Confidence Intervals for Parallel Systems with Covariates
Ayman Baklizi, Isa Daud and Noor Akma IbrahimDepartment of Mathematics
Faculty of Science and Environmental StudiesUniversiti Pertanian Malaysia
43400 UPM, Serdang, Selangor, Malaysia
Received: 10 April 1996
ABSTRAKLazimnya bagi model regresi dengan data tertapis selang keyakinannya yangtepat tidak mudah disingkap kembali dan dengan demikian hampiran kepadaselang ini diperlukan. Hampiran selang keyakinan ini lazimnya dapat dibinadengan menggunakan kaedah yang berdasarkan taburan normal asimptot daripenganggar kebolehjadian maksimum. Selang-selang seperti ini mudah dikiradan sering dimanfaatkan oleh kebanyakan pakej statistik berkomputer. Walaubagaimanapun selang-selang ini mempunyai beberapa kecacatan. Bagi saizsampel kecil atau sederhana kedl, selang-selang ini bersifat antikonservatifserta mempunyai kebarangkalian hujung atas dan hujung bawah yang tidaksimetri. Oleh yang demikian kaedah alternatif disarankan. Berdasarkan asimptotpenganggar kebolehjadian maksimum, kaedah alternatif ini menyarankan selanganggaran dibina dari ujian nisbah kebolehjadian songsang. Kemampuan selangini diselidik bagi model regresi yang merangkumi data rawak tertapis di sebelahkanan dengan kovariat. Pendekatan model adalah berlandaskan sistem selari.Daripada keputusan, selang-selang yang dibina berdasarkan ujian nisbahkebolehjadian songsang adalah lebih man tap. Selang-selang ini mempunyaikebarangkalian liputan yang hampir kepada nominalnya dengan kebarangkaliandi hujung atas dan di hujung bawah hampir simetri.
ABSTRACTExact confidence intervals for regression models with censored data are oftennot tractable, and hence approximate intervals are derived. The most commonmethod of obtaining these approximate intervals is based on the asymptoticnormal distribution of the maximum likelihood estimator. These intervals areeasy to compute and they are used in most computer statistical packages.However, these intervals have some limitations. When the sample size is smallor even moderate they tend to be anticonservative and have asymmetric upperand lower tail probabilities. An alternative method based on the asymptotics ofthe maximum likelihood estimator is to construct intervals from the invertedlikelihood ratio tests. The performance of these intervals is investigated for theregression models based on parallel systems with covariates, and with randomlyright censored data for finite samples. The simulation results show that theintervals based on the inverted likelihood ratio test have better performance.They have coverage probability that is close to the nominal one, and havenearly symmetric upper and lowel !;IiI probabilities.
Keywords: inverted likelihood tests, random censorship, simulation, asymptoticnormality
Ayman Baklizi, Isa Daud and oor Akma Ibrahim
INTRODUCTION
Recent research on confidence interval estimation based on the asymptotics ofthe maximum likelihood estimation shows that intervals derived from invertedlikelihood tests have desirable properties, and generally perform better thanthose based on the asymptotic normality of the maximum likelihood estimator(Meeker 1987; Doganaksoy and Schmee 1991; Doganaksoy 1991; Vander Wieland Meeker 1990).
We shall consider these intervals for the parameters of the regressionmodel based on parallel systems with covariates. A parallel system is a multicomponent system that fails only when all of its components fail. These systemsare used in industry to increase the reliability of certain products (Bain 1978).In a medical setting, lungs and kidneys are examples of such a parallel system(Elandt:Johnson and Johnson 1979). Parallel systems can be considered as aspecial kind of nomination sample which consists of independently distributedmaxima from subsamples with the same underlying distribution (Kvam andSamaniego 1993).
We shall assume that the life time of each component in the system has anexponential distribution, and the systems have equal numbers of components.The effect of the covariates will be incorporated in the model by expressing theparameters of the distribution as a function of these covariates (Elandtjohnsonand Johnson 1979). In this model, exact confidence in tervals based on themaximum likelihood estimator are not tractable, and therefore, large sampleapproximations are needed.
We shall consider two types of these approximations:1. Confidence intervals based on the asymptotic normality of the maximum
likelihood estimator.2. Confidence intervals based on inverted likelihood ratio tests.
Under the random censorship mechanism, the behaviour of these intervalswill be investigated in terms of their attainment of the nominal level,CI;mservativeness, and the symmetry of their tail probabilities.
