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					IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 1, MARCH 2004                                                                                        11




           Parametric Inference of Incomplete Data With
             Competing Risks Among Several Groups
                                                       Chanseok Park and K. B. Kulasekera



    Abstract—We develop parametric inferential methods for                                              I. INTRODUCTION
the competing risks problem where data arise due to multiple
causes of failure in several groups with censoring and possibly
missing causes. We provide the general likelihood method and the
closed-form maximum-likelihood estimators for the exponential
                                                                               I   N MANY engineering and medical studies, lifetime distri-
                                                                                   butions of items or individuals are of interest. Experimenters
                                                                               use various indexes associated with lifetime distributions to
model. Parametric tests are given for comparing different causes               evaluate systems. Typically, lifetime measurements are taken
and groups. An extensive numerical and graphical investigation
is presented to substantiate the proposed methods. A real-data                 from the relevant population(s) and statistical inferences on the
example is illustrated.                                                        corresponding indexes are conducted. Often the items or indi-
   Index Terms—Censored data, competing risks, exponential dis-
                                                                               viduals fail due to more than one failure mechanism, commonly
tribution, maximum likelihood, missing cause.                                  referred to as competing risks. In this setting, usually the cause
                                                                               of failure is known when the lifetime is observed. Furthermore,
                                                                               the items or individuals may also be grouped according to some
                                ACRONYMS1                                      criteria so that one has observations from multiple populations
               implies: statistical(ly)                                        where each observation is due to one of the failure mechanisms.
cdf            cumulative distribution function                                For example, one may wish to study the effect of the brand of
pdf            probability density function                                    air-conditioning systems which can fail either due to leaks of
CIF            cumulative incidence function                                   refrigerant or wear of drive belts. In these situations, an item
CSHF           cause-specific hazard function                                  of interest would be to know whether there are significant
MLE            maximum-likelihood estimator                                    differences due to brand(s) and the cause of failure. A typical
MSE            mean square error                                               lifetime data analysis problem of the above type is further
MVN            multivariate -normally distributed                              complicated due to possible censoring and unknown cause
SRE            simulated relative efficiency                                   of failure. In the example above, the experimenter fails to
                                                                               observe the exact lifetime of an air-conditioning system if the
                                 NOTATION                                      building or facility is destroyed or renovated thus discarding
               lifetime of the th subject in the th group due to the           all functioning systems. In such a situation one observes a
                 th cause                                                      right-censored lifetime value. In some situations, the exact
               censoring indicator variable                                    cause of failure may not be observed although the lifetime is
               vector parameters of the distribution of                        observed, thus masking the cause. We formally formulate the
               parameter matrix of                                             problem in the following required notation.
                                                                                  The traditional approach when dealing with competing risks
               pdf of
                                                                               is to consider the hypothetical latent lifetimes corresponding to
               cdf of
                                                                               each cause in the absence of the others [1]. Therefore, a subject
               survival function of
                                                                               in the th group,                  , is exposed to several potential
               hazard function of                                              causes of failure. Let there be a finite number of -independent
               likelihood function without missing causes                      causes of failure indexed by                 . Let          denote the
               likelihood function with missing causes                         latent lifetime of the th subject in the th group due to the th
                                                                               cause, where                   and                    . It is assumed
               Indicator function                                              that         are -independent for all           and are identically
               Fisher information matrix                                       distributed for all for a given          . The corresponding cdf,
                                                                               pdf, survival function, and hazard function of             are denoted
  Manuscript received March 7, 2003; revised June 22, 2003 and November        in general by                   ,                  ,                  ,
20, 2003. The work of C. Park was supported in part by Clemson RGC award.
The work of K. B. Kulasekera was supported in part by NIH grant R01 CA         and                 , respectively, where          is a vector of real
92504-02.                                                                      valued parameters, one for each           . Then the observed life-
  The authors are with the Department of Mathematical Sciences, Clemson
University, Clemson, SC 29634 USA (e-mail: cspark@ces.clemson.edu;
                                                                               time of the th subject in the th group is given by the random
kk@ces.clemson.edu).                                                           variable
  Digital Object Identifier 10.1109/TR.2003.821946
     1   The singular and plural of an acronym are always spelled the same.

