### Regression Modeling

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 16.3. An outlier identified by Nelson (1990) is omitted.

Table 16.3: Bearing Lifetime Data

Load

Life ( Revolutions)

0.87

1.67

2.2

2.51

3.00

3.90

4.70

7.53

14.7

27.76

37.4

0.99

0.80

1.0

1.37

2.25

2.95

3.70

6.07

6.65

7.05

7.37

1.09

0.18

0.2

0.24

0.26

0.32

0.32

0.42

0.44

0.88

1.18

0.073

0.098

0.117

0.135

0.175

0.262

0.270

0.350

0.386

0.456

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 ith 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
;


Figure 16.17 shows a listing of the bearing data.

Figure 16.17: Listing of the Bearing Data

Obs load Life lload
1 0.87 1.670 -0.13926
2 0.87 2.200 -0.13926
3 0.87 2.510 -0.13926
4 0.87 3.000 -0.13926
5 0.87 3.900 -0.13926
6 0.87 4.700 -0.13926
7 0.87 7.530 -0.13926
8 0.87 14.700 -0.13926
9 0.87 27.760 -0.13926
10 0.87 37.400 -0.13926
11 0.99 0.800 -0.01005
12 0.99 1.000 -0.01005
13 0.99 1.370 -0.01005
14 0.99 2.250 -0.01005
15 0.99 2.950 -0.01005
16 0.99 3.700 -0.01005
17 0.99 6.070 -0.01005
18 0.99 6.650 -0.01005
19 0.99 7.050 -0.01005
20 0.99 7.370 -0.01005
21 1.09 0.180 0.08618
22 1.09 0.200 0.08618
23 1.09 0.240 0.08618
24 1.09 0.260 0.08618
25 1.09 0.320 0.08618
26 1.09 0.320 0.08618
27 1.09 0.420 0.08618
28 1.09 0.440 0.08618
29 1.09 0.880 0.08618
30 1.18 0.073 0.16551
31 1.18 0.098 0.16551
32 1.18 0.117 0.16551
33 1.18 0.135 0.16551
34 1.18 0.175 0.16551
35 1.18 0.262 0.16551
36 1.18 0.270 0.16551
37 1.18 0.350 0.16551
38 1.18 0.386 0.16551
39 1.18 0.456 0.16551

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 16.18 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 16.19 shows a listing of the output data set RESIDUAL.

Figure 16.18: Analysis Results for the Bearing Data

The RELIABILITY Procedure

Model Information
Input Data Set WORK.BEARING
Analysis Variable Life
Distribution Weibull

Parameter Information
Parameter Effect
Prm1 Intercept
Prm2 lload
Prm3 EV Scale

 Algorithm converged.

Summary of Fit
Observations Used 39
Uncensored Values 39
Maximum Loglikelihood -51.77737

Weibull Parameter Estimates
Parameter Estimate Standard Error Asymptotic Normal
95% Confidence Limits
Lower Upper
Intercept 0.8323 0.1410 0.5560 1.1086
lload -13.8529 1.2333 -16.2703 -11.4356
EV Scale 0.8043 0.0999 0.6304 1.0260
Weibull Shape 1.2434 0.1545 0.9746 1.5862

Estimated Covariance Matrix
Weibull Parameters
Prm1 Prm2 Prm3
Prm1 0.01987 -0.04374 -0.00492
Prm2 -0.04374 1.52113 0.01578
Prm3 -0.00492 0.01578 0.00999

Estimated Correlation Matrix
Weibull Parameters
Prm1 Prm2 Prm3
Prm1 1.0000 -0.2516 -0.3491
Prm2 -0.2516 1.0000 0.1281
Prm3 -0.3491 0.1281 1.0000

