estimating medical care costs under conditions of censoring

Journal of Health Economics 23 (2004) 443–470

Estimating medical care costs underconditions of censoring

M. Raikoua,∗, A. McGuirea,b

a LSE Health and Social Care, London School of Economics and Political Science,Houghton Street, London WC2A 2AE, UK

b Division of Public Health King’s College, University of London, London, UK

Received 6 June 2002; accepted 31 July 2003

Abstract

A number of non-parametric estimators have been proposed to calculate average medical carecosts in the presence of censoring. This paper assesses their performance both in terms of biasand efficiency under extreme conditions using a medical dataset which exhibits heavy censoring.The estimators are further investigated using artificially generated data. Their variances are derivedfrom analytic formulae based on the estimators’ asymptotic properties and these are comparedto empirically derived bootstrap estimates. The analysis revealed various performance patternsranging from generally stable estimators under all conditions considered to estimators which becomeincreasingly unstable with increasing levels of censoring. The bootstrap estimates of variance wereconsistent with the analytically derived asymptotic variance estimates. Of the two estimators thatperformed best, one imposes restrictions on the censoring distribution while the other is not restrictedby the censoring pattern and on this basis the second may be preferred.© 2003 Elsevier B.V. All rights reserved.

JEL classification:C000; C100

Keywords:Cost analysis; Censoring; Survival analysis; Economic evaluation

1. Introduction

The issue of censoring has been addressed extensively both within the econometricsliterature and within the analysis of time to event data (Kiefer, 1988; Lancaster, 1990).As medical cost information is increasingly collected alongside clinical trials as the basis

∗ Corresponding author. Tel.:+44-207-955-6472.E-mail address:[email protected] (M. Raikou).

0167-6296/$ – see front matter © 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.jhealeco.2003.07.002

444 M. Raikou, A. McGuire / Journal of Health Economics 23 (2004) 443–470

for cost-effectiveness analysis the accommodation of censoring is becoming increasinglyimportant within this context. There are various types of censoring, such as right censor-ing or left censoring. Left censoring, which is not as common in clinical trials, involvesloss of information due to individual observations entering the study at different points ofprogression to endpoint. This paper does not address this issue. Right censoring occurswhenever some individuals are not observed for the full duration of interest which resultsin information being incomplete for these patients. In other words, an individual is rightcensored when his behaviour with respect to the variable under study is not observed forthe full time to the event of interest. Patients who are lost to follow-up, drop out of thestudy or are observed until the end of the study period without having reached the eventof interest are said to be right censored. Estimators of statistics of interest are biased if noaccount is taken of censoring with the bias increasing as the degree of censoring increases.Both parametric and non-parametric approaches have long been applied to the analysis oftime-to-event data yielding estimators that successfully account for the presence of cen-soring. It is only recently however that attention has turned to the issue of censored costdata.

Given the existence of consistent estimators of failure time statistics in the presence ofcensoring, initial attempts to adjust estimates of cost statistics for censoring were basedon application of traditional survival analysis techniques to cost data (e.g.Fenn et al.,1995, 1996). The assumption underlying the validity of these approaches is one of indepen-dence between the variable of interest and its censoring variable. This implies independencebetween time to event and time to censoring when failure time data are considered and in-dependence between cost at event and cost at censoring when cost data are analysed. Underrandom censoring, that is when censored data are censored at random, as is generally the casein medical studies, the assumption is valid with respect to time to event data but it is normallyviolated with respect to cost to event data. The violation is due to the lack of a common rate ofcost accrual over time among individuals, as patients who are in poorer health states generatehigher costs per unit of time and consequently are expected to generate higher cumulativecosts at both the failure time and the censoring time (Lin et al., 1997; Etzioni et al., 1999).This positive correlation implies that removal of certain observations from the sample due tocensoring affects the joint distribution of cost for the remaining observations in the sense thatat any point in time future cost expectation is statistically altered (from what it would havebeen in the absence of censoring) by censoring. As a result any analysis that does not modelthis dependency will lead to erroneous inferences. Consequently cost estimators based onsurvival analysis approaches such as the non-parametric Kaplan–Meier, the semiparametricproportional hazards regression or parametric models assuming distribution families such asthe Weibull and the exponential are normally inappropriate in the analysis of censored costdata.

Another two estimators have also been used to provide estimates of mean cost in thepresence of censoring which are referred to as “naıve” estimators in the literature becausethe first, referred to as the uncensored-cases estimator, only uses the uncensored cases inthe estimation of mean cost, while the second, referred to as the full-sample estimator,uses all cases but does not differentiate between censored and uncensored observations.Both these estimators will always be biased. The full-sample estimator is always biaseddownward because the costs incurred after censoring times are not accounted for whereas

M. Raikou, A. McGuire / Journal of Health Economics 23 (2004) 443–470 445

the uncensored-cases estimator is biased toward the costs of the patients with shortersurvival times because larger survival times are more likely to be censored (Lin et al.,1997).

Lin et al. (1997)acknowledge these difficulties and propose a method which attemptsto resolve these issues. They introduce two estimators of mean cost under conditions ofcensoring which rely on the study period being partitioned into a number of subintervalssuch that censored observations occur at the boundaries of these intervals. Under such cir-cumstances, the approach is shown to give consistent estimators of average cost and theassociated variances are analytically derived. Hence, the validity of the approach dependson the pattern of the censoring distribution being of such a form to allow censoring timesto correspond to the boundaries of the intervals of the partition. There is no a priori rea-son however to expect censoring to conform to any such pattern and therefore in mostapplications consistency will be violated to some degree. This limitation has led to a fur-ther set of estimators proposed byBang and Tsiatis (2000). Their estimators are shownto be consistent regardless of the censoring pattern and their variances are analyticallyderived.

This paper extends this earlier work. While the theoretical properties of these estimatorshave been studied byLin et al. (1997)and byBang and Tsiatis (2000), their applicationhas not been assessed under conditions of extreme censoring using real data.Etzioni et al.(1999), for example in their application of survival analysis techniques to censored costdata generated the censoring times according to an exponential distribution which resultedin 39% censoring in the data. In addition, the efficiency of the various estimators wasnot investigated and the Bang and Tsiatis estimators were not considered. In this paper, theestimators of mean cost proposed by both Lin et al. and Bang and Tsiatis and their variancesare investigated using a real clinical dataset which exhibits levels of censoring of 82%. Theestimators’ performance is further assessed using artificially generated data with varyingdegrees of censoring. In addition, the analytically derived variance estimators presented byLin et al. and Bang and Tsiatis are based on the assumption of asymptotic normality. To ourknowledge, no attempt has been made to determine the validity of this assumption whenthe estimators are applied to the smaller sample sizes observed in real medical data. Thisis undertaken in the present analysis by comparing variances derived using the bootstrapmethod1 with their respective asymptotically derived variances as presented by Lin et al.and Bang and Tsiatis.

The paper proceeds as follows. First, the set of non-parametric estimators proposedby Lin et al. and Bang and Tsiatis together with the assumptions underlying theirvalidity are presented. The main analysis, whose aim is to assess the estimators’ per-formance under extreme conditions, is presented in the following section which reportsthe results derived from the application of these estimators to a medical dataset withheavy censoring. A number of problems are identified within this part of the analysiswhich are subsequently investigated using subsets of the original data as well as an ar-tificially generated dataset. The final part of the analysis derives variance estimates us-ing the bootstrap approach as an alternative to the theoretically derived formulae for

1 The usual conditions for the validity of the bootstrap approach apply here as concern is with randomcensoring.


the asymptotic variance estimators. Some concluding remarks and suggestions are thengiven.

