The following example illustrates the analysis of an accelerated life test for Class B electrical motor insulation. The data are provided by Nelson (1990, p. 243). Forty insulation specimens were tested at four temperatures: , , , and C. The purpose of the test is to estimate the median life of the insulation at the design operating temperature of C.
The following SAS program creates the data listed in Figure 16.8. Ten specimens of the insulation were tested at each test temperature. The variable Time
provides a specimen time to failure or a censoring time, in hours. The variable Censor
is equal to 1 if the value of the variable Time
is a rightcensoring time and is equal to 0 if the value is a failure time. Some censor times and failure times are identical
at some of the temperatures. Rather than repeating identical observations in the input data set, the variable Count
provides the number of specimens with identical times and temperatures. The variable Temp
provides the test temperature in degrees centigrade. The variable Cntrsl
is a control variable specifying that percentiles are to be computed only for the first value of Temp
(C). The value of Temp
in the first observation (C) does not correspond to a test temperature. The missing values in the first observation cause the observation to be excluded
from the model fit, and the value of 1 for the variable Cntrl
causes percentiles corresponding to a temperature of C to be computed.
data classb; input hours temp count censor; if _n_ = 1 then cntrl=1; else cntrl=0; label hours='Hours'; datalines; . 130 . . 8064 150 10 1 1764 170 1 0 2772 170 1 0 3444 170 1 0 3542 170 1 0 3780 170 1 0 4860 170 1 0 5196 170 1 0 5448 170 3 1 408 190 2 0 1344 190 2 0 1440 190 1 0 1680 190 5 1 408 220 2 0 504 220 3 0 528 220 5 1 ;
Figure 16.8: Listing of the Class B Insulation Data
Obs  hours  temp  count  censor  cntrl 

1  .  130  .  .  1 
2  8064  150  10  1  0 
3  1764  170  1  0  0 
4  2772  170  1  0  0 
5  3444  170  1  0  0 
6  3542  170  1  0  0 
7  3780  170  1  0  0 
8  4860  170  1  0  0 
9  5196  170  1  0  0 
10  5448  170  3  1  0 
11  408  190  2  0  0 
12  1344  190  2  0  0 
13  1440  190  1  0  0 
14  1680  190  5  1  0 
15  408  220  2  0  0 
16  504  220  3  0  0 
17  528  220  5  1  0 
An Arrheniuslognormal model is fitted to the data in this example. In other words, the fitted model has the lognormal (base
10) distribution, and its location parameter depends on the centigrade temperature Temp
through the Arrhenius relationship