                                                               0018-9529/04$20.00 © 2004 IEEE
12                                                                             IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 1, MARCH 2004



  Typically, in reliability analysis problems, complete obser-         statistics for testing the above hypotheses. Our methods are the
vation of        may not be possible due to various censoring          first to address the competing risks failure data in several groups
schemes which are inherent in data collection processes. In this       with censoring and missing causes together.
work, it is further assumed that each     can be right-censored           When it is inappropriate or undesirable to assume a specific
by censoring times        which are -independent of lifetimes          parametric form and -independence in the competing risks
      for all and . Thus, one only observes                    ,       problem, one can use distribution-free methods. There is ex-
where                              , and      are censoring in-        tensive literature on nonparametric estimation and testing. Gray
dicator variables defined as                                           [8] proposed a class of -sample tests for comparing the CIF
                                                                       between groups for a given cause. Aly [9] proposed tests for
                                                                       testing the equality of two CSHF with censoring. Sun and Ti-
                                if                                     wari [10] provided a simple method of testing for comparing
                                                                (1)
                                if               .                     the CIF. Lam [11] tests whether the CSHF are the same when
                                                                       there are -dependence between multiple causes. Recently, Ku-
Our objectives are:                                                    lathinal and Gasbarra [12] extended the work of Lindkvist and
     1) Estimate the vector parameters        of the distribution      Belyaev [13] and Luo and Turnbull [14] by looking at compar-
        of      .                                                      ison of groups.
     2) Perform the following tests of hypotheses:                        We provide the general likelihood method in Section II.
           •      :                for all    ,                        Parameter estimation, asymptotic distributions and hypothesis
           •      :               for all for a given , and            testing for the exponential model are handled in Section III.
           •      :               for all for a given .                A real-data example is illustrated in Section IV and some
                                                                       simulation results in Section V.
Here, the alternative hypothesis in each case is taken to be the
negation of the statement given in the null hypothesis. The first
null states that there is no cause or group effect; the second stip-                   II. LIKELIHOOD CONSTRUCTION
ulates that there is no group effect for a given cause; and the          In this section, we develop the likelihood functions of the
third stipulates that there is no cause effect in a given group.       parameters of the underlying distributions       . Let     be
   The analysis of exponential data with two causes in a single        the indicator function of an event . For convenience, denote
group was studied by Cox [2]; and was extended to multiple                                    , and
causes by Herman and Patell [3]. The parametric estimation
problem for the case of a single group with two causes and
possible missing causes but without censoring has been dis-
cussed by Miyakawa [4]. Kundu and Basu [5] extended this                                           .
                                                                                                   .     ..       .
                                                                                                                  .
work to provide the approximate and asymptotic properties of                                       .          .   .
the parameter estimators, -confidence intervals, and bootstrap
  -confidence bounds. They provided the closed-form MLE for
the exponential case and constructed likelihood equations for          The likelihood function of the censored sample is
the Weibull case. However, they did not consider censoring.
Also, although they stated that their solutions extend to the mul-
tiple cause case, no explicit expressions were provided.
   An alternative to the traditional latent lifetime framework is
the parametric mixture-model approach adopted by Larson and
Dinse [6]. An attractive feature of this approach is that it re-
laxes the -independence assumption of the latent lifetime ap-
proach. However, the drawback is that finding the MLE in the
parametric mixture model is quite difficult; and requires intense
and often unstable numerical calculations. Recently, Maller and
Zhou [7] extended this model to allow the possibility that the
competing risks considered may not be exhaustive.
   In this article, we give the closed-form MLE for the exponen-
tial model with multiple groups, multiple causes, censored data,
and missing causes together. The proposed estimators include
the estimators given by Kundu and Basu [5] as a special case.
For the Weibull model, when the shape parameter is common
for all groups and causes, using the proposed estimators, one
can easily obtain the closed-form MLE for scale parameters
after the common shape parameter is estimated by the likeli-
hood method. These lead to the construction of reasonable test
PARK AND KULASEKERA: PARAMETRIC INFERENCE OF INCOMPLETE DATA                                                                        13



where                                                                A. Closed-Form MLE
                                                                       We assume that        is an exponential random variable with
                                                                     the rate parameter                . The pdf of       is



