Computational Statistics and Data Analysis 55 (2011) 1726–1735

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Computational Statistics and Data Analysis

journal homepage: www.elsevier.com/locate/csda

Estimating the mean of a mark variable under right censoring on thebasis of a state functionHong-Bin Fang a,∗, Jiantian Wang b, Dianliang Deng c, Man-Lai Tang d

a Division of Biostatistics, University of Maryland Greenebaum Cancer Center, Baltimore, MD 21201, USAb Department of Mathematics, Kent University, Union, NJ 07083, USAc Department of Mathematics and Statistics, University of Regina, Regina, SK S4S 0A2, Canadad Department of Mathematics, Hong Kong Baptist University, Hong Kong

a r t i c l e i n f o

Article history:Received 16 August 2010Accepted 30 October 2010Available online 5 November 2010

Keywords:Quality-adjusted lifetimeKaplan–Meier estimatorLifetime medical costMark variableMartingaleSurvival analysis

a b s t r a c t

A mark variable is a generalization of measurements such as lifetime medical costs andquality-adjusted lifetimes. Recently, analysis of themark variable has generated significantinterest as an important component in health treatment evaluation. In this paper, a novelapproach to estimating the mean of the mark variable under right censoring is proposed.The proposed estimator is of amuch simpler form thanmost existing estimators advocatedin the literature. Theoretical analysis and simulation studies indicate that the proposedestimator has practical applications due to its simplicity and accuracy. A real data set, froma Multicenter Automatic Defibrillator Implantation Trial (MADIT), is used to illustrate theproposed methodology.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

In the evaluation of clinical therapies, it has been increasingly recognized that a patient’s state history process is moreinformative than a simple endpoint measurement. Let T be the terminal event time (e.g. death) and {V (t), t ≥ 0} the statehistory process satisfying V (t) = 0 for t ≥ T . The measure Γ =

∞

0 V (t) dt is referred to as the mark variable, in line withthe terminology used in the theory of point processes (Huang and Louis, 1998). Let S(x) = P(Γ > x) and µ = E(γ ) bethe mark survival function and the mean of the mark variable Γ , respectively. The primary interest in many biostatisticalapplications is in estimating the mark survival function S(x) and the mean of the mark variable µ. Examples from theliterature include estimations of quality-adjusted survival functions and lifetime medical costs (Glasziou et al., 1990, 1998;Zhao and Tsiatis, 1997; Lin et al., 1997).

A patient’s state history {V (t), t ≥ 0} is observed until terminal time T or censoring time C (e.g. the patient is lost tofollow-up). Throughout this paper, we assume that censoring time C is independent of terminal time T and the state processV (t). The presence of censoring complicates the estimations of S(x) and µ. When V (t) = aI(T > t), where a is a constantand I(·) is the indicator function, mark variable Γ is proportional to the terminal time T , and the estimation of S(x) canbe readily solved by applying the classical Kaplan–Meier estimator. However, S(x) is generally not identifiable due to theinvolvement of an ‘‘induced’’ informative censoring mechanism (Lin et al., 1997; Huang and Lovata, 2002). Furthermore,µ is not identifiable even in a simple situation where V (t) = aI(T > t). For this reason, one has to choose a reasonableartificial endpoint L and focus on the truncated mark variable Γ (L) =

L0 V (t) dt , and µ(L) = E{Γ (L)}, the mean of the

corresponding mark variable up to time L.

∗ Corresponding author. Tel.: +1 410 706 4103; fax: +1 410 706 8548.E-mail address: hfang@som.umaryland.edu (H.-B. Fang).

0167-9473/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.csda.2010.10.028

H.-B. Fang et al. / Computational Statistics and Data Analysis 55 (2011) 1726–1735 1727

Let K(t) = P(C > t) be the survival function of censoring time C . The artificial endpoint L should be chosen such thatK(L−) > 0. Current estimating methods for µ(L) can be generally classified into two categories, namely, direct and indirectestimating methods. The indirect estimating method is to estimate the truncated mark survival function SL(x) = P{Γ (L) >x} first, and then obtain the estimation of µ(L) = E[Γ (L)] via the relationship E[Γ (L)] =

∞

0 SL(x) dx. On the basis of theInverse Probability of Censoring Weighting (IPCW) technique, various estimators of the truncated mark survival functionSL(x) have been proposed (e.g., Korn, 1993; Zhao and Tsiatis, 1997; Strawderman, 2000).

Direct estimating methods include the estimation of a mean quality-adjusted survival time under progressive statemodels, advocated by Glasziou et al. (1990), the estimation of amean lifetimemedical cost, proposed by Lin et al. (1997), andthe simpleweighted estimation and its refinements, suggested by Bang and Tsiatis (2000); Zhao and Tsiatis (2000); Zhao andTian (2001). It is noted that the estimator proposed by Glasziou et al. (1990) works well only when the data fit a progressivestate model. The estimator proposed by Lin et al. (1997) is consistent when censoring occurs solely at the boundaries of thepartition intervals. The simple naive estimator derived by Bang and Tsiatis (2000) is not efficient due to the fact that it isbuilt only on complete observations,while its refined versions inevitably increase theoretical and computational complexity.Zhao et al. (2007) showed the equivalent relationship among the estimators.