THE MODEL
Suppose that the life times of the components in the system, denoted by Xl' X2
,
... , X are independently and identically distributed as exponential withparam~ter A. The life time of the system is equivalent to the life time of thelongest-lasting component, and is therefore given by T = max (Xl' ... , X
m). The
density function of T is given by
78
f(t, A) {Ol~ ex{-i) (1 - exp(- i)r-')
t,A > o.otherwise
PertanikaJ. Sci. & Techno!. Vo!. 5 0.1,1997
[1]
Confidence Inter\'als for Parallel Systems with Co\'ariates
Putting A = exp(f3'Z), where Z = (1, Zl' ... , zp)' is a vector of covariatesand 13 = (13
0, 13, .. ·, f3
p)' is a vector of regression parameters. Transforming
U = log (T) we get.
g (u,Z) = m exp (u -:- 13'Z) exp (- e p(u - 13'Z))(1- exp(- exp(u - 13,z)))(m-I)
- 00 < u < 00, [2]
which defines a regression model given by u = 13' Z+ £ where £ is an error termwith density function given by
With m = 1 this model reduces to the exponential distribution with error termdistributed as an extreme value random variable (Kalbfleisch and Prentice1980) .
The observed data is of the form (y;' 3),i = 1, ... , n, where Yi is equal to(u
j' c), C" c
2' ... , c
nbeing the censor values, and 3 j is an indicator variable given
by
3.I
CONFIDENCE INTERVALS
[4]
This section describes two methods of obtaining approximate intervals for theregression parameters. These approximations are based on the asymptotics ofthe maximum likelihood estimator (Kalbfleisch and Prentice 1980; Lawless1982) .
APPROXIMATE INTERVALS BASED ON THE ASYMPTOTICNORMALITY
Intervals based on the asymptotic normality are widely used. Most of thecommon statistical packages use this kind of approximation. These intervals areeasy to calculate and they are reasonably accurate when the sample is large. Atwo-sided confidence interval for 13; is given by
[5]
where i = 0, 1, ... , P respectively, and where s(.) denotes the sample standarddeviation for the given estimator, and a denotes the nominal level of theconfidence interval.
PertanikaJ. Sci. & Techno!. Vo!. 5 No. I, 1997 79
Ayman Baklizi, Isa Daud and oor Akma Ibrahim
CONFIDENCE INTERVALS BASED ON INVERTEDLIKEUHOOD RATIO TESTS
These confidence intervals are constructed using the fact that, asymptotically,-2 log (R({3»); (i = 0,1, ... , p), has a chi-squared distribution with one degreeof freedom (Vander Wie1 and Meeker 1990), where for p = 1 with parameters{3o' {31 we have
[6]
[7]
where L denotes the likelihood function, and /30, A are the maximum
likelihood estimators of f30 and f31' The bounds of these intervals are givenas the solutions of
[8]
These intervals are somewhat complicated to compute; however, Venzon andMoolgavkar (1988) provide an efficient algorithm to compute the bounds.
THE SIMULATION STRUCTURE
The simulation structure adopted here has m = 3. It is restricted to the caseof two parameters, {3o and (3, equal to 1. The sample size is varied as 20, 30, and40. The level of significance a is taken to be 0.05. The covariate values aretaken on an equally spaced lattice, 0, 0.2, ... , 1.8. Four censoring mechanismsare considered: exponential censoring, uniform censoring, singly type 1censoring, and type 1 censoring, with three censoring proportions in each case,that is, 0.1, 0.3, and 0.5. There were 5000 replications for each simulation run, .as suggested by Piegorsch (1987).
The simulation program was written in FORTRAN with double precision.The maximum likelihood estimator was found using the Newton-Raphsonmethod; necessary and sufficient conditions for the existence of the maximumlikelihood estimator are given in Hamada and Tse (1988). The maximumlikelihood estimator is unique because the log-likelihood function is concave;this property follows from the concavity of log (g) where g is the densityfunction of u. See equation [2/], Burridge (1981). The method of Venzonand Moolgavkar (1988) was used to obtain the bounds of the intervals based oninverted likelihood ratio tests. Intervals based on the asymptotic normality wereobtained using the formula given in equation [5] with estimates of the standarddeviations obtained from the inverse of the observed information matrix. Thesimulation results are given in Tables 1, 2, 3, and 4.
