Gompertz model with time-dependent covariate in the presence of interval-, right- and left-censored data

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Gompertz model with time-dependentcovariate in the presence of interval-,right- and left-censored dataKaveh Kiani a & Jayanthi Arasan a ba Laboratory of Computational Statistics and Operations Research,Institute for Mathematical Research , Universiti Putra Malaysiab Department of Mathematics, Faculty of Science , Universiti PutraMalaysiaPublished online: 01 Mar 2012.

To cite this article: Kaveh Kiani & Jayanthi Arasan (2013) Gompertz model with time-dependentcovariate in the presence of interval-, right- and left-censored data, Journal of StatisticalComputation and Simulation, 83:8, 1472-1490, DOI: 10.1080/00949655.2012.662979

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Journal of Statistical Computation and Simulation, 2013Vol. 83, No. 8, 1472–1490, http://dx.doi.org/10.1080/00949655.2012.662979

Gompertz model with time-dependent covariate in the presenceof interval-, right- and left-censored data

Kaveh Kiania and Jayanthi Arasana,b*

aLaboratory of Computational Statistics and Operations Research, Institute for Mathematical Research,Universiti Putra Malaysia; bDepartment of Mathematics, Faculty of Science, Universiti Putra Malaysia

(Received 1 September 2011; final version received 30 January 2012)

In this paper, the Gompertz model is extended to incorporate time-dependent covariates in the presence ofinterval-, right-, left-censored and uncensored data. Then, its performance at different sample sizes, studyperiods and attendance probabilities are studied. Following that, the model is compared to a fixed covariatemodel. Finally, two confidence interval estimation methods, Wald and likelihood ratio (LR), are exploredand conclusions are drawn based on the results of the coverage probability study. The results indicate thatbias, standard error and root mean square error values of the parameter estimates decrease with the increasein study period, attendance probability and sample size. Also, LR was found to work slightly better thanthe Wald for parameters of the model.

Keywords: censored; confidence interval; covariate; Gompertz; interval; left; likelihood ratio; parametric;right; survival; time-dependent; Wald

AMS Subject Classification: 62N01; 62F10; 62F25

1. Introduction

The analysing and modeling of life-time data usually involves the use of parametric, semi-parametric or non-parametric models. The existences of censored data and time-dependentcovariates have motivated researchers to introduce new models and methods or extend exist-ing models to accommodate these components. In this paper, the Gompertz model was extendedto incorporate time-dependent covariates in the presence of interval-, right-, left-censored anduncensored data. The Gompertz model was introduced by Gompertz [1] as a model for humanmortality. Recently, it has found more application in fields such as biology and demography.

The hazard function of this distribution is

h(t) = λ exp(γ t), t ≥ 0, λ > 0, γ > 0. (1)

T is the non-negative continuous random variable which denotes the individual’s life time. Thescale parameter is λ and the shape parameter is γ . The survivor function of the model is

S(t) = exp

[λ

γ(1 − eγ t)

],

*Corresponding author. Email: [email protected]

© 2013 Taylor & Francis

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Journal of Statistical Computation and Simulation 1473

and the probability density function is

f (t) = λ exp(γ t) × exp

[λ

γ(1 − eγ t)

].

The properties of the Gompertz distribution are presented in Johnson et al. [2]. Garg et al.[3] obtained the maximum likelihood estimate for the parameters of this model. Wu et al. [4]proposed an unweighted and weighted least squares estimate for the parameters of the Gompertzdistribution for both uncensored and first failure-censored data. Others who have worked on theGompertz model are authors such as Makany [5] and Chen [6].

Works involving interval-censored data are few and most of the statistical software do notaccommodate these types of data. There are also many researchers who analyse interval-censoreddata by considering the midpoint of the time interval, (tLi, tRi ] as an estimate of the survival time.

Sun [7] presents up to date statistical models and methods for analysing interval-censored datamostly using non-parametric and semi-parametric models. There are some reviews on analysinginterval-censored data by authors such as, Huang and Wellner [8], Lindsey and Ryan [9], Gomezet al. [10] and Lesaffre et al. [11]. Odell et al. [12] presented a likelihood function that couldcontain interval-, right- and left-censored data using the Weibull regression model. Lindsey [13]studied nine parametric regression models with interval-censored data by using two methods, theexact likelihood and density estimation with intervals midpoint. Gomez et al. [14] described theimplementation of different methods for interval-censored data in R� using some parametric,non-parametric and semi-parametric survival models.

Fixed covariates are measured at the start of study and stay constant over the study period,for instance, gender or race. Time-dependent covariates, on the other hand, vary over time, forexample, age and blood pressure. Cox [15] suggests the use of time-dependent covariate in the PHregression models and gave the partial likelihood analysis and also generated the partial likelihoodfunction for censored data. Peterson [16] introduced an algorithm for estimating the parametersof parametric models in presence of time-dependent covariates. Seaman and Bird [17] extendedthe PH model with interval-censored data to accommodate time-dependent covariate. Sparlinget al. [18] proposed a parametric family of survival regression models for left-, right- and interval-censored data with both fixed and time-dependent covariates. Arasan and Lunn [19] extended thebivariate exponential model to incorporate a time-dependent covariate.

This paper consists of four sections. In Section 2, the Gompertz model with interval-, right-,left-censored and uncensored data is introduced and the performance of this model is studiedwhen the covariate is either time-dependent or fixed. Section 3 discusses two confidence interval(CI) estimation methods for the parameters of the proposed model. First method is the asymptoticnormality CI or Wald method and the second is the CI based on the likelihood ratio test or LRmethod. Finally, Section 4 consists of some discussions and conclusion.

2. Gompertz model with censored data and time-dependent covariate

As discussed earlier, Equation (1) represents the hazard rate of the Gompertz model withoutcovariate. The Gompertz model can be extended to incorporate covariates effect by allowingthe scale parameter λ to be a function of the covariates. Covariates in a study can be fixed,time-dependent or both.

In order to accommodate fixed covariate effects to the hazard function let zTi = (zi1, . . . , ziq) be

a vector of q fixed covariates for the ith subject, where i = 1, 2, . . . , n. Then, the λi could be

λi = exp(α0 + αzi),

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1474 K. Kiani and J. Arasan

and the hazard function of the Gompertz distribution can be expressed as

hη(ti|zi) = λi exp(γ ti) = exp(α0 + αzi + γ ti),

where α = (α1, α2, . . . , αq) and the vector of the parameters is η = (α0, α, γ ).If all observations are uncensored, then the likelihood function for the full sample is

L(η) =n∏

i=1

fη(ti|zi) =n∏

i=1

{hη(ti|zi) × exp

[λi

γ(1 − eγ ti)

]}.