The majority of existing estimators for µ(L) are based on the endpoint measure Γ (L) rather than the whole state historyprocess. The existing estimators may not take advantage of all observed information. In this paper, a novel approach toestimating µ(L), based upon the state history process, is proposed. Briefly, an estimator for µ(L) such as µ(L) =

L0 H(t)dt

where H(t) is a simpleweighted estimator ofH(t) = E{V (t)}, referred to as the state function, is proposed. As demonstratedby our theoretical results and numerical studies, the proposed estimator µ(L) establishes a good compromise betweensimplicity and efficiency.

The remainder of this paper is organized as follows: Section 2 proposes a new estimator with its asymptotic properties;Section 3 illustrates the proposed method with a real data set from aMulticenter Automatic Defibrillator Implantation Trial(MADIT); Section 4 gives comparisons with some existing estimators and simulation results; and Section 5 provides theconclusion and discussion. Theoretical proofs are presented in the Appendix.

2. The proposed mean estimation of a mark variable

With regard to the state process {V (t), 0 ≤ t ≤ L}, defineH(t) = E{V (t)} and refer to it as the state function. As for the nindividuals under study, let {Vi(t), t ≥ 0} be the ith individual’s state process and the ordered pair (Ti, Ci) be the individual’sterminal time and censoring time. Define the observation Xi = min(Ti, Ci) and the censoring indicator δi = I(Xi = Ti) fori = 1, 2, . . . , n. It is clear that the state of the ith individual at time t can be observed if, and only if, Ci ≥ t for i = 1, 2, . . . , n.Hence, a natural estimator for the state function H(t) can be given by

H(t) =1n

n−i=1

I(Xi ≥ t, δi = 0)

K(t)Vi(t) =

n−i=1

{I(Xi ≥ t, δi = 0)Vi(t)}/{nK(t)}, (2.1)

where K(t) is the Kaplan–Meier estimate for K(t), the survival function of censoring time C , based on the data {Xi, δi =

0; i = 1, . . . , n}. Define Γi =

∞

0 Vi(t)dt and Γ Li =

L0 Vi(t)dt where L is the endpoint to be chosen such that K(L−) > 0.

Recall S(x) = P(Γi > x) and µ(L) = E(Γ Li ). We propose an estimator for µ(L) as

µ(L) =

∫ L

0H(t) dt =

1n

n−i=1

∫ L

0

I(Xi ≥ t, δi = 0)

K(t)Vi(t)dt. (2.2)

Remark 1. Denote the survival function of the failure time T as ST (t). That is, ST (t) = P(T > t). If Vi(t) = I(Ti > t), thenH(t) = ST (t) and H(t) is simply the Kaplan–Meier estimator for ST (t).

Remark 2. It should be noted the entire state process is more informative than the cumulative measurement. For instance,spending $1000 in 10 days is not equivalent to spending $1000 in 1 day, even though the total costs are identical. Therefore,the state process is preferable to the cumulative measurement when modeling real situations. Accordingly, Estimator (2.2),which focuses on the state process (the price process in the context of medical costs), would be more useful than thosewhich focus only on a cumulative measurement such as Γ (L) =

L0 V (t)dt .

The consistency and asymptotic normality of the proposed estimator are asserted in the following theorems. The proofsare given in the Appendix.

Theorem 1. Let K(t) = P(C > t) and τ = inf{t : K(t) = 0}. Under the conventional assumption, censoring time C is indepen-dent of the failure time T and the state process V (t), and Estimator (2.2) is a consistent estimator of µ(L) for L < τ .

Theorem 2. Assume (i) L0 E(V 2(t))dt < ∞ and (ii) there exists a constant a > 0 such that K(L) > a. Then, as n → ∞,

√n[µ(L) − µ(L)] → N(0, σ 2

µ) in distribution, (2.3)

1728 H.-B. Fang et al. / Computational Statistics and Data Analysis 55 (2011) 1726–1735

Table 1Estimated mean medical cost up to Y years for the two therapy treatments (one year = 365 days).

Cost type Y = 1 Y = 2 Y = 3 Y = 4

Type 1 55325.6 60099.7 65808.7 83579.735566.6 45505.3 53977.7 58172.3

Type 2 119.72 242.56 344.81 441.6778.19 137.94 190.98 197.04

Type 3 844.94 1231.37 1712.36 3567.15651.31 1121.71 2542.38 2829.67

Type 4 1101.48 1970.85 2754.29 3472.88857.91 1428.12 1912.74 2269.13

Type 5 1823.28 3611.63 5583.42 7416.472365.89 4438.43 6121.71 7632.00

Type 6 322.21 419.01 747.75 1005.19312.94 398.23 1020.00 1882.19

The first number in each cell is the estimated mean cost for the ICD group while the second number is the estimated mean cost for the conventional group.

Table 2Estimated survival probability and estimated total mean medical cost up to Y years for the two therapy treatments (one year = 365 days).

Treatment Y = 1 Y = 2 Y = 3 Y = 4

ICD Survival probability 0.96341 0.90614 0.88457 0.79745Mean cost 59221.5 65746.7 70862.1 78934.3(Std. deviation) (45496) (47145) (48518) (48100)

Conventional Survival probability 0.76926 0.68971 0.58919 0.52607Mean cost 42248.3 57125.9 68538.1 73360.1(Std. deviation) (79013) (96416) (99541) (100530)

where

σ 2µ =

∫ L

0

∫ L

0

CV (s, t)K(t ∧ s)

+1 − K(t ∧ s)K(t ∧ s)

H(t)H(s)dtds, (2.4)

and CV (t, s) = E[V (t)V (s)] − H(t)H(s) is the covariance function of V (t) and V (s).

