R E S E A R C H A R T I C L E Open Access
Additive-multiplicative hazards
regression models for interval-censored semi-competing risks data with missing intermediate events
Jinheum Kim1, Jayoun Kim2and Seong W. Kim3*
Abstract
Background: In clinical trials and survival analysis, participants may be excluded from the study due to withdrawal, which is often referred to as lost-to-follow-up (LTF). It is natural to argue that a disease would be censored due to death;
however, when an LTF is present it is not guaranteed that the disease has been censored. This makes it important to consider both cases; the disease is censored or not censored. We also note that the illness process can be censored by LTF. We will consider a multi-state model in which LTF is not regarded as censoring but as a non-fatal event.
Methods: We propose a multi-state model for analyzing semi-competing risks data with interval-censored or missing intermediate events. More precisely, we employ the additive and multiplicative hazards model with log-normal frailty and construct the conditional likelihood to estimate the transition intensities among states in the multi-state model.
Marginalization of the full likelihood is accomplished using adaptive importance sampling, and the optimal solution of the regression parameters is achieved through the iterative quasi-Newton algorithm.
Results: Simulation is performed to investigate the finite-sample performance of the proposed estimation method in terms of the relative bias and coverage probability of the regression parameters. The proposed estimators turned out to be robust to misspecifications of the frailty distribution. PAQUID data have been analyzed and yielded somewhat prominent results.
Conclusions: We propose a multi-state model for semi-competing risks data for which there exists information on fatal events, but information on non-fatal events may not be available due to lost to follow-up. Simulation results show that the coverage probabilities of the regression parameters are close to a nominal level of 0.95 in most cases.
Regarding the analysis of real data, the risk of transition from a healthy state to dementia is higher for women;
however, the risk of death after being diagnosed with dementia is higher for men.
Keywords: Additive and multiplicative hazards model, Interval censoring; log-normal frailty, Missing intermediate event, Multi-state model, Semi-competing risks data
Background
In classical time-to-event or survival analysis, subjects are under risk for one fatal event. However, subjects do not fail from just one certain type of event in some appli- cations, but are under risk of failing from two or more mutually exclusive types of events. When an individual
*Correspondence:[email protected]
3Department of Applied Mathematics, Hanyang University, 15588 Ansan, South Korea
Full list of author information is available at the end of the article
is under risk of failing from two different types of event, these different event types are called competing risks. One of the events censors the other, and vice versa, in these competing risks frameworks [1–3]. However, many clini- cal trials have revealed that a subject can experience both a non-fatal event (e.g., a disease or relapse) and a fatal event (e.g., death), where the fatal event censors the non- fatal event but not vice versa. We call these types of data semi-competing risks data [4–6].
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In clinical trials, the occurrence of a non-fatal event can be detected in conjunction with possibly incessant mon- itoring during periodic follow-up. For illustration pur- poses of our methodologies, a dataset named PAQUID (Personnes Agées Quid) is analyzed to investigate the meaningful prognostic factors associated with dementia.
These data were initially analyzed by [7] using the con- ventional Cox model [8,9]. Complete descriptions of the PAQUID data can be found inConclusionssection. In this paper, we employ a semi-competing risks model where death may occur after dementia has occurred (i.e., been diagnosed), but death censors the disease. An illness- death model [10] is perhaps one of the most commonly and frequently used semi-competing risks models. Many studies have been conducted under semi-competing risks frameworks [4,5,11].
As shown in the PAQUID data, dementia can be cen- sored informatively by death. Furthermore, an additional informative censoring process can also occur. That is, participants may be excluded from the study due to with- drawal, which is often referred to as lost-to-follow-up (LTF). It is clear that dementia would be censored due to death; however, when an LTF is present it is not guar- anteed that dementia has been censored. This makes it important to consider both cases; dementia is censored or not censored. We also note that the illness process (dementia) can be censored by LTF. This forces us to con- sider a multi-state model in which LTF is not regarded as censoring but as a non-fatal event. Considerable stud- ies have utilized this multi-state model. For example, [12]
proposed a nonparametric method to estimate the sur- vival function associated with disease occurrences, while [6,13] used the Cox proportional hazards model [8,9] to estimate regression coefficients.
