On estimators of medical costs with censored data

Journal of Health Economics 23 (2004) 615–625

On estimators of medical costs with censored data Anthony O’Hagan a,∗ , John W. Stevens b a

Department of Probability and Statistics, Centre for Bayesian Statistics in Health Economics, University of Sheffield, Sheffield S3 7RH, UK b AstraZeneca R&D Charnwood UK Received 9 September 2002; accepted 11 June 2003

Abstract In the assessment of cost-effectiveness of alternative medical technologies, it is necessary to estimate the mean total cost per patient over the relevant patient population. Where information about costs comes from a clinical trial with censored data, care is needed to estimate mean total costs. We examine the theoretical connections between the two most widely used of a growing range of nonparametric estimators of costs under censoring. By clarifying the relationships between these simple methods we hope to make them more accessible and to facilitate the take-up of more sophisticated techniques. Recommendations are offered regarding the most appropriate of the available methods, but also on the potential for greater efficiency through parametric modelling. © 2004 Elsevier B.V. All rights reserved. JEL classification: C24 Keywords: Censoring; Lifetime; Kaplan–Meier estimator; Medical costs

1. Introduction We consider the evaluation of a medical intervention where patients are observed until either death or censoring occurs. For each patient there is a cost which accumulates over time until death. We wish to estimate the mean of this total cost over the relevant population of patients. In practice, a time limit T is set, so that a patient’s total cost is defined as their accumulated cost to time T or to death, whichever occurs first. It is rare for it to be possible to follow every patient in the trial to time T or to death, in order to observe each patient’s total cost. Censoring generally occurs in practice. Patients may be lost to follow-up for a variety of reasons. Also, administrative censoring occurs ∗ Corresponding author. Tel.: +44-114-222-3773; fax: +44-114-222-3759. E-mail address: [email protected] (A. O’Hagan).

0167-6296/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jhealeco.2003.06.006

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where patients are recruited to the trial over time, with a fixed end point for the trial. Whatever the nature of the censoring, care is needed in estimating the mean total cost, as pointed out by Lin et al. (1997). Even the simple estimator which is the sample mean of observed total costs for all those patients who were not censored will generally be biased and inconsistent. Costs are usually studied to inform a cost-effectiveness analysis, in which context it is the mean cost over the relevant population of present and future patients that is of primary importance. The principal current methods for estimation of the population mean cost are nonparametric. Key references are Lin et al. (1997) and Bang and Tsiatis (2000). We will refer to these as Lin and B&T, respectively. One motivation for the present paper is to clarify the two basic approaches, in order to promote understanding and consideration of these and more complex methods. In each of these two papers, the authors propose two forms of estimator, whose use depends on the detail with which cost information is available on individual patients in the trial. In one scenario, we only observe the total cost for each patient, up to the time when that patient died, was censored or reached time T . We will refer to this as minimal cost data. The second scenario is where accumulated patient costs are known at certain fixed elapsed times, such as annually or monthly. We will refer to this case as interval cost data. Both Lin and B&T present estimates that are appropriate in the case of minimal cost data and interval cost data. The two papers apply quite different approaches to estimation, and although B&T claim that their methods are generally more efficient, we are not aware of any attempt to relate one approach to the other theoretically. In Section 2, we review the Lin and B&T estimators, using a common notation, and in Section 3 we identify relationships between them. We conclude that if patient costs are only available at the time of death or censoring, then the approaches are equivalent because the Bang and Tsiatis complete-case estimator can be identified with a previously unreported limiting form of the first Lin estimator. Using this connection, we clarify the Bang and Tsiatis partitioned estimator and compare it with the second Lin estimator. Section 4 reviews other methods for cost estimation under censoring, with particular reference to parametric methods. Finally, some recommendations for practitioners are offered in Section 5.

