22 IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 1, MARCH 2004 Analysis of Grouped and Censored Data From Step-Stress Life Test Chengjie Xiong, Member, IEEE, and Ming Ji Abstract—This paper studies statistical analysis of grouped and Cdf of lifetime under stress censored data obtained from a step-stress accelerated life test. We Cdf of a test unit under step-stress testing assume that the stress change times in the step-stress life test are fixed and the lifetimes observed are type I censored. Maximum likelihood estimates and asymptotic confidence intervals for model I. INTRODUCTION parameters are obtained. We provide an asymptotic statistical test for the cumulative exposure model based on the grouped and type I censored data. We also present the optimum test plan for a simple step-stress test when the lifetime under constant stress is assumed exponential. Finally we give an application of our methods by ap- T HE ALTconsists of a variety of test methods for shortening the life of products or hastening the degradation of their performance. The aim of such testing is to quickly obtain data plying our analysis process to a real life data set. The proposed which, properly modeled and analyzed, yield desired informa- statistical methodology is especially useful when intermittent in- tion on product life or performance under normal use. ALT can spection is the only feasible way of checking the status of test units be carried out using constant stress, step-stress, or linearly in- during a step-stress test. creasing stress. The step stress scheme applies stress to test units Index Terms—Asymptotic variance, confidence interval, cumu- in the way that the stress setting of test units will be changed at lative exposure model, exponential distribution, Fisher informa- tion, maximum likelihood, optimum test plan. prespecified times. Generally, a test unit starts at a specified low stress. If the unit does not fail at a specified time, stress on it is raised and held a specified time. Stress is repeatedly increased ACRONYMS1 and held, until the test unit fails or a censoring time is reached. A accelerated life test simple SSALT uses only two stress levels. The problem of mod- cumulative distribution function eling data from ALT and making inferences from such data has confidence interval been studied by many authors. Chernoff  considered optimal maximum likelihood estimate/estimator life tests for estimation of model parameters based on data from - statistical(ly) ALT. Meeker and Nelson  obtained optimum ALT plans stress-response relationship for Weibull and extreme value distributions with censored data. step-stress ALT Nelson and Kielpinski  further studied optimum ALT plans for normal and lognormal life distributions based on censored NOTATION data. Nelson  considered data from SSALT and obtained MLE for the parameters of a Weibull distribution under the in- design stress verse power law using the breakdown time data of an electrical -th test stress, , insulation. Miller and Nelson  studied optimum test plans which minimized the asymptotic variance of the MLE of the number of step-stress test units mean life at a design stress for simple SSALT where all units -th stress change time, . were run to failure. Bai, Kim and Lee  further studied the similar optimum simple SSALT plan for the case where a pre- fixed censoring time, . specified censoring time was involved. Tyoskin and Krivolapov  presented a nonparametric approach for making inferences for SSALT data. Dorp, Mazzuchi, Fornell and Pollock  de- number of failure times observed during time veloped a Bayes model and studied the inferences of data from interval SSALT. Xiong  obtained inferences based on pivotal quanti- a strictly increasing Cdf ties for type II censored exponential data from a simple SSALT. asymptotic variance of Alhadeed and Yang  discussed the optimal simple step-stress plan for Khamis-Higgins model. Teng and Yeo  used the Manuscript received May 9, 2002; revised September 25, 2002. Responsible method of least-square to estimate the life-stress relationship in Editor: Min Xie. step-stress accelerated life tests. Hobbs  gave detailed discus- C. Xiong is with the Division of Biostatistics, Washington University, St. Louis, MO 63141 USA (e-mail: email@example.com). sion on Highly Accelerated Life Test (HALT) and Highly Accel- M. Ji is with the Division of Epidemiology and Biostatistics, San Diego State erated Stress Screens (HASS). Mann, Schafer and Singpurwalla University, San Diego, CA 92182 USA (e-mail: firstname.lastname@example.org).  