Parametric Inference of Incomplete Data With Competing Risks Among Several Groups

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					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 [4] considered optimal
               maximum likelihood estimate/estimator                              life tests for estimation of model parameters based on data from
 -             statistical(ly)                                                    ALT. Meeker and Nelson [12] obtained optimum ALT plans
               stress-response relationship                                       for Weibull and extreme value distributions with censored data.
               step-stress ALT                                                    Nelson and Kielpinski [17] further studied optimum ALT plans
                                                                                  for normal and lognormal life distributions based on censored
                                 NOTATION                                         data. Nelson [14] 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 [13] 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 [3] 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
                                                                                  [21] presented a nonparametric approach for making inferences
                                                                                  for SSALT data. Dorp, Mazzuchi, Fornell and Pollock [6] de-
                       number of failure times observed during time               veloped a Bayes model and studied the inferences of data from
                       interval                                                   SSALT. Xiong [22] 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 [2] discussed the optimal simple step-stress
                                                                                  plan for Khamis-Higgins model. Teng and Yeo [20] 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 [8] gave detailed discus-
  C. Xiong is with the Division of Biostatistics, Washington University, St.
Louis, MO 63141 USA (e-mail:                          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:                  [10] and Lawless [9] provided the general theory and applica-
  Digital Object Identifier 10.1109/TR.2004.824832
                                                                                  tions of lifetime data analysis. Meeker and Escobar [11] 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

[15], [16] 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 [14], [15], Miller and          ously accumulated fraction failed. Also, the change in stress has
Nelson [11], Yin and Sheng [23]). 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 [13] and Yin & Sheng [23], 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     [13] and Yin & Sheng [23].
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.
   During the step-stress life test, test units can be continuously
                                                                                                    for                ,
or intermittently inspected for failure. The latter type of test is
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, [7]). 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 [15]). 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

    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 [19] 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 [1] and Pearson [18].
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 , are given by Section II-B.
  Under      , a straightforward maximization of the likelihood
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 [15] 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
                                                                     statistical power. We choose to base our analysis on exponential
                                                                     failure time in the step-stress test.
                                                                        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
                                                                        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.

                                                                       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

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                                                                                 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-
     [1] A. Agresti, Categorical Data Analysis: John Wiley & Sons, 1990; 1982.   ventive Medicine, University of California, San Diego. He joined SDSU in Au-
     [2] 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.