2. Lin and B&T estimators 2.1. Notation Let the cumulative cost up to and including time t for a random patient in the population be denoted by M(t). It will usually be the case that M(t) is a non-decreasing function of t. M(t) will typically be a step function and hence not continuous, but by definition it is always right-continuous. The cumulative cost M(T) at time T is the principal random variable of interest, and we will refer to this as the final cost for a random patient. Inference will focus on the population mean final cost: µ = E(M(T)).

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Let V denote the time to death for a random patient. Formally, we suppose that M(t) = M(V) for all t ≥ V , since no further accumulation of costs occurs after death. Hence M(T) can be seen as the accumulated cost to time T or death, whichever occurs earlier. Let C denote the censoring time for a random patient. This will affect what we can observe about the patient. If C < V , for instance, we will not observe V , and in general we will not observe M(t) for any t > C. We will make the assumption that censoring is not informative about V or M(t). Then the actual censoring time itself provides no information, and is relevant only insofar as it affects what we observe for a given patient in a trial. We will suppose that in the trial we observe data on n patients, comprising • Xi = min{Vi , Ci }, the time for which patient i is observed, assumed to be be less than or equal to T , • di , an indicator variable that equals 1 if Xi = Vi and 0 if Xi = Ci , • cumulative costs at various points in time: {Mi (t) : t ∈ Hi }. In the case of minimal cost data, the set Hi comprises the single point Xi , whereas with interval cost data Hi comprises all the fixed interval endpoints prior to Xi , plus the point Xi . The notation is general, and would allow many other regimes of information, with the ideal case being full cost history data such that Hi = [0, Xi ], but the cases of minimal and interval cost data are the ones studied by Lin and B&T. As a general notational convention, the indicator function of a proposition A will be denoted by I(A), i.e. I(A) = 1 if A is true and otherwise I(A) = 0. For instance, di = I(Xi = Vi ). 2.2. Lin estimators Lin proposed two different estimators, for the cases of minimal cost data and interval cost data. Both estimators are proved to be consistent and asymptotically normally distributed under certain conditions. In both methods the Kaplan–Meier method is used to estimate the survival function SV (t) = P(V > t). This Kaplan–Meier estimate is denoted by Sˆ V (t), and given by Eq. (A.2) in the Appendix. Notice that the survivor function is defined here in terms of the probability P(V > t) of the event occurring after time t, whereas it was defined by Lin to be P(V ≥ t). Although the distribution of time to death may be continuous, its Kaplan–Meier estimate is not, and Sˆ V (t) is a step function. Our definition makes both functions continuous to the right, whereas Lin’s use of P(V ≥ t) would make them continuous to the left. The distinction is important because M(t) will generally not be continuous. We have chosen to reformulate Lin’s estimators in terms of this alternative definition for notational consistency with our cumulative measure M(t), for ease of comparison with the B&T estimators and for the more fundamental reason that P(V > t) is the proper complement of the cumulative distribution function. The first Lin estimator is constructed as follows, and is appropriate for interval cost data. Suppose that the time period from 0 to T is divided into K intervals by the observation points 0 = a1 < a2 < . . . < aK+1 = T . We think of the intervals being closed at the

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right, so that the kth interval is (ak , ak+1 ], whereas Lin interpreted them as closed at the left, consistently with their definition of the survivor function. Suppose also that for patient i we observe the cumulative cost at Xi and at each of the observation points ak < Xi . Let Mik = Mi (ak+1 ) − Mi (ak ) be the incremental cost for patient i over the kth time interval. This is actually observed for time intervals that are completed before Xi . To make use of the ˜ ik = Mik increment in cost over the last, typically incomplete, interval for patient i define M ˜ ik = Mi (Xi ) − Mi (ak ) if ak < Xi < ak+1 . Notice that M ˜ ik actually if ak+1 ≤ Xi and M only differs from Mik for a patient that is censored during the kth interval, since if Xi = Vi we have Mi (Xi ) = Mi (t) for all t > Xi . Then Lin’s first estimator is µ ˆ1 =

K


ˆ k, Sˆ V (ak )E

(1)

k=1

where n ˜ i=1 I(Xi > ak )Mik ˆk =  E . n i=1 I(Xi > ak )

