Biostatistics Advance Access originally published online on July 28, 2006
Biostatistics 2007 8(2):368-382; doi:10.1093/biostatistics/kxl016
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Regression analysis of mean quality-adjusted lifetime with censored data
Division of Biostatistics and Epidemiology, Department of Public Health Sciences, University of Virginia, Charlottesville, Virginia 22908, USA hkwang{at}virginia.edu
Department of Biostatistics and Computational Biology, University of Rochester, 601 Elmwood Avenue, Box 630, Rochester, New York 14642, USA
* To whom correspondence should be addressed.
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SUMMARY |
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 TOP SUMMARY 1. INTRODUCTION2. ESTIMATING EQUATIONS FOR...3. SIMULATION EXPERIMENTS4. APPLICATION5. CONCLUSION AND DISCUSSIONAPPENDIXREFERENCES |
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In clinical trials of chronic diseases such as acquired immunodeficiency syndrome, cancer, or cardiovascular diseases, the concept of quality-adjusted lifetime (QAL) has received more and more attention. In this paper, we consider the problem of how the covariates affect the mean QAL when the data are subject to right censoring. We allow a very general form for the mean model as a function of covariates. Using the idea of inverse probability weighting, we first construct a simple weighted estimating equation for the parameters in our mean model. We then find the form of the most efficient estimating equation, which yields the most efficient estimator for the regression parameters. Since the most efficient estimator depends on the distribution of the health history processes, and thus cannot be estimated nonparametrically, we consider different approaches for improving the efficiency of the simple weighted estimating equation using observed data. The applicability of these methods is demonstrated by both simulation experiments and a data example from a breast cancer clinical trial study.
Keywords: Counting process; Estimating equation; Martingale process; Quality of life; Survival analysis
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1. INTRODUCTION |
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TOPSUMMARY1. INTRODUCTION2. ESTIMATING EQUATIONS FOR...3. SIMULATION EXPERIMENTS4. APPLICATION5. CONCLUSION AND DISCUSSIONAPPENDIXREFERENCES |
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In studies that evaluate new therapies for chronic diseases such as cancer, acquired immunodeficiency syndrome, or cardiovascular diseases, extending overall survival time may not be the only goal. Improving patients’ quality of life is also important. Quality-adjusted lifetime (QAL) is a measure which combines patients’ quality of life and survival time together and provides a useful summary for evaluating the treatment effect.
QAL has been studied by Goldhirsch and others (1989), Glasziou and others (1990)
, and Gelber and others (1995). In their work, a patient's health history is partitioned into different health states, e.g. toxicity state during cancer treatment, period of good health, and disease relapse state. Each state is assigned a utility coefficient, usually ranging from 0 (death) to 1 (good health). The QAL, also called quality-adjusted time without symptoms and toxicity (Q-TWiST), is defined as the linear combination of the utility coefficients and the times spent in each health state. In a more general setting, the QAL is simply defined as the integration of utilities over a subject's survival time.
In most clinical trials, patients enter the study over a period of time, and we cannot always observe the QAL for every patient due to loss to follow-up and study termination. The inference on QAL thus has to be made using censored data. Censoring poses a unique problem for making inference on QAL, since even though we are willing to assume that censoring is independent of the health history process, the censored QAL is often correlated with the potential uncensored QAL due to the induced informative censoring problem (Gelber and others 1989). For example, people with poor quality of life will accrue QAL slowly, and when they are censored, they will have small censored QAL as well. Much research has been done on estimating the mean QAL (e.g. Glasziou and others 1990; Gelber and others 1991; Zhao and Tsiatis, 2000), or the survival distribution of QAL (e.g. Zhao and Tsiatis, 1997, 1999; Van der Laan and Hubbard, 1999) from censored data. However, in a real application, it is often of our interest to know how covariates affect the mean QAL.