2. Analytical framework

2.1. General setting

The aim of the approaches presented below is to derive an estimate of the mean total costµ = E(M) and its variance over a specified period when the data is right censored, wherethe random variableM denotes the total cost for a patient during some specified timeTandEdenotes expectation. The distribution of the random variableT is assumed continuous over(0, L] whereL denotes the upper bound ofT, i.e. the maximum time for which each patientis evaluated. In that case,M is the total cost incurred by a patient up to a maximum ofLunits of time. If all patients were observed for a minimum ofL units of time then completeinformation onM would be available and the mean cost would be estimated by the averageof the costs for each patient. In most cases however cost information is incomplete dueto censoring. Defining therefore a potential time to censoring denoted byU and lettingTdenote the time to death, the observables from a study in the presence of censoring areX = min(T, U), i.e. the last contact date;δ = I(T ≤ U), whereI(·) is the indicator functiontaking the value of 1 when the argument is true (i.e. if the observation is uncensored) and zerootherwise; the cost accrued up to timeXand other intermediate cost history for each subject,i.e.MH(t) = {M(u), u ≤ t}, whereMH(t) denotes the cost history up to timet,M = M(T),with M(u) being the known accumulated cost up to timeu andu denoting points in timeat which cost information becomes available. The observable data forn individuals arethen the independent and identically distributed random vectors{Xi = min(Ti, Ui), δi =I(Ti ≤ Ui), MH

i (Xi)}, i = 1, . . . , n wherei identifies an individual.Within this setting a total of six non-parametric estimators of average cost over the study

period in the presence of censoring are considered. Two proposed byLin et al. (1997)referred to as LIN1 & LIN2 in the following sections and four proposed byBang andTsiatis (2000)referred to as simple weighted, partitioned, improved simple and improvedpartitioned estimators. These estimators together with the assumptions underlying theirvalidity are outlined below.

2.2. Lin et al. estimators

Lin et al. (1997)present two approaches to estimate the mean total cost over the period(0, L]. The first requires information on a patient’s intermediate cost history whereas thesecond only uses the observed total costs at the last contact dates. In both approaches,the entire study period(0, L] is partitioned into subintervals [αk, αk+1), k = 1, . . . , K,(or k = 1, . . . , K + 1), whereα1 = 0 andαK+1 = L. The assumptions underlying bothapproaches are independence between time to failure and censoring time, an extension of theindependent censoring assumption to ensure that at no point in timet are patients censoredbecause they accrue unusually high or low costs, continuous distribution of failure time andcontinuous or discrete distribution of censoring time.


The estimator for mean cost when individual cost histories are recorded (referred to asLIN1) is given as

µLIN1 =K∑k=1

SkEk (1)

whereSk = pr(T ≥ ak) is the probability of surviving toak and it is consistently estimatedby the Kaplan–Meier method (Kaplan and Meier, 1958) asSk = ∏

u≤αk{1−(dN(u)/Y(u))},

whereNi(u) = I(Xi ≤ u, δi = 1) with N(u) = ∑ni=1Ni(u) counting the number of in-

dividuals dying over time,Yi(u) = I(Xi ≥ u) with Y(u) = ∑ni=1Yi(u) counting the

number of individuals at risk over time andEk = (∑n

i=1YkiMki/∑n

i=1Yki), k = 1, . . . , KwhereMki is the observed cost of individuali incurred in intervalk and in this caseYki =I(Xi ≥ αk). That is,Ek is an estimator for mean cost in intervalk and is derived fromthose individuals who are under observation at the start of the interval. For large sam-ples µLIN1 is shown to be asymptotically normal and its variance estimator is derivedusing the martingale version of the central limit theorem and is given inAppendix A byEq. (A.1).

The approach for estimating mean cost when individual cost histories are not recorded(referred to as LIN2) again entails partitioning the duration of the study into subintervals[αk, αk+1), but now only observed total costs are being used in the estimation process. Theestimator of mean cost is then given by

µLIN2 =K+1∑k=1

Ak(Sk − Sk+1) with αK+2 = ∞ (2)

where the survival probabilitiesSk are consistently estimated by the Kaplan–Meier method,with Sk − Sk+1 being the estimated Kaplan–Meier probability of death over the interval[αk, αk+1), andAk = (

∑ni=1YkiMi/

∑ni=1Yki), k = 1, . . . , K, where nowYki = I(αk ≤

Xi < αk+1, δi = 1) andMi is the observed total cost of individuali. That is, Ak isan estimator for mean cost for intervalk and is derived from those individuals who areobserved to die in the interval [αk, αk+1). With respect to the estimator of mean cost forthe last interval of the partition [αK+1, αK+2), this involves the observed total costs ofthe patients who are censored atL and is given asAK+1 = ∑n

i=1YK+1,iMi/∑n

i=1YK+1,iwhereYK+1,i = I(Xi ≥ L), that is estimation of the interval costAk (k = 1, . . . , K + 1)does not require cost information on those individuals who are censored before the largestobserved timeL. For largen, the estimator for the variance ofµLIN2 is derived using thesame theoretical framework as for the previous estimator and is given inAppendix AbyEq. (A.2).

Due to the consistency of the Kaplan–Meier estimator, the estimatorsµLIN1 andµLIN2are consistent as long as theEk ’s andAk ’s are consistent. Their consistency, as discussed byLin et al. (1997), is dependent on the censoring pattern and is ensured if censoring occurs atthe boundaries of the intervals of the partition. If the censoring distribution is discrete, theinterval boundaries can in theory be chosen to correspond to the possible censoring timesand therefore the estimators are still going to be consistent. If the censoring distributionis continuous, the shorter the interval length, that is the finer the partition of the study


period, the more unbiased the estimators. There is however a constraint associated withthis point with reference to Lin et al’s second approach (LIN2), which requires that thelength of the intervals of the partition is such that allows a reasonable number of deaths tobe observed in each subinterval. It may not be possible however to meet this requirementwhile simultaneously ensuring that the censoring times are confined to the boundaries ofthe intervals of the partition as required for consistency.

2.3. Bang and Tsiatis estimators

The set of estimators proposed byBang and Tsiatis (2000)do not impose any restric-tions on the distribution of censoring times. The idea underlying this class of estimatorsis the use of an inverse probability weight in the estimating equations through which cen-soring is appropriately accounted for. The same notation as above is adopted here and theassumptions underlying the Bang and Tsiatis estimators are continuous distribution forfailure time over(0, L], continuous distribution for censoring time with censoring arisingat random and the random variableU denoting time to censoring having survivor func-tion K(u) = pr(U > u), that is the survivor functionK(u) evaluated at a point in timeu gives the probability of an individual not being censored atu, and pr(Ui ≥ L) >

0 which ensures thatK(u) is bounded away from zero and that a number of patientsare still under observation atL to enable calculation of the cost over the defined period(0, L].

The first estimator proposed by Bang and Tsiatis uses cost information from uncensoredindividuals only and weights each complete cost observation by the inverse of the probabilityof not being censored evaluated at the point of the individual’s death. The simple weightedcomplete-case estimator is then defined as

µWT = 1

n

n∑i=1

δiMi

K(Ti)(3)

The idea underlying the use of this specific weight is that under conditions of independentcensoring, at timeTi, K(Ti) = pr(U > Ti) is the probability that individuali has survivedto Ti without being censored. Therefore, if individuali is observed to die atTi, then herepresents 1/K(Ti) individuals who might have been observed if there was no censoring.The unknown survivor functionK(·) is estimated by the Kaplan–Meier estimator based onthe data{Xi = min(Ti, Ui), 1 − δi, i = 1, . . . , n} asK(t) = ∏

u≤t {1 − (dNc(u)/Y(u))},whereNc

i (t) = I(Xi ≤ t, δi = 0) with Nc(t) = ∑ni=1N

ci (t) counting the number of

individuals censored over time andY(u) as defined above, i.e.Yi(u) = I(Xi ≥ u) withY(u) = ∑n

i=1Yi(u) counting the number of individuals at risk over time. For large sam-ples µWT is shown to be asymptotic normal and its variance estimator is derived us-ing the martingale version of the central limit theorem and is given inAppendix A byEq. (A.3).

The authors also propose a partitioned version of the simple weighted complete-caseestimator which makes use of the cost history for the censored observations that are notused by the simple weighted estimator. The idea underlying the partitioned estimator issimilar to that proposed by Lin et al. but the advantage of this method is that the consistency


and asymptotic normality of the proposed estimator, unlike Lin’s, does not depend on thechoice of the partition or the discreteness of the censoring times. The duration of analysis(0, L] is partitioned intoK subintervals(tj, tj+1], (j = 0, . . . , K−1), the simple weightedestimator is then used to derive the estimated cost incurred in each of theseK subintervalsand the final estimate of mean cost is derived by summing across these intervals. Thepartitioned estimator is therefore given as

µP = 1

n

n∑i=1

K∑j=1

δji {Mi(tj) − Mi(tj−1)}

Kj(Tji )

(4)

where for individuali: δji = I{min(Ti, tj) ≤ Ui}, Mi(tj) is the cumulative cost up to time

tj, Kj(Tji ) is the Kaplan–Meier estimator for the probability of not being censored based

on the dataset{Xji , δ

ji , i = 1, . . . , n} whereXj

i = min(Ttji , Ui) andT

tji = min(Ti, tj).