(2)

ˆ k is the sample mean increment in cost over To interpret this estimator, we note that E time interval k, averaged over all those patients who were still under observation at the start of the interval. It is seen as an estimator of the population conditional mean cost in time interval k given that a patient is still alive at the start of the interval, i.e. of Ek = E(M(ak+1 ) − M(ak )|V > ak ). The estimator Eq. (1) arises from the identity: µ=

K


SV (ak )Ek ,

(3)

k=1

which holds for any set of observation points, but depends on fact that E(M(ak+1 ) − M(ak )|V ≤ ak ) = 0, because no costs accumulate after death. Strictly, Eq. (1) and Eq. (3) assume that no cost accrues at time 0, since that time point is excluded from the first interval (a0 , a1 ] = (0, a1 ] in our formulation. (Lin correspondingly made an implicit assumption of no cost arising at time T , but did not discuss this detail.) If, as well may be the case, it is possible  that M(0) = 0, then it is simple to adjust ˆ 0 = n−1 ni=1 Mi (0), since Mi (0) is observed for all Eq. (1) by adding the sample mean E patients. Lin’s second estimator is appropriate for minimal cost data, i.e. when we only observe the cumulative cost at the time Xi . We again divide the period from 0 to T into K intervals by the points 0 = a1 < a2 < . . . < aK+1 = T , but we would not now refer to these as observation points, since cumulative costs are not now supposed to be observed at these points. Let Ak = E(µ|ak < V ≤ ak+1 ), then an alternative identity to Eq. (3) is µ=

K
k=1

{SV (ak ) − SV (ak+1 )}Ak .

(4)

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This suggests the estimator µ ˆ2 =

K


ˆ k, {Sˆ V (ak ) − Sˆ V (ak+1 )}A

(5)

k=1

where Sˆ V (t) is the Kaplan–Meier estimator of SV (t) as before, and where K I(ak < Xi ≤ ak+1 , di = 1)Mi (Xi ) ˆ . Ak = k=1 K k=1 I(ak < Xi ≤ ak+1 , di = 1) To interpret this estimator, notice that when di = 1 we have Xi = Vi and the observed ˆ k therefore averages the observed final Mi (Xi ) equals Mi (Vi ) = Mi (T). The estimate A costs for all those patients observed to die in the kth interval. This second estimator does not use any cost information at all from censored patients. In this respect it clearly may be inefficient unless the majority of patients are observed to die. Notice that Eq. (4) only holds as an identity if patients cannot survive beyond aK+1 = T . It is therefore necessary to treat all patients who survive to time T as if they died at that point, so that their complete costs can be included in the final, k = K, element of Eq. (5). 2.3. B&T estimators B&T propose a basic estimator that they call the weighted complete-case estimator. Like the second Lin estimator, it is applicable when we have only minimal cost data, so that for patient i the only cumulative measure that we observe is Mi (Xi ). Let SC (t) be the survival function for the censoring event, i.e. SC (t) = P(C > t). Notice that, consistently with the way we have defined SV (t), this definition makes SC (t) right-continuous. B&T propose to use a Kaplan–Meier estimator Sˆ C (t). This is in a sense the dual of Sˆ V (t). In the latter, survival to death is estimated by taking account of incomplete observations due to censoring by C. Sˆ C (t) estimates survival to the censoring time by taking account of incomplete observations due to ‘censoring’ by death, and is also constructed to be right-continuous. Formally, it is defined by Eq. (A.3) in the Appendix. The B&T weighted complete-case estimator is then n

µ ˆ3 =

1
di Mi (Xi ) . n Sˆ C (Xi )

(6)

i=1

B&T give a heuristic rationale for this estimator as follows. A patient who is observed to die at time Xi = Vi had a probability SC (Xi ) of not being censored. Hence, we can think of this patient as representing, on average, SC (Xi )−1 individuals who might have been censored. They prove that µ ˆ 3 is always a consistent estimator of µ. B&T go on to propose what they call a partitioned version of their estimator, using the same idea as Lin of partitioning the time period into K intervals and applying their weighted complete-case estimator within each interval. We found the precise form and derivation of their partitioned estimator unclear, and one motivation of the present article is to clarify their presentation by drawing connections with the Lin estimators.