It is common to assume that subjects have been event free at the time origin, t = 0. FollowingKalbfleisch and Prentice [20], time origin is the time from which survival is measured. Intervalcensoring occurs if instead of observing ti, only an interval (tLi , tRi ] is observed where ti ∈ (tLi , tRi ]and tLi ≤ tRi . The ith subject is left-censored when ti ∈ (0, li] or, the subject has met the event atunknown time prior to li and after time origin. The ith subject is right-censored when ti ∈ (ri, ∞]or, the subject has been event free at the last known time ri. Right- and left- censored data canbe regarded as special cases of interval-censored data with (tLi , tRi ] equal to (0, li] or (ri, ∞],see [7,20].

Finally, the ith subject has the exact survival time ti if the event of interest occurs at this timeand this information is available to the observer.

In this paper, we only consider the case when the censoring mechanism is independent(noninformative), see [7].

Let us define the following indicator variables in order to identify whether the ith subject isinterval-, right-, left-censored or uncensored.

δIi = 1 if subject is interval-censored, 0 otherwise;

δRi = 1 if subject is right-censored, 0 otherwise;

δLi = 1 if subject is left-censored, 0 otherwise;

δEi = 1 if exact survival time is observed, 0 otherwise.

Note that δEi = 1 − (δIi + δRi + δLi). Then, the likelihood function for the full sample will be

L(η) =n∏

i=1

[Sη(tLi |zi) − Sη(tRi |zi)]δIi [Sη(ri|zi)]δRi[1 − Sη(li|zi)

]δLi [fη(ti|zi)]δEi

=n∏

i=1

{exp

[λi

γ(1 − eγ tLi )

]− exp

[λi

γ(1 − eγ tRi )

]}δIi

×{

exp

[λi

γ(1 − eγ ri)

]}δRi{

1 − exp

[λi

γ(1 − eγ li)

]}δLi

×{

hη(ti|zi) exp

[λi

γ(1 − eγ ti)

]}δEi

.

In the model with time-dependent covariates we are dealing with covariates whose valueschange over time and not fixed throughout the study. Let Y1, Y2, . . . , Yq represent q time-dependentcovariates. Assume that the mth covariate of the ith subject has updated at a sequence of updatetimes τim0, τim1, . . . , τimkim . τim0 is the time origin, 0, and we consider it to be different from thestart of the study, which in our case is the first inspection time.

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{τimj} represents sequence of the update times, where j = 0, 1, . . . , kim. If kim = 0, this simplymeans that the covariate was not updated during the subject’s follow-up.

In order to accommodate covariate effects to the hazard function, let yim = (yim0, yim1, . . . , yimkim)

represents full history of the mth covariate for the ith subject which is updated at the {τimj}. It isclear that yim0 is the covariate’s baseline value, yim1 is the covariate’s value after first update andyimkim is the covariate’s value after kimth update.

The status of a subject’s covariate whether updated or not, can only be detected during aninspection. Although the covariate may have been updated several times between any two con-secutive inspections, we could only observe the update as occurring once and record yimj. It isclear that the time of this update, τimj, is usually unknown and has to be estimated. One way to dothis (which is used in this study) is by taking the midpoint of the subject’s consecutive inspectiontimes which the update has occurred in between. Another way is to simply consider the inspectiontime as an estimate of the update time.

In the case of left censoring (δLi = 1), the subject has already experienced the event of interestat the start of the study and usually there will be no prior record regarding the state of the subject’scovariate. So, to be more realistic it is assumed that all covariates were not updated.

We consider the case where covariate values follow a step function which means within theinterval [τimj, τim(j+1)), the value of the covariate stays constant at yimj and changes to yim(j+1) inthe following interval.

An example of this kind of covariate could be a change in a patient’s condition from one levelto another level during the study period. We should bear in mind that the covariate levels can beeither dependent or independent. For instance, a patient’s blood sugar level can change to a loweror higher level independently. On the other hand, a covariate such as stages of a disease may onlytake values that are either monotonically increasing or decreasing depending on the disease. Inthis case, we say that the covariate levels are dependent.

For the ith subject, let Yi[(ti)] denotes the complete history of the covariate values up to time ti.Yi[(ti)] = (yi1[(ti)] , yi2[(ti)] , . . . , yiq[(ti)]

) where yim[(ti)] is the vector of the mth covariate values up totime ti.

For the ith subject let Y i[ti] denotes vector of covariate values at time ti. YTi[ti] =

(yi1[ti], yi2[ti], . . . , yiq[ti]) where yim[ti] is the mth covariat’s value at time ti.The hazard function for the ith subject conditional on the given vector Y i[ti] can be expressed

as

hθ(ti|Y i[ti]) = λi[ti] exp(γ ti) = exp(β0 + βY i[ti] + γ ti), (2)

where β = (β1, β2, . . . , βq) and the vector of the parameters is θ = (β0, β, γ ).Following Sparling et al. [18], the likelihood function involving both censored and uncensored

subjects is given by

L(θ) =n∏

i=1

{[Sθ(tLi |Yi[(tLi)]) − Sθ(tRi |Yi[(tRi

)])]δIi [Sθ(ri|Yi[(ri)])]δRi

× [1 − Sθ(li|Yi[(li)])]δLi [fθ(ti|Yi[(ti)])]δEi }. (3)

Let us consider this model with a single covariate and at most one covariate update time soq = 1 and j = 0, 1. For simplicity of notations, yi10 = yi0, yi11 = yi1 and τi11 = τi1. For the ithsubject from Equation (2) the hazard functions evaluated at time ti when ti < τi1 and ti ≥ τi1 are

hθ(ti|yi0) = λi[ti] exp(γ ti) = exp(β0 + β1yi0 + γ ti) = λi0 exp(γ ti) = hi0(ti),

hθ(ti|yi1) = λi[ti] exp(γ ti) = exp(β0 + β1yi1 + γ ti) = λi1 exp(γ ti) = hi1(ti).

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The following indicator variables help us identify whether and when the covariate has beenupdated. For the ith subject,

δUi = 1 if covariate is updated, 0 otherwise;

δPi = 1 if covariate is updated before tLi , 0 otherwise.

So, under these assumptions the log-likelihood function of Equation (3) will be

l(θ) =n∑

i=1

δIi(1 − δUi)(1 − δPi) ln

{exp

[λi0 − hi0(tLi)

γ

]− exp

[λi0 − hi0(tRi)

γ

]}

+ δIiδUi(1 − δPi)ln

{exp

[λi0 − hi0(tLi)

γ

]− exp

[λi0 − hi0(τi1) + hi1(τi1) − hi1(tRi)

γ

]}+ δIiδUiδPi ln

{exp

[λi0 − hi0(τi1) + hi1(τi1) − hi1(tLi)

γ

]− exp

[λi0 − hi0(τi1) + hi1(τi1) − hi1(tRi)

γ

]}+ δRi(1 − δUi)

{λi0 − hi0(ri)

γ

}+ δRiδUi

{λi0 − hi0(τi1) + hi1(τi1) − hi1(ri)

γ

}+ δLi ln

{1 − exp

[λi0 − hi0(li)

γ

]}+ δEi(1 − δUi)

[ln(λi0) + γ ti + λi0 − hi0(ti)

γ

]+ δEiδUi

[ln(λi1) + γ ti + λi0 − hi0(τi1) + hi1(τi1) − hi1(ti)

γ

]. (4)

The MLE of the parameters can be obtained by using any iterative procedure such as theNewton–Raphson algorithm. See the appendix for the first and second derivatives of Equation (4).