An example that illustrates the use of regression models for QAL came from clinical studies conducted by the International Breast Cancer Study Group (IBCSG). The IBCSG Trial V (Cole and others 1993) was a randomized clinical trial investigating two treatments for the node-positive breast cancer: short-duration chemotherapy (1 month) and long-duration chemotherapy (6 or 7 months). One thousand two hundred and twenty-nine patients were enrolled in the study with 413 patients randomized to the short-term chemotherapy and 816 patients randomized to the long-term chemotherapy. The median follow-up for the study was 7 years. Six covariates were recorded from each patient upon enrollment in the trial, which included age, treatment, tumor size, tumor grades (medium or high), and number of nodes involved. It was of interest to learn how patients’ mean QAL might depend on these prognostic factors.
Different approaches have been proposed for the regression problems of QAL. Cole and others (1993) used a partitioned health state model and fitted Cox proportional hazards regression models for each transition time from the start of the study to the end of different health states. The mean QAL corresponding to a specific covariate value can be obtained by integrating the survival curves for that covariate value. With this approach, however, one cannot directly assess the covariate effects on mean QAL from the regression parameter estimates without additional calculations. In order to know how a covariate affects the mean QAL, one has to plug in different values for this covariate while fixing other covariates at some population averages. Fine and Gelber (2001) proposed an accelerated life model for the distribution of survival and quality-adjusted survival time. However, their interests are the distribution of QAL, not the mean QAL. The regression method related to mean QAL data was mentioned in Bang and Tsiatis (2002), but no semi-parametric efficiency study was performed.
In this paper, we will study the problem of regressing the mean QAL on the covariates. We will investigate how to construct estimating equations for the regression parameters and how to obtain more efficient estimators by using the semi-parametric theory developed by Robins and Rotnitzky (1992). We will assume a mean model for QAL, but will not make any additional assumption on the underlying distribution of the health history process. Censoring is assumed to be independent of the health history process. In Section 5, we will consider the situation when this condition is not met. Due to limited follow-up time, we only consider QAL accumulated up to a time limit L, where L is determined by the availability of data. The rest of this paper is organized as follows. In Section 2, we describe the regression model and discuss methods for obtaining efficient estimators for the regression parameters. It is followed by the simulation experiments in Section 3. The breast cancer data application is analyzed in Section 4 and finally, some concluding remarks are given in Section 5.
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2. ESTIMATING EQUATIONS FOR REGRESSION PARAMETERS |
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TOPSUMMARY1. INTRODUCTION2. ESTIMATING EQUATIONS FOR...3. SIMULATION EXPERIMENTS4. APPLICATION5. CONCLUSION AND DISCUSSIONAPPENDIXREFERENCES |
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For the ith individual in the study, let the health history process be represented by 
. V can include information on both discrete and continuous variables. Denote 
, the health history up to time t. Let 
be the survival time and q be a known utility function mapping 
to the interval 
; q is assumed to be known for our purpose. In Section 5, we discuss how to handle the situation when q is not known to us. The ith individual's QAL, denoted as 
, is equal to
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Denote the ith individual's censoring time by 
. Censoring is assumed to be independent of the health history process 
. The survival distribution of C is denoted as 
. Because of censoring, we cannot make inference on QAL over the entire health history; we only consider the QAL accumulated within a time limit L, which is chosen based on the study duration. In most cases, we would like to choose L as large as possible but not to exceed a limit beyond which few data are available to us. Consequently, the survival time of an individual will be truncated at L, that is, 
. For ease of notation, we still use T instead of 
. We assume that 
.
Let 
denote the 
vector of covariates associated with the ith individual, with the first covariate being the constant 1. The observed data for n individuals are the independently and identically distributed random quantities: 
. According to this definition, 
.
We describe the mean QAL as a function of the covariates of the form:
 | (2.1) |
where 
is a vector of parameters of interest. Special cases include 
, a linear regression model LRG and 
, a generalized LRG. Our goal is to make inference about in the mean model (2.1) for some pre-specified function g from the observed censored quality of life and survival data.
