Optimal Allocations of Stress Levels and Test Units in Accelerated Testing
[Editor's Note: In the online version of this article, we have corrected one equation used in the printed edition of Volume 5, Issue 1. The equation for the asymptotic variance of the ML estimate of Yp is given here in its entirety. In addition, we have modified the article to reflect the fact that the accelerated test design utility will be available in the upcoming release of ALTA 7, rather than ALTA 6 PRO.]
Before launching a new product, the manufacturer is always faced with decisions regarding the optimum method to estimate the reliability of the product or service. Accelerated testing (with accelerated time or accelerated stress) might be the recommended or required approach. Conducting a quantitative accelerated life test (QALT) requires the determination or development of an appropriate life-stress relationship model. Moreover, a test plan needs to be developed to obtain appropriate and sufficient information in order to accurately estimate reliability performance at operating conditions, significantly reduce test times and costs and achieve other objectives. Nelson (1990), Meeker and Escobar (1998) and Nelson (2003) provide a substantial review of the literature on how to develop optimum QALT plans. Such plans are becoming very popular and are starting to be used in engineering, materials science and manufacturing industries. However, the rate of increase of this popularity has been slow due to the limited tools available at this time for designing optimum accelerated life testing plans.
With the upcoming release of Version 7, ReliaSoft's ALTA will be the only software package capable of analyzing, modeling, planning and evaluating a quantitative accelerated life test. In this article, we will use ALTA's new accelerated life test planning module to investigate the procedure for designing QALT plans and apply the techniques to develop an example test plan for MOS capacitors.
Accelerated Life Testing Plans
1. The log-time-to-failure for each unit follows a location-scale distribution such that:
where μ and σ are the location and scale parameters respectively and Φ(·) is the standard form of the location-scale distribution.
2. Failure times for all test units, at all stress levels, are statistically independent. Without loss of generality, we follow the Nelson (1990) and Meeker and Escobar (1998) standardization of stress S by defining x = (S - SU)/(SH - SU), where SU is the design stress and SH is the highest test stress. S, SU and SH are the transformed stresses according to different life-stress relationships (linear, inverse power law and Arrhenius).
3. The location parameter μ is a linear function of stresses. Specifically, we assume that:
4. The scale parameter σ does not depend on the stress levels. All units are tested until η, a pre-specified test time.
5. Two of the most common models used in QALT are the linear Weibull and lognormal models. The Weibull model is given by:
where SEV denotes the smallest extreme value distribution. The lognormal model is given by:
That is, log life Y is assumed to have either an SEV or a normal distribution with location parameter μ(z), expressed as a linear function of z and constant scale parameter σ.
The ML estimate of the p quantile Yp at the normal stress Xo is:
where zp is the p percentile of the underlying standardized distribution. For SEV (Weibull), we have [zp = log[-log(1-p)] and for Normal (lognormal), zp is equal to the standard normal p percentile, Φnor-1(p). Thus, the asymptotic variance of the ML estimate of Yp is defined by:
ALTA Test Planning
Figure 1: Test Plans for a single accelerating stress
ALTA also provides two types of test plans for multiple accelerating stresses:
Figure 2: Test Plans for multiple accelerating stresses
Example: Application to
According to the specifications, the design stress level for this particular application is 323.16 K and the highest stress level is 523.16 K. After these input values have been entered in the ALTA test planning module, and assuming the failure data follow the Weibull distribution, it is possible to use the utility to develop a suitable QALT plan. We began with the 2 level statistically optimum plan. As shown in Figure 3, the optimized standardized low stress condition is x′L = 0.708. This translates into an actual stress of TL = 169.91° C (443.08 K). Therefore, 138 of the 200 test units would be assigned to 169.91° C (443.08 K) with the remaining 62 units tested at 250° C. 14.12%, or 19, of the 138 test units at 169.91° C (443.08 K) would be expected to fail during the 300 hour test.