Consistency follows by an argument similar to that used for the simple weighted estimatorand proof of asymptotic normality for this estimator and derivation of its variance for largesamples are again based on the theory of counting processes and the associated martingaleframework. The variance estimator forµP is given inAppendix Aby Eq. (A.4). The ad-vantage of this method over the simple weighted estimator is that individuali is considereduncensored in thejth interval wheneverUi > min(Ti, tj). Consequently, there is an in-crease in the cost information being used by this estimator, as individuals who were treatedas censored in the simple weighted estimator havingUi < Ti and whose cost informationwas thus not used in the estimation process will be now uncensored in some of the intervalsof the partition in which their costs will contribute to the estimates.

In an attempt to improve the efficiency of the simple weighted and partitioned estimators,the authors use the theory for missing data processes given byRobins and Rotnitzky (1992),andRobins et al. (1994). Estimation and the study of efficiency of the proposed estimatorsare based on the general theory for semiparametric models when data are missing at ran-dom. The idea is that efficiency will be improved through use of some functional of thecost history which will allow recovery of information lost due to censoring.2 To improvethe efficiency of their estimators, Bang and Tsiatis specify a fixed number of functionals[e1{MH(u)}, . . . , eJ {MH(u)}] of the cost history process and derive the simple improvedestimator as

µimp = 1

n

n∑i=1

δiMi

K(Ti)+ 1

nγ

n∑i=1

∫ ∞

0

dNci (u)

K(u)[e{MH

i (u)} − G∗(e{MH(u)}, u)] (5)

wheree{MHi (u)} is theJ×1 vector of the prespecified functionalsej{MH

i (u)} which the au-thors suggest taking asej{MH

i (u)} = Mij (u) if u > tj and zero otherwise, whereMij (u) isthe cost incurred in subinterval(tj−1, min(tj, u)] andG∗(e{MH(u)}, u) is theJ×1 vector ofG∗(ej{MH(u)}, u) whereG∗(·) is defined byG∗(e{MH(u)}, u) = ∑n

i=1e{MHi (u)}Yi(u)/

Y(u), and γ is given in Appendix A by Eq. (A.5). The estimator for the variance ofthe simple improved estimator for large samples is given byEq. (A.6) in the sameappendix.

2 SeeAppendix Afor a fuller explanation of the derivation of the Bang and Tsiatis improved estimators.


When the same methodology is applied to improve the efficiency of the partitionedestimator the resultant improved partitioned estimator is given by

µPimp = 1

n

n∑i=1

K∑j=1


Kj(Tji )

+ 1

nγ

n∑i=1

∫ ∞

0

dNci (u)

K(u)[e{MH

i (u)} − G∗(e{MH(u)}, u)] (6)

whereγ and the estimator for the asymptotic variance are given inAppendix AbyEqs. (A.9)and (A.11), respectively.

3. Methods and results

All six non-parametric estimators defined above, i.e. LIN1 (Eq. (1)), LIN2 (Eq. (2)), Bangand Tsiatis simple weighted (Eq. (3)), partitioned (Eq. (4)), simple improved (Eq. (5)) andimproved partitioned (Eq. (6)), and their associated variances given byEqs. (A.1)–(A.4),(A.6), and (A.11), respectively were applied to a medical dataset which exhibited extremelyhigh levels of censoring. The data were taken from a randomised controlled clinical trialand relate to a type 2 diabetic population of 3867 individuals allocated either to conven-tional policy (1138) or intensive policy (2729) with the aim of assessing the effectivenessof improved blood glucose control. The trial started in 1978 and ended in 1998 with amedian follow-up period to death, the last date at which clinical status was known, or tothe end of the trial period of 10 years. For each individual in the study the trial collectedinformation on both clinical effectiveness and resource use. The unit costs of hospitalisa-tion and treatment medication were attached to the volume of resources to calculate thetotal cost per patient per year directly from the trial data and these were then aggregatedto give a total cost per patient for the whole trial period. The analysis here aims at deriv-ing an estimate of average total cost over the trial period adjusting for censoring wherean observation was defined as censored if the patient was not observed for the full timeto death.

The failure event was all-cause mortality, resulting in 925 censored patients (81.3%censoring) and 213 failures in the conventional group and 2240 censored patients (82%censoring) and 489 failures in the intensive group by the end of the trial. Average follow-uptime was equal to 9.9 years reaching a maximum of 18.934 years for the conventional groupand 10.01 years reaching a maximum of 19.463 years for the intensive group. Despite thelong duration of the trial loss to follow-up and drop-out rates were negligible. The levelsof censoring witnessed in the trial largely reflect the low mortality rates in both arms at thetermination of the study. The assumption of independent censoring is valid in this data ascensoring was not related to any cost or medical reasons.

For each individual the observables were time to death or last contact, a variable takingthe values of 0 or 1 indicating censoring or failure respectively, the annual costs and thetotal cost from the start of follow-up to death or the last contact date. All estimators wereapplied to these trial data within each arm, i.e.n = 1138 for the conventional policy over


a period of (0, 18.934] years andn = 2729 for the intensive policy over a period of (0,19.463] years. Before presenting the results obtained from the various estimators some finalmethodological points with regards to the main analysis follow.

For each of the Lin et al. estimators, two sets of results are presented. The first was obtainedwhen the study duration was partitioned into yearly intervals, given that individual costswere available from the trial on an annual basis, while the second was obtained assuminga monthly interval length with the individual’s monthly cost calculated as the annual costdivided by twelve. This was undertaken to assess the impact that the interval length of thepartition has on the estimates as the validity of the Lin et al. approach relies on the pattern ofthe censoring distribution being such that censoring times can be confined to the boundariesof the intervals of the partition.

The Bang and Tsiatis partitioned and improved partitioned estimators are based on yearlysubintervals for the same reason as stated above, that is because intermediate cost history foreach subject was available on an annual basis. Also for this reason, for both simple improvedand partitioned improved estimators, the fixed number of functionals of the cost historyprocess were [e1{MH(u)}, . . . , eJ {MH(u)}], whereJ = 19 for conventional andJ = 20for intensive, i.e. annual subintervals were assumed in the recovery of cost information lostdue to censoring, and the set of prespecified functionals were defined in accordance to theauthors’ suggestion asej{MH

i (u)} = Mij (u) if u > tj and zero otherwise, whereMij (u) isthe cost incurred in subinterval(tj−1, min(tj, u)].

With regards to the stochastic integrals appearing in the Bang and Tsiatis estimatorspresented above, these are of the form

∫ L

0 f(u)dNc(u) = ∫(0,L] f(u)dNc(u) (or

∫∞0 f(u)d

Nc(u)), whereNc(u) is the counting process for censoring as defined above,f(·) is somefunction of time, 0 ≤ L ≤ ∞, and their calculation was undertaken as shown inAppendix B.

Finally, it should be noted that different versions of the Kaplan–Meier estimator of sur-vival have been proposed to define the estimator when the largest observed time correspondsto censoring. All versions of the Kaplan–Meier estimator equal

∏s≤t (1−$N(s))/Y(s) for

t ≤ Xmax, whereXmax denotes the largest observed time, and they are all equal to zero fort > Xmax if the event atXmax is a failure. In the original paper by Kaplan and Meier, theestimator was left undefined fort > Xmax if Xmax is a censored observation.Efron (1967)set the estimator equal to zero fort > Xmax even if the last observation was censored. Theversion adopted here was proposed byGill (1980)and sets the estimator equal toS(Xmax),that is equal to its value at the largest observed time, fort > Xmax even when the lastobservation is censored.

3.1. Main analysis

Table 1presents the estimates of mean cost and the associated standard errors for the con-ventional and the intensive policy groups over the study period as derived from applicationof the Lin et al. and Bang and Tsiatis estimators to the trial data.

Concentrating first on the estimates for mean cost obtained with each of the two methodsproposed by Lin et al, the results show that the length of the intervals of the partition doesnot have an impact on the estimates returned by either the LIN1 or the LIN2 estimators.This finding holds for both trial arms.


Table 1Results of the main analysis

Estimator Conventional Intensive

Mean S.E. Mean S.E.