where

is 1000 times the reciprocal absolute temperature. The lognormal (base e) distribution is also available.
The following SAS statements fit the Arrheniuslognormal model, and they display the fitted model distributions sidebyside on the probability and the relation plots shown in Figure 16.9:
proc reliability; distribution lognormal10; freq count; model hours*censor(1) = temp / relation=arr obstats( q=.1 .5 .9 control=cntrl ); rplot hours*censor(1) = temp / pplot fit=model noconf relation = arr plotdata plotfit 10 50 90 lupper = 1.e5 slower=120; run;
The PROC RELIABILITY statement invokes the procedure and specifies CLASSB
as the input data set. The DISTRIBUTION statement specifies that the lognormal (base 10) distribution is to be used for maximum
likelihood parameter estimation and probability plotting. The FREQ statement specifies that the variable Count
is to be used as a frequency variable; that is, if Count
=n, then there are n specimens with the time and temperature specified in the observation.
The MODEL statement fits a linear regression equation for the distribution location parameter as a function of independent
variables. In this case, the MODEL statement also transforms the independent variable through the Arrhenius relationship.
The dependent variable is specified as Time
. A value of 1 for the variable Censor
indicates that the corresponding value of Time
is a rightcensored observation; otherwise, the value is a failure time. The temperature variable Temp
is specified as the independent variable in the model. The MODEL statement option RELATION=ARR specifies the Arrhenius relationship.
The option OBSTATS requests statistics computed for each observation in the input data set. The options in parentheses following
OBSTATS indicate which statistics are to be computed. In this case, QUANTILE=.1 .5 .9 specifies that quantiles of the fitted
distribution are to be computed for the value of the variable Temp
at each observation. The CONTROL= option requests quantiles only for those observations in which the variable Cntrl
has a value of 1. This eliminates unnecessary quantiles in the OBSTATS table since, in this case, only the quantiles at the
design temperature of C are of interest.
The RPLOT, or RELATIONPLOT, statement displays a plot of the lifetime data and the fitted model. The dependent variable Time
, the independent variable Temp
, and the censoring indicator Censor
are the same as in the MODEL statement. The option FIT=MODEL specifies that the model fitted with the preceding MODEL statement
is to be used for probability plotting and in the relation plot. The option RELATION=ARR specifies an Arrhenius scale for
the horizontal axis of the relation plot. The PPLOT option specifies that a probability plot is to be displayed alongside
the relation plot. The type of probability plot is determined by the distribution named in the DISTRIBUTION statement, in
this case, a lognormal (base 10) distribution. Weibull, extreme value, lognormal (base e), normal, loglogistic, and logistic distributions are also available. The NOCONF option suppresses the default percentile
confidence bands on the probability plot. The PLOTDATA option specifies that the failure times are to be plotted on the relation
plot. The PLOTFIT option specifies that the 10th, 50th, and 90th percentiles of the fitted relationship are to be plotted
on the relation plot. The options LUPPER and SLOWER specify an upper limit on the life axis scale and a lower limit on the
stress (temperature) axis scale in the plots.
The plots produced by the preceding statements are shown in Figure 16.9. The plot on the left is an overlaid lognormal probability plot of the data and the fitted model. The plot on the right is a relation plot showing the data and the fitted relation. The fitted straight lines are percentiles of the fitted distribution at each temperature. An Arrhenius relation fitted to the data, plotted on an Arrhenius plot, yields straight percentile lines.
Since all the data at C are right censored, there are no failures corresponding to C on the probability plot. However, the fitted distribution at C is plotted on the probability plot.
Figure 16.9: Probability and Relation Plots for the Class B Insulation Data
The tabular output requested with the MODEL statement is shown in Figure 16.10. The “Model Information” table provides general information about the data and model. The “Summary of Fit” table shows the number of observations used, the number of failures and of censored values (accounting for the frequency count), and the maximum log likelihood for the fitted model.
The “Lognormal Parameter Estimates” table contains the Arrheniuslognormal model parameter estimates, their standard errors, and confidence interval estimates. In this table, INTERCEPT is the maximum likelihood estimate of , TEMP is the estimate of , and Scale is the estimate of the lognormal scale parameter, .
Figure 16.10: MODEL Statement Output for the Class B Data
Model Information  

Input Data Set  WORK.CLASSB  
Analysis Variable  hours  Hours 
Relation  Arrhenius( temp )  
Censor Variable  censor  
Frequency Variable  count  
Distribution  Lognormal (Base 10) 
Algorithm converged. 
Summary of Fit  

Observations Used  16 
Uncensored Values  17 
Right Censored Values  23 
Missing Observations  1 
Maximum Loglikelihood  12.96533 
Lognormal Parameter Estimates  

Parameter  Estimate  Standard Error  Asymptotic Normal  
95% Confidence Limits  
Lower  Upper  
Intercept  6.0182  0.9467  7.8737  4.1628 
temp  4.3103  0.4366  3.4546  5.1660 
Scale  0.2592  0.0473  0.1812  0.3708 
Observation Statistics  

Hours  censor  temp  count  Prob  Pcntl  Stderr  Lower  Upper 
.  .  130  .  0.1000  21937.658  6959.151  11780.636  40851.857 
.  .  130  .  0.5000  47135.132  16125.548  24106.685  92162.016 
.  .  130  .  0.9000  101274.29  42061.1  44872.401  228569.92 
The “Observation Statistics” table provides the estimates of the fitted distribution quantiles, their standard errors, and the confidence limits. These are given only for the value of C, as specified with the CONTROL= option in the MODEL statement. The predicted median life at C corresponds to a quantile of 0.5, and it is approximately 47,134 hours.
In addition to the MODEL statement output in Figure 16.10, the RELIABILITY procedure produces tabular output for each temperature that is identical to the output produced with the PROBPLOT statement. This output is not shown.