2.1. Simulation study

A simulation study using 1000 samples each with n = 50, 100, 150, 200, 250, 300 and 350 wasconducted for this model. The values of 0.06, 0.03 and 0.1 were chosen as the parameters of β0, β1

and γ . The update time or τi1 was generated from the exponential distribution with parameter ν.The value of ν can be adjusted to obtain larger or smaller values of τi1. Here we used the value ofν = 1.

As mentioned before the levels of the time-dependent covariate could be independent ordependent. So, two levels of the covariate were simulated using two different assumptions.

Firstly, the covariate levels are assumed to be independent. In order to achieve this we simulatedthe covariate values independently from the standard normal distribution.

Our second assumption is that the covariate value after update time depends on its value beforeupdate time. To achieve this, we condition the second level of the covariate to always be higherthan the first level of the covariate.

We believe this assumption to be more realistic and applicable to most time-dependent covari-ates such as changing stages of a disease. This was implemented by dividing the z scores of thestandard normal distribution to five consecutive equal probability intervals.

These intervals are made by 5-quantiles or quintiles of this distribution. The intervals are,(−∞, −1.15], (−1.15, −0.32], (−0.32, 0.32], (0.32, 1.15] and (1.15, +∞). Following that, for

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each subject we simulate at least two random variables from the standard normal distributionand select the first and second level of the covariate based on the following conditions:

(i) The second level of the covariate must always be higher than the first level.(ii) The two levels of the covariate must not fall in the same interval.

In real situation the two levels of the covariate may fall within the same interval and still beconsidered as an updated time-dependent covariate. However, for the purpose of data simulationwe only considered the covariate to be updated if its two levels fall in different intervals. Forinstance, if the value of the first level falls into the second interval, the second level must takeany value in the third or subsequent intervals. Random numbers, ui’s were generated from theuniform distribution, u(0, 1), to produce subjects’ actual survival times, ti’s.

ti =

⎧⎪⎪⎨⎪⎪⎩1

γln

[λi0 − λi0 exp(γ τi1) + λi1 exp(γ τi1) − γ ln ui

λi1

], ui < exp

{λi0

γ[1 − exp(γ τi1)]

};

1

γln

[1 − γ ln ui

λi0

], otherwise.

Interval-censored data mostly occurs in longitudinal studies with periodic follow-ups of sub-jects. But, it is common that some subjects miss some follow-ups. So, there are two sequences oftime for each subject, potential or scheduled inspection times and actual inspection times.

In this paper it is assumed that sequence of potential inspection times, P = {p1, . . . , ph}, is thesame for all subjects. Two study periods, 24 and 36 months, are assumed and scheduled follow-upsare monthly so, h = 24 and 36.

Actual inspection times are not necessarily the same for all subjects and depend on the eachsubject’s attendance pattern to the follow-up process. Sequence of actual inspection times of theith subject is Ci = {ci1, . . . , ciki}. It is clear that Ci ⊂ P and {tLi , tRi} ⊂ Ci. In order to simulateCi’s, the Bernoulli distribution with attendance probabilities 0.5 and 0.8 was utilized. It is clearthat tLi and tRi are actual inspection times for the ith subject where ti ∈ (tLi , tRi ]. It is assumed thatall subjects were observed from the beginning of the study, ci1 = p1, and have been event-free attime origin, t = 0.

Eight possible types of data can be generated by this simulation process where ε = 0.005.

(1) tLi < ti ≤ tRi < τi1, interval-censored and covariate is not updated,(2) tLi < ti ≤ tRi and tLi < τi1 ≤ tRi , interval-censored and covariate is updated,(3) τi1 ≤ tLi < ti ≤ tRi , interval-censored and covariate is updated,(4) 0 < ti ≤ p1, left-censored (li = p1).(5) tLi < ti < ∞ and tLi < τi1, right-censored and covariate is not updated (ri = tLi ),(6) τi1 ≤ tLi < ti < ∞, right-censored and covariate is updated (ri = tLi ),(7) (tRi − ε) ≤ ti ≤ tRi < τi1, exact survival time is observed and covariate is not updated,(8) τi1 ≤ (tRi − ε) ≤ ti ≤ tRi , exact survival time is observed and covariate is updated.

Table 1 shows the proportion of different types of data when the covariate levels are either indepen-dent or dependent. We also carried out a separate simulation study using a fixed covariate modelwith 24 potential inspection times to compare the performance of both models. All simulationstudies were carried out by using the FORTRAN� programming language.

2.2. Simulation results

The simulation study was done to assess the bias, standard error (SE) and root mean square error(RMSE) of the estimates at different study periods, attendance probabilities and sample sizes.

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Table 1. Average percentages of different data types for the time-dependentcovariate model.

Attendance probabilities 0.5 0.8

Updated covariates 47% 48%

Study periods 24 36 24 36

Interval (%) 77.6 86.1 77.5 84.9Right (%) 11.7 2.9 10.5 2.7Left (%) 8.1 8.1 7.9 7.9Exact (%) 2.6 2.9 4.1 4.5

From Table 1, we can clearly see that the 36 months study period generates more exact andinterval-censored data compared to the 24 months study period. This is obvious because longerstudy period will increase the chance of observing the event of interest, either exactly or in aninterval. A larger attendance probability will yield more uncensored data and interval-censoreddata with smaller intervals.

From Tables 2–4 we can clearly see that the bias, SE and RMSE values of the β̂0, β̂1 and γ̂

decrease with the increase in study period, attendance probability and sample size. All bias, SEand RMSE seem to be lower when the covariate levels are independent compared to when thecovariate levels are dependent. The results are similar for β̂0, β̂1 and γ̂ except for the RMSE of γ̂

which is slightly greater than the RMSE of the β̂1 and β̂0.Finally, Table 5 gives bias, SE and RMSE values of the α̂0, α̂1 and γ̂ for the fixed covariate

model when the study period is 24 months. The performance of the model indicates a very close

Table 2. Bias, SE and RMSE of β̂0 for the time-dependent covariate model.