Lin et al.Subintervals in years

LIN1 (µLIN1) 14006.2 897.73 13172 340.55LIN2 (µLIN2) 12428 636.93 16910.64 1010.87

Subintervals in monthsLIN1 (µLIN1) 13771.35 1025.60 13078.02 365.95LIN2 (µLIN2) 12530.39 668.21 16926.22 1012.53

Bang and TsiatisSimple weighted (µWT) 5732.735 840.8 9737.65 3043.5Partitioned (µP) 14639.48 1219.4 13839.67 445.6Simple improved (µimp) 3668.92 398.1 1620.97 1634Partitioned improved (µPimp) 334563.3 Variance< 0 −326298.3 Variance< 0

All estimators apart from LIN2 and the Bang and Tsiatis simple weighted display higherestimates for the conventional group compared to the intensive. One could argue that theconventional policy group incurring higher costs on average is probably indicative of the“true” result, as the intensive policy group were known to have significantly lower hospital-isation rates. In addition, the “naıve” estimators resulted in the same direction of differencein mean cost between the two groups3 which could be interpreted in support of the previousargument in the following manner. Despite the fact that the uncensored-cases estimator isbiased toward the costs of the individuals with shorter survival times as longer survivaltimes are more likely to be censored, the trial data has not shown a significant differencein survival, both with respect to the proportion dying and the length of survival time, be-tween the two groups and therefore one may assume that the degree of bias imparted inthe uncensored-cases estimator is similar between the two groups. Along similar lines, al-though the full-sample estimator is known to be biased downward as the costs incurred aftercensoring times are not accounted for, it could again be argued that the degree of bias inthe estimates is similar between the two arms on the basis that the trial data show the sameproportion of censoring in the two groups and that this similarity is also maintained overtime. All information from the trial is suggestive therefore of the conventional policy groupincurring higher costs than the intensive policy population.

On this basis, the fact that the LIN2 and the Bang and Tsiatis simple weighted estima-tors display lower estimates for the conventional group compared to the intensive group indirect contrast to the results obtained from all other estimators, gives a first indication ofpoor performance. This statement may be supported by the following observations. First,there is a similarity between the LIN2 and the Bang and Tsiatis simple weighted estimatorsin that they both use only the complete cost observations in estimating mean cost. Lin et al.

3 The full sample estimator was estimated atµFS = 8181.58 (S.E. = 305.62) for conventional andµFS =8029.86 (S.E. = 146.79) for intensive. The uncensored cases estimator was estimated atµU = 11901.01 (S.E. =1061.36) for conventional andµU = 10629.97 (S.E. = 510.00) for intensive.


explicitly state that their second approach relies on a “reasonable” number of deaths in eachsub-interval of the partition and suggest a minimum number of five deaths in each subinter-val. The number of deaths in our data is small in the majority of subintervals and decreasesmarkedly towards the end of the trial, resulting in a number of deaths below five or even zeroin the last intervals of the partition. The Bang and Tsiatis simple weighted estimator notonly displays the same pattern as stated above with respect to the direction of the differencein mean costs between the two arms, but it also results in low values of mean cost for botharms which are totally unlikely to be true since they are even lower than the respective meancosts estimated by the full-sample estimator. Although the Bang and Tsiatis simple weightedestimator does not rely on the pattern of the censoring distribution and therefore the smallnumber of complete cost observations does not affect the estimates in the same manner asin the Lin et al. second approach, the authors state however that caution should be exercisedwhen applying all their estimators in circumstances where there is heavy censoring in thetails of the distribution with small sample sizes which is precisely the case in our data.

The Bang and Tsiatis simple improved estimator gives even lower estimates of averagecost for both arms than the simple weighted estimator. Once again this probably reflectsthe heavy censoring experienced at the tails of the distribution. The Bang and Tsiatis parti-tioned improved estimator performs very poorly resulting in mean cost estimates of extrememagnitude including negative values for mean cost for the intensive arm and for variancesin both arms. Furthermore, although the same level of censoring affects both improved es-timators, the improved simple does not result in such extreme values as are observed in theimproved partitioned. This finding suggests that the high degree of censoring in particularat the tails of the distribution makes the improved partitioned much more unstable than thesimple improved. A direct consequence of heavy censoring at the tails is that the probabil-ity of an individual not being censored reaches very small values some of which approachzero. Thus, any quantity weighted by the inverse of such probabilities will be of extremelylarge absolute value. Partitioning the study period could amplify this problem. Noting thatthe covariance vector is the major difference between the two improved estimators, themost likely explanation for the observed pattern of results is that the degree of censoringespecially at the tails leads to extremely inflated quantities within this vector and leadsto the final estimator being extremely unstable. Given that the problem cannot be locatedprecisely, further investigation is undertaken below using artificially generated data.

This leaves two estimators which may be said on first indication to perform adequatelyin this particular dataset; the LIN1 estimator and the Bang and Tsiatis partitioned estimator.Not only do they both give estimates of a similar sensible magnitude with accompanying rea-sonable standard errors, but they also display the anticipated direction of difference in meancost between the two groups, with the conventional arm having higher average cost thanthe intensive arm. The similarities between these two estimators are the partitioning of thestudy duration into subintervals, the use of intermediate cost history for each subject and theuse of a probability weight to adjust cost in interval for censoring. The difference lies in thechoice of this weight and in the interval cost adjusted by it. In the LIN1 estimator the weightis defined as the probability of surviving to the beginning of each interval and this is usedto adjust estimates of mean cost in the interval. The consistency of this estimator, as statedabove, requires appropriate censoring conditions, so that censoring times correspond to theinterval boundaries of the partition. By contrast, in the Bang and Tsiatis partitioned estima-


tor the weight is defined as the inverse of the probability of an individual not being censoredat a given point in time and this is used to adjust individual observed costs in the interval.Moreover, consistency and asymptotic normality of the partitioned estimator does not de-pend on the choice of the intervals of the partition or the distribution of the censoring times,that is the asymptotic properties of this estimator are independent of the censoring pattern.

Generally the results of the main analysis support—under conditions of extremecensoring—the findings reported byLin et al. (1997)andBang and Tsiatis (2000). Evenunder moderate censoring conditions as are considered in these studies, the LIN1 estimatoris reported to perform better than LIN2 and is clearly preferred to LIN2 at higher levels ofcensoring if intermediate cost histories are available as it uses more cost information andrequires smaller sample sizes. Bang and Tsiatis show that the partitioned estimator per-formed better than their other proposed estimators with increasing censoring. The results ofthe main analysis here, however, indicate a number of potential difficulties which may arisewhen applying the estimators considered above to data with heavy censoring. Consequently,a number of additional analyses were undertaken to determine whether these difficultiesarose because of the characteristics of the specific dataset or the intrinsic properties of theestimators and thus empirically identify conditions under which the estimators perform asexpected from the theory.

3.2. Secondary analyses

The additional analyses presented below investigate further the Lin et al. and Bang andTsiatis estimators concentrating on the specific problems raised above. The estimators arethus assessed under the following circumstances. First, using the same trial data but exclud-ing the highest observed total costs. Secondly, using the same clinical trial data but varyingthe durations of analysis. Thirdly, using an “artificial” dataset constructed by randomlygenerating costs and survival times and varying the levels of censoring. Finally, using thebootstrap method to obtain estimates of the standard error for the estimators as an alternativeto the analytically derived asymptotic variance estimators (as given inAppendix A).

3.2.1. Sensitivity to high cost outliersIn both trial arms the distribution of cost was positively skewed with a very small number

of observations having extremely high values. To assess whether these high cost outliersinfluence the estimates, the extreme high cost observations in each arm were excludedfrom the analysis and the Lin estimates of mean cost based on these data are presented inTable 2.

Table 2Lin estimators excluding the highest observed costs from each group


Mean S.E. Mean S.E.

LIN1 13583.07 865.32 13058.34 334.46LIN2 12078.4 580.88 16821.76 1009.15


The resultant estimates are naturally slightly lower than the respective ones derived inthe main analysis, but the differences are not significant and the overall pattern of results isnot altered. The conclusions drawn from the main analysis results hold therefore regardlessof whether these extreme cost values are included in the analysis or not. The pattern of apositively skewed cost distribution with a small number of high outliers is also observed inthe administrative dataset used by Lin et al. and is likely to be a characteristic of any medicaldataset. The relevant results in Lin et al. give no indication that such a characteristic of costhas an impact on the performance of their estimators which is consistent with the findingreported above.