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3. Linking the estimators 3.1. Lin’s second estimator and B&T’s complete case estimator Whereas B&T demonstrate that both of their estimators are consistent estimators of µ, Lin acknowledge that µ ˆ 1 and µ ˆ 2 are consistent only under certain conditions. Considering first their second estimator, unless censoring only occurs at the ends of the intervals, µ ˆ 2 is not a consistent estimator of µ because those patients observed to die in the interval will not be a representative sample of those who actually die in the interval, and this leads to ˆ k being a biased estimator of Ak . However, in the case of their second estimator we noted A that the partitioning of [0, T ] into intervals by the points 0 = a1 < a2 < . . . < aK+1 = T is arbitrary. The ak s do not correspond to fixed observation times. Since they are arbitrary, we could simply define the set of ak s so that it includes all the censoring times. Suppose in fact that we go further and consider going to the limit of infinitesimally small intervals. As ak approaches ak+1 , Sˆ V (ak ) − Sˆ V (ak+1 ) becomes zero in all intervals except for times ak+1 where deaths are observed. To express this limit more clearly, it is helpful to introduce some further notation here. Let deaths be observed at the times t1d < t2d < . . . < tJd , and let the number of deaths observed at time tjd be mdj out of ndj patients under observation. We formally define tJd = T , so that all patients surviving uncensored to time T are treated as dying at that point, and thus mdJ = ndJ . Then from the formula Eq. (A.2) given for Sˆ V (t) in the Appendix, the limit of Sˆ V (tjd − ˆk δt) − Sˆ V (t d ) as δt tends down to zero is Sˆ V (t d )md /nd . At the same point t d in time, A j

j−1

j

j

j

clearly becomes the average of the final costs M(tjd ) out of those mdj observed deaths, and ˆ d ). So we obtain we denote it by A(t j

µ ˆ ∗2 =

J
mdj j=1

ndj

d ˆ jd ). )A(t Sˆ V (tj−1

(7)

For the first, j = 1, element of this sum we formally define Sˆ V (t0d ) = 1. This estimator is always available even with minimal cost data. Although we have constructed Eq. (7) by partitioning the time interval into arbitrarily fine intervals, we can use it with only minimal cost data. Furthermore, since we effectively have infinitesimal intervals, the assumption that censoring only occurs at the ends of intervals automatically holds. This version was not mentioned by Lin, although it is apparently superior to the form they gave for their second estimator. Now consider the B&T complete case estimator Eq. (6). Notice that the use of di means that the sum is essentially only over patients that are observed to die before being censored (including those who ‘die’ at T ). So using the notation of the Lin estimator we can rewrite Eq. (6) as µ ˆ3 =

J ˆ d)
mdj A(t j j=1

nSˆ C (tjd )

.

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We can now see a strong similarity to µ ˆ ∗2 . Both estimators are weighted averages of the ˆ d ) is the cumulative final measures observed for patients who are not censored. The A(t j d average of those cumulative final measures for the patients who die at time tj . These are d )md /nd in the expression Eq. (7) for µ weighted by Sˆ V (tj−1 ˆ ∗2 , and by mdj /{nSˆ C (tjd )} in µ ˆ 3. j j The similarity goes beyond this, however, because it is possible to show that these weights are actually equal, and hence that the B&T weighted complete-case estimator is identical to the limiting form Eq. (7) of the second Lin estimator. This follows from the general result Eq. (A.5) relating Sˆ C (t) to Sˆ V (t), proved in the Appendix. B&T’s estimator is always consistent, reflecting what we have already noted about the consistency of µ ˆ ∗2 . Since B&T’s µ ˆ 3 requires an unfamiliar use of Kaplan–Meier, people may find Eq. (7) a nicer formula to implement and understand in practice. 3.2. Lin’s first estimator and B&T’s partitioned estimator We now use the equivalence between these two estimators to establish a simpler version of B&T’s partitioned estimator. Following B&T’s informal description of this estimator, we will apply the estimator µ∗2 in a suitable form to replace the estimator Eq. (2) of Ek = E(M(ak+1 )−M(ak )|V > ak ) in Eq. (3). First consider partitioning the kth interval (ak , ak+1 ] k into K subintervals with boundaries ak = a0k < a1k < . . . < aK  +1 = ak+1 , and modifying Eq. (4) to give 