Covariate levels Independent Dependent


Attendance prob. 0.5 0.8 0.5 0.8 0.5 0.8 0.5 0.8

50 Bias −0.0233 −0.0216 −0.0221 −0.0196 −0.0238 −0.0235 −0.0258 −0.0273SE 0.2666 0.2555 0.2462 0.2445 0.2441 0.2373 0.2214 0.2194RMSE 0.2676 0.2564 0.2472 0.2453 0.2452 0.2385 0.2229 0.2211

100 Bias −0.0066 −0.0063 −0.0081 −0.0089 −0.0091 −0.0090 −0.0065 −0.0081SE 0.1805 0.1762 0.1629 0.1613 0.1799 0.1761 0.1629 0.1611RMSE 0.1807 0.1763 0.1631 0.1615 0.1801 0.1764 0.1631 0.1613

150 Bias −0.0040 −0.0044 −0.0048 −0.0055 −0.0068 −0.0073 −0.0041 −0.0058SE 0.1496 0.1469 0.1355 0.1346 0.1492 0.1464 0.1344 0.1339RMSE 0.1496 0.1469 0.1356 0.1347 0.1492 0.1465 0.1345 0.1341

200 Bias 0.0012 0.0001 −0.0002 −0.0007 −0.0026 −0.0030 0.0001 −0.0014SE 0.1287 0.1266 0.1163 0.1159 0.1288 0.1259 0.1160 0.1157RMSE 0.1287 0.1266 0.1163 0.1159 0.1288 0.1259 0.1160 0.1157

250 Bias 0.0011 0.0002 −0.0002 −0.0011 −0.0012 −0.0013 0.0002 −0.0012SE 0.1115 0.1096 0.1012 0.1008 0.1115 0.1095 0.1007 0.1009RMSE 0.1115 0.1096 0.1012 0.1008 0.1115 0.1095 0.1007 0.1009

300 Bias 0.0014 0.0007 −0.0004 −0.0011 −0.0008 −0.0006 0.0004 −0.0010SE 0.1050 0.1030 0.0963 0.0927 0.1040 0.1021 0.0925 0.0927RMSE 0.1050 0.1030 0.0963 0.0927 0.1040 0.1021 0.0925 0.0927

350 Bias 0.0008 0.0005 0.0004 −0.0003 −0.0016 −0.0009 0.0009 −0.0003SE 0.0985 0.0962 0.0862 0.0857 0.0986 0.0971 0.0864 0.0863RMSE 0.0985 0.0962 0.0862 0.0857 0.0986 0.0971 0.0864 0.0863

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Table 3. Bias, SE and RMSE of β̂1 for the time-dependent covariate model.



Attendance prob. 0.5 0.8 0.5 0.8 0.5 0.8 0.5 0.8

50 Bias 0.0013 0.0012 0.0013 0.0011 −0.0806 −0.0590 −0.0829 −0.0600SE 0.2179 0.2129 0.2179 0.2152 0.2064 0.2034 0.2033 0.2006RMSE 0.2179 0.2129 0.2179 0.2152 0.2216 0.2118 0.2196 0.2094

100 Bias −0.0070 −0.0063 −0.0041 −0.0067 −0.0608 −0.0414 −0.0689 −0.0455SE 0.1466 0.1437 0.1420 0.1386 0.1481 0.1451 0.1425 0.1403RMSE 0.1468 0.1439 0.1421 0.1387 0.1601 0.1509 0.1583 0.1475

150 Bias −0.0080 −0.0065 −0.0045 −0.0067 −0.0610 −0.0427 −0.0637 −0.0430SE 0.1187 0.1161 0.1126 0.1114 0.1175 0.1144 0.1109 0.1091RMSE 0.1189 0.1163 0.1127 0.1116 0.1323 0.1221 0.1279 0.1172

200 Bias −0.0068 −0.0059 −0.0065 −0.0063 −0.0609 −0.0433 −0.0636 −0.0442SE 0.1016 0.0947 0.0996 0.0946 0.1036 0.1016 0.0974 0.0970RMSE 0.1018 0.0948 0.0998 0.0948 0.1202 0.1105 0.1163 0.1066

250 Bias −0.0003 −0.0008 −0.0009 0.0001 −0.0538 −0.0386 −0.0596 −0.0401SE 0.0903 0.0885 0.0855 0.0835 0.0875 0.0861 0.0834 0.0819RMSE 0.0903 0.0885 0.0855 0.0835 0.1027 0.0943 0.1025 0.0912

300 Bias −0.0034 −0.0021 −0.0032 −0.0033 −0.0574 −0.0411 −0.0636 0.0452SE 0.0835 0.0828 0.0781 0.0779 0.0796 0.0777 0.0778 0.0797RMSE 0.0836 0.0829 0.0781 0.0780 0.0981 0.0879 0.1005 0.0916

350 Bias −0.0008 −0.0007 −0.0019 −0.0011 −0.0584 0.0416 −0.0623 −0.0427SE 0.0772 0.0767 0.0744 0.0731 0.0779 0.0760 0.0737 0.0738RMSE 0.0772 0.0767 0.0744 0.0731 0.0974 0.0866 0.0965 0.0853

Table 4. Bias, SE and RMSE of γ̂ for the time-dependent covariate model.



Attendance prob. 0.5 0.8 0.5 0.8 0.5 0.8 0.5 0.8

50 Bias 0.0410 0.0390 0.0410 0.0400 0.0804 0.0704 0.0874 0.0783SE 0.3449 0.3141 0.3449 0.2894 0.3346 0.3193 0.2545 0.2511RMSE 0.3473 0.3165 0.3473 0.2922 0.3441 0.3270 0.2691 0.2630

100 Bias 0.0157 0.0156 0.0204 0.0208 0.0470 0.0386 0.0469 0.0384SE 0.2270 0.2163 0.1644 0.1626 0.2359 0.2262 0.1732 0.1725RMSE 0.2275 0.2169 0.1657 0.1639 0.2406 0.2295 0.1794 0.1768

150 Bias 0.0096 0.0102 0.0127 0.0130 0.0396 0.0334 0.0373 0.0307SE 0.1834 0.1758 0.1309 0.1305 0.1901 0.1838 0.1382 0.1378RMSE 0.1837 0.1761 0.1315 0.1312 0.1942 0.1868 0.1431 0.1412

200 Bias 0.0028 0.0047 0.0064 0.0067 0.0334 0.0275 0.0309 0.0247SE 0.1605 0.1548 0.1157 0.1153 0.1685 0.1615 0.1226 0.1216RMSE 0.1605 0.1548 0.1158 0.1155 0.1718 0.1638 0.1264 0.1240

250 Bias 0.0000 0.0015 0.0035 0.0041 0.0252 0.0194 0.0259 0.0194SE 0.1414 0.1355 0.1014 0.1011 0.1482 0.1424 0.1053 0.1052RMSE 0.1414 0.1355 0.1014 0.1011 0.1504 0.1438 0.1084 0.1070

300 Bias −0.0008 0.0007 0.0039 0.0043 0.0264 0.0195 0.0274 0.0217SE 0.1328 0.1278 0.0940 0.0926 0.1354 0.1312 0.0978 0.1005RMSE 0.1328 0.1278 0.0940 0.0927 0.1380 0.1326 0.1016 0.1028

350 Bias 0.0015 0.0021 0.0028 0.0036 0.0288 0.0209 0.0266 0.0200SE 0.1234 0.1191 0.0860 0.0851 0.1310 0.1272 0.0929 0.0919RMSE 0.1234 0.1191 0.0861 0.0852 0.1341 0.1289 0.0967 0.0940

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Table 5. Bias, SE and RMSE of the estimates for the fixed covariate model (study period = 24).