3.2.2. Impact of varying the duration of analysisAs mentioned previously, the main analysis results indicated that the Lin et al. second

approach—not using individual cost histories—gave inconsistent estimates with respectto the direction of the difference in average cost between the two trial arms. Given thereliance of this method on the number of uncensored individuals in each subinterval andon the number who are censored at the largest observed time, the duration of analysis wasrestricted to 18, 17, 16, 15 and 12 years and both Lin et al. estimators were applied to thesedata.

As shown inTable 3, the number of uncensored individuals decreases towards the endof the study and is equal to zero for the last two intervals in the conventional group andthe last interval in the intensive group, thus falling below the minimum of five deaths in

Table 3Total number of individuals and number of uncensored cases in each interval of the partition

Interval Conventional Intensive

Number enteringinterval

Number dyingwithin interval

Number enteringinterval

Number dyingwithin interval

1 1138 8 2729 242 1125 10 2700 223 1112 11 2673 284 1097 18 2632 235 1076 12 2596 376 1050 16 2539 517 1017 19 2442 358 917 19 2233 439 816 20 1985 39

10 695 12 1681 3711 561 22 1347 3612 433 9 1062 3313 323 16 818 3014 214 9 556 1815 129 7 326 1316 69 3 187 817 53 2 127 818 32 0 70 319 12 0 18 120 2 0


Table 4Lin et al. estimators for different durations of analysis


Mean S.E. Mean S.E.

L = 18 yearsLIN1 13564.66 798.29 12752.36 340.01LIN2 15409.63 2930.63 12597.07 1118.68

L = 17 yearsLIN1 12831.22 665.14 12295.81 324.57LIN2 14785.19 1661.96 14031.57 842.77

L = 16 yearsLIN1 11884.79 583.97 11434.42 245.7LIN2 13683.87 1215.49 12206.42 583.56

L = 15 yearsLIN1 11258.03 512.54 10750.22 210.09LIN2 12381.43 987.25 11434.13 480.74

L = 12 yearsLIN1 8869.46 357.95 8642.84 163.97LIN2 9230.87 458.12 8858.89 220.52

each interval of the partition suggested byLin et al. (1997). In addition, in both groupsthere is only one individual censored at the maximum observed time which implies that theestimate of average cost in theK + 1 interval (withαK+2 = ∞) is determined on the basisof this one individual. Restricting the duration of analysis effectively results in increasingthe number of individuals who are censored at the upper bound of the analysis time, andmore importantly eliminates the impact of the last intervals of the partition on the estimatesin which the number of uncensored individuals is very small.Table 4reports the impact ofdiffering durations of analysis on the two Lin et al. estimators.

The initial point to be made here is that the first estimator by Lin et al. (LIN1) remainsstable for all different durations of analysis. That is, its absolute magnitude decreases asduration decreases since it is estimating average costs over a shorter time period and the rateof decrease appears to be reasonable in both trial arms. More importantly, as in the mainanalysis results, the conventional group is shown to incur higher costs on average than theintensive group for all time periods of analysis.

With respect to the second estimator by Lin et al. (LIN2), the results show that whenduration of analysis was restricted to 17 years or less, this estimator became stable resultingin the expected estimates, that is the estimator resulted in conventional policy having ahigher mean cost than intensive policy. These results indicate that LIN2 is indeed sensitiveto the number of deaths in the intervals of the partition and to the number of individualscensored at the largest observed time. More specifically, increasing these numbers to a“reasonable” level results in obtaining less biased estimates of mean cost in each of thesubintervals, as the greater the number of individuals who contribute cost information ineach interval the more representative are the estimates of mean cost in the correspondinginterval.


With regards to the Bang and Tsiatis estimators, the problems identified in the mainanalysis were the very low estimates resulting from the simple weighted and the improvedsimple estimators and the extreme estimates resulting from the improved partitioned esti-mator. As the authors point out, very heavy censoring in the tails of the distribution mayrender the estimators unstable with small sample sizes. As already stated, the trial data werevery heavily censored reaching 82% in both conventional and intensive policy groups bythe trial end. In addition, as shown inTable 3, towards the end of the study the number ofindividuals still under observation decreases substantially falling to 12 in the conventionalgroup at the last year of follow-up and to two in the intensive group at the last year offollow-up. To assess the impact that an increase in the number of individuals being underobservation at the end of the analysis time has on the Bang and Tsiatis estimators, theduration of analysis was restricted to 18, 17, 16, 15 and 12 years and the estimators werecomputed for the conventional policy group.

The results are presented inTable 5and show the same pattern as observed in the mainanalysis. That is, the partitioned estimator gives estimates of average cost very similarto the LIN1 estimator for the various time durations of analysis, the simple weightedand the improved simple estimators still give low estimates compared to the partitioned

Table 5Bang and Tsiatis estimators for different durations of analysis for the conventional policy group

Estimator Mean S.E.

L = 18 years (censoring 81.3%)Simple 5732.73 840.77Partitioned 14639.48 1219.37Simple improved 3668.92 398.10Partitioned improved 334562.5 Variance< 0

L = 17 years (censoring 81.3%)Simple 5732.73 840.77Partitioned 13410.59 731.83Simple improved 3668.92 398.10Partitioned improved Mean< 0 (mean= −15906.16) Variance< 0

L = 16 years (censoring 81.5%)Simple 5481.43 829.84Partitioned 12519.31 683.96Simple improved 5064.38 384.93Partitioned improved Mean< 0 (mean= −25535.95) Variance< 0




and LIN1 and the improved partitioned again results in extreme values. In other words,the problems identified in the main analysis do not appear to be resolved by restrict-ing total analysis time. This is probably due to the fact that although the total numberof individuals under observation towards the end of the analysis period is slightly in-creased as the period is restricted, the proportion of patients who are censored remainshigh (between 81 and 82% until 15 years), even increasing slightly as duration decreases(84.5% at 12 years). This gives a strong indication that the issue of heavy censoring es-pecially in the tails of the distribution is primarily responsible for the estimators’ poorperformance.

3.2.3. SimulationAs the level of censoring is directly related to the performance of all estimators consid-

ered and more specifically as Bang and Tsiatis state very heavy censoring in the tails ofthe distribution could result in their estimators becoming unstable, the performance of thevarious estimators was assessed for different levels of censoring. An artificial dataset wasconstructed as described below with censoring set at varying levels in order to simulate theimpact of censoring that could be encountered within a real analysis and explicitly test thevarious degrees that the degree of censoring has on the estimators of interest while at thesame time ensuring that individual costs vary in a predefined manner.4 As well as having theadvantage that different levels of censoring can be set and the impact of censoring can beisolated, an artificial dataset also allows estimation of the “true” mean cost, that is the meancost if censoring was not present in the data. A direct assessment of the performance of thevarious estimators is thus achieved through comparison of the estimated means to the “true”mean.

A sample size of 1138 individuals was chosen for this “artificial” dataset to equal thesmaller sample size of the clinical trial data used in the main analysis—since one of theconcerns for the validity of the methods is related to the sample size. Survival times weregenerated from a uniform distribution on [0, 10] years. The average 10-year cost is theparameter of interest, with the total cost for individuali being

Mi = Mi(0) + biTLi +

10∑j=1

τij (min[{TLi − (j − 1)}+, 1]) + diI(Ti ≤ 10)

whereMi(0) is the initial diagnostic cost,bi is the deterministic annual cost,τij is the ran-dom annual cost for thejth year,di is the terminal death cost andα+ = max(0, α). Forthe distribution of each cost element,Mi(0), bi, τij, di are assumed uniformly distributedon [5000, 15000], [1000, 2600], [0, 400] and [10000, 30000], respectively. Various levelsof censoring were considered with the censoring times being uniformly distributed on [0,

4 Although the design of the artificial dataset is similar to that used in the simulations undertaken by Bangand Tsiatis (and Lin et al.) the objective is different. Multiple replications are not undertaken as this would notsimulate “real” data. Replications were undertaken in the earlier studies as there the objective was to conceptuallydemonstrate the degree of bias imparted by the different estimators. The analysis here is intended primarily toexplore the source of the problems associated with the Bang and Tsiatis estimators identified in the main analysisand more generally to illustrate the cost estimates that would be derived in practice from application of thealternative estimators under varying degrees of censoring.


