A Bayesian Method for Planning Accelerated Life Testing - IEEE Xplore

IEEE TRANSACTIONS ON RELIABILITY, VOL. 64, NO. 4, DECEMBER 2015

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A Bayesian Method for Planning Accelerated Life Testing Ancha Xu and Yincai Tang

Abstract—In this paper, a Bayesian criterion is proposed based on the expected Kullback-Leibler divergence between the posterior and the prior distributions of the parameters of interest. We call the Bayesian criterion the reference optimality criterion, which is to find an optimal plan to maximize the amount of information from the data. A large-sample approximation is utilized to simplify the formula to obtain optimal plans numerically. Because optimal plans based on reference optimality criterion do not depend on the sample size, a modified reference optimality criterion is proposed. We give numerical examples using the Weibull distribution with type I censoring to illustrate the methods, and to examine the influence of the prior distribution, censoring time, and sample size. We also compare our methods with other criteria through Monte Carlo simulation. Index Terms—Accelerated life testing, Kullback-Leibler divergence, reference prior, Weibull distribution, censored data.

ACRONYMS AND ABBREVIATIONS ALT

accelerated life test(ing)

ML

maximum likelihood

ROC

reference optimality criterion

MROC

modified reference optimality criterion

SMSE

squared root of mean square errors NOTATION a specific design the observed data the model parameter vector the likelihood function the prior of the expected Fisher information matrix of the observed data and the candidate plan

for

Manuscript received April 28, 2014; revised October 28, 2014 and March 02, 2015; accepted May 18, 2015. Date of publication June 01, 2015; date of current version November 25, 2015. This work was supported in part by the Natural Science Foundation of China (11271136 and 11201345), Program of Shanghai Subject Chief Scientist (14XD1401600), the 111 Project (B14019), Natural Science Foundation of Zhejiang province (LY15G010006), and China Postdoctoral Science Foundation (2015M572598). Associate Editor: S. Bae. A. Xu is with the College of Mathematics and Information Science, Wenzhou University, Zhejiang 325035, China, and also with the Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, 710072, China (e-mail: [email protected]). Yincai Tang is with the School of Finance and Statistics, East China Normal University, Shanghai 200241, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TR.2015.2436374

the lifetime of a product number of levels in constant-stress ALT stress level for the th stress environment number of units tested at stress level censoring time use stress condition the variance-covariance matrix of the prior distribution for I. INTRODUCTION

I

T is hard to collect failure time data for highly reliable items under typical operating conditions. The most often used method to assess highly reliable components is to test them under extreme operating conditions. Such tests are referred to as accelerated life tests (ALTs). There are two main issues in ALT: statistical inference, and optimal test plans. Statistical inference has been widely studied. For details, see Nelson [1], Meeker and Escobar [2], and the references therein. However, the precision of statistical inference greatly depends on the design of the ALT plans. Planning ALT is an important practical problem, which needs to be paid attention to when collecting lifetime data. Chernoff [3] was the first one to study ALT plans. After that, many authors extended Chernoff's model for lognormal, Weibull, or other life distributions. Nelson [4], [5] gave a very comprehensive review about the area of planning ALT, and listed 159 references. Choosing an optimal criterion is a basic part of the design of ALT plans. The V-optimality criterion is the most widely used one, which minimizes the asymptotic variance of the maximum likelihood (ML) estimator of a specified quantile lifetime, or a function of the model parameters at the use stress condition. Based on this criterion, Pascual [6] presented a method for constant-stress ALT planning when there are two or more failure modes. Liu and Qiu [7] considered step-stress ALT planning with statistically independent failure modes. Guan and Tang [8] studied step-stress ALT planning under type-I censoring when the lifetime of the series system followed a Marshall-Olkin multivariate exponential distribution. Ye et al. [9] considered test planning when heterogeneities exist in the operating environment. These references describe the methodology with one accelerating variable. For more than one experimental variable, see Li and Fard [10], Xu and Fei [11], and Zhu and Elsayed [12]. Escobar and Meeker [13] presented D-optimality criterion that minimized the determinant of the covariance matrix of the model parameter estimates. Guan et al. [14] compared these two criteria in planning constant-stress ALTs when the lifetime of the unit follows