Parameter estimates α̂0 α̂1 γ̂

Attendance probabilities 0.5 0.8 0.5 0.8 0.5 0.8

50 Bias −0.0028 −0.0211 −0.0023 −0.0019 0.0465 0.0443SE 0.2600 0.2508 0.2217 0.2219 0.3417 0.3253RMSE 0.2610 0.2515 0.2217 0.2219 0.3448 0.3284

100 Bias −0.0073 −0.0075 0.0014 0.0016 0.0206 0.0210SE 0.1814 0.1769 0.1523 0.1516 0.2276 0.2168RMSE 0.1815 0.1771 0.1523 0.1516 0.2286 0.2178

150 Bias −0.0044 −0.0051 −0.0003 −0.0002 0.0124 0.0140SE 0.1489 0.1462 0.1176 0.1165 0.1814 0.1748RMSE 0.1490 0.1462 0.1176 0.1165 0.1814 0.1748

200 Bias 0.0009 0.0002 −0.0052 −0.0056 0.0050 0.0067SE 0.1286 0.1265 0.1018 0.1009 0.1590 0.1527RMSE 0.1286 0.1265 0.1019 0.1528 0.1591 0.1528

250 Bias 0.0006 0.0000 0.0037 0.0034 0.0021 0.0034SE 0.1117 0.1094 0.0895 0.0888 0.1413 0.1349RMSE 0.1117 0.1094 0.0896 0.0889 0.1413 0.1349

300 Bias 0.0015 0.0010 −0.0034 −0.0032 0.0003 0.0011SE 0.1047 0.1029 0.0811 0.0809 0.1319 0.1280RMSE 0.1047 0.1029 0.0811 0.0810 0.1319 0.1280

350 Bias 0.0010 0.0005 0.0008 0.0007 0.0017 0.0029SE 0.0989 0.0965 0.0797 0.0794 0.1235 0.1187RMSE 0.0989 0.0965 0.0797 0.0794 0.1235 0.1187

similarity to the time-dependent covariate model. Thus, we can say the time-dependent covariatemodel performs as well as the fixed covariate model despite its complexity.

3. Confidence interval estimates

In this section, performance of two CI estimates when applied to the parameters of the time-dependent covariate model are compared and analysed. The first method is the Wald and thesecond is the LR, see Arasan and Lunn [19] and Arasan [21]. For discussions in Sections 3.1and 3.2, we will use β1 as our example and similar procedure would then apply for the rest of theparameters.

3.1. Asymptotic normality confidence intervals (Wald)

Let θ̂ be the maximum likelihood estimator for the vector of parameters θ and l(θ) the log-likelihood function of θ. Following Cox and Hinkley [22], Under mild regularity conditions, θ̂ isasymptotically normally distributed with mean θ and covariance matrix I−1(θ), where I(θ) is theFisher information matrix evaluated at the true value of the θ. The matrix I(θ) can be estimatedby the observed information matrix I(θ̂). The v̂ar(β̂1) is the (2, 2)th element of matrix I−1(θ̂). The100(1 − α)% CI for β1 is

β̂1 − z1−α/2

√v̂ar(β̂1) < β1 < β̂1 + z1−α/2

√v̂ar(β̂1).

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3.2. Likelihood ratio statistics confidence intervals (LR)

For a parameter of interest β1, the likelihood ratio statistics for testing the null hypothesis, H0 :β1 = b1 versus H1 : β1 �= b1 is given as (b1) = −2[l(b1, η̃) − l(β̂1, η̂)], where l is the log-likelihood function, η is the vector of nuisance parameters, (β̂1, η̂) is the maximum likelihoodestimator of (β1, η) and η̃ is the restricted maximum likelihood estimator of η under H0.

For large sample size is approximately χ2(1). An approximate 100(1 − α)% CI for β1 can be

obtained by finding two values of b1 for which H0 is not rejected at (1 − α) level of significance,those are values that satisfy, l(b1, η̃) = l(β̂1, η̂) − 1/2χ2

(1;1−α) with the lower bound b1L < β̂1 and

Figure 1. Estimated error probabilities of Wald and LR methods where covariate levels are independent.

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upper bound b1U > β̂1. The LR interval requires considerable computational effort but it is knownto work very well with censored data even for small sample sizes.

3.3. Coverage probability study

A coverage probability study was conducted using N = 1500 samples of sizes n =50, 100, 150, 200, 250, 300 and 350 to compare the performance of the CI estimates at α = 0.05and 0.1 where α is the nominal error probability. The values of 0.03, 0.06 and 0.1 were chosen asthe parameters of β0, β1 and γ . The study period was assumed 24 month with monthly follow-ups

Figure 2. Estimated error probabilities of Wald and LR methods where covariate levels are dependent.

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and attendance probability were assumed equal to 1 and 0.5, for all patients. Following that, wecalculated the estimated total error probabilities by adding the number of times in which an inter-val did not contain the true parameter value divided by the total number of samples. The estimatedtotal error probability consists of the left error probability and right error probability. Left erroroccurs when the true value of the parameter is less than the lower bound of a CI and right erroroccurs when the true value of the parameter is more than the upper bound of a CI. Left and righterror probabilities for 100(1 − α)% CI of the Wald and LR intervals are given in what follows.

For the Wald intervals,

left = #{β̂1 − z1−α/2

√v̂ar(β̂1) > β1}

1500, right = #{β̂1 + z1−α/2

√v̂ar(β̂1) < β1}

1500.

For the LR intervals,

left = #{(β1) > χ2(1,α) and β̂1 > β1}1500

, right = #{(β1) > χ2(1,α) and β̂1 < β1}1500

.

Following Doganaksoy and Schmee [23], if the total error probability is greater than α + 2.58 ×SE(α̂), then the method is termed anticonservative, if the total error probability is less thanα − 2.58 × SE(α̂), then the method is termed conservative, and if the larger error probability ismore than 1.5 times the smaller one, then the method is termed asymmetrical. Standard error ofestimated error probability is approximately SE(α̂) = √

α(1 − α)/N .

3.4. Coverage probability results and discussion

The coverage probability of a CI is the probability that the interval contains the true parametervalue and should preferably be equal or close to the nominal coverage probability, 1 − α. Figures 1and 2 illustrate the estimated left and right error probabilities for parameters β0, β1 and γ attwo levels of the nominal error probabilities, α = 0.05 and 0.1 when attendance probability isequal to 1.

Table 6 shows the complete summary of the results obtained from the coverage probabilitystudy. The overall performances of the different methods were judged based on the total numberof anticonservative, conservative and asymmetrical intervals.

