This example is an illustration of a Weibull regression model that uses a load accelerated life test of rolling bearings, with data provided by Nelson (1990, p. 305) . Bearings are tested at four different loads, and lifetimes in of revolutions are measured. The data are shown in Table 12.3. An outlier identified by Nelson (1990) is omitted.

These data are modeled with a Weibull regression model in which the independent variable is the logarithm of the load. The model is

where is the location parameter of the extreme value distribution and

for the th bearing. The following statements create a SAS data set containing the loads, log loads, and bearing lifetimes:

   data bearing;                                             
      input load life @@;                                    
      lload = log(load);                                     
      datalines;                                             
    .87 1.67    .87 2.2    .87 2.51    .87 3.0    .87 3.9    
    .87 4.7     .87 7.53   .87 14.7    .87 27.76  .87 37.4   
    .99  .8     .99 1.0    .99 1.37    .99 2.25   .99 2.95   
    .99 3.7     .99 6.07   .99 6.65    .99 7.05   .99 7.37   
   1.09  .18   1.09  .2   1.09  .24   1.09  .26  1.09  .32   
   1.09  .32   1.09  .42  1.09  .44   1.09  .88  1.18  .073  
   1.18  .098  1.18  .117 1.18  .135  1.18  .175 1.18  .262  
   1.18  .270  1.18  .350 1.18  .386  1.18  .456             
   ;                                                         
   run;                                                      

Figure 12.17 shows a listing of the bearing data.

The following statements fit the regression model by maximum likelihood that uses the Weibull distribution:

ods output modobstats = Residual;
proc reliability data=bearing;
   distribution weibull;
   model life = lload /  covb
                         corrb
                         obstats
                         ;
run;

The PROC RELIABILITY statement invokes the procedure and identifies BEARING as the input data set. The DISTRIBUTION statement specifies the Weibull distribution for model fitting. The MODEL statement specifies the regression model, identifying Life as the variable that provides the response values (the lifetimes) and Lload as the independent variable (the log loads). The MODEL statement option COVB requests the regression parameter covariance matrix, and the CORRB option requests the correlation matrix. The option OBSTATS requests a table that contains residuals, predicted values, and other statistics. The ODS OUTPUT statement creates a SAS data set named RESIDUAL that contains the table created by the OBSTATS option.

Figure 12.18 shows the tabular output produced by the RELIABILITY procedure. The "Weibull Parameter Estimates" table contains parameter estimates, their standard errors, and 95% confidence intervals. In this table, INTERCEPT corresponds to , LLOAD corresponds to , and SHAPE corresponds to the Weibull shape parameter. Figure 12.19 shows a listing of the output data set RESIDUAL.