In the meantime, most non-fatal events are observed periodically. That is, the event time is not observed exactly but lies on an interval of the form(L,R], whereLis the last time a subject visited without possessing a disease andR is the first time that the subject was diagnosed with a dis- ease. This type of censoring is called interval censoring.
We could emulate what [6] did and assume that a non- fatal event of a subject occurs uniformly on the interval (L,R]. However, using the methods proposed by [14,15], we instead partition the interval (L,R] into a few sub- intervals, in which a non-fatal event can occur. Ultimately, different weights can be assigned to each sub-interval.
Thus, the former method corresponds to an uncondi- tional probability approach with equal weight on all of the sub-intervals, whereas the latter utilizes a conditional probability approach with a specific weight, depending on the sub-interval.
In our study, we use the latter method to deal with non-fatal events that are interval-censored on an inter- val. In addition, we propose an additive-multiplicative
model by combining the Cox proportional hazards model with the additive risk model of [16], in accordance with a multi-state model. The additive-multiplicative model was initially introduced by [17] and has since been devel- oped by a number of researchers. Scheike and Zhang [19]
incorporated time-varying covariates for the additive part and time-independent covariates for the multiplicative part. On the other hand, [18] estimated relevant parame- ters by considering time-varying covariates for both addi- tive and multiplicative parts. We also consider the frailty effect as a latent variable to incorporate possible connec- tions between events; this is done because each individual is exposed to several events, including the occurrence of illness, LTF, and death.
The rest of the paper is organized as follows. First, we explain notations and procedures for parameter estima- tion along with the proposed models. Second, extensive simulation studies are presented to investigate the model performances in terms of the relative bias and coverage probability of the proposed estimates. We also provide the results of real data analysis. Finally, we present a summary and concluding remarks, including some of the drawbacks of the proposed models and directions for future research.
Methods Models
As depicted in Fig. 1, the proposed model in this study consists of five states: healthy (H), non-fatal (NF), fatal (F), lost-to-follow-up (LTF), and unobserved non-fatal (NF(LTF)). Each state is denoted by numbers 0 through 4, respectively. A total of seven possible transitions exists in the model: 0 → 1, 0 → 2, 0 → 3, 1 → 2, 3 → 2, 3 → 4, and 4 → 2. However, among these transitions, both 3→4 and 4→2 (displayed as dotted lines in Fig.1 are unobservable and should be regarded as potential transitions.
Let t be the time from study entry. Additionally, St is defined as the state that each subject can take at t ≥ 0. Then, St ∈ {0, 1, 2, 3, 4}. Let A = {(r,s) : (r,s) = (0, 1),(0, 2),(0, 3),(1, 2),(3, 2),(3, 4),(4, 2)}. Also, defineλrs(t)to be the transition intensity from statesrto sat timet. That is,
λrs(t)= lim
dt→0
Pr(St+dt=s|St=r)
dt for (r,s)∈A, andλrs(t)=0 for(r,s) /∈A. As mentioned above, the data corresponding to transitions 3 → 4 and 4 → 2 are not observable, requiring the following assumptions forλ34(t) andλ42(t):
λ34(t) = λ01(t), t≥0, (1) λ42(t) = λ12(t), t≥0. (2) Assumptions (1) and (2) imply that the transition intensities of H to NF and NF to F may be the same,
Fig. 1Five-state model
irrespective of the occurrence of LTF. As mentioned in Backgroundsection, given covariatesz =
z1,z2,. . .,zp andw = (w1,w2,. . .,wd), along with frailtyu, we con- sider additive and multiplicative models defined as λrs(t|z,w,u)=η
βrsz+exp αrsw
θrsγrstγrs−1
for (r,s)∈A, (3) where θrs(> 0) and γrs(> 0) are the scale and shape parameters of a Weibull distribution, respectively, andβrs andαrs are vectors of the regression coefficients for the additive and multiplicative parts, respectively. Moreover, η = exp(u) is the frailty for a log-normal distribution and u is assumed to follow a normal distribution with a mean of zero and varianceσ2. Thus, we use a Weibull distribution as a baseline transition intensity and impose a multiplicative frailty effect on the transition intensities.
