Reliability HotWire

Issue 105, November 2009

Reliability Basics

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 life-stress 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)
  • Cox-Snell 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]:

Equation

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 life-stress 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 life-stress 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].

Equation

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 life-stress 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.

Cox-Snell Residuals (SR)
The Cox-Snell residuals are given by:

Equation

where is the calculated reliability value at failure time . Cox-Snell 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 Cox-Snell 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 .

Equation

Using the transform method, can be obtained by:

Equation

For example, for an exponential distribution:

Equation

We can see that the random variable is generated from the random uniform variable .

For the Cox-Snell 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 Cox-Snell residuals should follow an exponential distribution with .

For the data in Table 1, if we use the Arrhenius/Weibull model, the Cox-Snell residuals plotted on an exponential paper will be:



Figure 5: Cox-Snell 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]:

Equation

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 two-sided 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.

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