Figure 16.19: Listing of Data Set Residual

Obs Life lload Xbeta Surv Resid SRESID Aresid
1 1.67 -0.139262 2.7614742 0.9407681 -2.248651 -2.795921 -2.795921
2 2.2 -0.139262 2.7614742 0.9175782 -1.973017 -2.453205 -2.453205
3 2.51 -0.139262 2.7614742 0.9036277 -1.841191 -2.289296 -2.289296
4 3 -0.139262 2.7614742 0.8811799 -1.662862 -2.067565 -2.067565
5 3.9 -0.139262 2.7614742 0.8392186 -1.400498 -1.741347 -1.741347
6 4.7 -0.139262 2.7614742 0.8016738 -1.213912 -1.50935 -1.50935
7 7.53 -0.139262 2.7614742 0.6721971 -0.742579 -0.923306 -0.923306
8 14.7 -0.139262 2.7614742 0.4015113 -0.073627 -0.091546 -0.091546
9 27.76 -0.139262 2.7614742 0.1337746 0.562122 0.6989298 0.6989298
10 37.4 -0.139262 2.7614742 0.0542547 0.8601965 1.069549 1.069549
11 0.8 -0.01005 0.971511 0.7973909 -1.194655 -1.485407 -1.485407
12 1 -0.01005 0.971511 0.741702 -0.971511 -1.207955 -1.207955
13 1.37 -0.01005 0.971511 0.6427726 -0.6567 -0.816526 -0.816526
14 2.25 -0.01005 0.971511 0.4408692 -0.160581 -0.199663 -0.199663
15 2.95 -0.01005 0.971511 0.3175927 0.1102941 0.1371372 0.1371372
16 3.7 -0.01005 0.971511 0.2186832 0.3368218 0.4187966 0.4187966
17 6.07 -0.01005 0.971511 0.0600164 0.8318476 1.0343005 1.0343005
18 6.65 -0.01005 0.971511 0.0428027 0.9231058 1.147769 1.147769
19 7.05 -0.01005 0.971511 0.0337583 0.9815166 1.2203956 1.2203956
20 7.37 -0.01005 0.971511 0.0278531 1.0259067 1.2755892 1.2755892
21 0.18 0.0861777 -0.361531 0.8303684 -1.353268 -1.682623 -1.682623
22 0.2 0.0861777 -0.361531 0.809042 -1.247907 -1.55162 -1.55162
23 0.24 0.0861777 -0.361531 0.7665749 -1.065586 -1.324925 -1.324925
24 0.26 0.0861777 -0.361531 0.7455451 -0.985543 -1.225402 -1.225402
25 0.32 0.0861777 -0.361531 0.6837688 -0.777904 -0.967228 -0.967228
26 0.32 0.0861777 -0.361531 0.6837688 -0.777904 -0.967228 -0.967228
27 0.42 0.0861777 -0.361531 0.5868036 -0.50597 -0.629112 -0.629112
28 0.44 0.0861777 -0.361531 0.5684693 -0.45945 -0.57127 -0.57127
29 0.88 0.0861777 -0.361531 0.2625812 0.2336973 0.290574 0.290574
30 0.073 0.1655144 -1.460578 0.7887184 -1.156718 -1.438237 -1.438237
31 0.098 0.1655144 -1.460578 0.7101313 -0.86221 -1.072052 -1.072052
32 0.117 0.1655144 -1.460578 0.6526714 -0.685003 -0.851717 -0.851717
33 0.135 0.1655144 -1.460578 0.6006317 -0.541902 -0.673789 -0.673789
34 0.175 0.1655144 -1.460578 0.4946523 -0.282391 -0.351119 -0.351119
35 0.262 0.1655144 -1.460578 0.3126729 0.1211675 0.1506569 0.1506569
36 0.27 0.1655144 -1.460578 0.2991233 0.1512449 0.1880546 0.1880546
37 0.35 0.1655144 -1.460578 0.1889073 0.4107561 0.5107249 0.5107249
38 0.386 0.1655144 -1.460578 0.1522503 0.5086604 0.6324568 0.6324568
39 0.456 0.1655144 -1.460578 0.0987061 0.6753158 0.8396724 0.8396724

The value of the lifetime Life and the log load Lload are included in this data set, as well as statistics computed from the fitted model. The variable Xbeta is the value of the linear predictor

for each observation. The variable Surv contains the value of the reliability function, the variable Sresid contains the standardized residual, and the variable Aresid contains a residual adjusted for right-censored observations. Since there are no censored values in these data, Sresid is equal to Aresid for all the bearings. See Table 16.32 and Table 16.33 for other statistics that are available in the OBSTATS table and data set. See the section Regression Model Statistics Computed for Each Observation for Lifetime Data for a description of the residuals and other statistics.

If the fitted regression model is adequate, the standardized residuals have a standard extreme value distribution. You can check the residuals by using the RELIABILITY procedure and the RESIDUAL data set to create an extreme value probability plot of the residuals.

The following statements create the plot in Figure 16.20:

proc reliability data=residual;
distribution ev;
probplot sresid;
run;


Although the estimated location is near zero and the estimated scale is near one, the plot reveals systematic curvature, indicating that the Weibull regression model might be inadequate.