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the generalized exponential distribution. Because the Fisher information matrix of the model parameters is related to the unknown parameters, determining the optimal plans needs the planning values based on the two criteria. The planning values could be obtained from previous experience with similar products and materials, or engineering judgment. Consequently, the robustness of such optimal plans should be studied according to different planning values. See Fard and Li [15], and Tsai et al. [16]. Besides, the two criteria are based on a large-sample approximation, while ALTs are usually subject to the constraint that the available number of test units has to be small either because of high cost of the units, or availability of prototype units. The Bayesian method can then be used to deal with the case of small sample size. Incorporating prior information into test planning has been considered by some authors. Verdinelli et al. [17] maximized the Shannon information between the marginal distribution of the data and the posterior predictive distribution, then obtained ALT plans. Erkanli and Soyer [18] proposed a simulation method, and used curve-fitting techniques for ALT planning. Zhang and Meeker [19] presented a Bayesian criterion that minimized the pre-posterior variance of a quantity of interest at use stress conditions, and illustrated the method by a constant-stress ALT example. Yuan et al. [20] utilized the Bayesian criterion, and developed two algorithms for planning optimal simple step-stress ALTs. In this paper, we propose a Bayesian criterion based on the expected Kullback-Leibler divergence between the posterior and the prior distributions of the parameters of interest. The criterion is to maximize the expected information from an experiment. Such a criterion has many advantages as we show in the simulation. The first advantage is that the optimal plans based on the criterion are robust for the choice of priors as we show in Section III. The second advantage is that the estimates based on the ALT lifetime data under the optimal plans have smaller standard deviations; even the prior information is vague or distorted as we show in Section VI. This paper proceeds as follows. In Section II, a Bayesian criterion called reference optimality criterion is proposed, and a large-sample approximation is utilized to simplify the formula. Section III gives a numerical example to study the influence of censoring time and priors. Because optimal plans based on reference optimality criterion are statistically independent of sample size, a modified reference optimality criterion is presented in Section IV. Then in Section V we use an example to illustrate the methods, and to examine the influence of the prior distribution, censoring, and sample size. The proposed criteria are compared with other criteria in Section VI. Finally, we give a conclusion. II. REFERENCE OPTIMALITY CRITERION In discussing the Bayesian approach to optimal design problems, Erkanli and Soyer [18] indicated that it is required to

specify three components: (i) a utility (loss) function that reflects the consequences of selecting a specific design, (ii) a probability model, and (iii) a prior distribution reflecting a designer's prior beliefs about all unknown quantities. Thus, a Bayesian criterion should include these components. Assume that is a specific design, and that is the observed data. The likelihood function is denoted as , where is the unknown parameter vector. Assume that the prior of is . Then, according to Bayes' theorem, the posterior density function of the model parameters is

where

is the marginal density function of given by

If is the parameter of interest, which is usually a function of the model parameters, then we define the utility function as the Kullback-Leibler divergence of the prior density function of from the marginal posterior density function of . That is, (1) is the prior of , and is the posterior where density function of which can be described as

Equation (1) is a non-symmetric measure of the difference between and . See Kullback [21]. But for a given plan , the Kullback-Leibler divergence (1) depends on the unobserved data . The pre-posterior expectation of over can be used to obtain a Bayesian planning criterion. Thus the optimal plan is the one that maximizes (2) at the bottom of the page. From Lindley [22], is the expected information about to be provided by the data when the prior density of is . Thus, given the prior, the Bayesian planning criterion is to find an optimal plan to maximize the amount of information about from the data. Bernardo [23] utilized it to develop a new noninformative prior referred to as the reference prior, and now the reference prior is very popular in Bayesian analysis. See Berger and Bernardo [24], Ghosal [25], Berger et al. [26], Xu and Tang [27], [28], and Guan et al. [29]. Thus, we refer to the Bayesian planning criterion as the reference optimality criterion (ROC). Because (2) involves the observed data should depend on the sample size. Therefore, the optimal plans based on defined in (2) may depend on the sample size. This dependency is the first difference from the classical optimality criteria. Besides, the Bayesian criterion maximizing is from an information perspective, which is much different from V-optimal and Bayesian criterion (Zhang and Meeker [19]) based on the inference perspective.