20] years, i.e. 25% censoring, [0, 12.5] years, i.e. 41% censoring, [0, 10] years, i.e. 51%censoring, [0, 9.5] years, i.e. 55% censoring and [0, 9] years, i.e. 57.5% censoring. The com-ponents were generated independently and the simple weighted estimator, the partitionedestimator, the improved simple, the improved partitioned and the LIN1 estimator (whichuses individual cost history) were calculated for all levels of censoring. The Bang and Tsi-atis partitioned and improved partitioned estimators were based on yearly subintervals andfor both simple improved and partitioned improved estimators, annual subintervals wereassumed in the recovery of cost information lost due to censoring, and the set of prespec-ified e-functionals were defined as in the analysis of the real trial data. The LIN1 estimatorbased an annual subintervals was also estimated using this artificial dataset as under all cir-cumstances considered in the real data analyses it remained stable and generally performedwell.

The resultant estimates of the average 10-year cost and its asymptotic variance based onthe artificial data are reported inTable 6. The “true” average cost obtained when completeinformation was assumed on all individuals in these data was equal to 41144.50 and servesas the reference cost to be compared with all other estimates under different levels of censor-ing. As expected, as the level of censoring increases all estimators generally exhibit higherdegrees of bias. The LIN1 and the Bang and Tsiatis partitioned estimators performed wellat all levels of censoring. The Bang and Tsiatis improved partitioned estimator performedequally well up to a level of censoring of 51%. It resulted in negative estimates for the vari-ance when censoring reached 55%. The simple weighted and simple improved estimatorsappear to give increasingly lower estimates as censoring increases, with the estimates beingclose to the others only up to 41% censoring. At 55% censoring the average cost derived bythe simple weighted estimator was approximately half the “true” mean cost value and theone derived by the simple improved estimator was even lower. Overall, the findings fromthis analysis support the findings of the analysis based on the real clinical data. The LIN1and Bang and Tsiatis partitioned estimators appear stable at all levels of censoring whereasthe simple weighted and both improved estimators appear extremely sensitive to the levelof censoring reflecting a similar pattern, only less extreme, to the one observed in the trialdata.

3.2.4. Bootstrap estimates of the variancesThe derivation of the standard errors for all the estimators proposed by Lin et al. and

Bang and Tsiatis is based on the large sample properties of these estimators. Study oftheir asymptotic properties has shown that the estimators converge to a normal distributionand use of the martingale version of the central limit theorem allows estimators for theirvariances to be formulated. While efficiency is therefore shown to hold conceptually apotential problem relates to the validity of the assumption of asymptotic normality whenthe approaches are applied to any particular dataset. Although asymptotic statistics is of boththeoretical and practical importance, it is a theory of approximations. Such approximationsare particularly useful in studying theoretically the efficiency of the statistics of interest butare of questionable value if the statistical procedure which has been shown to function forn → ∞ is to be applied to a finite sample. In most situations the theory itself does notprovide a means for assessing the magnitude of the approximation errors and it is usuallythe case that the accuracy of the asymptotic results is judged by simulation studies.


Table 6Estimates based on the “artificial” dataset

Estimator Mean S.E.

Censoring 25%Simple weighted 41348.2 475Simple improved 40452.9 433.8Partitioned 41654.9 342.1Improved partitioned 40876 316.2LIN1 39545.6 311

Censoring 41%Simple weighted 37228.3 1713.2Simple improved 34724.4 854.7Partitioned 40000.2 734.4Improved partitioned 38575.4 366.3LIN1 37367.4 355.8

Censoring 51%Simple weighted 29284.3 3340.4Simple improved 25334.2 1627.8Partitioned 37242.8 1306.4Improved partitioned 34683.7 514LIN1 35456.3 354

Censoring 55%Simple weighted 21037.6 684.6Simple improved 15922.8 536.2Partitioned 33839.1 315.2Improved partitioned 32446.4 Variance< 0LIN1 34280 296.1

Censoring 57.5%Simple weighted 18921.7 605Simple improved 13361.8 549.4Partitioned 33048 271.6Improved partitioned 32787.7 Variance< 0LIN1 33686 271.9

The true mean cost (no censoring) is 41144.5.

To test the validity of the estimators’ asymptotic results, empirical standard errors forboth Lin et al. estimators and for the Bang and Tsiatis simple weighted and partitioned esti-mators were derived using the bootstrap method. The bootstrap estimates were obtained bydrawing random samples with replacement of sizen = 1138 from the observed distributionfor the conventional group andn = 2729 for the intensive group and calculating the LIN1,LIN2 and the Bang & Tsiatis simple weighted and simple partitioned estimates of averagecost across a large number of replications. All sets of bootstrap estimates were obtainedfor 200 and 1000 bootstrap replications which are deemed adequate for the calculationof standard errors. The standard errors derived from the bootstrap method are reported inTable 7.

With respect to the Lin et al. estimators the bootstrap estimates were also derived for aduration of analysis of 17 years as this was the point at which the LIN2 estimator became


Table 7Bootstrap estimates of the standard error

Conventional Intensive

Lin et al: when the time of analysis was the complete follow-up period (L = 18.9 years for conventional andL = 19.5 years for intensive)

Replications 200LIN1 823.49 333.51LIN2 8085.00 3784.31


Lin et al: when the time of analysis was 17 years for both conventional and intensive



Bang and Tsiatis: when the time of analysis was the complete follow-up period (L = 18.9 years forconventional andL = 19.5 years for intensive)

Replications 200Simple weighted 786.13 3098.45Partitioned 1207.90 439.87

Replications 1000Simple weighted 830.24 3132.54Partitioned 1379.02 451.28

stable. Comparison of the empirically derived variance estimates using the bootstrap methodwith their respective asymptotic variance estimates reported inTables 1 and 4shows thatfor all estimators the bootstrap estimates of the standard error confirm those obtained fromthe formulae (this being the case for LIN2 under the conditions where this became stableas expected).

This finding therefore supports the validity of the assumptions underlying the estimators’asymptotic properties. Conversely, the bootstrap method gives a reasonable approximationto the theoretically derived variances.

4. Discussion

This paper has concentrated on non-parametric estimators of cost statistics under con-ditions of right censoring. As such estimators are free of assumptions regarding the dis-tribution of cost and can easily incorporate the presence of censoring in the cost obser-vations they can be particularly appealing. The Kaplan–Meier estimator has been proveninappropriate in the analysis of cost-to-event data due to the violation of independence


between the random variable of interest and its censoring variable. Consequently, a num-ber of alternative non-parametric estimators have been proposed recently that require in-dependence between time-to-event and time-to-censoring but not independence betweencost-at-event and cost-at-censoring. Although these estimators are free of assumptions withrespect to the distribution of cost, they are not entirely free of restrictions. More specif-ically, consistency of the estimators proposed by Lin et al. depends on the pattern of thedistribution of censoring times and although the asymptotic properties of the estimatorsproposed by Bang and Tsiatis are independent of the censoring pattern, the estimatorscan become unstable under conditions of heavy censoring at the tails of the distribution.In theory, provided that their respective assumptions are valid, each of the Lin et al. andBang and Tsiatis estimators considered in this paper will provide consistent estimators ofaverage cost. From the theory it is also expected that the degree of censoring will havea direct impact on the estimators’ performance with this deteriorating as censoring in-creases although this impact will vary among the approaches. While the estimators’ desir-able properties, that is consistency and efficiency, have been shown to hold conceptuallythe degree to which these properties are retained in practice will depend on the particularapplication.

Within the context of the analysis presented here the first estimator proposed by Linet al. which uses information on intermediate individual cost histories appeared stable un-der a wide variety of conditions as opposed to their second estimator which only usesinformation on total costs from individuals who are either observed for the full time toevent or are censored at the upper bound of analysis time and was shown to be sensi-tive to the number of individuals contributing cost information. With respect to the set ofestimators proposed by Bang and Tsiatis, the simple weighted estimator using only com-plete cost information and both improved estimators appeared extremely sensitive to thelevel of censoring and became increasingly unstable as censoring increased. In contrast,their partitioned estimator which uses information on intermediate individual cost historiesperformed well under all circumstances. Concentrating on the two most stable estimators,these are similar in that they both partition the study period into subintervals and makeuse of individual intermediate cost history within each subinterval and in that they bothuse a weight to adjust interval costs for censoring. They are different both in the choiceof this weight and in the interval costs that are adjusted by it. In LIN1 the weight is theKaplan–Meier probability of survival to the start of the interval that adjusts estimates ofmean cost in the interval, whereas the Bang and Tsiatis partitioned estimator uses the in-verse of the probability of an individual not being censored evaluated at a given point intime to adjust individual observed costs in the interval. On the basis that both approachesrequire the same amount of cost information, but the second approach is not restrictedby the pattern of the censoring distribution and is therefore more general, it might bepreferred.