80 PertanikaJ. Sci. & Techno\. Vol. 5 o. 1,1997
TABLE 1Observed error rates of confidence intervals for regression
parameters based on 5000 samples under singly type 1 censoring
SS = sample size, CP = censoring proportion, M = method, L = lowertail error probability, U = upper tail error probability, T = total errorprobability, LR = likelihood ratio, AN= asymptotic normality
TABLE 2Observed error rates of confidence intervals for regression
parameters based on 5000 samples under uniform censoring
SS = sample size, CP = censoring proportion, M = method, L = lowertail error probability, U = upper tail error probability, T = total errorprobability, LR = likelihood ratio, AN = asymptotic normality
81
TABLE 3Observed error rates of confidence intervals for regression
parameters based on 5000 samples under exponential censoring
f3. f3,
55
20
30
40
CP M
0.1 LR
0.3 LRAN
0.5 LRAN
0.1 LRAN
0.3 LRAN
0.5 LRAN
0.1 LRAN
0.3 LR
0.5 LRN
L
0.02360.0] 780.02500.0] 720.02480.0122
0.02520.02080.02540.02020.02580.0178
0.02780.02200.03000.02440.02700.0190
u
0.02720.03240.02480.03000.02340.0284
0.02720.03260.02800.03260.02700.0338
0.02560.02940.02520.02840.02700.0324
T
0.05080.05020.04980.04720.04820.0406
0.05240.05340.05340.05280.05280.0516
0.05340.05140.05520.05280.05400.05]4
L
0.02960.02980.02940.02900.03020.0262
0.02460.02460.02320.02300.02360.0222
0.02680.02680.02600.02560.02760.0270
u
0.02660.02660.02820.02700.02780.0240
0.02500.02500.02720.02620.02420.0228
0.02780.02800.02940.02900.03080.0290
T
0.05620.05640.05760.05600.05800.0502
0.04960.04960.05040.04920.04780.0450
0.05460.05480.05540.05460.05840.0560
55 = sample size, CP = censoring proportion, M = method, L = lowertail error probability, U = upper tail error probability, T = total errorprobability, LR = likelihood ratio, = asymptotic normality
TABLE 4Observed error rate of confidence intervals for regressionparameters based on 5000 samples under type 1 censoring
5520
30
40
CP M0.] LR
AN
0.3 LRAN
0.5 LRAN
0.1 LRAN
0.3 LRAN
0.5 LRAN
0.] LRAN
0.3 LR
0.5 LRAl'l
L
0.02120.0]540.02520.01640.02860.0] ]2
0.02640.02180.02740.01940.03180.0188
0.02760.02280.02680.02100.03100.0234
u0.02720.03220.02800.03300.02620.0320
0.02560.03080.02800.03280.02780.0356
0.02560.03020.02560.03020.02780.0326
T
0.04830.04760.05320.04940.05480.0432
0.05200.05260.05540.05220.05960.0544
0.05320.05300.05240.05120.05880.0560
L
0.02880.02880.03]60.02980.02740.0234
0.02880.02840.02480.02400.02700.0248
0.02660.02660.02600.02480.02960.0268
u0.03060.03060.03180.03000.02860.0234
0.02780.02760.02600.02560.02780.0262
0.02840.02860.02920.02820.03210.0308
T
0.05940.05940.06340.05980.05600.0474
0.05660.05600.05080.04960.05480.0510
0.05500.05520.05520.05300.06]80.0576
82
55 = sample size, CP = censoring proportion, M = method, L = lowertail error probability, U = upper tail error probability, T = total errorprobability, LR = likelihood ratio, = asymptotic normality
Confidence Intervals for Parallel Systems with Covariates
DISCUSSION AND CONCLUSIONS
To judge the adequacy of a confidence interval in a simulation study, twoimportant observations have to be made: (1) the attainment of the observederror probability to the nominal one, or at least, conservativeness and, (2) thedegree of symmetry of the observed lower and upper tail probabilities Oennings1987).
Tables 1, 2, 3 and 4 show that both intervals tend to achieve the nominallevel. They tend to be symmetric for the slope parameter. However, intervalsbased on the asymptotic normality of the maximum likelihood estimator for theintercept parameter have an observed upper and lower tail probabilities thatare highly asymmetric, especially for small samples. On the other hand,intervals based on inverted likelihood ratio tests have observed upper and lowertail probabilities that are symmetric even for sample size as small as 20, whichis an important consideration for one sided confidence limits (Doganaksoy1991) .
As the censoring proportion increases, the intervals tend to get shorter andhave less coverage probabilities. The form of the censoring mechanism and theproportion of censored cases do not appear to have a clear effect on the relativeperformance of the two kinds of confidence intervals.
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Ayman Baklizi, Isa Daud and 001' Akma Ibrahim
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