## 2019

STATISTICS ROUNDTABLE

# Timely Reliability Assessment

## Using destructive degradation tests

by Necip Doganaksoy, Gerald J. Hahn and William Q. Meeker

Often, we must demonstrate long-term product or component reliability within a relatively short time window. For example, it is not uncommon to need to assess 10-year reliability from a six-month duration test program. In such situations, basing life data analyses on appropriately selected degradation data (that is, monitoring change of one or more relevant quality or performance characteristics) when coupled with test acceleration can be highly useful.

In our last Statistics Roundtable column^{1} and an earlier one,^{2} we illustrated the benefits of using
degradation data in reliability analyses. These case studies dealt with battery
and gallium arsenide laser life, respectively. In both cases, it was possible
to obtain degradation measurements on the same unit over time.

In contrast, in this column we will consider the frequently encountered case in which the degradation measurement is destructive and, consequently, only a single degradation measurement can be obtained on each test unit. In such cases, the reliability assessment may be based on an accelerated destructive degradation test (ADDT). We explain ADDTs using an example dealing with a seal to be used in a new product.

### Seal strength application

A severely weakened seal adversely affects product performance and could lead to safety concerns. The seal’s strength is measured as the force required to bend under a predefined loading system and typically is initially between 20 to 30 newtons. The seal is expected to degrade slowly over time and is designated as failed when its strength falls below 0.5 newtons. It was desired to demonstrate with 95% confidence that 10-year seal reliability in the use environment of 100°C exceeds 99% (that is, less than 1% of the seals have a 10-year strength below 0.5 newtons or, equivalently, the probability of seal failure after 10 years is less than 0.01). This demonstration, moreover, needed to be achieved after only 25 weeks of testing.

Conducting time-to-failure testing at 100°C for 25 weeks will not provide the desired reliability demonstration because the anticipated result of no failures after 25 weeks tells us nothing about what will happen after 10 years. Therefore, the team turned to an ADDT for the desired reliability assessment.

### Developing an ADDT

In developing an ADDT, we first identify a physically and empirically appropriate acceleration variable, as well as a degradation characteristic to be measured. We also seek a physically meaningful engineering model to relate test duration and the accelerating variable to the measured degradation. Next, a test plan is developed to obtain data to fit the assumed model and to assess goodness of fit of the model to the data. After validation, the fitted model is used to conduct the desired reliability evaluations. All of this must be accomplished within the designated elapsed time limitations.

### Accelerating variable and degradation measurement

Fortunately, the
development team had a great deal of experience with ADDTs on seals and similar
materials, including the construction of underlying models. In addition, the
team was able to use the results of earlier research.^{3-5}

Based on this knowledge, temperature and seal strength were chosen as the accelerating variable and the relevant degradation measurement, respectively, in the ADDT. Moreover, it was expected that temperature would affect degradation rates via an Arrhenius relationship.

### Test plan

The basic goal of the test plan was to conduct 25-week life tests at temperatures more severe than the use temperature of 100°C to be able to fit and assess the assumed relationship between temperature and strength, then extrapolate the fitted model to estimate 10-year seal reliability at 100°C.

The resulting plan involved testing 50 randomly selected seal specimens at each of four elevated temperatures (200°C, 250° C, 300°C and 350°C) in a test chamber that reflected the key characteristics of the application environment (except for the elevated temperature). Next, every five weeks up to 25 weeks, 10 seal specimens at each temperature were removed from the chamber for strength measurement. This test caused physical damage and, consequently, the specimen could no longer be used for further testing. Baseline strength measurements were obtained on an additional 10 unexposed seal samples at the beginning of the test (week 0).

### Test results

Figure 1 provides lognormal probability plots of the 10 strength measurements at each measurement time at each of the five test temperatures. Figure 2 displays the mean strengths (on a logarithmic scale) of the 10 units at each of the temperature and test time combinations. The horizontal dashed line shows the failure threshold. The fitted lines shown in the plots are covered later.