Since the parameters θ34,θ42,γ34,γ42,α34,α42,β34, and β42related to transitions 3→4 and 4→2 should satisfy the assumptions in (1) and (2), the parameter vector estimated for the model in (3) is ζ =
θ∗,γ∗,α∗,β∗,σ2 , where θ∗ = (θ01,θ02,θ03,θ12,θ32), γ∗ = (γ01,γ02,γ03,γ12,γ32),
α∗ =
α01,α02,α03,α12,α32
, and β∗ =
β01,β02,β03,β12,β32
. According to the model in (3), the cumulative hazard functionsHk(t1,t2) (k =0, 1, 3, 4) for leaving statekbetweent1andt2can be represented as
H0(t1,t2|z,w,u) = t2 t1
{λ01(s|z,w,u)+λ02(s|z,w,u)+λ03(s|z,w,u)}ds
= 3 r=1
η
β0rz
(t2−t1)+exp α0rw
θ0r
t2γ0r−tγ10r ,
H1(t1,t2|z,w,u) = t2
t1
λ12(s|z,w,u)ds
= η β12z
(t2−t1)+exp α12w
θ12
t2γ12−tγ112 ,
H3(t1,t2|z,w,u)= t2 t1
{λ32(s|z,w,u)+λ34(s|z,w,u)}ds
=η β32z
(t2−t1)+exp α32w
θ32
tγ232−t1γ32 +
β34z
(t2−t1)+exp α34w
θ34
tγ234−t1γ34 , H4(t1,t2|z,w,u)= t2
t1
λ42(s|z,w,u)ds
=η β42z
(t2−t1)+exp α42w
θ42
tγ242−t1γ42 . Based on Assumptions (1) and (2), we note thatβ34=β01, β42 = β12,α34 = α01,α42 = α12,θ34 = θ01,θ42 = θ12, γ34=γ01, andγ42=γ12in the equations ofH3andH4. Parameter estimation
As shown in Fig.1, a total of six routes can be experienced by a subject from the beginning to the end of the study.
These are route 1 (0→0), route 2 (0→2), route 3 (0→ 1), route 4 (0→1→2), route 5 (0→3), and route 6 (0
→3→2). In particular, route 5 can be classified into two paths, i.e., 0→3 and 0→3→4, depending on whether or not a subject experiences the unobservable NF state. Sim- ilarly, route 6 can be classified into two paths: 0→3→ 2 and 0→3→4→2. We introduce notations to define the likelihood associated with each route. Consider three random variablesR,L, andT, each of which represents a time from the start of the study until the occurrence of a non-fatal event, LTF, and a fatal event, respectively. Fur- thermore, letH0(s)be the set of subjects staying in state 0 at times. That is,
H0(s)= {R∧L∧T>s}.
LetH3,f(s)be the set of subjects who have already expe- rienced LTF at time f and remain in state 3 at time s.
Then,
H3,f(s)= {L=f,R∧T >s,f ≤s}.
