# The UNIVARIATE Procedure

### Example 4.22 Fitting Lognormal, Weibull, and Gamma Curves

To determine an appropriate model for a data 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 distribution and display the density curves on a 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. The following statements save the measurements (Gap) 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:

title 'Distribution of Plate Gaps';
ods graphics on;
ods select Histogram ParameterEstimates GoodnessOfFit FitQuantiles;
proc univariate data=Plates;
var Gap;
histogram / midpoints=0.2 to 1.8 by 0.2
lognormal
weibull
gamma
odstitle = title;
inset n mean(5.3) std='Std Dev'(5.3) skewness(5.3)
/ pos = ne  header = 'Summary Statistics';
run;


The ODS SELECT statement restricts the output to the "ParameterEstimates," "GoodnessOfFit," and "FitQuantiles" tables; see the section ODS Table Names. The LOGNORMAL, WEIBULL, and GAMMA primary options request superimposed fitted curves on the histogram in Output 4.22.1. Note that a threshold parameter is assumed for each curve. In applications where the threshold is not zero, you can specify with the THETA= secondary option.

The LOGNORMAL, WEIBULL, and GAMMA options also produce the summaries for the fitted distributions shown in Output 4.22.2 through Output 4.22.4.

Output 4.22.2 provides three EDF goodness-of-fit tests for the lognormal distribution: the Anderson-Darling, the Cramér-von Mises, and the Kolmogorov-Smirnov tests. At the significance level, all tests support the conclusion that the two-parameter lognormal distribution with scale parameter and shape parameter provides a good model for the distribution of plate gaps.

Output 4.22.1: Superimposing a Histogram with Fitted Curves

Output 4.22.2: Summary of Fitted Lognormal Distribution

 Distribution of Plate Gaps

The UNIVARIATE 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 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

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 4.22.3: Summary of Fitted Weibull Distribution

 Distribution of Plate Gaps

The UNIVARIATE 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 p Value
Cramer-von Mises W-Sq 0.15937281 Pr > W-Sq 0.016
Anderson-Darling A-Sq 1.15693542 Pr > A-Sq <0.010

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 4.22.3 provides two EDF goodness-of-fit tests for the Weibull distribution: the Anderson-Darling and the Cramér–von Mises tests. The p-values for the EDF tests are all less than 0.10, indicating that the data do not support a Weibull model.

Output 4.22.4: Summary of Fitted Gamma Distribution

 Distribution of Plate Gaps

The UNIVARIATE 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 p Value
Kolmogorov-Smirnov D 0.09695325 Pr > D >0.250
Cramer-von Mises W-Sq 0.07398467 Pr > W-Sq >0.250
Anderson-Darling A-Sq 0.58106613 Pr > A-Sq 0.137

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 4.22.4 provides three EDF goodness-of-fit tests for the gamma distribution: the Anderson-Darling, the Cramér–von Mises, and the Kolmogorov-Smirnov tests. At the significance level, all tests support the conclusion that the gamma distribution with scale parameter and shape parameter provides a good model for the distribution of plate gaps.

Based on this analysis, the fitted lognormal distribution and the fitted gamma distribution are both good models for the distribution of plate gaps.

A sample program for this example, uniex13.sas, is available in the SAS Sample Library for Base SAS software.