(2)

XU AND TANG: A BAYESIAN METHOD FOR PLANNING ACCELERATED LIFE TESTING

Thus, the maximized is a measure of the amount of information from the experiment, and cannot be used in the statistical inference. However, when the data are collected under the optimal plans based on maximizing , the estimates would have smaller standard deviations as we show in Section VI.A. Criterion (2) involves calculation of high-dimension integration. Usually, there is no closed form, and exact numerical computation is intractable. Thus, approximations or simulations need to be utilized. We use a large-sample approximation to provide a useful basis for optimization. Let denote the expected Fisher information matrix of for the observed data , and the candidate plan . From Ghosh and Mukerjee [30], under some regularity conditions, we have as the sample size that

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. However, just as with classical optimality criteria, the approximated can not be used to assess the effects of the sample size on the optimal plans. III. NUMERICAL EXAMPLE In this section, we present a numerical example to illustrate our method. A -level constant-stress ALT is considered. Let be the lifetime of a product, and follow a Weibull distribution with cumulative distribution function (6)

where we see the equation at the bottom of the page, and is the expected Fisher information matrix of with held fixed. Let

where , and are the location, and scale parameters of the log lifetime, and . In a -level constant-stress ALT, units are tested at stress level until censoring time , where . The sample size is . Let denote the use stress condition. The location parameter is assumed to be a linear function of the stress (or a transformation of the stress), i.e.,

(4)

(7)

(3)

Then we have

Let (5) Then, after some algebra calculations, we have

Thus, maximizing is equivalent to maximizing . In the following sections, we will refer to the optimum criteria base on as the ROC. If has no closed form, then Monte Carlo simulation could be used. The procedure is as follows. 1. Generate from , where , and is large enough. 2. Approximate

by

.

Remark: For a specific ALT problem, the optimal test plans based on the approximated are statistically independent of sample size, which can be easily shown from the Fisher information matrix . For ALTs, the Fisher information matrix has the form

where is statistically independent of the sample size . See Escobar and Meeker [13] for more details. Thus, the sample size in does not affect the choice of the optimal plans. The large-sample approximation can obtain a closed form of

are unknown parameters, and is a given where and decreasing function of stress level . In particular, for the Arrhenius model, and for the inverse power law model. For simplicity, we standardize the stress levels as , so that . Then, , where , and . Notice that is negative, indicating that failure probability increases with . In this setting, we have , and . Suppose that the engineers want to estimate , where is the quantile of the life distribution at the use stress condition. Then, we have

We assume that the true values of , and are 12, , and 0.8, respectively. Furthermore, let . Thus, . When the sample size, the censoring time , and the highest stress level are pre-specified, the planning problem is to find the optimum number of variable levels , the corresponding stress levels , and the proportions of test units allocated to each stress level . The example extends example 20.1 of Meeker and Escobar [2], and has been considered by Nelson and Kielpinski [31], Nelson and Meeker [32], and Zhang and Meeker [19]. We will revisit this problem by using our Bayesian method. A. The Prior Distribution The Bayesian method has many advantages in analyzing data, because valuable prior information can be incorporated into sta-