Now, leteibe the entry time of study,aibe the last time the ithsubject visited before a non-fatal event was observed, andbi be the first time a non-fatal event is observed by the ith subject fori = 1, 2,. . .,n. Consider an indicator functionIij, which is 1 if subjectifollows routejand zero otherwise forj = 1, 2,. . ., 6. LetBj = {i : Iij = 1}. For subjecti ∈ B1∪B2, we haveai,bi ≥ ti; this is the case because a non-fatal event has not been observed before timeti. For subjecti∈B3∪B4, we haveai<bi≤ti; this is the case because a non-fatal event has occurred between ai andbi. When subject iis a member of B5∪B6, LTF has occurred at time ai, which yields ai < ti; however, bi<tiorbi ≥ti, depending on whether an unobservable non-fatal event has occurred. Thus,tiwould be a censor- ing time fori∈B1∪B3∪B5, whereas it would be a time of death fori∈B2∪B4∪B6. Therefore, likelihood func- tions Q1andQ2 can be constructed for routes 1 and 2, respectively. These are given as follows:
Qi1 = Pr(Ri∧Li∧Ti>ti|H0(ei),zi,wi,ui)
= exp{−H0(ei,ti|zi,wi,ui)}, i∈B1. (4) Qi2 = Pr(T=ti,R∧L>ti|H0(ei),zi,wi,ui)
= Qi1×λ02(ti|zi,wi,ui), i∈B2. (5) Likelihood functions can also be constructed for routes 3 and 4:
Q∗i3 = Pr(Ri∈(ai,bi] ,Li>ti,Ti>ti|H0(ei),zi,wi,ui)
= bi
ai
exp{−H0(ei,s|zi,wi,ui)}λ01(s|zi,wi,ui)
×exp{−H1(s,ti|zi,wi,ui)} ds. (6)
Q∗i4= Pr(Ri∈(ai,bi] ,Li>ti,Ri<Ti=ti|H0(ei),zi,wi,ui)
= Q∗i3×λ12(ti|zi,wi,ui), i∈B4. (7) Equations (6) and (7) are derived by assuming that a non- fatal event of subject i in the set B3 ∪ B4 can occur uniformly on the interval(ai,bi] [6]. However, we parti- tion the interval(ai,bi] into several sub-intervals where non-fatal events could occur and assign different weights to each interval [15].
• LetRi ∈(ai,bi]be an interval for the occurrence of non-fatal events associated with subjects in routes 3 or 4. Lets1be the smallest value among allbi’s for subjects in the setB3∪B4. Lets2be the smallest value among allbi’s corresponding to subjects having aigreater than or equal tos1. This process is
repeated until we have no subjects withaigreater than or equal tosm(m=1, 2,. . .). Thus, we can have a refined set of time points
0=s0<s1<s2<· · ·<sl<sl+1= ∞.
• We can define the weightwimat timesm (m=1, 2,. . .) for subjectiin the setB3∪B4:
wim= dimexp{−H0(ei,sm|wi,zi,ui)}λ01(sm|wi,zi,ui) l
m=1dimexp{−H0(ei,sm|wi,zi,ui)}λ01(sm|wi,zi,ui), (8) wheredim = I(sm ∈ (ai,bi]). Subsequently, likelihood functions incorporated with weightwimin (8) for routes 3 and 4 are given by
Qi3= l m=1
dimwimexp{−H0(ei,sm|zi,wi,ui)}λ01(sm|zi,wi,ui)
×exp{−H1(sm,ti|zi,wi,ui)}, i∈B3.
(9) Qi4=Qi3×λ12(ti|zi,wi,ui), i∈B4. (10) Finally, likelihood functions for routes 5 and 6 are given by
Qi5 = Pr(Ri∧Ti>ti|H3,ai(ai),zi,wi,ui)
+Pr(Ri∈(ai,ti] ,Ti>ti|H3,ai(ai),zi,wi,ui)
= exp{−H0(ei,ai|zi,wi,ui)}λ03(ai|zi,wi,ui)
×
exp{−H3(ai,ti|zi,wi,ui)}
+ ti
ai
exp{−H3(ai,s|zi,wi,ui)}λ34(s|zi,wi,ui)
×exp{−H4(s,ti|zi,wi,ui)}ds , i∈B5, (11)
Qi6=Pr(Ri>Ti,Ti=ti|H3,ai(ai),zi,wi,ui)
+Pr(Ri∈(ai, ti] , Ri<Ti=ti|H3,ai(ai),zi,wi,ui)
=exp{−H0(ei,ai|zi,wi,ui)}λ03(ai|zi,wi,ui)
×
exp{−H3(ai,ti|zi,wi,ui)}λ32(ti|zi,wi,ui) + ti
ai
exp{−H3(ai,s|zi,wi,ui)}λ34(s|zi,wi,ui)
×exp{−H4(s,ti|zi,wi,ui)}ds
λ42(ti|zi,wi,ui) ,i∈B6. (12) Therefore, based on Eqs. (4)-(5) and (9)-(12), the likeli- hood function for the parameter vectorζis
L(ζ)= n i=1
⎧⎨
⎩ 6 j=1
QIijij
⎫⎬
⎭φ
0,σ2;ui
, (13)
where φ(·) is the probability density function of a nor- mal distribution with a mean of zero and variance σ2. In our analysis, we use the NLMIXED procedure
of the SAS software to estimate ζ. For the sake of parameter estimation procedures, we define the marginal likelihood as
m(ζ)=
· · ·
L(ζ)du1· · ·dun.
