Example 29.3: Gamma Distribution Applied to Life Data

The GENMOD Procedure

Example 29.3: Gamma Distribution Applied to Life Data

Life data are sometimes modeled with the gamma distribution. Although PROC GENMOD does not analyze censored data or provide other useful lifetime distributions such as the Weibull or lognormal, it can be used for modeling complete (uncensored) data with the gamma distribution, and it can provide a statistical test for the exponential distribution against other gamma distribution alternatives. Refer to Lawless (1982) or Nelson (1982) for applications of the gamma distribution to life data.

The following data represent failure times of machine parts, some of which are manufactured by manufacturer A and some by manufacturer B.

   data A;
      input lifetime@@ ;
      mfg = 'A';
      datalines;
   620  470  260  89   388  242
   103  100  39   460  284  1285
   218  393  106  158  152  477
   403  103  69   158  818  947
   399  1274 32   12   134  660
   548  381  203  871  193  531
   317  85   1410 250  41   1101
   32   421  32   343  376  1512
   1792 47   95   76   515  72
   1585 253  6    860  89   1055
   537  101  385  176  11   565
   164  16   1267 352  160  195
   1279 356  751  500  803  560
   151  24   689  1119 1733 2194
   763  555  14   45   776  1
   ;

   data B;
      input lifetime@@ ;
      mfg = 'B';
      datalines;
   1747 945  12   1453 14   150
   20   41   35   69   195  89
   1090 1868 294  96   618  44
   142  892  1307 310  230  30
   403  860  23   406  1054 1935
   561  348  130  13   230  250
   317  304  79   1793 536  12
   9    256  201  733  510  660
   122  27   273  1231 182  289
   667  761  1096 43   44   87
   405  998  1409 61   278  407
   113  25   940  28   848  41
   646  575  219  303  304  38
   195  1061 174  377  388  10
   246  323  198  234  39   308
   55   729  813  1216 1618 539
   6    1566 459  946  764  794
   35   181  147  116  141  19
   380  609  546
   ;

   data lifdat;
      set A B;
   run;

The following SAS statements use PROC GENMOD to compute Type 3 statistics to test for differences between the two manufacturers in machine part life. Type 3 statistics are identical to Type 1 statistics in this case, since there is only one effect in the model. The log link function is selected to ensure that the mean is positive.

   proc genmod data = lifdat;
      class mfg;
      model lifetime = mfg / dist=gamma
                             link=log
                             type3;
   run;

The output from these statements is displayed in Output 29.3.1.

Output 29.3.1: Gamma Model of Life Data

The GENMOD Procedure

Model Information
Data Set	WORK.LIFDAT
Distribution	Gamma
Link Function	Log
Dependent Variable	lifetime
Observations Used	201

Class Level Information
Class	Levels	Values
mfg	2	A B

Criteria For Assessing Goodness Of Fit
Criterion	DF	Value	Value/DF
Deviance	199	287.0591	1.4425
Scaled Deviance	199	237.5335	1.1936
Pearson Chi-Square	199	211.6870	1.0638
Scaled Pearson X2	199	175.1652	0.8802
Log Likelihood		-1432.4177

Analysis Of Parameter Estimates
Parameter		DF	Estimate	Standard Error	Wald 95% Confidence Limits		Chi-Square	Pr > ChiSq
Intercept		1	6.1302	0.1043	5.9257	6.3347	3451.61	<.0001
mfg	A	1	0.0199	0.1559	-0.2857	0.3255	0.02	0.8985
mfg	B	0	0.0000	0.0000	0.0000	0.0000	.	.
Scale		1	0.8275	0.0714	0.6987	0.9800

NOTE:

The scale parameter was estimated by maximum likelihood.

LR Statistics For Type 3 Analysis
Source	DF	Chi-Square	Pr > ChiSq
mfg	1	0.02	0.8985

The p-value of 0.8985 for the chi-square statistic in the Type 3 table indicates that there is no significant difference in the part life for the two manufacturers.

Using the following statements, you can refit the model without using the manufacturer as an effect. The LRCI option in the MODEL statement is specified to compute profile likelihood confidence intervals for the mean life and scale parameters.

   proc genmod data = lifdat;
      model lifetime = / dist=gamma
                         link=log
                         lrci;
   run;

Output 29.3.2: Refitting of the Gamma Model: Omitting the mfg Effect

The GENMOD Procedure

Analysis Of Parameter Estimates
Parameter	DF	Estimate	Standard Error	Likelihood Ratio 95% Confidence Limits		Chi-Square	Pr > ChiSq
Intercept	1	6.1391	0.0775	5.9904	6.2956	6268.10	<.0001
Scale	1	0.8274	0.0714	0.6959	0.9762

NOTE:

The scale parameter was estimated by maximum likelihood.

The intercept is the estimated log mean of the fitted gamma distribution, so that the mean life of the parts is

$\mu = \exp({\rm INTERCEPT}) = \exp(6.1391) = 463.64$

The SCALE parameter used in PROC GENMOD is the inverse of the gamma dispersion parameter, and it is sometimes called the gamma index parameter. See the "Response Probability Distributions" section for the definition of the gamma probability density function. A value of 1 for the index parameter corresponds to the exponential distribution . The estimated value of the scale parameter is 0.8274. The 95% profile likelihood confidence interval for the scale parameter is (0.6959, 0.9762), which does not contain 1. The hypothesis of an exponential distribution for the data is, therefore, rejected at the 0.05 level. A confidence interval for the mean life is

(exp(5.99), exp(6.30)) = (399.57, 542.18)

Chapter Contents
Previous
Next
Top