3. An actual example

In Section 3, we use the Multicenter Automatic Defibrillator Implantation Trial (MADIT) data to illustrate our methodproposed in Section 2. This experiment was fully sequential and randomized in order to evaluate the effectiveness of anImplantable Cardiac Defibrillator (ICD) in preventing sudden death in high risk patients. In MADIT, a total of 181 patientsfrom 36 centers in the United States were enrolled. Of the 181 patients, 89 were assigned to the ICD group and 92 wereassigned to the conventional therapy group. The data were heavily censored. Specifically, 57 out of 89 patients in the ICDgroupwere censoredwhile 77 out of 92 patients in the conventional therapy groupwere censored (seeMushlin et al. (1998)for more details). The MADIT data set consists of a patient’s ID code, treatment code (1 for ICD and 0 for conventional),observed survival time in days, death indicator (1 for death, 0 for censored), cost type, and daily cost from the start to thecompletion of the trial. There are six cost types: Type 1 is for hospitalization and emergency department visits; Type 2 isfor outpatient tests and procedures; Type 3 is for physician/specialist visits; Type 4 is for community services; Type 5 is formedical supplies, and Type 6 is for medications.

We use daily cost as the price process V (t). The initial day in this data set was coded as 0, and the days prior to the initialday were coded as negative. Regarding patients whose starting days were negative, we use time translation to adjust theirstarting days to 0. Using Estimator (2.1), we are able to obtain the estimate for the mean daily cost H(t) for t = 0, 1, . . . , L(L ≤ τ ) with τ = 1812. By accumulating the daily costs up to time L, we obtain the estimatedmean cost µ(L). Table 1 showsthe results.

As observed in Table 1, among the six cost types, Type 1 cost (initialization cost) is dominant. The initial cost of anICD treatment is much higher than that for a conventional treatment. Also, an ICD treatment is generally more expensivethan a conventional treatment during hospitalization and emergency department visits, outpatient tests and procedures,physician/specialist visits, and community services (i.e., Types 1–4). It is interesting to note the cost ofmedical supplies withregard to (i.e., Type 5 cost) a conventional treatment is higher than the cost of an ICD treatment. The cost of medications(i.e., Type 6) in the ICD treatment is higher than that of the conventional treatment during short term treatment (e.g., ≤2years). However for long term treatment (e.g., ≥3 years), the cost of medications during a conventional treatment is higherthan that during an ICD treatment.

Moss et al. (1996) showed that the adoption of an implantable cardiac defibrillator would lead to improved survival, ascompared to conventional medical therapy. Table 2 presents survival probabilities and the total medical costs for one year,two years, three years and four years. ICD treatment significantly increases a patients’ survival probability by approximately

H.-B. Fang et al. / Computational Statistics and Data Analysis 55 (2011) 1726–1735 1729

20% over conventional treatment while the cost difference between the two groups over a period of 4 years, on average,is around $5600. Taking into consideration costs saved per life year, we have reason to believe that the ICD treatment iseconomically attractive. Because clinical conditions among patients were different, it is not surprising that the estimatedvariations of medical costs are relatively large. However, Table 2 shows that the standard deviations of medical costs foran ICD treatment have been reduced by half, as compared to conventional treatment, which will be of benefit to medicaldecision makers and insurance policy providers.

4. Comparison of existing estimators

In Section 4, we provide comparisons between the proposed estimator and some existing estimators in order to estimatethe mean of a mark variable. Simulation studies are conducted following a review of the existing estimators.

4.1. Review of existing estimators

In the literature, themajority ofmethods for estimating themark survival function S(x) and themean of themark variableµ(L) are based on weighted estimation. For a given n, and individual observations {Vi(t), (Xi, δi); i = 1, 2, . . . , n}, with theIPCW technique, the weighted estimators for S(x) and µ(L) can be expressed, respectively, as

Sw(x) =1n

n−i=1

I{Xi ≥ ξi(x), δi = 0}

K{ξi(x)}I(Γi > x), (4.1)

and

µw(L) =1n

n−i=1

I{Xi ≥ ξi(L), δi = 0}

K{ξi(L)}Γ Li , (4.2)

where Γi =

∞

0 Vi(t)dt and Γ Li =

L0 Vi(t)dt , where L is the endpoint to be chosen such that K(L−) > 0. K(·) is the Kaplan–

Meier estimate for K(t), the survival function of the censoring time variable C . Functions ξi(x) and ξi(L) are chosensuch that I{Xi ≥ ξi(x), δ1 = 0}I(Γi > x) and I{Xi ≥ ξi(L), δi = 0}Γ L

i are observable. When ξi(x) = Ti and ξi(L) = Ti, theestimators in (4.1) and (4.2) are called naive weighted estimators, denoted by Snw(x) and µnw(L), respectively. Since the naiveweighted estimator µnw(L) is identical to the estimator proposed by Bang and Tsiatis (2000), we denote it as µbt(L) and callit the simple Bang–Tsiatis estimator. To improve upon the efficiency of the simple Bang–Tsiatis estimator, Bang and Tsiatis(2000) proposed a partition version of the simple Bang–Tsiatis estimator (denote it as µbtp(L)), and Zhao and Tian (2001)added a complicated adjusted term. However, the resulting estimators become far more complicated and, thus, diminishthe practicality of the application.