If complete data are observed, a consistent estimator 
for in the mean model (2.1) can be obtained from the following estimating equation:
 | (2.2) |
where 
is 
-dimensional vector of functions of , and the superscript F represents models for full data. From the semi-parametric theory (Robins and Rotnitzky, 1992), we know that the most efficient estimating equation for the complete data case is the one with
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where 
is the true value of the parameters.
In the special case of a linear model where and 
is assumed to be a constant, the most efficient estimating equation is obtained by setting 
, and hence
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This equation is the same as the ordinary least squares estimating equation for the LRGs.
When censoring is present, cannot be observed for everybody so the estimating equation (2.2) cannot be used. However, using the idea of inverse probability weighting, which was originally proposed by Horvitz and Thompson (1952), we can construct a simple weighted estimating equation for in our mean model (2.1) with censored data:
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where 
and 
is the survival probability for the censoring variable C at time . The consistency of the simple weighted estimating equation is shown by
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Since is not known to us, we can estimate it using the Kaplan–Meier estimator 
(Kaplan and Meier, 1958). Hence, our simple weighted estimating equation becomes
 | (2.3) |
In the special case when and 
, the estimating equation (2.3) has a closed-form solution for given by
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where we use the notation 
, 
, for vectors a and b.
The simple weighted estimator is easy to calculate, however, only the QAL for the subjects with observed failures are used in the estimating equation. Therefore, it cannot be efficient.
To develop the theory on efficiency study of the estimating equations, we use the counting processes and the associated martingale theory as described by Fleming and Harrington (1991). Let the filtration 
be the increasing sequence of 
-algebras generated by
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We consider the martingale process 
, where 
, 
, 
is the hazard function for the censoring distribution.
From the semi-parametric theory for missing data processes developed by Robins and Rotnitzky (1992) and Robins and others (1994), the influence function for the estimating equation for any regular asymptotic linear (RAL) estimators of can be written as
 | (2.4) |
where 
is the influence function for the complete data, 
for any random variable or functional W, 
, and 
is any -dimensional vector of functionals of the health history 
. It should be noted that the influence function for the simple weighted estimating equation (2.3) is simply the first two terms of (2.4) (Zhao and Tsiatis, 1997, equation A.7).
From Robins and Rotnitzky (1992), the most efficient estimating equation is obtained by choosing
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where 
, and
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 | (2.5) |
From the above results, the most efficient estimating equation can be formed by
 | (2.6) |
where 
is a consistent estimator for 
, for any functional W.
In theory, the asymptotic variance of from solving (2.6) should achieve the semi-parametric efficiency bound, which means that from (2.6) has the smallest variance among the class of all regular asymptotically linear estimators. However, it is not useful to use (2.6) for data analysis, since 
and 
depend on the unknown true population parameters which are difficult to estimate nonparametrically.
Due to the difficulty in obtaining the most efficient estimating equation nonparametrically, we wish to find an estimating equation which can be obtained from the observed data, and which can be more efficient than the simple weighted estimating equation for any choice of 
. In the subsequent section, we will discuss the issues of choosing .
We first consider a method for obtaining the improved estimating equation, which is similar to the approach appeared in Zhao and Tsiatis (1999)
for obtaining improved estimators of mean QAL, and in Bang and Tsiatis (2002) for median regression of medical costs. We will call this approach the best-coefficient approach and denote it as BC.
For any chosen , if we multiply the third term in (2.4) by a constant 
 | (2.7) |
where is equal to 
, with 
and 
being the second and third terms in (2.4), i.e.
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then the variance of this influence function (2.7) will always be smaller than that of the simple weighted estimating equation. In practice, is not known, so it has to be estimated from available data, which will result in some additional variability for finite sample sizes. We will examine its finite sample performance in our simulation study.
We can derive an explicit formula for the BC estimator in the special case when 
and 
. If we choose 
, for example, we can get
 | (2.8) |
 | (2.9) |
for any functionals X and Y. If X or Y involves , then 
will be replaced by the 
function
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The detail of the derivation is given in the Appendix.
and 
in place of 
.Our second strategy is to choose
, and use in place of in the formula for 
, i.e. we choose
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The corresponding estimating equation, named the improved estimating equation and denoted as IMP, has the following form:
 | (2.10) |
This estimating equation (2.10) is not guaranteed to be always more efficient than the simple weighted estimating equation (2.3). However, due to the usual correlation between and , we expect this estimator to perform well in most realistic settings.