There is a long history related to the use of the inverse of the probability of inclusionin adjusting estimates for missingness. The same inverse probability weight was first usedby Horvitz and Thompson (1952)in the context of sample surveys, byKoul et al. (1981)in studying censored failure times using a linear regression methodology, byRobins andRotnitzky (1992)in the context of recovering information missing due to censoring, byLinand Ying (1993)in non-parametric estimation of the bivarate survival function under uni-


variate censoring, byRobins et al. (1994)in adjusting estimates of regression coefficientsfor missingness in the data, byRotnitzky and Robins (1995)in studying semiparametricregression models in the presence of censoring dependent on covariates, byRobins et al.(1995)in studying semiparametric regression models for repeated outcomes in the presenceof censoring dependent on covariates, byZhao and Tsiatis (1997)in deriving a consistentestimator for the distribution of quality adjusted survival time under conditions of censor-ing, and recently byLin (2000) in adjusting medical cost estimates for censoring usinga linear regression approach. In all these applications use of this weight results in con-sistent estimators for the statistics of interest while adjusting for missingness. The samegeneral finding emerges from the analysis undertaken in this paper but at the same timethe performance of the corresponding estimators appears to be subject to the amount ofcost history information entering the estimating equations. This is why the simple weightedand the partitioned estimators yield such different estimates of mean cost. That is, al-though the same general definition of the probability weight underlies both estimators,the points in time at which the individual probabilities are evaluated differ between theapproaches in a manner that is determined by the points at which information on indi-vidual cost histories becomes available. The implication is that the weight alone is notsufficient to adjust the estimates for the loss of information when the level of missingness istoo high.

Nevertheless, despite the limitations associated with the assumptions underlying theestimators’ validity and their dependence on the data under consideration, the present anal-ysis has identified estimators whose performance is deemed satisfactory under extremecensoring conditions. Consequently, their application to the analysis of censored cost datais appropriate when estimates of mean cost over the study period are sought. When interestextends however beyond the maximum time for which data is available or when questionsregarding the effect of covariates on cost arise, parametric models become a necessary alter-native. It is clearly important that such parametric models make adjustment for censoring.An emerging area of research is thus the investigation of parametric models which incor-porate the types of adjustment made for censoring described in this paper. Provided thatcensoring is appropriately accounted for and that the distributional assumptions imposedby a specific parametric approach are justified, the within study estimates derived by such amodel could be compared to non-parametric estimates as a means of assessing the validityof the parametric approach before this was used to extrapolate beyond the end of the studyperiod or to a different population setting.

Acknowledgements

This research was supported in part by an NHS London Executive R&D Research Fel-lowship. We also wish to thank David Cox for comments on an earlier draft, Heejung Bangfor her response to specific questions, Alastair Gray for comments on the data collectionand collation and participants at seminar presentations given at the University of York andthe LSE. The usual disclaimer applies. We dedicate this paper to the memory of RobertTurner.


Appendix A.

A.1. Variance estimators forµLIN1 andµLIN2

For large samplesµLIN1 is shown to be asymptotically normal and its variance estimatoris derived using the martingale version of the central limit theorem5 and is given as

var(µLIN1) =n∑

i=1

K∑k=1

K∑l=1

WkiWli (A.1)

where

Wki = SkYki(Mki − Ek)∑nj=1Ykj

− SkEk

I(Xi ≤ ak)δi

Ri

−∑

j:Xj≤min(ak,Xi)

δj

R2j

,

andRi =n∑

l=1

I(Xl ≥ Xi)

For largen, the estimator for the variance ofµLIN2 is derived using the same theoreticalframework as for the previous estimator and is given as

var(µLIN2) =n∑

i=1

K+1∑k=1

K+1∑l=1

WkiWli (A.2)

where

Wki = (Sk − Sk+1)Yki(Mi − Ak)∑nj=1Ykj

+ Ak(Sk+1Dk+1,i − SkDki),

Dki = I(Xi ≤ ak)δi

Ri

−∑

j:Xj≤min(ak,Xi)

δj

R2j

, andRi =n∑

l=1

I(Xl ≥ Xi)

5 The underlying martingale process is (t) = N(t) − ∫ t

0I(X ≥ u)λ(u)du, whereλ(u) denotes the hazard

function for the failure time distribution thus making the process given byA(t) = ∫ t

0I(X ≥ u)λ(u)du thecompensator of the processN(t) with respect to the filtration representing the increasing information over time onthe individuals’ survival and censoring up to and including timet. In general when the compensator is known or canbe computed as is the case in the applications we are concerned with, the martingale approach to statistical modelsfor counting processes forms the basis for studying the statistical properties of the estimators in the presence ofcensoring and deriving explicit expressions for their variance estimators for large sample sizes with the use ofthe martingale version of the central limit theorem (Gill, 1980; Fleming and Harrington, 1991; Andersen et al.,1993).


A.2. Variance estimators forµWT andµP

For large samplesµWT is shown to be asymptotic normal and its variance estimator isderived using the martingale version of the central limit theorem6 and is given as

var(µWT) = 1

n

[1

n

n∑i=1

δi(Mi − µWT)2

K(Ti)+ 1

n

∫ L

0

dNc(u)

K2(u){G(M2, u) − G2(M, u)}

]

(A.3)

where

G(M, u) = 1

n

1

S(u)

n∑i=1

δiMiI(Ti ≥ u)

K(Ti)

andS(u) is the Kaplan–Meier estimator forS(u) = pr(T > u).The variance estimator forµP is given by

var(µP)= 1

n

1

n

n∑i=1

δi(Mi − µP)2

K(Ti)+∫ L

0

K∑j=1

K∑l=1

Sj∧l(u){Gj∧l(MjMl, u)

− Gj∧l(Mj, u)Gj∧l(Ml, u)} dNc(u)

Y(u)K(u)

]

(A.4)

where

Gj∧l(Ml, u) = 1

n

1

Sj∧l(u)

n∑i=1

δj∨li Mil I(T

j∧li ≥ u)

Kj∨l(Tj∨li )

Gj∧l(MjMl, u) = 1

n

1

Sj∧l(u)

n∑i=1

δj∨li MijMil I(T

j∧li ≥ u)

Kj∨l(Tj∨li )

j ∨ l = max(j, l), j ∧ l = min(j, l), Mij = Mi(tj) − Mi(tj−1), Ttji = T

ji , andSj(u) is the

Kaplan–Meier estimator of pr{min(Ti, tj) ≥ u}.

A.3. Bang and Tsiatis improved estimators

Estimation and the study of efficiency of the improved estimators are based on thegeneral theory for semiparametric models when data are missing at random. Due to ef-

6 The associated martingale process is` c(t) = Nc(t) − ∫ t

0I(X ≥ u)λc(u)du and the respective filtration isnow given asσ {I(Ui ≤ t), t ≤ u; I(Ti ≤ x), Mi(x), 0 ≤ x < ∞, i = 1, . . . , n} and represents theincreasing information over time on the censoring times up to timeu and survival times and cost histories over allnon-negative times. Consequently the process given byAc(t) = ∫ t

0I(X ≥ u)λc(u)du, whereλc(u) is the hazardfunction for the censoring distribution, is the compensator for the counting processNc(t).


ficiency requirements7 and on the premise that most estimators are asymptotically linearand their asymptotic properties are directly related to their influence function (Newey,1990), the issue in the application of interest is to identify the class of influence functionsfor regular asymptotically linear estimators when the data may be censored. To improvethe efficiency of their estimators, Bang and Tsiatis specify a fixed number of function-als [e1{MH(u)}, . . . , eJ {MH(u)}] of the cost history process thus restricting the class ofinfluence functions8 to

Mi − µ −∫ ∞

0

d` ci (u)

K(u){Mi − G(M, u)}

+J∑

j=1

γj

∫ ∞

0

d` ci (u)

K(u)[ej{MH

i (u)} − G(ej{MH(u)}, u)]

and determine the set of constantsγj, j = 1, . . . , J which minimise the variance of theabove expression. The optimal set of constants are derived asγopt = cov(yi, Zi)var(Zi)

−1,whereZi is a 1× J vector andzij andyi are scalars given by

yi =∫ ∞

0

d` ci (u)

K(u){Mi − G(M, u)}

zij =∫ ∞

0

d` ci (u)

K(u)[ej{MH

i (u)} − G(ej{MH(u)}, u)], j = 1, . . . , J

7 Assessment of the asymptotic efficiency of any given semiparametric estimator is performed by compar-ing the estimator’s asymptotic variance with the semiparametric efficiency bound. To ensure the existence of asemiparametric efficiency bound the estimator must be regular. Regularity conditions can be easily derived forasymptotically linear estimators. Furthermore, establishing regularity conditions for asymptotically linear estima-tors not only ensures the existence of a semiparametric efficiency bound but also allows calculation of the bound(Newey, 1990).