If ξi(x) = Ti ∧ si(x) and

si(x) =

infs :

∫ s

0Vi(t)dt ≥ x

if

∫ Xi

0Vi(t)dt ≥ x,

∞ otherwise,

then the estimators in (4.1) and (4.2) become sharp weighted estimators, denoted by Ssw(x) and µsw(L). It is noted that naiveweighted estimators weigh complete observations while sharp weighted estimators weigh sufficient information. For thisreason, a sharpweighted estimator is usuallymore efficient than its naive counterpart (see, Zhao and Tsiatis, 1997). It shouldbe pointed out that a weighted estimator cannot generally guarantee its consistency. Oneway to ensure the consistency of aweighted estimator is to truncate the failure time, that is, replace T by T = T ∧L, where L is an artificial endpoint. Denote thetruncated version of the sharp weighted estimator Ssw(x) by SLsw(x). Also note that the sharp weighted estimator SLsw(x) maynot be appropriate as it does not guarantee that SLsw(x) ≤ 1. To overcome this deficiency, Zhao and Tsiatis (1997) proposedan estimator for K(t) based on the data {X ′

i , δ′

i = 0; i = 1, . . . , n}, where X ′

i = min(T ′

i (x), Ci), δ′

i = I{X ′

i = T ′

i (x)}, andT ′

i (x) = Ti ∧ s′i(x) with s′i(x) = inf{s : s∧L0 Vi(t)dt ≥ x}. Hence, the corresponding estimators for SL(x) and µ(L) are

SLzt(x) =1n

n−i=1

I{X ′

i ≥ T ′

i (x), δ′

i = 0}

Kx{T ′

i (x)}I(Γ L

i > x), (4.3)

and

µzt(L) =

∫∞

0SLzt(x)dx =

∫∞

0

1n

n−i=1

I{X ′

i ≥ T ′

i (x), δ′

i = 0}

Kx{T ′

i (x)}I(Γ L

i > x) dx, (4.4)

respectively. In the expression for estimator SLzt(x), the estimated censoring survival function is indexed by x, indicating thatthe estimator of K(t) depends upon x.

1730 H.-B. Fang et al. / Computational Statistics and Data Analysis 55 (2011) 1726–1735

When the clinical course canbe expressed in a progression of states, themark variable canbewritten asΓ =∑J

j=1 wjT (j),where wj is the health index and T (j) is the time from the beginning of the study to the end of health state j (with T (J) theoverall survival time). Glasziou et al. (1990) proposed the following estimator for µ(L):

µG(L) =

J−j=1

wjµj(L), (4.5)

where µj(L) = L0 S(j)(t)dt and S(j)(t) is the Kaplan–Meier estimator for S(j)(t) = P(T (j) ≥ t).

On the basis of the partition of the observed interval of a patient’s lifetime, Lin et al. (1997) proposed three estimatorsfor estimating the mean medical cost. Let Am be the average terminal cost for those subjects who have been observed asdeceased within the mth interval and Sm be the Kaplan–Meier estimator for the survival function of terminal time T at thebeginning of themth interval. As in Zhao et al. (2007), the three estimators, µLinT, µLinA, and µLinB, respectively, are given asfollows:

µLinT =

M+1−m=1

Am(Sm − Sm+1), (4.6)

and

µLinA =

M+1−m=1

EmSm, (4.7)

where Em is the average cost during the mth interval for the subjects who are alive at the beginning of the mth interval; ifthose who are censored during the mth interval are excluded in the calculation of Em, then the resulting estimator in (4.7)is µLinB.

When the state process V (t) = Rδ(t − T ), that is, a Dirichlet type process with a random variable R and a Dirichletfunction δ(t), then in such an extreme situation, the five estimators µnw(L), µsw(L), µzt(L), µbt(L) and µ(L) are all identical.In this case, a random variable governs the entire state process. In the context of a lifetime medical cost, the medical cost isimposed only at the terminal time. In such a situation, the estimation of the mean medical cost, or in general, for the meanof the mark variable, can be readily solved. And among the estimators, the simple naive Bang–Tsiatis estimator, µbt(L), ispreferred due to its simpler form. Unfortunately, in actual clinical situations, medical costs are usually imposed during theentire treatment period, not limited to the terminal time.

With regard to the progressive state model, Glasziou’s estimator µG(L) is obviously the most efficient estimator due tothe optimality of the Kaplan–Meier estimator. In a situation where V (t) = cI(T > t), which implies that the medical cost isproportional to the hospitalization cost in the context of a lifetime medical cost, our proposed estimator µ(L) is identical toµG(L). As for the general progressive state model, if the subtle difference induced by different estimations of the censoringsurvival function K(t) is negligible, then µ(L) = µG(L). Therefore, in the progressive state model, the proposed estimator isjust about the most efficient.

To ensure consistency, the estimators µbt(L), µbtp(L), and µzt(L) truncate the actual terminal time T to T ∧ L. Suchtruncation treats all patients as deceased after time L and, thus, is not reasonable and will create inefficiency. Notice thatGlasziou’s estimator µG(L) and the three estimators proposed by Lin et al. (1997) do not involve such truncation. Theaforementioned estimators are the most efficient if the data fit the prescribed model and the partition of the time linematches the censoring times perfectly.