Ek =

K
k =1

{SVk (akk ) − SVk (akk +1 )}Akk ,

where SVk (t) = P(V > t|V > ak ) = SV (t)/SV (ak ) and Akk = E(Ek |akk < V ≤ akk +1 ). We then estimate SVk (t) by Sˆ Vk (t) = Sˆ V (t)/Sˆ V (ak ) and Akk by the mean of the incremental cost Mik = Mi (ak+1 ) − Mi (ak ) amongst all patients observed to die within the subinterval k (akk , akk +1 ] or, in the case of the last element of the sum, to survive to aK  +1 = ak+1 uncensored. Now taking the limit as the subintervals become infinitesimally small, we obtain the estimator   K
n
Sˆ V− (Xi ) Sˆ V− (ak+1 ) µ ˆ4 = di I(ak < Xi < ak+1 ) Mik , (8) + I(Xi ≥ ak+1 ) ni n∗k+1 k=1 i=1

where ni is the number of patients under observation immediately preceding Xi , n∗k is the number of patients under observation immediately preceding ak , and Sˆ V− (t) is the value of the Kaplan–Meier estimate immediately preceding t. Note that Sˆ V− (t) is the left-continuous form of the Kaplan–Meier estimate, estimating P(V ≥ t), which is the form used by Lin. The estimator Eq. (8) is, we believe, B&Ts partitioned estimator, but we hope that the description here is clearer than the original formulation by B&T. It also employs the usual survivorship function rather than the complementary function SC (t) used by B&T. The inner summation is a weighted sum of the observed increments Mik for patient i in interval k,

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for all those patients who are observed to die strictly within the interval and for all those patients who are observed uncensored to the end of the interval. Notice that the estimator Eq. (7) reduces to the simple average of total costs observed for all patients, in the case where there is no censoring. In the same way, Eq. (8) reduces to Eq. (1) if no patients are censored within intervals. This is the situation in which Eq. (1) is known to be consistent, whereas Eq. (8) is always consistent. We can now see clearly the relationship between B&T’s partitioned estimator Eq. (8) and Lin’s first estimator Eq. (1). B&T’s partitioned estimator is consistent, but Lin’s estimator makes use of information about costs for patients who are censored during the interval, whereas B&T’s partitioned estimator loses that information. A measure of this loss of information might be the total of the lost time, expressed as a proportion of the total observed time: n − maxak ≤Xi (ak ) i=1 Xi . (9) n i=1 Xi 4. Other methods We have considered only the simplest methods in a growing literature concerning cost estimation in the presence of censoring. Despite the existence of many more sophisticated techniques, the adoption of such methods in the practice of health economics seems to be limited almost exclusively to the Lin estimators. Our objective has been to clarify the simplest methods with a view of increasing understanding and encouraging the consideration of more efficient methods. For instance, B&T present a much wider class of estimators than those considered here and further estimators of the same type are proposed by Zhao and Tian (2001). These estimators fall into a general class originally discussed by Robins and Rotnitzky (1992) and van der Laan and Hubbard (1998), which were also applied to estimating quality adjusted survival time by Zhao and Tsiatis (2000) and further extended by Strawderman (2000). Methods for incorporating covariate information are proposed by Lin (2000a,b) and Jain and Strawderman (2002). All of these methods rely exclusively upon nonparametric estimation, leading to estimators that are consistent regardless of the true underlying survivor function and regardless of the way in which costs are accumulated over time. This robustness to the underlying model is often regarded as a strong recommendation for nonparametric methods. However, consistency is a rather weak property for an estimator, and there are several reasons to consider alternative methods based on plausible parametric assumptions. For instance, cost distributions are typically highly skew, and costs accumulate at highly uneven rates over time. O’Hagan and Stevens (2003) argue in the uncensored case that nonparametric methods may be inefficient for estimating mean costs, and the same is likely to be true in the more complex censored case. Both Lin and B&T present approximate standard errors for their estimators and arguments for asymptotic normality, and similar results are given in the other papers cited, but the accuracy of these approximations and the rate of approach to normality will also be very strongly influenced by the degree of skewness in the component costs. In addition to accommodating skewness in the cost data, parametric methods could model the cost-accrual and survival processes over time, and so allow extrapolation beyond the