From Figure 1 and Table 6, we can observe that both Wald and LR methods work rather wellwhen the covariate levels are independent, especially for parameters β0 and β1. There were noconservative intervals and only 1 anticonservative intervals produced by each method. There werehowever, many asymmetrical intervals produced by the Wald method.

Table 6. Performance of the Wald and LR methods.

Attendance prob. 1 0.5

Covariate levels Dependent Independent Dependent Independent

α 0.05 0.1 0.05 0.1 0.05 0.1 0.05 0.1

Conservative Wald 0 0 0 0 0 0 1 1LR 0 0 0 0 6 0 1 1

Anticonservative Wald 4 2 1 0 9 9 0 0LR 8 2 1 0 10 7 0 0

Asymmetric Wald 14 14 5 2 14 14 4 1LR 11 11 5 0 20 14 4 1

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1484K

.Kianiand

J.Arasan

Table 7. Estimated error probabilities of Wald and LR methods where attendance probability is 1 and α = 0.05.


Left error Right error Total error Left error Right error Total error

Parameters Sample sizes Wald LR Wald LR Wald LR Wald LR Wald LR Wald LR

β0 50 0.0280 0.0260 0.0287 0.0393 0.0567 0.0653 0.0327 0.0233 0.0293 0.0400 0.0620 0.0633100 0.0293 0.0267 0.0193 0.0287 0.0486 0.0553 0.0313 0.0293 0.0213 0.0353 0.0526 0.0647150 0.0253 0.0200 0.0273 0.0313 0.0526 0.0513 0.0240 0.0213 0.0273 0.0307 0.0513 0.0520200 0.0300 0.0267 0.0287 0.0360 0.0587 0.0627 0.0273 0.0247 0.0280 0.0393 0.0553 0.0640250 0.0253 0.0233 0.0193 0.0253 0.0446 0.0487 0.0227 0.0193 0.0200 0.0300 0.0427 0.0493300 0.0253 0.0220 0.0253 0.0233 0.0506 0.0453 0.0207 0.0193 0.0233 0.0420 0.0440 0.0613350 0.0267 0.0233 0.0260 0.0220 0.0527 0.0453 0.0247 0.0240 0.0273 0.0353 0.0520 0.0593

β1 50 0.0287 0.0293 0.0253 0.0333 0.0540 0.0627 0.0133 0.0247 0.0493 0.0267 0.0620 0.0513100 0.0233 0.0233 0.0247 0.0233 0.0480 0.0467 0.0153 0.0153 0.0440 0.0240 0.0593 0.0393150 0.0227 0.0227 0.0293 0.0293 0.0520 0.0520 0.0067 0.0067 0.0413 0.0307 0.0480 0.0373200 0.0187 0.0167 0.0207 0.0293 0.0394 0.0460 0.0100 0.0113 0.0547 0.0313 0.0647 0.0427250 0.0313 0.0313 0.0173 0.0227 0.0486 0.0540 0.0087 0.0100 0.0487 0.0220 0.0574 0.0320300 0.0227 0.0233 0.0227 0.0273 0.0454 0.0507 0.0107 0.0160 0.0573 0.0300 0.0680 0.0460350 0.0260 0.0253 0.0253 0.0287 0.0513 0.0540 0.0067 0.0173 0.0707 0.0293 0.0774 0.0467

γ 50 0.0440 0.0393 0.0233 0.0247 0.0673 0.0640 0.0507 0.0400 0.0173 0.0347 0.0680 0.0747100 0.0300 0.0287 0.0187 0.0247 0.0487 0.0433 0.0367 0.0353 0.0193 0.0413 0.0560 0.0767150 0.0320 0.0313 0.0200 0.0287 0.0520 0.0600 0.0320 0.0307 0.0133 0.0413 0.0453 0.0720200 0.0373 0.0360 0.0233 0.0207 0.0606 0.0567 0.0400 0.0393 0.0207 0.0540 0.0607 0.0933250 0.0267 0.0253 0.0200 0.0180 0.0467 0.0433 0.0307 0.0300 0.0167 0.0480 0.0474 0.0780300 0.0247 0.0233 0.0213 0.0227 0.0460 0.0460 0.0347 0.0420 0.0207 0.0547 0.0554 0.0967350 0.0233 0.0220 0.0227 0.0253 0.0460 0.0473 0.0360 0.0353 0.0147 0.0520 0.0507 0.0873

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JournalofStatisticalCom

putationand

Simulation

1485

Table 8. Estimated error probabilities of Wald and LR methods where attendance probability is 0.5 and α = 0.05.


Left error Right error Total error Left error Right error Total error

Parameters Sample sizes Wald LR Wald LR Wald LR Wald LR Wald LR Wald LR

β0 50 0.0347 0.0260 0.0293 0.0393 0.0640 0.0653 0.0235 0.0233 0.0241 0.0427 0.0476 0.0660100 0.0313 0.0260 0.0193 0.0273 0.0506 0.0533 0.0287 0.0213 0.0233 0.0427 0.0520 0.0640150 0.0273 0.0260 0.0213 0.0260 0.0486 0.0520 0.0240 0.0253 0.0227 0.0427 0.0467 0.0680200 0.0300 0.0247 0.0247 0.0353 0.0547 0.0600 0.0294 0.0193 0.0240 0.0427 0.0534 0.0620250 0.0200 0.0167 0.0213 0.0220 0.0413 0.0387 0.0213 0.0200 0.0200 0.0427 0.0413 0.0627300 0.0207 0.0193 0.0233 0.0247 0.0440 0.0440 0.0227 0.0240 0.0200 0.0427 0.0427 0.0667350 0.0267 0.0253 0.0227 0.0240 0.0494 0.0493 0.0260 0.0247 0.0247 0.0427 0.0507 0.0673

β1 50 0.0260 0.0293 0.0240 0.0333 0.0500 0.0627 0.0031 0.0113 0.0501 0.0240 0.0532 0.0353100 0.0233 0.0233 0.0247 0.0213 0.0480 0.0447 0.0053 0.0060 0.0607 0.0240 0.0660 0.0300150 0.0233 0.0240 0.0253 0.0260 0.0486 0.0500 0.0020 0.0027 0.0660 0.0240 0.0680 0.0267200 0.0160 0.0160 0.0193 0.0267 0.0353 0.0427 0.0047 0.0107 0.0814 0.0240 0.0861 0.0347250 0.0267 0.0267 0.0247 0.0233 0.0514 0.0500 0.0000 0.0060 0.0807 0.0240 0.0807 0.0300300 0.0193 0.0193 0.0213 0.0273 0.0406 0.0467 0.0033 0.0040 0.0907 0.0240 0.0940 0.0280350 0.0227 0.0227 0.0213 0.0253 0.0440 0.0480 0.0020 0.0020 0.1167 0.0240 0.1187 0.0260