and Lawless  provided the general theory and applica- Digital Object Identifier 10.1109/TR.2004.824832 tions of lifetime data analysis. Meeker and Escobar  briefly 1The singular and plural of an acronym are always spelled the same. surveyed optimum test plans for different types of ALT. Nelson 0018-9529/04$20.00 © 2004 IEEE XIONG AND JI: ANALYSIS OF GROUPED AND CENSORED DATA FROM STEP-STRESS LIFE TEST 23 ,  provided an extensive and comprehensive source for stress, regardless of how the fraction is accumulated. Moreover, theory and examples for ALT and SSALT. if held at the current stress, survivors will fail according to the The cumulative exposure model has been used by many au- cumulative distribution for that stress but starting at the previ- thors to model data from SSALT (Nelson , , Miller and ously accumulated fraction failed. Also, the change in stress has Nelson , Yin and Sheng ). This model assumes that the no effect on life, only the level of the stress does. As pointed out remaining lifetime of a test unit depends on the current cumula- by Miller & Nelson  and Yin & Sheng , this model has tive fraction failed and current stress—regardless how the frac- many applications in industrial life testing. For a detailed de- tion is accumulated. Moreover, if held at the current stress, sur- scription of cumulative exposure model, see Miller & Nelson vivors will fail according to the cumulative distribution for that  and Yin & Sheng . stress but starting at the previously accumulated fraction failed. Mathematically, the cumulative distribution of time In addition, the change in stress has no effect on life—only the to failure from a step-stress test described above is level of stress does. Nelson [15, Chapter 10] extensively studied cumulative exposure models when the stress is changed at pre- specified times. for During the step-stress life test, test units can be continuously for , or intermittently inspected for failure. The latter type of test is (1) frequently used since it generally requires less testing effort and can be administratively more convenient. In some other cases, intermittent inspection is the only feasible way of checking the where is the equivalent start time at step satisfying status of test units (see, for example, ). The data obtained from intermittent inspections are called grouped data and con- sist of only the number of failures in the inspection intervals. Although many authors studied the inferences and optimum test By Assumption 1, plan of step-stress life test when individual lifetimes are ob- served, we are not aware of any work in the literature which is specifically designed for these problems when only grouped and censored data are available in a step-stress test. The problem we consider in this paper is the statistical inference of model parameters and optimum test plans based on only grouped and for . We further assume: type I censored data obtained from a step-stress life test. The 4. The stress-response relationship (SSR) is a estimation of model parameters and a statistical test for cumu- log-linear function of stress . That is, lative exposure model based on grouped and censored data are discussed in Section II. Section III presents an optimum simple (2) step-stress plan that minimizes the asymptotic variance of the maximum likelihood estimator of the logarithm of SRR. Finally, where and are unknown model parameters Section IV gives a real life application to our analysis procedure. which typically depend on the nature of the product Throughout the paper we make following assumptions: and the test method. Although the specification of 1) At any constant stress , , the Cdf of a looks rather restrictive, it covers some of the test unit lifetime is most important models used in industry, such as the power law model, the Eyring model and the Arrhenius model (Nelson ). With the above spec- ifications, it is straightforward to find that , where the SSR is a function of stress . . Thus, the distribution function of 2) The stresses are applied in the order . step stress failure time is 3) The lifetimes of test units under SSALT are –independent. II. CUMULATIVE EXPOSURE MODEL AND ESTIMATION for A. Cumulative Exposure Model We denote the cumulative life distribution function at time for , for a test unit at each constant stress , , by . For the step-stress life test, there is a probability distribution (3) of time to failure on test. Data from this distribution are observed during the test. The cumulative exposure model of Similar results as in Section II-A still hold when time assumes that the remaining life of a test unit depends more general lifetime distributions such as only on the current cumulative fraction failed and the current are used. 24 IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 1, MARCH 2004 B. MLE and Confidence Interval Estimate The expected second partial derivatives of the log likelihood We first consider the case when all test units are subject to the function at are same censoring time and the same stress-change patterns with the same set of stresses and the same stress change-times. We assume that data obtained from such step-stress test described in 2.1 are grouped and type I censored. More specifically, we as- sume that the intermittent inspection times during the step-stress test coincide with the stress-change times and the censoring time. Suppose that test units begin at low stress . If the unit does not fail at a specified time , stress on it is raised to a higher stress . The stress is repeatedly increased and held in this fashion, until the test unit fails or the fixed censoring time ( ) is reached. Assume that units fail during Let be the by diagonal matrix with , the inspection time interval , . , as its diagonal elements. Let , ( ). To simplify the notations, we denote, for , , be the by 2 Jacobian matrix , of with respect to , i.e., for . Because Let be the step-stress lifetime under SSALT and , the expected Fisher information for . The cu- matrix , , 2, is given by mulative exposure model (3) implies that for , Let be the MLE of