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TABLE I PRIOR DISTRIBUTIONS

tistical inference. A prior distribution can be elicited from the experts' opinions or historical data. In the -level constant-stress ALT model, are the unknown parameters, and is the parameter of interest. Zhang and Meeker [19] had argued for the statistical independence among , and ; and specified statistically independent priors for them. Following Zhang and Meeker [19], we use the three sets of prior distributions for , and , given in Table I. Continuous uniform distributions are used to express our uncertainty in the values of the parameters. For example, the prior for indicates that the log 1% quantile lifetime at the use stress condition is believed to be between 8 and 8.64 with no preference on any specific value within the interval. Other distributions such as lognormal and Gamma can also be used according to experts' opinions; see Zhang and Meeker [19], and Liu and Tang [33]. The prior means of Prior I to Prior III are around the true values. However, from Prior I to Prior III, we become more uncertain in the values of the parameters. Especially, by using for in Prior III, we believe the lifetime interval of is roughly 17 days to 4.8 years. B. Optimal Plans Based on ROC The criterion (5) is related to the model, the prior, and the test plan . The closed form of cannot be obtained in our model. Thus, a Monte Carlo algorithm is used to generate statistically independent prior samples of size . Because the parameters are assumed to be statistically independent, is large enough to provide precision in the Monte Carlo approximation; see Robert and Casella [34]. For assessing the effects of censoring time, we choose , and hours. Because there is no theoretical upper bound for the optimal number of stress levels , we do a sequence of maximization, increasing until no improvement is obtained. When , only two variables ( and ) need to be optimized. The results are listed in Table II. From Table II, we see that a smaller censoring time leads to higher and more units allocated to , and larger censoring time results in lower and less units allocated to . With the more uncertainty of the prior information, higher and less units allocated to are preferred. Besides, with fixed censoring time, the optimal plans are quite robust for different priors. For example, when

, the optimal changes from 0.791 to 0.868, and the optimal changes from 0.404 to 0.425, although the priors are quite different. When , the optimization involves finding the optimal lower and middle points ( and ) and their respective allocations ( and ). However, for all the three priors, the optimal plans are , which means is optimal. This result is the same as that based on the V-optimality criterion and Bayesian criterion presented by Zhang and Meeker [19]. Thus, compromise test plans are considered. We assume the middle point , and 20% units are allocated to the stress level . Of course one may assume other percentages; see Yuan et al. [20]. Table III provides 20% optimized compromise plans for different censoring time and priors, where we just list the optimal , and , because , and . From Table III, we see that the criterion value from this compromise plan is very close to the optimum plan in Table II. For example, when , and prior I is used, based on a compromise plan, which is only 6.1% less than the optimum. However, three level test plans are more robust, and more practical than the two-level test plans. See Ma and Meeker [35]. Table IV gives the relative efficiencies of optimized compromise plans with different proportions of units constrained to the middle level of the accelerating variable. We can see that more prior information or a larger censoring time will lead to less loss of efficiency. IV. MODIFIED REFERENCE OPTIMALITY CRITERION Because practical applications are usually focused on tests with small or moderate sample sizes, ROC as well as classical criteria are based on large-sample approximation. Thus, the optimal test plans based on these criteria may be misleading. Ma and Meeker [35] also indicated that the large-sample approximation may not be adequate when the sample size is small. Different sample sizes should lead to different test plans. However, the optimal test plans based on ROC are statistically independent of sample size. Thus, we propose a modified ROC in this section. Let denote the variance-covariance matrix of the prior distribution for . Then is the prior precision matrix for . From Berger [36], we have that, when the sample size is large enough, given , the posterior precision matrix for is . Similar to (4), we write as (8) by , and by Then, replacing in (5) leads to a new optimality criterion, called the modified reference optimality criterion (MROC). Thus, the MROC is to find an optimal plan by maximizing (9) at the bottom of the page. Similarly, if has no closed form, Monte Carlo simulation

(9)


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TABLE II OPTIMAL PLANS BASED ON ROC WHEN

TABLE III COMPROMISE TEST PLANS BASED

ON

ROC

TABLE IV RELATIVE EFFICIENCY OF A COMPROMISE ALT PLAN TO ESTIMATE

could be used to approximate . According to the forms of and , we see that the sample size cannot be separated as that in . Thus, the sample size will affect the optimal plans based on . V. NUMERICAL EXAMPLE (CONTINUED) In this section, we will study the numerical example in Section III using MROC. Two scenarios are considered. For the first scenario, we fix the censoring time at hours, then assess the influence of sample size , choosing 18 values from 30 to . Figs. 1 and 2 respectively show the optimal test plans based on MROC when , and compromise plans when . We can see that both the sample size and the prior play important roles in test planning. For example, when the sample size is small, say , the optimal lower stress will be large for prior I and prior II. However, increases slightly as the sample size increases for prior III. Among the three priors, prior III has the least information, and only a few