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length of the trial. Fenn et al. (1996) point out the importance of this to meet the long-term perspective needed for practical economic evaluation. Carides et al. (2000) fit both linear and nonparametric regression relationships between cost and therapy time, for uncensored patients. 5. Recommendations We offer the following recommendations on appropriate estimation of average costs in the presence of censoring. 1. Naive estimation of average costs can lead to serious biases in the presence of censoring. The methods of Lin and B&T presented here are demonstrably better and are simple to use. 2. When only total costs are available on each patient, the B&T complete case estimator is recommended. We have shown its equivalence to a limiting form of Lin’s second estimator, and the resulting expression Eq. (7) is probably the simplest way to apply this method. 3. More efficiency can be obtained from having more information on cost accrual, and we recommend that costs per patient in each of a number of time periods should always be recorded in trials with censoring. If enough time periods are used, then the relative loss of information Eq. (9) from the B&T partitioned estimator will be small and this estimator is then recommended for its consistency. Our expression Eq. (8) is probably the simplest way to apply this estimator. If Eq. (9) is not small, Lin’s first estimator Eq. (1) may be preferred because it uses more information. 4. Other nonparametric estimators may be more efficient, but have been little used in practice. It is not clear in general that their gains outweight the extra complexity of using them. However, where covariate adjustment is needed, the methods of Lin (2000a,b) and Jain and Strawderman (2002) should be considered 5. Parametric modelling has the potential to address the skewness in the cost data and to extract more information from censored data by modelling cost accrual. Parametric modelling of the survivor function would also permit extrapolation of conclusions beyond the length of the trial. We are not aware of any general work of this kind in the literature, but suggest that this is an important direction for research. Acknowledgements The authors thank the referees for their careful and thoughtful reading of an earlier version of this article. The finished version has benefited materially from their constructive comments. Appendix A. Relationship between estimated survivor functions In this appendix, we prove a result relating the Kaplan–Meier estimates Sˆ V (t) and Sˆ C (t) of the two survivor functions. The general result is quoted by B&T in a left-continuous form

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as their Eq. (1). Although clearly not a new result, it is proved here using our notation, in various forms that are needed in the main text. In the notation of Section 3.1, consider the sequence of observed deaths and censoring events in time. We begin at time 0 with n patients alive and uncensored. There follows a sequence of times t1d , t2d , . . . at which deaths are observed to occur, where mdj deaths are d and tjd we suppose that cjd patients are censored, and observed at time tjd . Between tj−1 ndj is the number of patients under observation at time tjd . Strictly, it is the number under observation at the instant before time tjd , i.e. ndj

=

n
i=1

I(Xi ≥ tjd ).

The distinction is important because at the time tjd itself one or more patients are observed to die. In contrast, we will now define the function n(t) =

n


I(Xi > t)

(A.1)

i=1

to denote the number of patients alive and under observation at time t. Note that this is another right-continuous function. The estimator Sˆ V (t) now has the familiar form Sˆ V (t) =

j  nd − m d =1

nd

for all

d tjd ≤ t < tj+1 .