obtained from solving (5). can be consistently estimated by , where , The likelihood function based on data vector , 2, is obtained by replacing in by its MLE ( ) is (up to a constant): . Based on the asymptotic normality of with estimated (4) covariance matrix , we can set up the asymptotic confidence intervals for , , the SRR of lifetime at design stress where is specified by SRR (2) and . Thus, the log , and the reliability function at design likelihood function is a function of unknown parameter and : stress . Let , , 2, be the estimated asymptotic covariance matrix of . It is straightforward that an asymptotic CI for is , and an asymptotic CI for is , where is the point of the standard normal To find the maximum likelihood estimators (MLE) for and , distribution. The asymptotic variance for we maximize over and . The maximization of is given by requires the solution to the system: An asymptotic CI for is . Finally, because is a strictly increasing function of , an asymptotic CI for at a given time is . (5) When the step-stress testing is a simple SSALT, i.e., when where is the probability density function of , there exist closed form MLE for and . The MLE of . Generally, the solution of (5) requires a numerical method and solves such as Newton-Raphson. Seo and Yum  proposed several approximate ML estimators and compared with the MLE by a Monte-Carlo simulation when the lifetime distribution is as- sumed exponential. XIONG AND JI: ANALYSIS OF GROUPED AND CENSORED DATA FROM STEP-STRESS LIFE TEST 25 The solutions are Therefore, an asymptotic likelihood ratio test of significance level rejects if where is the upper percentile of the dis- where is the inverse function of . tribution with degrees of freedom. In the more general situation when different test units are Because subject to different censoring times and different stress-change patterns with different sets of stresses and even different stress change-times, the likelihood function for each test unit can be given by (4) for each test unit with . The likeli- hood function for a sample of size test units is the product of is stochastically equivalent to all individual likelihood functions by Assumption 3. The MLE of and can be obtained by maximizing this likelihood func- tion using a numerical method such as Newton-Raphson. Al- though the lifetime distributions of test units are independent, but they are not identical. The asymptotic confidence interval es- timates for various model parameters given above, however, are another asymptotically equivalent test of significance level still valid when the Fisher information matrix is replaced is the well known goodness-of-fit test which by the average information matrix to take into account of the rejects if difference in the lifetime distributions. The detailed theoretical justification can be found in Cox and Hinkley [5, Chapter 9]. C. A Statistical Test for Cumulative Exposure Model When The mathematical verification of these tests can be found in We only consider the case when all test units are subject to Agresti  and Pearson . the same censoring time and the same stress-change patterns with the same set of stresses and the same stress change-times III. OPTIMUM TEST PLANS in this section. Let be the step-stress lifetime under SSALT We next discuss the optimum test plan for choosing in and for . As pointed a particular case. Suppose that test units are tested under a out in Section II.B, if the cumulative exposure model (3) is the simple SSALT which uses the same censoring time and the correct model for , then for , same stress-change patterns with same set of stresses ( ) and same stress change-times . Assume that censoring time is given. Suppose that the lifetimes at constant stresses and are exponential with means and , respectively, where A statistical test for the cumulative exposure model can be ob- , , 2. Thus, for tained by testing the null hypothesis . As in Section II, we assume that units failed during , against the alternative : the inspection interval , , 2, 3, . there is no constraint on . When grouped and The expected Fisher information matrix given in Sec- type I censored data are available from test units, the likeli- tion II-B is now simplified as hood function of , is The MLE of , under are given by where where , are given by Section II-B. Under , a straightforward maximization of the likelihood and function gives the MLE of , as 26 IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 1, MARCH 2004 Because TABLE I STEP-STRESS PATTERN AFTER STEP 4 on a specimen ( ) is the natural logarithm of the ratio between the voltage and the insulation thickness. we find that the asymptotic variance of , where The original data were observed as exact failure times. In , are given by Section II-B, is given by order to demonstrate our estimation process, we grouped the failure time data according to the intervals formed by consecu- tive stress-change times. There were five censored failure times in the data set. The grouped and censored data are summarized in the following Table II. Because of the different thickness for different specimens and different voltages at different steps in the testing, each spec- imen has its own stress pattern and censoring time. A likelihood function can be written for each specimen according to (4) in where is the amount