Maximizing          with respect to is equivalent to separately
                                                                     Then the pdf of      is obtained by
maximizing              with respect to        for
and                . Thus we have reduced the joint maximum-
likelihood problem for the parameters of          distributions to
        separate estimation problems for the parameters of each
distribution.                                                        where                      . The likelihood function of        is
   When lifetimes of subjects are observed with causes of failure
being unknown (missing), we have to add the sub-density func-
tions of the time     for each th cause into the likelihood func-
tion. The CIF for each th cause is



Then we have the corresponding sub-density function

                                                                     where                           . With                      , the
                                                                     log-likelihood function becomes

Therefore the pdf of       is given by




   Denote              if cause of failure is unknown. Then the
likelihood is given by
                                                                       Define                       for                        . Then
                                                                     we have




where




and                                                                  Because                    , we have



Maximizing          with respect to      is equivalent to maxi-
mizing           with respect to        for               . Thus                                                                   (2)
we have reduced the joint maximum-likelihood problem for the
parameters of    distributions to     separate estimation prob-
lems for the parameters of each distribution.                          The MLE of         are obtained by solving the following:

                  III. EXPONENTIAL MODEL
  In this section, we provide the closed-form MLE for the ex-                                                                      (3)
ponential model and the asymptotic distribution of the proposed      From the above, we have
MLE. Parametric tests are also given for comparing different
causes and groups.
14                                                                                           IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 1, MARCH 2004




Fig. 1. Empirical CIF with 95% s -confidence interval. (a) Cause 1 in Group I. (b) Cause 2 in Group I. (c) Cause 1 in Group II. (d) Cause 2 in Group II. (e) Cause 1
in Group III. (f) Cause 2 in Group III.


where                           . It follows that                                  B. Asymptotic Distributions of the MLE
                                                                                     We provide the asymptotic distribution of the proposed
                                                                                   MLE for the exponential model. Using this result, the
                                                                                   asymptotic -confidence intervals can be obtained. First we
Substituting this into (3), we have the following closed-form                      obtain the Fisher information matrix of the parameters
MLE                                                                                                    which is denoted by
PARK AND KULASEKERA: PARAMETRIC INFERENCE OF INCOMPLETE DATA                                                                                   15



for                 and                            . Then we obtain                                       TABLE I
                                                                                                 PARAMETER ESTIMATES WITHOUT
                                                                                                       ANY RESTRICTIONS


                                                              and

                                                              and
                                                              for all       .
                                                                                                            TABLE II
Because      has a binomial distribution                                                   PARAMETER ESTIMATES UNDER   H ,H   , AND   H
for              , we have                                           . Then
we have

                                                           and
                                                           and
                                                           for all      .
Using this, we have the following partitioned Fisher information
matrix of the parameters
                                                                                                            TABLE III
                                                                                               TEST STATISTICS AND CRITICAL VALUES



                                              ..
                                                   .



                                     ..                                         where                      . Here,
                                          .


where                                                     and
                                                                                where                                   . In practice, we usually
                . Note that                            whenever             .
Hence we have                                                                   estimate           by                  . Using this, we can find
                                                                                approximate -confidence intervals for    by taking             to
                                                                                be MVN with the mean and the covariance matrix                   .
                                          ..
                                               .                                C. Hypothesis Testing
                                                                                  Here, we provide a method for hypothesis testing based on
The matrix           is diagonally partitioned, so its inverse is               the maximum-likelihood method. A likelihood ratio statistic by
given by                                                                        Neyman and Pearson is



where                      . Using Theorem 8.3.3 in Graybill                    It is well-known that under the null hypothesis
[15] and doing some algebra, we have