samples can dominate the prior. Thus, as increases, increases to generate more failures, so that more information can be obtained from the ALT. While the information in prior I is the largest, it needs the most samples to eliminate the influence of prior information. Thus, as increases, decreases first, then increases. When changes, will also change. More samples will be allocated to a smaller, lower stress level to collect more failure information. Thus, increases first, then decreases when increases. For the other scenario, we fix the sample size , then assess the influence of censoring time , choosing 18 values from 300 to hours. Figs. 3 and 4 respectively show the optimal test plans based on MROC when , and compromise plans when . We see that decreases when increases. This relation happens because a larger censoring time leads to more failures even at lower stress levels, and so more information will be obtained from the experiment. is determined by both the prior and the censoring time. For example, increases when increases for prior I; but for priors II and III, the story

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Fig. 1. Optimal test plans based on MROC with

Fig. 2. Compromise test plans based on MROC with


, and different sample sizes when

.

, and different sample sizes.

is different. However, will go to 1 or 0.8 (compromise plan) when censoring time is large enough; that is, all the

units are tested at the use stress condition. As Zhang and Meeker [19] indicated, with diffuse prior information, test plan results


Fig. 3. Optimal test plans based on MROC with

Fig. 4. Compromise test plans based on MROC with

, and different censoring time when

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.

, and different censoring time.

are highly sensitive to the particular form of the diffuse prior. This result explains why the shapes of in Figs. 3 and 4 are

abnormal. Thus, test planning with a highly diffuse prior is not practical.

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Fig. 5. Histogram of the posterior modes of

based on different criteria and priors.

VI. COMPARISON WITH OTHER CRITERIA Once an ALT plan is determined, the engineers may be interested in the possible outcomes (such as an estimate of a quantile lifetime, and its variance) from the test. For evaluation purposes, we can repeatedly simulate possible experimental data for one set of true model parameters, and see how the possible estimation results behave under the specific plan. We will compare ROC and MROC with the V-optimality criterion and the Bayesian criterion (Zhang and Meeker [19]) through simulations. In the following, we consider a two-level constant-stress ALT with a Weibull distribution and type-I censoring as in Section III. The true values of , and are assumed to be 12, , and 0.8, respectively. The quantity of interest is to estimate . We choose , and . When Bayesian methods are used, we use the three sets of priors given in Table V. The means of prior I are the true values. In the second set, the range of is larger than that in the first set, and the mean is 8.5. The last prior set has the least amount of information. However, the true value is in the prior's support. The aim of choosing these priors is to assess the performance of the

criteria when the prior information is different. Based on these priors, we obtain the optimal plans based on ROC, MROC, and the Bayesian criterion of Zhang and Meeker [19]. See Table VI. A. Simulations Based on Fixed Model Parameters A simulation study of the optimal plans in Table VI is conducted. Besides, when using the true values as the planning values, the optimal plans based on the V-optimality criterion are , and . Using the true model parameters, we generate a dataset of 100 units with a censoring time of hours according to the test levels and allocations. A total of 50,000 datasets are simulated. For each simulated dataset, we computed the posterior mode of . Then we calculated the mean, and the square root of the mean square errors (SMSE) based on the 50,000 posterior modes. For the V-optimality criterion, the mean, and SMSE of are 8.356, and 0.479, respectively. Fig. 5 shows the results based on ROC, MROC, and the Bayesian criterion for different priors. We can see two important results. • When prior I is used, MROC performs the best, because the SMSE is the smallest. The Bayesian criterion takes second


Fig. 6. Histogram of the posterior standard deviation of

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based on different criteria and priors.

place. ROC, MROC, and the Bayesian criterion are better than the V-optimality criterion. • When we do not have enough information about the model parameters, the V-optimality criterion is hard to implement because the planning values cannot be obtained easily. Thus, Bayesian methods are preferred, and we can choose priors, e.g., priors II and III are used. In such a case, MROC still performs the best. Remark: Based on the simulation, we recommend Bayesian methods for planning ALT. The reason for this recommendation is that, if the planning values are available, the prior distribution can also be obtained easily, e.g., prior I. As we see, optimal plans based on ROC, MROC, and Bayesian criterion perform much better than that based on the V-optimality criterion. For the three Bayesian criteria, MROC, and Bayesian criterion are recommended, because the optimal plans based on the two criteria depend on the sample size, and the estimates of the model parameters based on the ALT data under such plans perform much better.