of stress ex- Section II-B. The likelihood function for the whole sample is trapolation. Our optimum criterion is to find the optimum stress the product of all these individual likelihood functions in the change time such that the is sample. By using exact failure times instead of grouped count minimized. Because data, Nelson  fitted the Weibull model to the step-stress data and presented the MLE of model parameters on Nelson [15, Table 2.2 of Chapter 10]. The MLE estimate for the Weibull shape parameter is 0.755 97 with an asymptotic 95% confidence the minimum of is attained at some between interval as from 0.18 to 1.33. Because the confidence interval 0 and based on the fact that is a continuous contains the value 1, there is no significant evidence against the function of when is between 0 and . The minimization hypothesis that the failure times of these specimens follow an of over solves the equation exponential distribution when tested against the larger family of Weibull distributions based on the standard normal test at a significance level of 5%, although it could be due to the lack of (6) statistical power. We choose to base our analysis on exponential failure time in the step-stress test. where The analysis provided by Nelson [15, Chapter 10] assumed that the SRR is an inverse power law model and used the stress as the ratio between the voltage and the insulation thickness. In our set-up of log-linear SRR, the inverse power law model trans- lates into , where is the mean of the exponential distribution at stress , and stress now becomes the natural logarithm of the ratio between the voltage and the insulation thickness. The design stress is at 400 V/mil, there- fore, . Table III presents the MLE and CI of various parameters. To demonstrate how to find the optimum design under The uniqueness of the solution to (6) is given in the Appendix . a simple step-stress life test, we assume that the voltage In general, the solution to (6) is not in a closed form and there- levels from step 5 (26 kilovolts) and step 6 (28.5 kilovolts) fore requires numerical method such as Newton-Raphson. in the step-stress pattern are used to conduct a future simple step-stress life test. We also assume that the test uses the cable IV. EXAMPLE insulation with thickness equal to 30 mils (one of the four We use a real data set reported in Nelson [15, Table 2.1 of types used in the study). Therefore, the two stress levels for Chapter 10] to demonstrate our estimation and testing proce- this simple step-stress test is dure. The data set was obtained from a step-stress test of cryo- and . We still use design stress genic cable insulation. Each specimen was first stressed for 10 . The amount of stress extrapolation is . minutes each at voltages of 5 kV, 10 kV, 15 kV, and 20 kV be- We assume that the simple step-stress test has to stop after fore it went into step 5. Thereafter one group of specimens was 1800 minutes (censoring time ). Using the MLE of and stressed 15 minutes at each step given in Table I. obtained from the grouped and censored data in Table II, we nu- Three other groups were held 60, 240, and 960 minutes at merically solved (6) and found that the optimum stress-change each step. Thus there were four step-stress patterns. The stress time is after 1191.6 minutes ( ) of testing under stress . XIONG AND JI: ANALYSIS OF GROUPED AND CENSORED DATA FROM STEP-STRESS LIFE TEST 27 TABLE II COUNT DATA TABLE III a SSALT is to study the statistical analysis and the optimum PARAMETER ESTIMATES design when both the stress change times and the censoring time are random variables; such as the order statistics at the current stress levels, and when only these order statistics (stress change times and type II censoring time) are observed during the testing. APPENDIX The proof of the uniqueness of the solution to (6) V. DISCUSSION Taking the second derivative of with respect We give some final discussions about the results in this to , we obtain paper. First, we have presented a statistical methodology to analyze grouped and censored data obtained from a step-stress accelerated life test. This methodology will be especially useful when intermittent inspection is the only feasible way of checking the status of test units during a step-stress test. Second, although the cumulative exposure model has been a popular choice for the analysis of step-stress life test data, its validity in various situations remains to be tested. We have pre- sented a goodness-of-fit test for this model based on grouped and censored data under certain assumptions. Third, for a simple step-stress test, we have derived the optimum choice for the stress change time which minimizes under an exponential distribution. This optimum design is particularly important at the designing stage of the test as it gives a guide to experimenters about when the intermittent inspection and the stress change should be carried out during where and are two functions defined by the test. On the other hand, because the results of the paper are based on the lifetime Cdf of the form for , our methods have some limitations. A more general lifetime Cdf would be , which contains important distributions including Weibull. Although parts of the results such as those in Section II-A still hold when this more general family of Cdf is used, other parts do not when and is unknown. The results on the optimum design are based on the exponential lifetime distribution; its sensitivity on the assumption of exponential distribution is not addressed in the paper. These questions are also the direction of future research Because in this area. Another important and interesting variation for future research associated with grouped and censored data from 28 IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 1, MARCH 2004 it follows that for any ,  D. S. Bai, M. S. Kim, and S. H. Lee, “Optimum simple step-stress ac- celerated life tests with censoring,” IEEE Trans. Reliability, vol. 38, pp. 528–532, Dec 1989.  