                                                                                where the degrees of freedom of the limiting distribution is the
                                                                                difference between the number of free parameters under the null
                                                                                hypothesis      and the number of free parameters under the al-
This inverse always exists unless          ( i.e.,, each subject                ternative    . We develop hypothesis tests for each of the fol-
in the th group is observed only with censoring or missing                      lowing:
causes) but this condition is extremely unrealistic in practice.                    •      :                for all     ,
Then we have the following asymptotic distribution of                               •     :               for all for a given , and
                  from the property of MLE                                          •     :                for all for a given .
                                                                                Here, the alternative hypothesis in each case is taken to be the
                                                                                negation of the statement given in the null hypothesis.
16                                                                         IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 1, MARCH 2004



                                                            TABLE IV
                                       FAILURE TIMES AND CAUSES FOR 139 ELECTRICAL APPLIANCES




  Here, we provide the MLE for the exponential model under           The following test statistics , ,         are used to test the
each of the above null hypotheses.                                 null hypotheses    ,      ,    , respectively:
     • Under      :
          We have restrictions on scale parameters
       for all      . Using these and (2), we obtain the MLE of                                                       and
             :



                                                                   where                   ,                   ,                    , and
                                                                                         . Here, , , and           have asymptotic
                                                                   distributions with the degrees of freedom                 ,           ,
     • Under      :
                                                                   and               , respectively.
          We have restrictions on scale parameters
       for all for a given . Using these and (2), we obtain the       It is worth noting that        is obtained by taking the average of
       MLE of        :                                                      with respect to ;        is obtained by taking the weighted
                                                                   harmonic mean of               with respect to with the weights
                                                                                                                              ; and
                                                                   is obtained by taking the average of           with respect to .

                                                                                                IV. EXAMPLE
     • Under       :                                                 The data in this example were first presented by Nelson [16]
          We have restrictions on scale parameters                 and have since then been used frequently for illustration in com-
       for all for a given . Using these and (2), we obtain the    peting risks literature including Crowder [17] and Lawless [18].
       MLE of          :                                           The data consist of failure or censoring times for 139 appliances
                                                                   (36 in Group I, 51 in Group II, and 52 in Group III) subjected to
                                                                   a manual lifetime test. Failures were classified into 18 different
                                                                   modes. Among the 67 observed failures in Groups I, II, and III,
                                                                   only mode 11 appears more than twice in all three groups. We
PARK AND KULASEKERA: PARAMETRIC INFERENCE OF INCOMPLETE DATA                                                                               17



                                                                   TABLE V
                                                    THE BIAS AND THE MSE OF THE ESTIMATORS




Fig. 2. (a) S vs.  (8); (b) S vs.  (6); and (c) S vs.  (6).


shall focus on failure mode 11 by coding the causes as follows:         able. We also can consider some other methods for this model
        (mode 11),         (other modes), and          (censored).      validity. For formal goodness-of-fit tests for exponentiality, the
We provide the data in Table IV.                                        reader is referred to Spurrier [20] and Akritas [21].
   For the exponential distribution, the cumulative hazard func-           We estimated the rate parameters under the exponential
tion is              . Therefore, when the empirical cumulative         model without any restrictions, and under the null hypotheses
hazard function         is plotted against , the resultant graph            ,        ,      . The estimates without any restrictions are
should give an approximately straight line passing through the          shown in Table I; and the estimates under the null hypotheses
origin for an exponential model. This is a common graphical                 ,      ,       are shown in Table II. We also tested the null hy-
technique for checking exponentiality. However, with competing          potheses         ,     ,    . Table III shows these results denoted
risks, we have to check the validity of a specific model for each       by , ,              under      ,     ,     , respectively; and also
cause. One way to do this is to compare the empirical CIF with the      provides asymptotic critical values at the -significance level
parametric CIF, where the CIF of the exponential model is given         of             . The results indicate that the null hypotheses
by                                                                      and       should be rejected at the -significance level of
                                                                             . There is evidence of a cause effect in at least one group.