Then, we find the value of ζ that minimizes f(ζ) =
−logm(ζ), which is referred to as ζˆ. Consequently, the inverse of the Hessian matrix evaluated atζˆis defined as the estimated variance-covariance matrix ofζˆ. Numerical integration is required for the frailty distribution. For this purpose, we use the adaptive importance sampling [20].
Finally, we employ quasi-Newton optimization, which uti- lizes the gradient vector and the Hessian matrix off(ζ), to achieve the optimal solution ofζ.
Results
Simulation studies
Extensive simulation is performed to investigate the finite-sample properties of the estimators proposed in Methods section. As mentioned earlier, we assume a Weibull distribution with a shape parameter ofγrs =1 as the baseline transition intensity and a log-normal distri- bution for frailtyη = exp(u), whereuis generated from a normal distribution with a mean of zero and a variance of 0.01. Furthermore, we use a binary covariate forz(gen- erated from a Bernoulli trial with a success probability of 0.5) and a continuous covariate forw(generated from a standard normal distribution). We fix the sample sizenat 200 and the censoring timeCat 365. A total of 500 repli- cations is used in our simulations. The following presents the details related to the generation of random variates for theith(i=1, 2,. . .,n)subject.
• Step 0: We may allow the total number of
occurrences for non-fatal events to be 24 times in a 12-month period, such as15, 31,. . ., 349, 365days.
However, the actual visiting time of each subject can be different from the designated times. Hence, we add random numbers, generated from a normal distribution with a mean of zero and a variance of 9, to each designated time point. Subsequently, the actual observed time points will be defined as
0=l0<l1i<· · ·<l23,i<l24=366.
Letu01i,u02i, andu03ibe random numbers generated from a uniform distribution on the interval(0, 1). Additionally, letRi,Ti, andLibe, respectively, the rootss of the equations:
01(s|zi,wi,ui)+log(1−u01i)=0, 02(s|zi,wi,ui)+log(1−u02i)=0, and
03(s|zi,wi,ui)+log(1−u03i)=0,
where
0j(s|zi,wi,ui)=ηi
β0jzi
s+exp{α0jwi}θ0jsγ0j
forj=1, 2, 3.
• Step 1: IfC≤Ri∧Ti∧Li, then theithsubject is defined as being censored without experiencing a non-fatal event, i.e.,i∈B1. IfTi=Ri∧Ti∧Li, then theithsubject is defined as being dead without experiencing a non-fatal event, i.e.,i∈B2. However, ifRi=Ri∧Ti∧Li, proceed to Step 2, and if Li=Ri∧Ti∧Li, proceed to Step 3.
• Step 2: Letu12ibe a random number generated from a uniform distribution on the interval
(1−exp{12(Ri|zi,wi,ui)}, 1), where 12(s|zi,wi,ui)=ηi
(β12zi)s+exp{α12wi}θ12sγ12 . RedefineTias the roots of the equation,
12(s|zi,wi,ui)+log(1−u12i)=0.
IfC≤Ti, then theithsubject is defined as being censored after experiencing a non-fatal event, i.e., i∈B3. Otherwise, theithsubject is defined as being dead at timeTiafter experiencing a non-fatal event, i.e.,i∈B4. Moreover,
- IfRi∈(0,l1i), letai=0andbi=l1i. If Ri∈
lk−1,i,lki
, letai=lk−1,iandbi=lkifor k=2, 3,. . ., 23.
- However, ifRi∈(l23,i,C), the type of path should be redefined because a non-fatal event for the subject did not occur before the time of the last observation.
Thus, ifC≤Ti, theithsubject is defined as being censored without experiencing a non-fatal event, i.e., i∈B1. Otherwise, theithsubject is defined as being dead at timeTiwithout experiencing a non-fatal event, i.e.,i∈B2.