(A.2)

Note that it is a right-continuous step function with steps at the points tjd . This notation is ideal for defining Sˆ V (t) but less convenient for defining Sˆ C (t), for which we need to identify the times at which censoring occurs, rather than simply how many patients are censored between death times. Starting with Sˆ C (0) = 1, suppose that the first censoring time is t1c < t1d , and that mc1 patients are censored at that time, then at that point Sˆ C (t1c ) = (n − mc1 )/n. Now suppose that the second censoring time t2c is also less than t1d , and that mc2 patients are censored at that time. Then at that point we have Sˆ C (t2c ) = {(n − mc1 )/n}{(n − mc1 − mc2 )/(n − mc1 )} = (n − mc1 − mc2 )/n. In general, regardless of how many patients are censored before the first death, we can see that for all t < t1d we have Sˆ C (t) = n(t)/n. However, no patients are actually censored at time t1d , and hence Sˆ C (t1d ) = (n − c1d )/n = nd1 /n. A similar argument can now be applied for t1d < t < t2d , to show that over that range we have Sˆ C (t) = Sˆ C (t1d )n(t)/(nd1 − md1 ). Also, Sˆ C (t2d ) = (nd1 /n){nd2 /(nd1 − md1 )}. In general, we have Sˆ C (t) =

n(t)

j 

nd

ndj − mdj

=1

nd−1 − md−1

where we define nd0 − md0 = n.

,

(A.3)

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We therefore have the following general identity to link the two survivor functions: 1 Sˆ C (t)

=

n ˆ SV (t), n(t)

for all

d tjd ≤ t < tj+1 .

(A.4)

At times of observed deaths, noting that n(tjd ) = ndj − mdj , we obtain 1 n n d ), Sˆ (t d ) = d Sˆ V (tj−1 = d d V j ˆSC (t d ) n − m n j j j j

(A.5)

where the second form arises from simple manipulation using Eq. (A.2). References Bang, H., Tsiatis, A.A., 2000. Estimating medical costs with censored data. Biometrika 87, 329–343. Carides, G.W., Heyse, J.F., Iglewicz, B., 2000. A regression-based method for estimating mean treatment cost in the presence of right-censoring. Biostatistics 1, 299–313. Jain, A.K., Strawderman, R.L., 2002. Flexible hazard regression modeling for medical cost data. Biostatistics 3, 101–118. van der Laan, H.J., Hubbard, A.E., 1998. Locally efficient estimation of the survival distribution with right-censored data and covariates when collection of data is delayed. Biometrika 85, 339–348. Lin, D.Y., 2000a. Linear regression analysis of censored medical costs. Biostatistics 1, 35–47. Lin, D.Y., 2000b. Proportional means regression for censored medical costs. Biometrics 56, 775–778. Lin, D.Y., Feuer, E.J., Etzioni, R., Wax, Y., 1997. Estimating medical costs from incomplete follow-up. Biometrics 53, 419–434. O’Hagan, A., Stevens, J.W., 2003. Assessing and comparing costs: how robust are the bootstrap and methods based on asymptotic normality? Health Economics 12, 33–49. Robins, J.M., Rotnitzky, A., 1992. Recovery of information and adjustment for dependent censoring using surrogate markers. In: Jewell, N., Dietz, K., Farewell, V. (Eds.), AIDS Epidemiology—Methodological Issues. Birkhauser, Boston, pp. 297–331. Strawderman, R.L., 2000. Estimating the mean of an increasing stochastic process at a censored stopping time. Journal of the American Statistical Association 95, 1192–1208. Zhao, H., Tian, L., 2001. On estimating medical cost and incremental cost-effectiveness ratios with censored data. Biometrics 57, 1002–1008. Zhao, H., Tsiatis, A.A., 2000. Estimating mean quality adjusted lifetime with censored data. Sankhya B62, 175– 188.