8 Following Robins and Rotnitzky (1992), Bang and Tsiatis show that the entire class of influence functionsunder random censoring with unspecified distribution is

Mi − µ −∫ ∞

0

d` ci (u)

K(u){Mi − G(M, u)} +

∫ ∞

0

d` ci (u)

K(u)[e{MH

i (u)} − G(e{MH(u)}, u)] (∗)

wheree{MHi (u)} is an arbitrary functional of the cost history and

G(e{MH(u)}, u) = 1

S(u)E[e{MH

i (u)}I(Ti ≥ u)]

The estimator of mean cost whose influence function is given by (∗) is then of the form

µgen = 1

n

n∑i=1

δiMi

K(Ti)+ 1

n

n∑i=1

∫ ∞

0

d` ci (u)

K(u)[e{MH

i (u)} − G∗(e{MH(u)}, u)]

where

G∗(e{MH(u)}, u) =∑n

i=1e{MHi (u)}Yi(u)

Y(u).

Determining the optimal set of functionals of cost historyeopt{MHi (u)} will result in deriving the most efficient

estimator within this class.


The simple improved estimator is then given by

µimp = 1

n

n∑i=1

δiMi

K(Ti)+ 1

nγ

n∑i=1

∫ ∞

0

dNci (u)

K(u)[e{MH

i (u)} − G∗(e{MH(u)}, u)]

where

γ = cov(yi, Zi)var(Zi)−1 (A.5)

e{MHi (u)} is theJ × 1 vector of the prespecified functionalsej{MH

i (u)} which the authorssuggest taking asej{MH

i (u)} = Mij (u) if u > tj and zero otherwise, whereMij (u) is thecost incurred in subinterval(tj−1, min(tj, u)] andG∗(e{MH(u)}, u) is theJ × 1 vector ofG∗(ej{MH(u)}, u) whereG∗(·) is defined by

G∗(e{MH(u)}, u) =∑n

i=1e{MHi (u)}Yi(u)

Y(u)

For large samples, the variance is estimated by

var(µimp)= 1

n

[1

n

n∑i=1

δi(Mi − µimp)2

K(Ti)+ 1

n

∫ ∞

0

dNc(u)

K2(u){G(M2, u)

− G2(M, u)} − cov(yi, Zi)var(Zi)−1cov(yi, Zi)

′]

(A.6)

The jth element in the 1× J vector of the estimator of cov(yi, Zi) is

1

n

∫ ∞

0

dNc(u)

K2(u)

(1

n

1

S(u)

n∑i=1

[δi

K(Ti){Mi

− G(M, u)}{Mij (u) − G(Mj(u), u)}I(Ti ≥ u)

])(A.7)

and a consistent estimator of var(Zi) has its (j, l)th element as

1

n

∫ ∞

0

dNc(u)

K2(u)

(1

n

1

S(u)

n∑i=1

[δi

K(Ti){Mij (u)

−G(Mj(u), u)}{Mil (u) − G(Ml(u), u)}I(Ti ≥ u)])

(A.8)

for j, l = 1, . . . J .When the same methodology is applied to improve the efficiency of the partitioned

estimator the resultant improved partitioned estimator is given by

µPimp = 1

n

n∑i=1

K∑j=1


Kj(Tji )

+ 1

nγ

n∑i=1

∫ ∞

0

dNci (u)

K(u)[e{MH

i (u)} − G∗(e{MH(u)}, u)]


where

γ = cov(yi, Zi)var(Zi)−1 (A.9)

the vector var(Zi) and the set of functionals are as defined for the improved simple estimatorand thejth element of the 1× J vector of cov(yi, Zi) is

1

n

∫ ∞

0

dNc(u)

K2(u)

(K∑l=1

1

n

1

Sl(u)

n∑i=1

[δl∨ji

Kl∨j(Tl∨ji )

{Mil

− Gl(Ml, u)}{Mij (u) − G(Mj(u), u)}I(T li ≥ u)

])(A.10)

The asymptotic variance is estimated by

var(µPimp) = 1

n

1

n

n∑i=1

δi(Mi − µPimp)2

K(Ti)+∫ L

0

K∑j=1

K∑l=1

Sj∧l(u){Gj∧l(MjMl, u)

− Gj∧l(Mj, u)Gj∧l(Ml, u)} dNc(u)

Y(u)K(u)

− n−1cov(yi, Zi)var(Zi)−1cov(yi, Zi)

′ (A.11)

Appendix B

The stochastic integrals appearing in the Bang and Tsiatis estimators are of the form∫ L

0f(u)dNc(u) =

∫(0,L]

f(u)dNc(u) (or∫ ∞

0f(u)dNc(u))

whereNc(·) is the counting process for censoring defined above,f(·) is some functionof time and 0≤ L ≤ ∞. Both f(·) andNc(·) satisfy the required properties to ensurethat the above integrals are well defined as Lebesgue–Stieltjes integrals.9 Moreover, giventhatNc(u) = ∑n

i=1Nci (u) = ∑n

i=1I(Xi ≤ u, δi = 0) and as a step function it has at mostcountably many jumps at{u1, u2, . . . } with$Nc(uk) = Nc(uk)−Nc(uk−) > 0, the aboveintegrals were evaluated as follows:

9 In general, for any two stochastic processesX andY the integral∫X dY , which is a stochastic process itself, is

well defined as a Lebesgue–Stieltjes integral over a given time interval(s, t] if XandYsatisfy certain measurabilityand sample path properties, that is, the sample paths ofY are assumed right-continuous with left-hand limitsand of locally bounded variation on(s, t], i.e.

∫ t

s|dY(u)| is finite for all t > 0 and the sample path ofX is a

measurable function on interval(s, t]. Any bounded variation process can be written as the differenceY1 − Y2

of two non-decreasing processes and consequently, all sample path properties of the integral∫ t

sX dY (considered

as a function oft) follow from properties of integrals with respect to non-decreasing functions. Most processesencountered within the context of survival analysis are of bounded variation on finite intervals. Any integral∫∞s

X dY over an infinite interval is well defined as Lebesgue–Stieltjes integral as it is a limit of integrals overfinite intervals (Fleming and Harrington, 1991).


∫ L

0f(u)dNc(u) =

n∑i=1

∫ L

0f(u)dNc

i (u)

=∫ L

0f(u)dNc

1(u) +∫ L

0f(u)dNc

2(u) + · · · +∫ L

0f(u)dNc

n(u)

=∑

k:0<uk≤L

f(uk)$Nc1(uk) +

∑k:0<uk≤L

f(uk)$Nc2(uk)

+ · · · +∑

k:0<uk≤L

f(uk)$Ncn(uk)

= f(u)|u=X1 andδ1=0 + f(u)|u=X2 andδ2=0

+ · · · + f(u)|u=Xn andδn=0 =n∑

i=1

f(u)|u=Xi andδi=0

where for each individuali, the Stieltjes integral∫ L

0 f(u)dNci (u) = ∑

k:0<uk≤L f(uk)$Nci −

(uk) represents the sum of the values off(·) at the jump times(uk) of Nci (u) in the interval

(0, L] where the jumps of the paths of the processNci (u) are of size+1 at the time of

censoring for individuali, i.e. atu = Xi with δi = 0.

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