• Step 3: Letu32iandu34ibe random numbers
generated from uniform distributions on the intervals (1−exp{32(Li|zi,wi,ui)}, 1)and
(1−exp{34(Li|zi,wi,ui)}, 1),respectively, where 3j(s|zi,wi,ui)=ηi
(β3jzi)s+exp{α3jwi}θ3jsγ3j
,forj=2, 4.
Now redefineRiandTias the rootss of the equations:
32(s|zi,wi,ui)+log(1−u32i)=0 and
34(s|zi,wi,ui)+log(1−u34i)=0,
respectively. IfC≤Ri∧Ti, theithsubject is defined as being censored without experiencing a non-fatal event after LTF, i.e.,i∈B5. IfTi≤Ri, then theith subject is defined as being dead without experiencing a non-fatal event after LTF, i.e.,i∈B6. However, if Ri<Ti, move to Step 4.
Table1Empiricalresultsfortheaveragesoftherelativebias(r.Bias)andstandarderrors(SEM),standarddeviation(SD),andcoverageprobability(CP)fortheregressionparameters andvarianceparameterofthelog-normalfrailtybasedonthe‘imputed-by-the-right-endpoint’and‘proposed’methodswhenthetypeofregressioncoefficientsis‘even’under threetypesofLTFproportions(‘low’,‘moderate’,and‘high’) Low(LTF(%)=22.6)Moderate(LTF(%)=34.2)High(LTF(%)=47.3) ParameterTruevaluer.Bias(%)SD ×105 SEM ×105 CP(%)r.Bias(%)SD ×105 SEM ×105 CP(%)r.Bias(%)SD ×105 SEM ×105 CP(%) Imputed-by-the-right-endpointmethod α010.0181.7150261510496.873.3151051497595.220.6144501518696.2 α020.01176.2222972296195.4324.5244082406694.828.8276432742695.4 α030.01133.2270062642995.486.1162231708696.275.3132911325095.2 α120.01-145.7284812584995.2-113.4299962627993.674.1273092562096.2 α320.01109.7654895561593.277.1360873374796.850.8266802521994.8 β010.004-10.1818793.8-9.4808891.6-8.6878892.6 β020.0041.1929294.6-1.4979694.4-0.410310194.0 β030.004-7.2878691.2-6.410510593.4-9.313613391.6 β120.0043.511511597.05.711211695.63.411611595.8 β320.004-0.321719996.87.617417396.45.314815297.2 σ20.01904.58164878293.8815.57848808492.4771.67626707690.4 Proposedmethod α010.0132.2152361513696.264.8153441508495.020.7144961527996.2 α020.0170.2226922295195.0286.1240102413395.2100.4280302756995.2 α030.01289.4275252634995.0121.3167401716195.086.3134651339195.0 α120.01-24.6267182566195.8-83.4293552623094.0-37.0271022573696.2 α320.01123.2671885612293.693.4362913357795.654.0265912520494.2 β010.004-5.5899095.0-5.0859193.8-5.3889194.4 β020.0045.9959796.24.09910094.24.810710694.8 β030.004-1.7929193.2-1.710711095.6-3.314113993.4 β120.004-1.910911096.40.910811295.20.111011196.0 β320.0041.021419996.68.417917596.24.415015397.0 σ20.011016.29069907492.2897.87743819891.0874.17616742387.2
Table2Empiricalresultsfortheaveragesoftherelativebias(r.Bias)andstandarderrors(SEM),standarddeviation(SD),andcoverageprobability(CP)fortheregressionparameters andvarianceparameterofthelog-normalfrailtybasedonthe‘imputed-by-the-right-endpoint’and‘proposed’methodswhenthetypeofregressioncoefficientsis‘dec’underthree typesofLTFproportions(‘low’,‘moderate’,and‘high’) Low(LTF(%)=19.4)Moderate(LTF(%)=31.4)High(LTF(%)=43.5) ParameterTruevaluer.Bias(%)SD ×105 SEM ×105 CP(%)r.Bias(%)SD ×105 SEM ×105 CP(%)r.Bias(%)SD ×105 SEM ×105 CP(%) Imputed-by-the-right-endpointmethod α010.0136.6155481497393.8-110.5158901513794.2-97.0153271542594.6 