### Assumed model

Fitting a statistical model to the data to allow extrapolating seal strength to a temperature of 100°C after 520 weeks of exposure requires assuming a scientifically justifiable and empirically supported engineering model. This called for assumptions about:

- The statistical distribution of strength measurements at each temperature and test time combination.
- The relationship between average strength and temperature over time.

**Statistical distribution of strength measurements at each temperature
and test time: **It was assumed that the seal strength measurements
at each time and temperature combination follow a lognormal distribution (that
is, the log transformed strengths are normally distributed).

In addition, it is assumed that σ, the standard deviation of log strength, (characterizing the spread due to seal variability and measurement error) is the same at each temperature-test time combination. Using mathematical terminology:

log*Y _{tTi}*

*∼*

_{ }*N(µ(t,T),*σ

*), in which:*

^{2}*Y*is the measured strength on seal sample_{tTi}*i*at time*t*at temperature*T*.*µ(t,T)*is the mean log strength at time*t*and temperature*T*.- σ is the (assumed constant) standard deviation of log strength.

The lognormal probability plots of the data shown in Figure 1 at each test condition allow us to assess the preceding assumptions. The straight lines on these plots are maximum likelihood (ML) fits to the data, assuming a lognormal distribution, at each test condition. These lines appear to fit the data well, suggesting that a lognormal distribution provides a reasonable description of the data at the individual test conditions. (A lognormal distribution appeared to provide a better fit visually than a Weibull distribution. A statistical likelihood ratio test, however, did not find a significant difference between these two models.)

Moreover, the nearly parallel lines in Figure 1 suggest a common (lognormal) scale parameter (σ) at all test conditions. This observation also was supported by a statistical likelihood ratio test.

**Relationship between average strength and temperature over time:** Again, based on physical considerations and past experience, the
following form relationship was assumed to relate test time and temperature to
mean strength:

*µ(t,T)
= *β_{0 } *- te*^{β}^{1}^{-}^{β}^{2}^{(11,605/}^{T}^{)}, in which

*t*is the test time in weeks.*T*is the exposure Kelvin temperature (that is, degrees Celsius + 273.15).*µ(t,T)*is the mean log strength at time*t*(in weeks) and temperature*T*.- β
_{0}, β_{1}and β_{2 }are parameters to be estimated from the data. - 11,605
is the reciprocal of Boltzmann’s constant in units of eV.

### Implications of assumed model

The preceding two-part model implies:

- A simple linear relationship between mean log strength and time on test at a given temperature. The slope of this relationship at a particular temperature represents the degradation rate at that temperature.
- A simple linear
relationship between the logarithm of the estimated degradation rate at a
particular temperature and the transformed temperature (11,605/T) (that is,
degradation rate has an Arrhenius dependence on temperature).

### Procedure for model estimation and software

The parameters σ, β_{0}, β_{1} and β_{2} are estimated using
ML. Several commercially available software packages, including JMP and SAS,
offer built-in functionality for ML estimation of the preceding model.
Alternatively, any software package that allows nonlinear least-squares
estimation can be adapted for such estimation.

### Fitted model

The preceding model was fitted to the ADDT data using the JMP software. The resulting estimates, including the estimated standard errors of each of the estimated parameters, are shown in Table 1.

### Assessment of adequacy of fit

The preceding ML fits to the data are super-imposed on the plots in Figure 2. The plotted points, representing the observed data, seem to scatter randomly around the fitted line, suggesting that the assumed model adequately represent the data. Residuals plots (not shown here) did not reveal any unusual patterns, also suggesting that the fitted model is adequate.

### Extrapolation to use condition

As stated earlier, the major purpose of this investigation was to hopefully demonstrate with 95% confidence that 10-year seal reliability at the use condition of 100°C exceeds 99%.

Extrapolating the fitted model to 100°C and 10 years’ exposure leads to
an estimated median strength of 8 newtons, with a 95%
lower confidence bound of 6.5 newtons. Our major
interest is not in median strength at the use condition, but in estimates of
the lower quantiles in general, and the 0.01 quantile, in particular, of the strength distribution. For
the assumed model, the α quantile of the lognormal strength distribution at time* t* and temperature *T* is:

*y*_{α} = exp [*µ(t,T)+z*_{α }σ], in which *z*_{α} is the α_{ }quantile of the standard normal distribution.