Let Ti = Ti ∧ L. In terms of the state process Vi(t), the simple Bang–Tsiatis estimator can be written as

µbt(L) =1n

n−i=1

I(Xi ≥ Ti, δi = 0)

K(Ti)

∫ Ti

0Vi(t) dt. (4.8)

Even though the theoretical comparison of the efficiency of the proposed estimator µ(L) with µzt(L) proves to be elusive,Theorem 3 shows that our proposed estimator µ(L) is more efficient than the simple Bang–Tsiatis estimator µbt(L).

Theorem 3. Let AV denote asymptotic variance. Assume that failure time T and the mark variable Γ (L) = L0 V (t)dt are

continuous. Then we have

AV [√n{µ(L) − µ(L)}] ≤ AV [

√n{µbt(L) − µ(L)}]. (4.9)

4.2. Numerical studies

Zhao et al. (2007) showed that the simple Bang–Tsiatis estimator µbt(L) is equivalent to µLinT while estimators µzt(L),µLinA, µLinB, andµbtp(L) are equivalent under various conditions. Consequently, in Section 4.2 we conduct numerical studiesfor two scenarios. The first scenario compares the proposed estimator µ(L)with the Zhao–Tsiatis type estimator µzt(L), and

H.-B. Fang et al. / Computational Statistics and Data Analysis 55 (2011) 1726–1735 1731

Table 3Comparison of the proposed estimator and the Zhao–Tsiatis type estimator.

L True Estimated Bias MSE Time cost

40 8.9250 8.9569 0.3019 0.4882 4459.0362 0.1111 0.5225 41469

60 18.0408 18.1262 0.0854 1.3001 64818.2785 0.2377 1.5485 61715

80 28.6276 28.7727 0.1451 2.3687 82228.9607 0.3331 3.1512 60968

90 33.5537 33.7275 0.1688 3.1655 89733.8296 0.2759 4.4968 57789

The first number in each cell is the value obtained from the proposed Estimator (2.2) and the second number is the corresponding value obtained from theZhao–Tsiatis type Estimator (2.7). The numbers in the ‘Time cost’ column are the computational time cost (in seconds) obtained through C++ by usingthese estimators. The results are obtained from 1000 simulations with sample sizes of 400.

the second scenario compares the proposed estimator with the simple Bang–Tsiatis estimator µbt(L). Two different statemodels (namely, progressive and nonprogressive state models) are examined, respectively, in the following scenarios:

Scenario 1. This study compares the proposed estimator µ(L) and the Zhao–Tsiatis type estimator µzt(L) using aprogressive state model presented in Gelber et al. (1989). It is assumed that patients entering the study initially experiencesome toxicity and then enjoy a period of good health until they relapse. Let T1 and T2 represent the times from treatmentinitiation to the end of toxicity and to disease relapse, respectively. The quality-adjusted lifetime, up to endpoint L in thiscase, is Γ (L) = max(T2 ∧ L − T1, 0). Assume T1 is uniformly distributed on [0, Tox] ([0, 72]), T2 is exponentially distributedwith hazard λ = 1/120, and censoring time C follows a uniform distribution on [0, 94]. Further, assume the quality of lifefunction V (t) = I(t < max(T2 ∧ L− T1, 0)) as defined in Gelber et al. (1989). That is, the quality of life in the period withouttoxicity is regarded as perfect. Thus via a few simple calculations, we have

µ(L) =

1

Tox1λ2

(1 − e−λL) −L

λToxe−λL, if L < Tox,

1Tox

1λ2

(1 − e−λTox) −1λe−λL, if L ≥ Tox.

The comparison results for estimators µ(L) and µzt(L), for this data set, are reported in Table 3. Table 3 shows that theperformances of the two estimators are nearly identical. However, the time cost for µzt(L) is significantly greater than thatfor µ(L). For each value of x, the estimated censoring survival function K(·) in µzt(L) must be calculated on the basis of thedata {X ′

i , δ′

i = 0; i = 1, . . . , n}, where X ′

i = min(T ′

i (x), Ci) and δ′

i = I{X ′

i = T ′

i (x)}. In other words, the estimator µzt(L) isfar more computationally complicated than the estimator µ(L).

Scenario 2. In this scenario, we examine a simple nonprogressive state model. Assume patients come from two differentpopulations. The quality-adjusted life in population 1 can be described by progressive model D1 and the quality-adjustedlife in population 2 by progressive model D2. We assume the quality of life functions V (t) = I(T1 < t < T2) in D1 andV (t) = 1/2I(0 ≤ t < T3) + I(T3 ≤ t < T4) in D2, where T1 to T4 are random variables. When T1 > T2 in D1 or T3 > T4in D2, V (t) is defined to be zero. Suppose a patient comes from population 1 with a probability of 1/3 and another patientcomes from population 2 with a probability of 2/3. Assume T1 ∼ U[0, 60], T2 ∼ exp(λ), T3 ∼ exp(β) and T4 ∼ U[48, 96].According to this data model, the true state function H(t) is given by

H(t) =

13

t60

e−λt−

13e−βt

+23

−1

3 · 48 · β(e−48β

− e−96β), t < 48

13

t60

e−λt−

23

−t

3 · 48−

13 · 48 · β

e−βt

+2396 − t48

−1

3 · 48 · βe−96β , 48 ≤ t ≤ 60

13e−λt

−

23

−t

3 · 48−

13 · 48 · β

e−βt

+2396 − t48

−1

3 · 48 · βe−96β , 60 < t ≤ 96

13e−λt , t > 96.