H. Chernoff, “Optimal accelerated life designs for estimation,” Techno- metrics, vol. 4, pp. 381–408, 1962. Therefore  D. R. Cox and D. V. Hinkley, Theoretical Statistics. London: Chapman & Hall, 1974.  J. R. Dorp, T. A. Mazzuchi, G. E. Fornell, and L. R. Pollock, “A Bayes approach to step-stress accelerated life test,” IEEE Trans. Reliability, vol. 45, pp. 491–498, Sep 1996. Because for any ,  S. Ehrenfeld, “Some experimental design problems in attribute life testing,” Journal of American Statistical Association, vol. 57, pp. 668–679, 1962.  G. K. Hobbs, Accelerated Reliability Engineering: John Wiley & Sons, 2000.  J. F. Lawless, Statistical Models and Methods for Lifetime Data: John Wiley & Sons, 1982.  N. R. Mann, R. E. Schafer, and N. D. Singpurwalla, Methods for Statis- tical Analysis of Reliability and Life Data: John Wiley & Sons, 1974. and  W. Q. Meeker and L. A. Escobar, “A review of recent research and cur- rent issues in accelerated testing,” International Statistical Review, vol. 61, pp. 147–168, 1993.  W. Q. Meeker and W. B. Nelson, “Optimum accelerated life tests for Weibull and extreme value distributions and censored data,” IEEE Trans. Reliability, vol. R-24, pp. 321–332, Dec 1975.  R. W. Miller and W. B. Nelson, “Optimum simple step-stress plans for accelerated life testing,” IEEE Trans. Reliability, vol. R-32, pp. 59–65, Apr 1983.  W. B. Nelson, “Accelerated life testing – Step-stress models and data analysis,” IEEE Trans. Reliability, vol. R-29, pp. 103–108, Jun 1980.  , Accelerated Life Testing, Statistical Models, Test Plans, and Data Analysis: John Wiley & Sons, 1990.  , Applied Life Data Analysis: John Wiley & Sons, 1982.  W. B. Nelson and T. J. Kielpinski, “Theory for optimum censored ac- celerated life tests for normal and lognormal life distributions,” Techno- metrics, vol. 18, pp. 105–114, 1976.  K. Pearson, “On a criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can it follows that for any , by the fact be reasonably supposed to have arisen from random sampling,” Philos. . Therefore, for any , Mag., ser. 5, vol. 50, pp. 157–175, 1900.  S. K. Seo and B. J. Yum, “Estimation methods for the mean of the ex- ponential distribution based on grouped & censored data,” IEEE Trans. Reliability, vol. 42, pp. 87–96, Mar 1993.  S. L. Teng and K. P. Yeo, “A least-square approach to analyzing life- stress relationship in step-stress accelerated life tests,” IEEE Trans. Re- which then implies that for any , liability, vol. 51, pp. 177–182, Jun 2002.  O. I. Tyoskin and S. Y. Krivolapov, “Nonparametric model for step-stress accelerated life test,” IEEE Trans. Reliability, vol. 45, pp. 346–350, Jun 1996. It follows that  C. Xiong, “Inferences on a simple step-stress model with type II cen- sored exponential data,” IEEE Trans. Reliability, vol. 47, pp. 142–146, Jun 1998.  X. K. Yin and B. Z. Sheng, “Some aspects of accelerated life testing by progressive stress,” IEEE Trans. Reliability, vol. R-36, pp. 150–155, Apr 1987. and Chengjie Xiong received his BS (1983) in Mathematics from Xiangtan University. He taught Mathematics at Zhangjiako Telecommunication College The uniqueness of the solution to (6) follows by the strict con- (1983–1986). He received his MS (1989) in Mathematics from Peking Uni- versity, PhD (1997) in Statistics from Kansas State University. He has joined vexity of with respect to . Washington University in St. Louis since 2001. He is a member of the ASA. His research interests are in survival analysis and reliability, mixed models and ACKNOWLEDGMENT order restricted statistical inferences. The authors are greatly indebted to the editor and the three anonymous referees for their invaluable comments and sugges- Ming Ji received his BS in Mathematics and MS in Control Theory from East tions. China Normal University in 1985 and 1988, respectively. He received a MS in Mathematics from Kansas State University in 1995 and then a PhD in Sta- REFERENCES tistics from University of California, Davis in 1999. From 1999 to 2001, he worked as an Assistant Adjunct Professor in the Department of Family and Pre-  A. Agresti, Categorical Data Analysis: John Wiley & Sons, 1990; 1982. ventive Medicine, University of California, San Diego. He joined SDSU in Au-  A. A. Alhadeed and S. S. Yang, “Optimal simple step-stress plan for gust, 2001. He is member of the ASA and International Biometrics Society. His Khamis-Higgins model,” IEEE Trans. Reliability, vol. 51, pp. 212–215, research interest includes: reliability, longitudinal data analysis, missing data, Jun 2002. causal inference, medical screening tests and microarray data analysis.
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