TABLE V PRIOR DISTRIBUTIONS USED IN THE SIMULATIONS

B. Simulations Based on Simulated Model Parameters Usually, exact values of the model parameters are not known. Thus, we consider simulations with the uncertainty of the model parameters generating from the specified prior distribution. For each prior set in Table V, we generate 10,000 sets of model parameters from the prior distribution. For each simulated set of model parameters, we simulate 100 experimental datasets according to optimal plans in Table VI. Then the posterior mode of

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TABLE VI OPTIMAL PLANS FOR DIFFERENT BAYESIAN CRITERIA

is computed for each dataset. Based on 100 posterior modes, we calculate the standard deviation. These standard deviations based on different priors are shown in Fig. 6. We can see that ROC and MROC perform better than Bayesian criterion, because the average standard deviation are smaller, which means the optimal plans based on ROC and MROC are more robust. VII. CONCLUSION In this paper, we have presented two Bayesian methods for planning ALT. We use ROC based on the expected Kullback-Leibler divergence between the posterior and the prior distributions of the parameters of interest. A large-sample approximation provides a useful simplification. Because optimal plans based on ROC are statistically independent of sample size, MROC is introduced. We illustrate the proposed methods using a numerical example. Simulations show MROC is better than the Bayesian criterion proposed by Zhang and Meeker [19] when there is little prior information. The Bayesian methods described in this article could also be applied in step-stress ALT planning problems using other lifetime distributions, as well as with accelerated degradation models. ACKNOWLEDGMENT The authors would like to thank the managing editor, Dr. Jason Rupe, the associate editor, Dr. Suk Bae, and the two referees for their helpful comments and suggestions which have led to an improvement of this paper. REFERENCES [1] W. B. Nelson, Accelerated Testing. New York, NY, USA: Wiley, 1990. [2] W. Q. Meeker and L. A. Escobar, Statistical Methods for Reliability Data. New York, NY, USA: Wiley, 1998. [3] H. Chernoff, “Optimal accelerated life designs for estimation,” Technometrics, vol. 4, pp. 381–408, 1962. [4] W. B. Nelson, “A bibliography of accelerated test plans,” IEEE Trans. Rel., vol. 54, pp. 194–197, 2005. [5] W. B. Nelson, “A bibliography of accelerated test plans Part II-references,” IEEE Trans. Rel., vol. 54, pp. 370–373, 2005. [6] F. Pascual, “Accelerated life test planning with independent Weibull competing risks with known shape parameter,” IEEE Trans. Rel., vol. 56, pp. 85–93, 2007. [7] X. Liu and W. Qiu, “Modeling and planning of step-stress accelerated life tests with independent competing risks,” IEEE Trans. Rel., vol. 60, pp. 712–720, 2011. [8] Q. Guan and Y. Tang, “Optimal step-stress test under Type-I censoring for multivariate exponential distribution,” J. Statist. Plan. Inf., vol. 142, pp. 1908–1923, 2012. [9] Z. S. Ye, Y. Hong, and Y. Xie, “How do heterogeneities in operating environment affect field failure predictions and test planning?,” Ann. Appl. Statist., vol. 7, pp. 2249–2271, 2013. [10] C. Li and N. Fard, “Optimum bivariate step-stress accelerated life test for censored data,” IEEE Trans. Rel., vol. 56, pp. 77–84, 2007. [11] H. Xu and H. Fei, “Planning step-stress accelerated life tests with two experimental variables,” IEEE Trans. Rel., vol. 56, pp. 569–579, 2007.

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