The true value of µ(L) can be easily obtained via the relation µ(L) = L0 H(t)dt . Let λ = 1/120, β = 1/12 and C ∼

U[50, 100]. We conduct a numerical study with 1000 replications and sample sizes of 200. The results are shown in Table 4.For small endpoint L (e.g., L ≤ 50), there is no censoring. In this case, both the simple Bang–Tsiatis estimator and the

proposed Estimator (2.2) reduce to an empirical estimator. In terms of bias, we notice from the simulation that theperformance of the proposed estimator would be better than the performance of the simple Bang–Tsiatis estimator.Moreover, the Mean Square Error (MSE) of our proposed estimator is much smaller. As L increases, it is expected that theMSEs of both estimators will increase as well. As expected, the MSE of the simple Bang–Tsiatis estimator is much largerand increases much faster than that of the proposed estimator, since the simple Bang–Tsiatis estimator is based only on theobserved total patientmedical costs, while the proposed estimator is based on the entiremedical cost history. Asmentioned,spending $1000 in 10 days is not the same as spending $1000 in one day. Hence, there is reason to believe that our proposed

1732 H.-B. Fang et al. / Computational Statistics and Data Analysis 55 (2011) 1726–1735

Table 4Comparison of the proposed estimator and the Bang–Tsiatis estimator.

L True Estimated Bias MSE Comp. time (in second)

50 34.5869 34.5250 −0.08180 1.75344 22134.5570 −0.02997 4.32175 23

65 45.5042 45.5107 0.00749 3.33623 24645.5218 0.01656 11.4496 32

80 53.1306 53.0576 −0.00732 5.55524 26752.9144 −0.21621 25.4262 41

90 56.3001 56.3424 0.04220 8.22867 27256.7138 0.41361 55.5133 47

95 57.3143 57.3072 −0.00709 8.15427 26357.4594 0.14512 87.6991 51

The first number in each column is the estimated mean of the corresponding value obtained by using the proposed estimator and the second number isthe corresponding value obtained by using the Bang–Tsiatis estimator. The results are obtained from 1000 simulations with sample sizes of 200.

estimator retains far more efficiency, especially when L is large. Since the simple Bang–Tsiatis estimator has a simple formand does not involve integration, we are not surprised that it requires less computation time.

5. Discussion

The estimations of the lifetime medical costs and the quality-adjusted lives are crucial in the treatment evaluation ofchronic diseases (see, e.g., Gelber et al., 1995). Glasziou’s Estimator (4.5) is optimal if the data are attained from a progressivemodel. Therefore, under progressive health state models Glasziou’s estimator is highly recommended (see, Zhao and Tsiatis,2000). Yet, not all clinical data follow progressive models and, thus, it is necessary to continually search for other reliableestimators.

Already, several weighted estimators have been proposed. To guarantee the consistency of a weighted estimator, theterminal time T is usually truncated by T ∧ L. However, such truncation may result in target deviation and loss of efficiency.Therefore, the simple Bang–Tsiatis estimator is not very efficient. Its revised versions, such as the partition version µbtp(L)and the version adjusted by Zhao and Tian (2001), are quite cumbersome. Some sophisticated weighted estimators, such asthe Zhao–Tsiatis type estimator µzt(L), are still too complicated for practical application.

Obviously, the entire health state history is more informative than the summary of measurements at the endpoint. Asshown in Laan and Hubbard (1999), the most efficient estimator is the estimator which extensively utilizes the state historyinformation. It is worth noting that our proposed estimator is (i) based on the entire health state history; (ii) of a simpleform; and (iii) truncation-free. Hence, in view of the aforementioned reasons, the proposed estimator is better than manyexisting estimators. Our numerical studies are quite supportive of such a conclusion.

Quality-adjusted lifetime is a prototype of amark variable. In a synthesis of survival and quality-adjusted lifetime (qualityof life), Cox et al. (1992) raised the following question, ‘‘Can we keep it simple?’’. The answer for the estimation of the marksurvival function may, in general, be negative. However, the answer for the estimation of the mean of the mark variablemay be positive and our proposed estimator is a possible solution. The main concern of medical cost analysis is the averagemedical cost over time. The proposed estimator, as shown in the numerical studies and the theoretical analysis, performswell under most circumstances and is reasonable for practical applications.

Acknowledgements

The authors are sincerely grateful to Dr. W. J. Hall and Dr. Hongwei Zhao for providing the MADIT cost data, and toDr. Yingfu Li for supplying references. The authors also thank the associate editor and reviewers for their constructivecomments.

Appendix

Proof of Theorem 1. The consistency of Estimator (2.2) follows that of Estimator (2.1), which can be proven in astraightforward fashion. In fact, for t < τ ,

H(t) − H(t) =1n

n−i=1

I(Xi ≥ t, δi = 0)

K(t)Vi(t) − H(t)

=1

K(t)1n

n−i=1

{Vi(t)I(Xi ≥ t, δi = 0) − H(t)K(t)} +1n

n−i=1

Vi(t)I(Xi ≥ t, δi = 0)

1

K(t)−

1K(t)

= I + II.

