HISTOGRAM Statement: CAPABILITY Procedure

Example 5.9 Fitting Lognormal, Weibull, and Gamma Curves

Note: See Superimposing Fitted Curves on a Histogram 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 @@;
   datalines;
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:

ods graphics on;
proc capability data=Plates;
   var Gap;
   specs  lsl = 0.3 usl = 0.8;
   histogram /
       midpoints=0.2 to 1.8 by 0.2
       lognormal
       weibull
       gamma
       nospeclegend;
   inset n mean(5.3) std='Std Dev'(5.3) skewness(5.3)
          / pos = ne  header = 'Summary Statistics';
run;

The LOGNORMAL, WEIBULL, and GAMMA options superimpose fitted curves on the histogram in Output 5.9.1. 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 5.9.1: Superimposing a Histogram with Fitted Curves

Superimposing a Histogram with Fitted Curves


The LOGNORMAL, WEIBULL, and GAMMA options also produce the summaries for the fitted distributions shown in Output 5.9.2, Output 5.9.3, and Output 5.9.4.

Output 5.9.2: Summary of Fitted Lognormal Distribution

The CAPABILITY Procedure
Fitted Lognormal Distribution for Gap (Plate Gap in cm)

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
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 5.9.2 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 5.9.3: Summary of Fitted Weibull Distribution

The CAPABILITY Procedure
Fitted Weibull Distribution for Gap (Plate Gap in cm)

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 15.0252997 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
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 5.9.3 provides two EDF goodness-of-fit tests for the Weibull distribution: the Anderson-Darling and the Cramer-von Mises tests. (See Table 5.23 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 5.9.4: Summary of Fitted Gamma Distribution

The CAPABILITY Procedure
Fitted Gamma Distribution for Gap (Plate Gap in cm)

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
Kolmogorov-Smirnov D 0.0969533   Pr > D >0.250
Cramer-von Mises W-Sq 0.0739847   Pr > W-Sq >0.250
Anderson-Darling A-Sq 0.5810661   Pr > A-Sq 0.137
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
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 5.9.4 provides four goodness-of-fit tests for the gamma distribution. 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:

\[  \begin{array}{rcl} \Pr [\mbox{gap} > \mbox{USL}] &  = &  \Pr \left[Z > \frac{1}{\sigma } (\log (\mbox{USL}-\theta )-\zeta ) \right] \\ &  = &  1-\Phi \left[\frac{1}{\sigma } (\log (\mbox{USL}-\theta )-\zeta ) \right] \end{array}  \]

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 $\Pr [\mbox{gap} > 0.8] = 0.2352$. This value is expressed as a percent (Est Pct > USL) in Output 5.9.2.