a process distribution, you should consider curves from several distribution families. As shown in this example, you can use the HISTOGRAM statement to fit more than one type of distribution and display the density curves on the same histogram. The gap between two plates is measured (in cm) for each of 50 welded assemblies selected at random from the output of a welding process assumed to be in statistical control. The lower and upper specification limits for the gap are 0.3 cm and 0.8 cm, respectively. The measurements are saved in a data set named PLATES.

   data plates;
      label gap='Plate Gap in cm';
      input gap @@;
      cards;
   0.746  0.357  0.376  0.327  0.485  1.741  0.241  0.777  0.768
   0.409  0.252  0.512  0.534  1.656  0.742  0.378  0.714  1.121
   0.597  0.231  0.541  0.805  0.682  0.418  0.506  0.501  0.247
   0.922  0.880  0.344  0.519  1.302  0.275  0.601  0.388  0.450
   0.845  0.319  0.486  0.529  1.547  0.690  0.676  0.314  0.736
   0.643  0.483  0.352  0.636  1.080
   ;

The following statements fit three distributions (lognormal, Weibull, and gamma) and display their density curves on a single histogram:

   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 = 3   clsl=black
             usl = 0.8  lusl = 20  cusl=black;
      histogram /
          midpoints=0.2 to 1.8 by 0.2
          lognormal (l=1  color=red)
          weibull   (l=2  color=blue)
          gamma     (l=8  color=yellow)
          nospeclegend
          vaxis   =  axis1
          cframe  = ligr
          legend  = legend1;
      inset n mean(5.3) std='Std Dev'(5.3) skewness(5.3)
             / pos = ne  header = 'Summary Statistics' cfill = blank;
      axis1  label=(a=90 r=0);
   run;

The LOGNORMAL, WEIBULL, and GAMMA options superimpose fitted curves on the histogram in Output 4.2.1. The L= options specify distinct line types for the curves. Note that a threshold parameter $\theta=0$ is assumed for each curve. In applications where the threshold is not zero, you can specify $\theta$ with the THETA= option.

Output 4.2.1: Superimposing a Histogram with Fitted Curves

The LOGNORMAL, WEIBULL, and GAMMA options also produce the summaries for the fitted distributions shown in Output 4.2.2, Output 4.2.3, and Output 4.2.4.

Output 4.2.2: Summary of Fitted Lognormal Distribution

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	7.51762213	6	Pr > Chi-Sq	0.276

Percent Outside Specifications for Lognormal Distribution
Lower Limit		Upper Limit
LSL	0.300000	USL	0.800000
Obs Pct < LSL	10.000000	Obs Pct > USL	20.000000
Est Pct < LSL	10.719540	Est Pct > USL	23.519008

Quantiles for Lognormal Distribution
Percent	Quantile
Percent	Observed	Estimated
1.0	0.23100	0.17449
5.0	0.24700	0.24526
10.0	0.29450	0.29407
25.0	0.37800	0.39825
50.0	0.53150	0.55780
75.0	0.74600	0.78129
90.0	1.10050	1.05807
95.0	1.54700	1.26862
99.0	1.74100	1.78313

Output 4.2.2 provides four goodness-of-fit tests for the lognormal distribution: the chi-square test and three tests based on the EDF (Anderson-Darling, Cramer-von Mises, and Kolmogorov-Smirnov). See "Chi-Square Goodness-of-Fit Test" and "EDF Goodness-of-Fit Tests" for more information. The EDF tests are superior to the chi-square test because they are not dependent on the set of midpoints used for the histogram.

At the $\alpha=0.10$ significance level, all four tests support the conclusion that the two-parameter lognormal distribution with scale parameter $\hat{\zeta}=-0.58$ , and shape parameter $\hat{\sigma}=0.50$ provides a good model for the distribution of plate gaps.