The independence between the censoring time and health states implies E{Vi(t)I(Xi ≥ t, δi = 0)−H(t)K(t)} = 0. Accordingto the Law of Large Numbers, the first term converges to zero in probability and almost surely. Due to the consistency of the

H.-B. Fang et al. / Computational Statistics and Data Analysis 55 (2011) 1726–1735 1733

Kaplan–Meier estimator, the second term converges to zero in probability and also almost surely by Theorem 1.1 in Stuteand Wang (1993). Thus, the Egoroff Theorem (Bartle, 1995, p. 74) implies H(t) converges to H(t) almost uniformly in [0, L]with probability 1 for L < τ . Therefore, µ(L) converges to µ(L) in probability. �

Proof of Theorem 2. Note that

H(t) =1n

n−i=1

I(Xi ≥ t, δi = 0)

K(t)Vi(t)

and

µ(L) =1n

n−i=1

∫ L

0

I(Xi ≥ t, δi = 0)

K(t)Vi(t)dt =

1n

n−i=1

µi(L)

where

µi(L) =

∫ L

0

I(Xi ≥ t, δi = 0)

K(t)Vi(t)dt, i = 1, 2, . . .

are random variables.Now suppose survival function K(s) is known. Then, we have

var(µi(L)) = E[µi(L)]2 − [Eµi(L)]2

= E∫ L

0

∫ L

0

I(C ≥ t)K(t)

V (t)I(C ≥ s)K(s)

V (s)dtds

−

E

∫ L

0

I(C ≥ t)K(t)

Vi(t)dt2

=

∫ L

0

∫ L

0E

I(C ≥ t)K(t)

I(C ≥ s)K(s)

V (t)V (s)dtds −

∫ L

0E

I(C ≥ t)K(t)

Vi(t)dt

2

=

∫ L

0

∫ L

0E

I(C ≥ t ∨ s)K(t)K(s)

V (t)V (s)dtds −

∫ L

0

EI(C ≥ t)

K(t)EV (t)

dt

2

=

∫ L

0

∫ L

0

K(t ∨ s)K(t)K(s)

E(V (t)V (s))dtds −

∫ L

0

∫ L

0

K(t)K(t)

K(s)K(s)

EV (t)EV (t)dtds

=

∫ L

0

∫ L

0

K(t ∨ s)K(t)K(s)

CV (s, t) +

K(t ∨ s)K(t)K(s)

− 1H(t)H(s)

dtds

=

∫ L

0

∫ L

0

CV (s, t)K(t ∧ s)

+

1

K(t ∧ s)− 1

H(t)H(s)

dtds.

Next, if survival function K(t) is unknown, it can be estimated by using the Kaplan–Meier estimator K(t) and

µ(L) =1n

n−i=1

∫ L

0

I(Xi ≥ t, δi = 0)

K(t)Vi(t)dt

=1n

n−i=1

∫ L

0

I(Xi ≥ t, δi = 0)

K(t)Vi(t)dt −

1n

n−i=1

∫ L

0

I(Xi ≥ t, δi = 0)K(t)

Vi(t)dt

+1n

n−i=1

∫ L

0

I(Xi ≥ t, δi = 0)K(t)

Vi(t)dt

=1n

n−i=1

∫ L

0

I(Xi ≥ t, δi = 0)

K(t)−

I(Xi ≥ t, δi = 0)K(t)

Vi(t)dt +

1n

n−i=1

∫ L

0

I(Xi ≥ t, δi = 0)K(t)

Vi(t)dt

=1n

n−i=1

∫ L

0

K(t) − K(t)

K(t)

I(Xi ≥ t, δi = 0)K(t)

Vi(t)dt +1n

n−i=1

∫ L

0

I(Xi ≥ t, δi = 0)K(t)

Vi(t)dt.

Therefore,

√n[µ(L) − µ(L)] = n−1/2

n−i=1

∫ L

0

K(t) − K(t)

K(t)

I(Xi ≥ t, δi = 0)K(t)

Vi(t)dt

+ n−1/2n−

i=1

∫ L

0

I(Xi ≥ t, δi = 0)K(t)

Vi(t)dt − E∫ L

0

I(Xi ≥ t, δi = 0)K(t)

Vi(t)dt

= I1 + I2.

1734 H.-B. Fang et al. / Computational Statistics and Data Analysis 55 (2011) 1726–1735

Note that L0

I(Xi≥t,δi=0)K(t) Vi(t)dt, i = 1, 2, . . . , are i.i.d. random variables with mean µ(L) and variance σ 2

µ. According to theclassical Central Limit Theorem, we have

I2 → N(0, σ 2µ) in distribution.

