Residual Plots in Accelerated Life Testing Data Analysis
In linear regression, it is assumed that the residuals are normally distributed
with mean of 0 and have a constant standard deviation.
In DOE++, ReliaSoft’s Design
of Experiments (DOE) software, different residual plots are available for validating these
assumptions [1]. In accelerated life data analysis, residual plots
also play an important role in identifying the right lifestress relationship
and the right lifetime distribution. In addition to validating a model, residual
plots also can help users to identify outliers. In this article, we discuss
several residual plots in ALTA,
ReliaSoft’s accelerated life data analysis software package [2].
There are three different types of residual plot in
ALTA 7:
 Standardized Residuals (SR)
 CoxSnell Residuals
 Standardized vs. Fitted Values
Standardized Residuals (SR) For the Weibull distribution, once the parameters have been estimated, the
standardized residuals can be calculated by [2]:
where:
 is the residual for the
ith observation.
 is either the failure
time or the suspension time. If it is a suspension
time, then is also a
suspension.
 is the estimated shape
parameter for the Weibull distribution.
 is the estimated life
characteristic at a given
stress .
If the Weibull distribution and the assumed lifestress relationship can describe
the data well, then the
residuals should look like a sample from a standard smallest extreme value
distribution (SEV or Gumbel distribution). In ALTA, the standardized
residuals are plotted on a smallest extreme value distribution probability paper.
If the assumed model adequately fits the data, then the residuals should appear
to follow a straight line on such a probability plot.
Example: For the data given in the following table,
the Weibull distribution
and the Arrhenius lifestress relationship are used.
Failure Time 
Temperature (K) 
248 
406 
456 
406 
528 
406 
731 
406 
813 
406 
164 
416 
176 
416 
289 
416 
319 
416 
340 
416 
543 
416 
92 
426 
105 
426 
155 
426 
184 
426 
219 
426 
235 
426 
The estimated model parameters are shown in Figure 1.
Figure 1: Results for the Arrhenius/Weibull model
The residual plot is shown in Figure 2.
Figure 2: Standardized Residual plot for the
Arrhenius/Weibull model
For the lognormal distribution, once the parameters have been estimated,
the standardized residuals can be calculated by [2].
where:
 is the residual for the
ith observation.
 is either the failure
time or the suspension time. If it is a suspension
time, then is also a
suspension.
 is the estimated shape
parameter for the lognormal distribution.
 is the estimated life
characteristic at a given
stress .
If the assumed lifestress relationship and the lognormal distribution can fit the
data well, then the standardized residuals should be normally distributed with a
mean of zero and a standard deviation of 1. When the residuals are plotted on a
normal probability paper, it should fall on the straight line given that the
assumed model is good for the data. For the data in Table 1, if the lognormal
distribution is used, the results are shown in Figure 3,
while Figure 4 shows the plot of the data.
Figure 3: Results for the Arrhenius/Lognormal
model
Figure 4: Standardized Residual Plot for the
Arrhenius/Lognormal model
Figures 2 and 4 both show that the residuals fall on the straight line.
Therefore, for this data set, either the lognormal or the Weibull distribution
can be used. Checking the logarithm of the likelihood value, it can be seen that the
likelihood value is 103.3880 for the Weibull distribution and 103.5459 for
the lognormal distribution. So the Weibull distribution is slightly better since
it has a slightly larger likelihood value.
CoxSnell Residuals (SR) The CoxSnell residuals are given by:
where is the calculated
reliability value at failure time .
CoxSnell residuals are more general than the standardized residuals since it can be
applied to any distribution. If the underlying model can fit the data well, the
CoxSnell residuals follow an exponential distribution
with .
To understand why it is an exponential distribution
with , some basic Monte Carlo
simulation knowledge is necessary. To generate a random
variable for any distribution,
a uniform random variable within (0, 1) is generated first,
say , which is treated as the
reliability (or unreliability) at time .
Using the transform method, can be obtained by:
For example, for an exponential distribution:
We can see that the random variable
is generated from the random uniform
variable .
For the CoxSnell residuals, we know
that , which
is , is the estimated
reliability at failure time .
If the assumed model is good and the estimated model parameters are accurate,
then should appear to be
sampled from a uniform distribution within (0, 1). Therefore, the CoxSnell residuals
should follow an exponential distribution with .
For the data in Table 1, if we use the Arrhenius/Weibull model, the CoxSnell
residuals plotted on an exponential paper will be:
Figure 5: CoxSnell Residual Plot for the
Arrhenius/Weibull model
In Figure 5, the last point is off the straight line. It is the residual for the
failure time of 543 at stress of 416 K. Compared to other failure times at this stress
level, it is a little bit high. Further investigation should be conducted on this
observation.
Standardized vs. Fitted Values The fitted values are the
predicted values for the
Weibull distribution and the
predicted for the lognormal
distribution. For the data in Table 1, the standardized residuals vs. fitted values
plot is shown in Figure 6.
Figure 6: Standardized vs. Fitted Values Plot for the Arrhenius/Weibull mode
From Figure 6, two points are out of the average and
seem to be outliers:
 (337.927, 1.407), the 6th failure time at stress level 416
 (635.966, 2.793), the smallest failure time at stress level of 426
We can test them statistically. Since
the standard residuals follow a standard SEV distribution
with cdf [3]:
The 0.05 and 0.95 percentiles can be calculated using the above equation. They
are 2.970 and 1.097. Therefore, point (337.927, 1.407) is the only one that is out
of the twosided bounds of [2.970, 1.097] at a significance level of 90%. This is
similar to what we got from Figure 5 where the 6th failure at stress level 416
seems abnormal.
Conclusion In this article, three different residual plots in
ALTA 7 were discussed. Using residual plots, we can test
whether a distribution
is good for a data set or not. Residual plots also can help us identify outliers
in the observed failure times.
References
[1] ReliaSoft Corporation, Experiment Design and
Analysis Reference, Tucson, AZ: ReliaSoft Publishing, 2008.
[2] ReliaSoft Corporation, Accelerated Life Testing
Reference, Tucson, AZ: ReliaSoft Publishing, 2007.
[3] ReliaSoft Corporation, Life Data
Analysis Reference, Tucson, AZ: ReliaSoft Publishing, 2007.