α020.02-8.8239902269796.02.5237082512297.044.4297322845295.2 α030.01400.1279262676895.89.1187911713693.8-80.3143851350093.6 α120.01100.0278982637194.6199.2282712583392.8172.0291322638793.8 α320.01274.6662615653893.6121.0355623382994.4125.5298792619394.2 β010.004-6.6999790.0-8.7899693.4-6.3959795.2 β020.008-0.913713994.0-2.812914095.8-0.314314996.0 β030.004-2.8969893.0-5.712111691.0-9.514214494.4 β120.0046.412012597.26.212012798.45.812412596.8 β320.0040.623421896.87.620418896.49.517716995.0 σ20.01631.47287854597.6705.26851798296.2714.87217754090.4 Proposedmethod α010.0121.6155861513494.0-56.8162681528594.0-92.4155301548794.2 α020.02-0.6243162283395.8-18.9241162516296.6104.0300082845895.0 α030.01337.6279432692495.4-9.4186291726594.0-83.6142791360094.2 α120.0143.2276532645294.6175.4276502597393.8102.5284812625193.2 α320.01296.4671965702893.4140.3365133426194.8170.0300372614593.8 β010.004-2.510610191.8-4.5959994.6-2.310010095.8 β020.0083.814614695.62.013314898.63.614815597.0 β030.0041.310210293.4-0.312412194.0-3.014915095.4 β120.0040.311211996.00.811612298.01.111712196.4 β320.0042.523222297.28.520418996.49.717817095.6 σ20.01838.48153921794.2852.37505844693.8791.27447774589.2
Table3Empiricalresultsfortheaveragesoftherelativebias(r.Bias)andstandarderrors(SEM),standarddeviation(SD),andcoverageprobability(CP)fortheregressionparameters andvarianceparameterofthelog-normalfrailtybasedonthe‘imputed-by-the-right-endpoint’and‘proposed’methodswhenthetypeofregressioncoefficientsis‘acc’underthree typesofLTFproportions(‘low’,‘moderate’,and‘high’) Low(LTF(%)=22.3)Moderate(LTF(%)=34.2)High(LTF(%)=47.6) ParameterTruevaluer.Bias(%)SD ×105 SEM ×105 CP(%)r.Bias(%)SD ×105 SEM ×105 CP(%)r.Bias(%)SD ×105 SEM ×105 CP(%) Imputed-by-the-right-endpointmethod α010.013.3148481491595.2-20.8163131502992.0-63.6156511514495.0 α020.01140.0229102236094.023.4248522438395.09.0261692802097.2 α030.0179.0253962662296.2115.9181321719492.8-4.9142621314893.0 α120.0125-100.2272532639794.410.5301462671593.074.9258812584495.4 α320.01479.4598365092992.0-108.0371183360995.2125.6260692502794.6 β010.004-11.6898689.4-10.8928788.6-11.3868789.2 β020.004-1.2839296.4-3.9899495.2-0.510210192.8 β030.004-8.6838591.4-10.310910390.8-11.614913389.4 β120.0053.713313096.02.312512696.23.013112996.0 β320.0040.419719995.86.617217195.23.413915196.2 σ20.01878.98926882292.6734.86989775491.4775.57514723187.6 Proposedmethod α010.0129.1151611502495.211.9164491511191.8-99.5157041516994.8 α020.01126.2229202243593.8-49.7251792448895.2-34.2264952821897.4 α030.019.4267682677995.898.7179481728793.81.1140131319193.8 α120.012521.2268322640495.244.7299342656792.852.4257812586195.6 α320.01280.8607295175391.4-181.5368593379695.4173.3268192505693.2 β010.004-7.5959092.2-6.8959090.8-8.1899093.0 β020.0044.1889697.21.8949996.24.610610593.4 β030.004-3.9888993.6-4.111610892.8-6.315413890.6 β120.005-0.912412496.4-2.011712297.0-1.212312495.4 β320.0042.319520095.86.917117295.63.813715297.2 σ20.01999.29282912493.0936.48781821988.6873.08002740485.6