γ 50 0.0407 0.0393 0.0233 0.0247 0.0640 0.0640 0.0361 0.0427 0.0122 0.0593 0.0483 0.1020100 0.0280 0.0273 0.0173 0.0220 0.0453 0.0493 0.0380 0.0367 0.0133 0.0600 0.0513 0.0967150 0.0280 0.0260 0.0207 0.0247 0.0487 0.0507 0.0360 0.0347 0.0120 0.0640 0.0480 0.0987200 0.0367 0.0353 0.0227 0.0200 0.0594 0.0553 0.0420 0.0407 0.0153 0.0767 0.0573 0.1173250 0.0253 0.0220 0.0180 0.0247 0.0433 0.0467 0.0400 0.0387 0.0173 0.0800 0.0573 0.1187300 0.0253 0.0247 0.0253 0.0213 0.0506 0.0460 0.0360 0.0347 0.0147 0.0907 0.0507 0.1253350 0.0253 0.0240 0.0213 0.0213 0.0466 0.0453 0.0500 0.0480 0.0167 0.1147 0.0667 0.1627

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From Figure 2 and Table 6, we can observe that when the covariate levels are dependent, bothmethods only work well for the parameter β0, The LR works slightly better than the Wald forparameters β1 and γ but overall there were many anticonservative and asymmetrical intervalsproduced by both methods.

From Table 6, when attendance probability decrease to 0.5 and covariate levels are dependent,performance of the LR method significantly decrease while Wald method only produce moreanticonservative intervals.

From Tables 7 and 8, we can see that the estimated total error probabilities of both methodsare close to the 0.05 when the covariate levels are independent. When the covariate levels aredependent, the estimated total error probabilities of both methods for parameter β0 are close tothe 0.05, but for parameter β1 and γ the errors are far from the 0.05.

4. Conclusion

In this paper, the maximum likelihood estimate for the parameters of the Gompertz model withboth fixed and time-dependent covariates in the presence of interval-, right-, left-censored anduncensored data were analysed. It was shown that the bias, SE and RMSE values decrease whenthe study period, attendance probability and sample size increase. We also proposed two CIestimation methods for the parameters of the time-dependent covariate model. Both Wald andLR performed reasonably well when the covariate levels are independent. However the methodsfailed to perform as well when the covariate levels are dependent.

The discussion in this paper was restricted to two covariate levels. Thus, it would be possibleto carry out further work to include more covariate levels. Also, the model could be extended toinclude larger number of both fixed and time-dependent covariates to see their performance. TheGompertz model could also be extended further to include additional parameters, if necessary.This paper only focused on CI estimation methods that depend on the asymptotic normality of theMLE method. Thus, other alternative CI estimation methods such as the bootstrap and jackknifecould be studied in the future.

Acknowledgement

The authors would like to thank the editor, the associate editor and the referee for their comments, which greatly improvedthe paper.

References

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function, J. Roy. Stat. Soc. Ser. C. Appl. Stat. 19 (1970), pp. 152–159.[4] J.W. Wu, W. Hung, and C. Tsai, Estimation of parameters of the Gompertz distribution using the least squares

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Appendix

Let

Ai = δIi (1 − δUi )(1 − δPi ), Bi = δIi δUi (1 − δPi ),

Ci = δIi δUi δPi , Di = δRi (1 − δUi ),

Ei = δRi δUi , Fi = δLi ,

Gi = δEi (1 − δUi ), Hi = δEi δUi ,

W1i = λi0 − hi0(tLi )

γ, W2i = λi0 − hi0(tRi )

γ,

W3i = hi0(tLi ) − hi0(tRi )

γ, W4i = hi1(tLi ) − hi1(tRi )

γ,

W5i = λi0 − hi0(τi1) + hi1(τi1) − hi1(tLi )

γ, W6i = λi0 − hi0(τi1) + hi1(τi1) − hi1(tRi )

γ,

W7i = hi0(tLi ) − hi0(τi1) + hi1(τi1) − hi1(tRi )

γ, W8i = λi0 − hi0(τi1) + hi1(τi1) − hi1(ti)

γ,

W9i = λi0 − hi0(ti)

γ, W10i = λi0 − hi0(ri)

γ,

W11i = λi0 − hi0(li)

γ, W12i = λi0 − hi0(τi1) + hi1(τi1) − hi1(ri)

γ.

The first and second derivatives of Equation (4) would be as follows:

∂l(θ)

∂βj=

n∑i=1

(Aiy

ji0

{W1i − W3i eW3i

1 − eW3i

}+ Bi

{y j

i0W1i − eW7i [y ji0hi0(tLi ) − y j

i0hi0(τi1) + y ji1hi1(τi1) − y j

i1hi1(tRi )]γ (1 − eW7i )

}

+ Ci

{y j

i0λi0 − y ji0hi0(τi1) + y j

i1hi1(τi1) − y ji1hi1(tLi )

γ− y j

i1W4i eW4i

1 − eW4i

}

+ Diyj

i0{W10i} + Ei

{y j


i1hi1(τi1) − y ji1hi1(ri)

γ

}

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+ Fiyj

i0

{W11i eW11i

1 − eW11i

}+ Giy

ji0{W9i + 1}

+Hi

{y j


i1hi1(τi1) − y ji1hi1(ti)

γ+ y j

i1

})j = 0, 1.

∂l(θ)

∂γ=

n∑i=1

(Ai

{−tLi hi0(tLi )

γ− W1i

γ− eW3i [tLi hi0(tLi ) − tRi hi0(tRi ) − W3i]

γ (1 − eW3i )

}

+ Bi

{−tLi hi0(tLi ) − W1i

γ− eW7i

1 − eW7i×

[tLi hi0(tLi ) − τi1hi0(τi1) + τi1hi1(τi1) − tRi hi1(tRi ) − W7i

γ

]}

+ Ci

{−τi1hi0(τi1) + τi1hi1(τi1) − tLi hi1(tLi ) − W5i

γ− eW4i [tLi hi1(tLi ) − tRi hi1(tRi ) − W4i]

γ (1 − eW4i )

}+ Di

{−rihi0(ri) − W10i

γ

}+ Ei

{−τi1hi0(τi1) + τi1hi1(τi1) − rihi1(ri) − W12i

γ

}

+ Fi

{− eW11i [−lihi0(li) − W11i]

γ (1 − eW11i )

}+ Gi

{−tihi0(ti) − W9i

γ+ ti

}+Hi

{−τi1hi0(τi1) + τi1hi1(τi1) − tihi1(ti) − W8i

γ+ ti

}).