In the case when and , this improved estimator has an explicit form 
, where
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Our last strategy for improving efficiency is to choose
and estimate by regressing 
on observed covariates, using only those observations with 
(Robins and Rotnitzky, 1992). The resulting estimating equation will be in the same form as (2.10) except that is replaced by the estimate of from the regression approach.
To implement this idea, we may choose an LRG, regressing on some covariates that are predictive of QAL; or we may use a generalized additive model (Hastie and Tibshirani, 1990; Van der Laan and Hubbard, 1999), which accommodates a nonparametric regression of on some functions of the health history process , e.g. . We will compare the performance of these choices in our simulation studies.
Compared to the best choice of for the complete data case
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the optimum choice of for the incomplete data case is
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where 
is defined as (2.5). It is equivalent to down-weight the influence of , due to the added uncertainty about the variance of given for the censored observation. Since it is harder to estimate the second moment than the first moment given the high dimensional health history process, and secondly, using an incorrect model could potentially increase the variability of the estimating equation, we choose not to attempt to estimate and use instead the best choice of for the complete data case.
In this section, we derive the variance estimators for the regression parameters in our various estimating equations. Suppose is the solution to an estimating equation and is the true value of the parameters. From Taylor's expansion, we have
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where
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In the special case when and , we have
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Based on the general influence function (2.4), 
is equal to
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where 
, 
, and 
are defined similarly as 
, , and , with true parameters plugged in.
Using derivations similar to those in the Appendix, we can show that for large n, , the asymptotic variance of 
, can be estimated by
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where
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for any vectors of functionals of X and Y. Similar as the definition for 
, 
will be replaced by the function if X or Y involves the random variable .
The variance of the simple weighted estimating equation (2.3) is just the first two terms in (2.12). Due to the special coefficient used in the BC estimating equation (2.7), its variance can be easily shown to be
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3. SIMULATION EXPERIMENTS |
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TOPSUMMARY1. INTRODUCTION2. ESTIMATING EQUATIONS FOR...3. SIMULATION EXPERIMENTS4. APPLICATION5. CONCLUSION AND DISCUSSIONAPPENDIXREFERENCES |
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In this section, we conduct some simulation experiments to evaluate our proposed estimating equations for the parameters in the regression models. Similar to the IBCSG Trial V example which is to be presented in Section 4, we consider patients entering the study first experiencing toxicity for a certain time, then a period of good health (TWiST), then their disease relapse followed by death. We use TOX to represent the time from the treatment initiation to end of toxicity, TR the time from treatment initiation to disease relapse, and OS the time from treatment initiation to death. The QAL is defined as:
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where 
is the utility coefficient for TOX, 
the utility coefficient for the REL (the period between disease relapse and death). The utility coefficient for TWiST is assumed to be 1, and 
.
We generate 5000 simulations, each consisting of two groups of censored health status data with sample sizes varying from 100 to 400 for each group. Two scenarios are considered here. In the first scenario, TOX is uniformly distributed on [0, 
] for group 1 (
) and uniformly distributed on [0, 
] for group 2 (
); TR is exponentially distributed with hazard 
for group 1 and hazard 
for group 2, and both are truncated at 
. OS is exponentially distributed with hazard 
and truncated at 
for both groups. The censoring variables for both groups are uniform on [70, 116] and are independent of TOX, TR, and OS, which results in the amount of censoring to be about 35% for group 1 and 36% for group 2. For each group, if TR is greater than OS, we set TR = OS. Similarly, if TOX is greater than TR, we set TOX = TR. The true mean QAL for group k (
) is
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Plugging in the parameter values, we can obtain that the true mean QAL is 47.91 for group 1 and 43.89 for group 2. Using an LRG with treatment as a covariate, and group 2 as the reference group, the intercept and slope parameters in our regression model are 43.89 and 4.02, respectively. In the second scenario, group 2 is generated the same way as group 1 in the first scenario, resulting in an intercept of 47.91 and a slope of 0.