Figure 3 shows the ML estimates of the 0.01 quantile of the seal strength degradation distribution at 100°C (solid line) and the approximate 95% lower confidence bounds, based on the approximate normal distribution of ML estimators (dashed line) for different exposure times. (The horizontal dashed line again shows the failure threshold.) The estimated 0.01 quantile after 10 years of use at 100°C is 5.2 newtons with an approximate 95% lower confidence bound of 4.2 newtons. This value is comfortably above the failure threshold of 0.5 newtons, thus providing the desired reliability demonstration.

### Inclusion of batch-to-batch variation

The
seal example is a simplification—using computer-generated data—of
an application described in a more technical article.^{6} That article
examines the fitted model (based on residual plotting) and reveals an
additional complication—namely, differences in mean strength between the
batches of 10 seals removed at each temperature at each test time. The article
describes how the statistical model was extended to consider such
batch-to-batch variation. This source of variation was excluded from the data
generated for this column.

### Extreme caution advised

The major purpose of this column is to show the use of accelerated degradation testing for situations in which the degradation measurement is destructive. The method is illustrated by a study dealing with seal strength degradation. The results are clearly comforting to the investigators because they demonstrated statistically the desired level of reliability.

We must remember, however, this demonstration was based on some strong assumptions, including a model that used tests with a maximum duration of 25 weeks at a minimum use temperature of 200°C to draw conclusions about performance after 520 weeks at a temperature of 100°C.

We gained some reassurance of the validity of the results by the fact that various analyses of fit adequacy provided no evidence to question our assumptions. However, the assumed model adequately fitting the given data within the region of experimentation is no guarantee that the model will provide good estimates when severely extrapolated.

An additional extrapolation results from the fact that engineering-developed units were used to represent future high-volume production units. We plan to discuss these concerns—and what we might do to alleviate them—in a future column.

### References

- Necip Doganaksoy, Gerald J. Hahn
and William Q. Meeker, "Providing Better Insights: Improved Life Analyses Using
Degradation Testing,"
*Quality Progress,*November 2013, pp. 54-56. - William
Q. Meeker, Necip Doganaksoy
and Gerald J. Hahn, "Using Degradation Data for Product Reliability Analysis,"
*Quality Progress,*June 2001, pp. 60-65. - Wayne
B. Nelson, "Analysis of Performance Degradation Data from Accelerated Tests,"
*IEEE Transactions on Reliability*, June 1981, Vol. R-30, No. 2, pp. 149-155. - Wayne
B. Nelson,
*Accelerated Testing: Statistical Models, Test Plans, and Data Analyses,*Wiley, 1990. - Luis
A. Escobar, William Q. Meeker, D.L. Kugler, and L.L.
Kramer, "Accelerated Destructive
Degradation Tests: Data, Models, and Analysis," which appeared as chapter 21 in
*Mathematical and Statistical Methods in Reliability,*B.H. Lindqvist and K. A. Doksum, eds.,*World Scientific*, 2001. - Ming
Li and Necip Doganaksoy,
"Batch Variability in Accelerated Degradation Testing,"
*Journal of Quality Technology,*2014, Vol. 46, No. 2, pp. 171-180.

**Necip Doganaksoy** is principal statistician at GlobalFoundries in Malta, NY. He has a doctorate in
administrative and engineering systems from Union College in Schenectady. Doganaksoy is a fellow of ASQ and the American Statistical
Association.

**Gerald J. Hahn** is a retired manager of statistics at the GE
Global Research Center in Schenectady, NY. He has a doctorate in statistics and
operations research from Rensselaer Polytechnic Institute in Troy, NY. Hahn is
a fellow of ASQ and the American Statistical Association.

**William Q. Meeker** is
professor of statistics and distinguished professor of liberal arts and
sciences at Iowa State University in Ames, IA. He has a doctorate in
administrative and engineering systems from Union College in Schenectady, NY.
Meeker is a fellow of ASQ and the American Statistical Association.

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