Figure 3: 2 level statistically optimum test plan results
After reviewing these results, it was determined that the statistically optimum plan was not intuitively satisfying because it limits the test program to merely two temperatures and because of the relatively high value for the lowest stress condition, 169.91° C (443.08 K). Instead, it seemed appropriate to trade off some of the 19 expected failures at the lower stress for the sake of reducing the temperature and permitting testing at a middle stress condition. This led to the optimized 4:2:1 plan, which is shown in Figure 4. With this plan, the ratio of the asymptotic variance of the estimator of the 100pth percentile of the time-to-failure distribution, Ratio(p), is shown in Figure 4 to be 1.22 relative to that for the statistically optimum plan. Therefore, this approach increases the variance by 22%. This is the price that one may be paying for using the more robust and intuitively appealing optimized 4:2:1 plan.
Figure 4: 3 level 4:2:1 allocation plan results
After reviewing the results for the second plan, the low test temperature, 156° C (429 K), was thought to involve too much stress extrapolation relative to the design temperature of 50° C and hence a lower temperature seemed desirable for the standardized low stress condition. Thus, we adjusted the optimized 4:2:1 plan by reducing the low stress value to some fraction of the low stress value in the optimum plan. We can use different fractions to adjust the low stress value as long as the selected plan results in at least 3.33% ( (100p/3)% ) failures and at least five expected failures at the low stress (Ref. 1). As shown in Figure 5, a plan with 0.9 fraction low stress was selected. The probability of failure at the low stress level is 0.0533 (which satisfies the minimum requirement of 3.33%) and the expected number of failures is six.
Figure 5: Adjusted 3 level 4:2:1 allocation plan results
ALTA also provides a "Sensitivity Analysis" option, which allows the analyst to evaluate the test plans under consideration. This includes necessary sample size determination, robust analysis to misspecified models and sensitivity analysis to the guess value.
Up to this point, we have assumed that the number of available test specimens was predetermined by economic or other practical constraints. When this is not strictly so, it may be possible to choose a sample size that is large enough to provide a specified degree of precision. In ALTA, the number of test units that is required in the entire test program to estimate the 100pth percentile at the design stress to within of (1 + γ ) (i.e., an error of less than 100 γ %) with probability (1 + α) is approximately:
where z(1 - α/2) is the percentage point of the standard normal distribution, σ is a guess value for the scale parameter and V is the asymptotic variance of the estimate of the 100pth percentile of the time to failure distribution at the design stress multiplied by n/σ2.
Accelerated life testing plans developed under an assumed model are suitable only if the model is correct. Although these plans perform well under some models, they may or may not perform equally well under other models. For robust analysis to a misspecified model, ALTA uses R(WL/LL) and R(LW/WW) to analyze the possibility of bias due to model misspecification. If Weibull is the assumed distribution, R(WL/LL) is calculated. R(WL/LL) denotes the ratio of the variance for the plan obtained with an assumed Weibull distribution to that for the actual lognormal distribution, both being evaluated under the actual lognormal. R(LW/WW) is defined similarly for an assumed lognormal distribution and an actual Weibull distribution.
On the other hand, the calculated accuracy or sample size for the optimum plan also depends on the assumed values of the model parameters (the guess values of the failure probabilities). Sensitivity analysis will be investigated with respect to the guess values of the failure probabilities to test the robustness of the theoretical models. Traditionally, the model parameters are either estimates from the preliminary experiments or based on the experience of the experimenters and these assumed values differ from the true ones. Thus the calculated accuracy or sample size differs from the correct one. It is useful to re-evaluate a plan using other assumed values, changing one parameter at a time. If the plan or accuracy is sensitive to a parameter value, then one must consider changing the plan. Such an analysis can also be carried out on other characteristics of the plan, such as the probability of no failures at the lower test stress level.
ALTA uses the ratio of the variance of a plan generated under new modified guess values and the original guess value to analyze the sensitivity of plans to the guess value of the failure probabilities. Figure 6 shows these results for the MOS capacitor test plan, which include sample size calculation and some other sensitivity analysis.
Figure 6: Sensitivity Analysis in ALTA
2. Meeker, W. Q. & Escobar, L. A. (1995).
Planning Accelerated Life Tests with Two or More
3. Meeker, W. Q. & Escobar, L. A. (1998).
Statistical Methods for Reliability Data. New York:
4. Nelson, W. (1990). Accelerated
Testing - Statistical Models, Test Plans, and Data
5. Nelson, W. (2003). “Bibliography of
Accelerated Test Plans,” available from the author at