We calculate the estimates for the intercept and slope, using the WT equation (2.3), the BC equation (2.8), the IMP equation (2.10), and the regression method. In the regression method, we consider three different approaches to estimate : (1) using the sample average of only from observations with 
, conditioning on the treatment at each time u (denoted as AVE); (2) fitting an LRG for only from observations with , combining all the censoring points and using treatment as the covariate (denoted as LRG); (3) fitting a generalized additive model for only from observations with by smoothing on the (denoted as SM).
Tables 1 and 2 are results from the two simulation experiments, respectively. The sample standard errors (SSE), the estimated standard errors (ESE), and the sample coverage probabilities (CP) of the true parameters by the 95% confidence intervals of those estimators are given. We also calculate the estimates if we use the true (denoted as TrueE), since in the simulation we know the true distributions hence can be obtained. However, in practice, this estimator cannot be used since we do not know the true expectation of given the health history process .
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From the results of our simulation studies, we can see that the biases for all the estimators are rather small, indicating that all the estimating equations give consistent estimates of the regression coefficients. The empirical sample variances are very close to the estimated variances from formula (2.12). The estimators LRG and AVE have bigger SSE than the simple weighted estimator. Using the smoothing approach, smoothing model (SM) does not improve the efficiency. As expected from the theory, plugging in the true expectation gives us the most efficient estimator. The IMP estimator performs the best among all the estimators not using the true expectation. When the sample size is 100, the coverage probability for the BC estimator is not very accurate, but it improves considerably as sample size increases. The coverage probabilities for all other estimators are very close to 0.95 even for small sample sizes.
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4. APPLICATION |
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TOPSUMMARY1. INTRODUCTION2. ESTIMATING EQUATIONS FOR...3. SIMULATION EXPERIMENTS4. APPLICATION5. CONCLUSION AND DISCUSSIONAPPENDIXREFERENCES |
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In the IBCSG Trial V, each patient experienced in sequence three health states: TOX (toxicity), TWiST (perfect health), and REL (disease relapse). We illustrate our methods with the quality-of-life coefficients
and the time limit 
(months). The amount of censoring is 29.3%. Similar to Cole and others (1993), we consider six covariates recorded from each patient upon enrollment in the study: treatment group (
short duration, 
long duration); tumor size (less than 2 cm, at least 2 cm); logarithm of age; medium tumor grade (not medium grade, medium grade); high tumor grade (not high grade, high grade); number of nodes involved (fewer than 4, at least 4). As in Cole and others (1993), 94 patients are removed from the data due to missing values for tumor grade. Thus, there are 1135 patients included in the analysis and at months, 363 patients are still at risk.
We first considered a linear model with the six covariates and the interaction terms between treatment (treat) and the other covariates—tumor size (tsize), medium tumor grade (mgrade), high tumor grade (hgrade), and number of nodes involved (nodegrp). However, none of the interaction terms are significant; thus, they are excluded from our final model. We calculate the estimators using the WT estimating equation (2.3), the IMP estimating equation (2.10), the BC estimating equation (2.8), and other regression approaches for estimating : the linear regression approach (LRG), the sample average approach conditioning on the treatment (AVE), and the generalized additive model with being the only regressor in the SM. Table 3 shows the results.
Concentrating on the covariate estimates and their standard errors first, we see that the AVE approach which estimates the expectation of 
by using the sample average for each treatment group at each censoring time does not perform well. The estimated standard error is bigger than all other approaches. The LRG approach, which is similar to AVE but combines all the censoring points together, performs slightly better. For some covariates, the SM approach produces a smaller standard error, but it is bigger for other covariates. BC estimator is consistently better than WT, but the improvement is not very big. The best performing estimator is the IMP estimator, similar as what we see in the simulation study. We have also considered two other generalized additive models: smoothing with both treatment and as the regressors in the model, and smoothing with all the six covariates and as the regressors in the model. We found out that adding more regressors in the smoothing method did not make much difference, so the results from those two models are not included in Table 3.