Fig. 1 shows the empirical CIF based on Aalen [19] with point-                               V. SIMULATION RESULTS
wise approximate 95% -confidence limits                                    To evaluate the performance of the proposed estimators and
                                                                        tests, an extensive simulation study was carried out using lan-
                                                                        guage [22].
                                                                           We generated the data from -independent exponential dis-
The parametric CIF with the MLE are also superimposed on the            tributions. Let        denote the lifetime of the th subject in
plot and it is seen that they lie reasonably well within -confi-        the th group due to the th cause according to the exponential
dence bands. This indicates that an exponential model is reason-        distribution with the parameter         . Then              are
18                                                                               IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 1, MARCH 2004




Fig. 3. (a) S vs.  (8); (b) S vs.  (6); and (c) S vs.  (6).




Fig. 4. (a) S vs.  (8); (b) S vs.  (6); and (c) S vs.  (6).

                                                                  TABLE VI
                                                          OBSERVED LEVELS OF THE TESTS




                                                                 TABLE VII
                                                         OBSERVED POWERS OF THE TESTS




given by                                         , where                 A. Parameter Estimation
are censoring times and      are censoring indicator variables              We considered the case of              and            using the
defined as (1). Denote                               ,                   following parameter matrices
                    , and                      . We censored
the lifetimes using an exponential sample        with the rate
under consideration.
PARK AND KULASEKERA: PARAMETRIC INFERENCE OF INCOMPLETE DATA                                                                                                    19




Fig. 5.   The observed powers P (), P ( ), P ( ) using the test statistics S , S , S , respectively. 3 ( ) is used for (a) and (b); 3 ( ) for (c) and (d); and
3 () for (e) and (f).


We generated random samples of sizes                         for                      In Table V, we have presented the bias and the MSE of the
the first and second groups; and censored the lifetimes using an                   estimators of       with 5000 iterations. To help compare the
exponential sample        with the rate 2.5. Then we masked the                    MSE, we also find the SRE which is defined as
causes of failure with different missing percentages of causes,
0% (no missing causes), 10%, 20%, and 30%.
20                                                                                     IEEE TRANSACTIONS ON RELIABILITY, VOL. 53, NO. 1, MARCH 2004



To examine the impact of missing causes on the parameter                      Therefore, this statistic    does not change when the lifetimes
estimation, we compare the estimates with the complete data                   with missing causes are deleted. Table VII displays the ob-
set without missing causes (denoted by ), the data set with                   served powers of the test statistics ,        ,    using        ,
missing causes      , and the data set after deleting the lifetimes                  ,        with            , respectively. The table shows
with missing causes         . The results in Table V indicate                 that deleting the data with missing causes results in a loss of
that the proposed method outperforms the ad hoc method of                     power. To make the above observations graphically explicit, we
deleting the lifetimes with missing causes.                                   can change the value of and plot the observed powers of the
                                                                              three data sets—the complete data set (0% missing), the data
B. Hypothesis Tests                                                           set including missing causes, and the data set after deleting the
                                                                              data with missing causes—on the same graph. This graph is
   We considered the case of                  and               using the
                                                                              displayed in Fig. 5 over a fine grid of from 0.6 to 1.5.
following parameter matrices                with              .
   1)                                                     , where                                       ACKNOWLEDGMENT
                                                                                 Dr. Park is grateful to Dr. M. Leeds for his help and encour-
                                                                              agement throughout this research. The authors thank the ref-
                                                                              erees for their useful suggestions.
     2)                                                   , where
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With          , only      has an asymptotic          distribution.                  1998.
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PARK AND KULASEKERA: PARAMETRIC INFERENCE OF INCOMPLETE DATA                                                                                                      21



Chanseok Park is an Assistant Professor of Mathematical Sciences at Clemson           K. B. Kulasekera is a Professor of Mathematical Sciences at Clemson Univer-
University, Clemson, SC. He received his B.S. in Mechanical Engineering from          sity. He received his B.S. in 1979 from the University of Sri Lanka; his M.A.
Seoul National University; his M.A. in Mathematics from the University of             in Statistics from the University of New Brunswick; and his Ph.D. in Statistics
Texas at Austin; and his Ph.D. in Statistics in 2000 from the Pennsylvania State      from the University of Nebraska, Lincoln, NE. His research interests include
University. His research interests include survival analysis, competing risks, sta-   survival analysis, nonparametric regression, and multivariate methods.
tistical inference using quadratic inference function, robust inference, and sta-
tistical computing and simulation.