∂l(θ)

∂γ 2=

n∑i=1

(Ai

{−t2Li

hi0(tLi )

γ+ 2(tLi hi0(tLi ) − W1i)

γ 2

− eW3i

1 − eW3i

[(t2Li

hi0(tLi ) − t2Ri

hi0(tRi )

γ− 2(tLi hi0(tLi ) − tRi hi0(tRi ) − W3i)

γ 2

)

+(

tLi hi0(tLi ) − tRi hi0(tRi ) − W3i

γ

)2

+ eW3i

1 − eW3i

(tLi hi0(tLi ) − tRi hi0(tRi ) − W3i

γ

)2]}

+ Bi

{−t2Li

hi0(tLi )

γ+ 2tLi hi0(tLi ) + 2W1i

γ 2− eW7i

1 − eW7i

×[

t2Li

hi0(tLi ) − τ 2i1hi0(τi1) + τ 2

i1hi1(τi1) − t2Ri

hi1(tRi )

γ

− 2(tLi hi0(tLi ) − τi1hi0(τi1) + τi1hi1(τi1) − tRi hi1(tRi ) − W7i)

γ 2

]

− eW7i

1 − eW7i

[tLi hi0(tLi ) − τi1hi0(τi1) + τi1hi1(τi1) − tRi hi1(tRi ) − W7i

γ

]2

−[

eW7i

1 − eW7i

]2 [tLi hi0(tLi ) − τi1hi0(τi1) + τi1hi1(τi1) − tRi hi1(tRi ) − W7i

γ

]2}

+ Ci

{−τ 2i1hi0(τi1) + τ 2

i1hi1(τi1) − t2Li

hi1(tLi )

γ− 2(−τi1hi0(τi1) + τi1hi1(τi1) − tLi hi1(tLi ) − W5i)

γ 2

− eW4i

1 − eW4i

[t2Li

hi1(tLi ) − t2Ri

hi1(tRi )

γ− 2(tLi hi1(tLi ) − tRi hi1(tRi ) − W4i)

γ 2

]

− [tLi hi1(tLi ) − tRi hi1(tRi ) + W4i]2 eW4i

γ 2(1 − eW4i )−

[(tLi hi1(tLi ) − tRi hi1(tRi ) + W4i)eW4i

γ (1 − eW4i )

]2}

+ Di

{−r2i hi0(ri) + 2W10i

γ+ 2rihi0(ri)

γ 2

}+ Ei

{−τ 2

i1hi0(τi1) + τ 2i1hi1(τi1) − r2

i hi1(ri)

γ

− 2(−τi1hi0(τi1) + τi1hi1(τi1) − rihi1(ri) + 2W12i)

γ 2

}

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2013


+ Fi

{− eW11i

1 − eW11i

[(−l2i hi0(li)

γ− 2(−lihi0(li) − W11i)

γ 2

)+

(−lihi0(li) − W11i

γ

)2

+ eW11i

1 − eW11i


γ

)2]}

+ Gi

{−t2i hi0(ti)

γ− 2(tihi0(ti) − W9i)

γ 2

}

+ Hi

{−τ 2

i1hi0(τi1) + τ 2i1hi1(τi1) − t2

i hi1(ti)

γ− 2(−τi1hi0(τi1) + τi1hi1(τi1) − tihi1(ti) + W8i)

γ 2

}).

∂l(θ)

∂βj∂γ=

n∑i=1

(Aiy

ji0

{−tLi hi0(tLi ) − W1i

γ− eW3i

1 − eW3i

×[

tLi hi0(tLi ) − tRi hi0(tRi ) − W3i

γ+ W3i


γ

)

+W3i


γ

)eW3i

1 − eW3i

]}+ Bi

{−y j

i0(tLi hi0(tLi ) + W1i)

γ− eW7i

1 − eW7i

×[

y ji0tLi hi0(tLi ) − y j

i0τi1hi0(τi1) + y ji1τi1hi1(τi1) − y j

i1tRi hi1(tRi )

γ

+ y ji0hi0(tLi ) − y j


i1hi1(tRi )

γ

− y ji0hi0(tLi ) − y j


i1hi1(tRi )

γ

× tLi hi0(tLi ) − τi1hi0(τi1) + τi1hi1(τi1) − tRi hi1(tRi ) − W7i

γ

− y ji0hi0(tLi ) − y j


i1hi1(tRi )

γ× eW7i

1 − eW7i

× tLi hi0(tLi ) − τi1hi0(τi1) + τi1hi1(τi1) − tRi hi1(tRi ) − γ W7i

γ 2

]}

+ Ci

{−y j


i1tLi hi1(tLi )

γ− y j


i1hi1(τi1) − y ji1hi1(tLi )

γ 2

− y ji1 eW4i

1 − eW4i

[tLi hi1(tLi ) − tRi hi1(tRi ) − W4i

γ+ W4i


γ

)

+ W4i eW4i

1 − eW4i


γ

)]}

+ Diyi0

{−rihi0(ri) − W10i

γ

}+ Ei

{−y j


i1rihi1(ri)

γ

− y ji0λi0 − y j


i1hi1(ri)

γ 2

}

+ Fiyj

i0

{− eW11i

1 − eW11i×

[−lihi0(li) − W11i

γ+ W11i


γ

)

+W11i


γ

)eW11i

1 − eW11i

]}

+ Giyj

i0

{−tihi0(ti) − W9i

γ

}+ Hi

{−y j


i1tihi1(ti)

γ

− y ji0λi0 − y j


i1hi1(ti)

γ 2

})j = 0, 1.

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2013


∂l(θ)

∂βj∂βk=

n∑i=1

(Aiy

(j+k)

i0

{W1i − W3i eW3i

1 − eW3i

[1 + W3i + W3i eW3i 1 − eW3i

]}

+ Bi

{y(j+k)

i0 W1i − eW7i [y(j+k)

i0 hi0(tLi ) − y(j+k)

i0 hi0(τi1) + y(j+k)

i1 hi1(τi1) − y(j+k)

i1 hi1(tRi )]γ (1 − eW7i )

− W2−j−k7i eW7i [yi0hi0(tLi ) − yi0hi0(τi1) + yi1hi1(τi1) − yi1hi1(tRi )](j+k)

γ (1 − eW7i )

− W2−j−k7i [(yi0hi0(tLi ) − yi0hi0(τi1) + yi1hi1(τi1) − yi1hi1(tRi ))](j+k) ×

[eW7i

γ (1 − eW7i )

]2}

+ Ci

{y(j+k)

i0 λi0 − y(j+k)



i1 hi1(tLi )

γ

−y(j+k)

i1W4i eW4i

1 − eW4i

[1 + W4i + W4i eW4i

1 − eW4i

]}+ Diy

(j+k)

i0 {W10i}

+ Ei

{y(j+k)

i0 λi0 − y(j+k)



i1 hi1(ri)

γ

}

+ Fiy(j+k)

i0

{− W11i eW11i

1 − eW11i

[1 + W11i + W11i eW11i

1 − eW11i

]}+ Giy

(j+k)

i0 {W9i}

+Hi

{y(j+k)

i0 λi0 − y(j+k)



i1 hi1(ti)

γ

})j = 0, 1 k = 0, 1.

Dow

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Mos

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Sta

te U

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Bib

liote

] at

13:

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7 Se

ptem

ber

2013

Documents

Gompertz model with time-dependent covariate in the presence of interval-, right- and left-censored data