Output 4.2.3: Summary of Fitted Weibull Distribution

Distribution of Plate Gaps

The CAPABILITY Procedure

Fitted Weibull Distribution for gap

Parameters for Weibull Distribution
Parameter	Symbol	Estimate
Threshold	Theta	0
Scale	Sigma	0.719208
Shape	C	1.961159
Mean		0.637641
Std Dev		0.339248

Goodness-of-Fit Tests for Weibull Distribution
Test	Statistic		DF	p Value
Cramer-von Mises	W-Sq	0.1593728		Pr > W-Sq	0.016
Anderson-Darling	A-Sq	1.1569354		Pr > A-Sq	<0.010
Chi-Square	Chi-Sq	15.0252996	6	Pr > Chi-Sq	0.020

Percent Outside Specifications for Weibull Distribution
Lower Limit		Upper Limit
LSL	0.300000	USL	0.800000
Obs Pct < LSL	10.000000	Obs Pct > USL	20.000000
Est Pct < LSL	16.473319	Est Pct > USL	29.165543

Quantiles for Weibull Distribution
Percent	Quantile
Percent	Observed	Estimated
1.0	0.23100	0.06889
5.0	0.24700	0.15817
10.0	0.29450	0.22831
25.0	0.37800	0.38102
50.0	0.53150	0.59661
75.0	0.74600	0.84955
90.0	1.10050	1.10040
95.0	1.54700	1.25842
99.0	1.74100	1.56691

Output 4.2.3 provides two EDF goodness-of-fit tests for the Weibull distribution: the Anderson-Darling and the Cramer-von Mises tests. (See Table 4.15 for a complete list of the EDF tests available in the HISTOGRAM statement.) The probability values for the chi-square and EDF tests are all less than 0.10, indicating that the data do not support a Weibull model.

Output 4.2.4: Summary of Fitted Gamma Distribution

Distribution of Plate Gaps

The CAPABILITY Procedure

Fitted Gamma Distribution for gap

Parameters for Gamma Distribution
Parameter	Symbol	Estimate
Threshold	Theta	0
Scale	Sigma	0.155198
Shape	Alpha	4.082646
Mean		0.63362
Std Dev		0.313587

Goodness-of-Fit Tests for Gamma Distribution
Test	Statistic		DF	p Value
Chi-Square	Chi-Sq	12.3075959	6	Pr > Chi-Sq	0.055

Percent Outside Specifications for Gamma Distribution
Lower Limit		Upper Limit
LSL	0.300000	USL	0.800000
Obs Pct < LSL	10.000000	Obs Pct > USL	20.000000
Est Pct < LSL	12.111039	Est Pct > USL	25.696522

Quantiles for Gamma Distribution
Percent	Quantile
Percent	Observed	Estimated
1.0	0.23100	0.13326
5.0	0.24700	0.21951
10.0	0.29450	0.27938
25.0	0.37800	0.40404
50.0	0.53150	0.58271
75.0	0.74600	0.80804
90.0	1.10050	1.05392
95.0	1.54700	1.22160
99.0	1.74100	1.57939

Output 4.2.4 provides a chi-square goodness-of-fit test for the gamma distribution. (None of the EDF tests are currently supported when the scale and shape parameter of the gamma distribution are estimated; see Table 4.15.) The probability value for the chi-square test is less than 0.10, indicating that the data do not support a gamma model.

Based on this analysis, the fitted lognormal distribution is the best model for the distribution of plate gaps. You can use this distribution to calculate useful quantities. For instance, you can compute the probability that the gap of a randomly sampled plate exceeds the upper specification limit, as follows:

$\Pr[{gap} \gt {USL}] & = & \Pr[Z \gt \frac{1}{\sigma} (\log({USL}-\theta)-\zeta) ] \ & = & 1-\Phi[\frac{1}{\sigma} (\log({USL}-\theta)-\zeta) ]$

where Z has a standard normal distribution, and $\Phi(\cdot)$ is the standard normal cumulative distribution function. Note that $\Phi(\cdot)$ can be computed with the DATA step function PROBNORM. In this example, USL = 0.8 and Pr[gap > 0.8] = 0.2352. This value is expressed as a percent (Est Pct > USL) in Output 4.2.2.

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