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Using the estimates from the IMP approach, we find that all six covariates are significant. A subject who is older, who has smaller tumor size, smaller number of nodes involved, lower tumor grade, and who is on the long-duration arm, has a longer expected QAL. This finding agrees with the description provided in the caption of Table 1 of Cole and others (1993)
. Similar as Cole and others (1993), a sensitivity analysis can be carried out with varying values of and . For any fixed values for covariates, a treatment option can be chosen based on different quality-of-life utility values. |
5. CONCLUSION AND DISCUSSION |
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TOPSUMMARY1. INTRODUCTION2. ESTIMATING EQUATIONS FOR...3. SIMULATION EXPERIMENTS4. APPLICATION5. CONCLUSION AND DISCUSSIONAPPENDIXREFERENCES |
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In this paper, we have developed methods on how to estimate the covariate effects on the mean QAL, and how to obtain more efficient estimating equations. The theory developed by Robins and Rotnitzky (1992)
provides the key to finding the form of the most efficient estimating equation, however, it is a function of the health history and cannot be estimated nonparametrically. We have examined different approaches for obtaining consistent estimators for regression coefficients. We have assumed that the quality-of-life coefficient q is fixed in our methods. However, in a real application, q is often not known and has to be chosen from the quality-of-life questionnaires. A lot of research has been devoted to this area. There are instruments developed which can translate health states into quality-of-life coefficients. In the cases similar to our example when the number of health states are limited, we can perform a sensitivity analysis and find out the treatment advantages for each set of utility values.
The simulation studies show that the best performing estimator is the IMP estimator using in place of . The estimators using regression method (linear regression, or additive models) do not perform well. The BC estimator should always have smaller variance than the simple weighted estimator from the large sample theory, however, the improvement is not very big from our simulation studies.
We have assumed that censoring is independent of the health history process. If this assumption is not true, and censoring depends on some known covariates, we can accommodate this situation by fitting a Cox proportional hazards model to estimate the censoring distribution. If the Cox regression model is true, we can still get consistent estimators for the regression coefficients.
From Robins and Rotnitzky (1992), can be optimized to improve the efficiency of the estimating equation. However, optimizing h involves estimating the second moments of which will introduce some extra variability. How much more efficiency we may gain if we try to obtain the optimized h will be a subject of future research.
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APPENDIX |
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TOPSUMMARY1. INTRODUCTION2. ESTIMATING EQUATIONS FOR...3. SIMULATION EXPERIMENTS4. APPLICATION5. CONCLUSION AND DISCUSSIONAPPENDIXREFERENCES |
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In the special case when , , and , we can construct the following estimating equation based on the influence function (2.7):
 | (A.1) |
where 
is a consistent estimator for 
and can be obtained as follows.
Since
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it can be estimated consistently by
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where for any random variables X and Y is defined in (2.9).
Next,
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It can be estimated consistently by
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Using these results, the estimating equation (A.1) can be written as
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from which the BC estimator can be obtained easily and shown in (2.8).
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ACKNOWLEDGMENTS |
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We are very thankful to Dr. Andrea Rotnitzky and Dr. Xing Qiu for helpful discussions on our paper. We are also very grateful to the IBCSG for making the IBCSG Trial V data available to us. Hongwei Zhao was supported by Department of Defense Breast Cancer Research Program DAMD170210558. Conflict of Interest: None declared.
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TOPSUMMARY1. INTRODUCTION2. ESTIMATING EQUATIONS FOR...3. SIMULATION EXPERIMENTS4. APPLICATION5. CONCLUSION AND DISCUSSIONAPPENDIXREFERENCES |
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Received February 2, 2006; revised June 22, 2006; accepted for publication December 21, 2006.
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