Finally, according to the Cauchy–Schwartz Inequality, we have

E|I1| = E

n−1/2n−

i=1

∫ L

0

K(t) − K(t)

K(t)

I(Xi ≥ t, δi = 0)K(t)

Vi(t)dt

≤ n−1/2

n−i=1

∫ L

0E

K(t) − K(t)

K(t)

I(Xi ≥ t, δi = 0)K(t)

Vi(t)

dt≤ n−1/2

n−i=1

∫ L

0[E(K(t) − K(t))2]1/2

E

I(Xi ≥ t, δi = 0)

K(t)K(t)Vi(t)

21/2

dt

≤ Mn−1/2n−

i=1

∫ L

0n−1

EV 2(t)K 3(t)

1/2

dt

≤ Mn−3/2n−

i=1

∫ L

0EV 2(t)dt (under assumption (ii) that K−1(t) is bounded)

≤ Mn−1/2→ 0 (under assumption (i))

where [E(K(t) − K(t))2] = O(n−2) is obtained from Phadia and Van Ryzin (1980). Therefore, the theorem follows. �

In order to prove Theorem 3, we require the following lemmas:

Lemma A.1. Define the observation Xi = min(Ti, Ci) and the censoring indicator δi = I(Xi = Ti) for i = 1, 2, . . . , n. Let K(t)and λc(t) be the survival function and hazard function of C. Set Fu = σ {I(Xi ≤ x, δi = 1), I(Xi ≤ u, δi = 0), x ≤ u, i =

1, . . . , n}. Then, Mci (s) = I(Xi ≤ s, δi = 0) −

s0 I(Xi ≥ u)λc(u)du is an Fs-martingale and

I(δi = 0)K(Ti)

= 1 −

∫∞

0

dMci (u)

K(u).

Lemma A.2. Let Mci (s) be defined as in Lemma A.1 and Mc(s) =

∑ni=1 M

ci (s). Let S(t) and K(t) be the Kaplan–Meier estimators

for S(t) = Pr(T > t) and K(t) = Pr(C > t), respectively. Then,

K(t) − K(t)K(t)

=

∫ t

0

dMc(u)

nK(u)S(u−).

Lemma A.3. Assume the continuous functions g(L, t) and f (L, t) are defined on L ∈ [0, τ ), t ∈ [0, L], and satisfy g(L, 0) ≤

f (L, 0), and g(L, L) ≤ f (L, L). Then, for t ∈ [0, L], g(L, t) ≤ f (L, t).

The proofs for Lemmas A.1 and A.2 can be found in Strawderman (2000) and Gill (1981). The proof for Lemma A.3 isstraightforward, and thus omitted.

Proof of Theorem 3. For any fixed t , let C ti (u) = I{Xi ∧ t ≤ u, δi = 0}. Then, using Lemma A.2 in Strawderman (2000) we

have that

M(c,t)i (u) = C t

i (u) −

∫ u

0I(Xi ≥ v)I(t ≥ v)λc(v) dv

is an Fu martingale satisfying dM(c,t)i (u) = I(t ≥ u)dMc

i (u). Denote Vi(t)I(Xi > t, δi = 0) as Hi(t) and define Wi(u) = u0 Vi(t)dt . By applying Lemmas A.1 and A.2, and using arguments similar to those in the proof of Theorem 2 in Strawderman

(2000), we are able to write µ(L) = n−1 ∑ni=1

L0 Vi(t)dt − n−1 ∑n

i=1 Di, where

Di =

∫ L

0Vi(t)dt

∫∞

0

dM(c,t)i (u)K(u)

−

∫ L

0

1n

n−j=1

I(Xi > t, δi = 0)

K(t)Vj(t)dt

∫ t

0

dMci (u)

K(u)S(u−)

≈

∫ L

0Vi(t)dt

∫∞

0I(t ≥ u)

dMci (u)

K(u)−

∫ L

0H(t)dt

∫ t

0

dMci (u)

K(u)S(u)

H.-B. Fang et al. / Computational Statistics and Data Analysis 55 (2011) 1726–1735 1735

=

∫ L

0Vi(t)dt

∫ t

0

dMci (u)

K(u)−

∫ L

0H(t)dt

∫ t

0

dMci (u)

K(u)S(u)

=

∫ L

0

∫ L

uVi(t)dt

dMc

i (u)K(u)

−

∫ L

0

∫ L

uH(t)dt

dMc

i (u)K(u)S(u)

=

∫ L

0

∫ L

uVi(t)dt −

∫ L

uH(t)dt/S(u)

dMc

i (u)K(u)

.

SinceWi(L) is F0 measurable,Wi(L) and Di are uncorrelated. Furthermore, the asymptotic variance of 1√n

∑ni=1{

L0 Vi(t)dt −

µ(L)} is var(Wi(L) − µ(L)). Hence,

AV{√n(µ(L) − µ(L))} − var(Wi(L) − µ(L))

= E∫ L

0

∫ L

uVi(t)dt −

∫ L

uH(t)dt/S(u)

2

I(Xi ≥ u, δi = 1)λc(u)K(u)

du

= E∫ Xi∧L,δi=1

0

Wi(L) − Wi(u) − S−1(u){µ(L) − µ(u)}

2 λc(u)K(u)

du.

On the other hand, using Theorem 2 in Strawderman (2000) we have

AV{√n(µbt(L) − µ(L))} − var(Wi(L) − µ(L))

= E∫

∞

0

[∫ L

0Vi(t)dt − S−1(u)E

∫ L

0Vi(t)dtI(Xi ≥ u, δi = 1)

]2

I(Xi ≥ u, δi = 1)λc(u)K(u)

du

= E∫ Xi∧L,δi=1

0

Wi(L) − S−1(u)µ(L)

2 λc(u)K(u)

du.

Using Lemma A.3, we haveWi(L) − Wi(u) − S−1(u){µ(L) − µ(u)}

2≤

Wi(L) − S−1(u)µ(L)

2.

Hence, we achieve the desired conclusion, AV{√n(µ(L) − µ(L))} ≤ AV{

√n(µbt(L) − µ(L))}. �

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