Biostatistics Advance Access originally published online on February 14, 2006
Biostatistics 2006 7(4):515-529; doi:10.1093/biostatistics/kxj023
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Attributable risk function in the proportional hazards model for censored time-to-event
Program in Biostatistics, Division of Public Health Sciences, Fred Hutchinson Cancer Research Center, Seattle, WA 98109, USA yqchen{at}scharp.org
Department of Biostatistics, Harvard University, Boston, MA 02115, USA
Division of Biostatistics, School of Public Health, University of California, Berkeley, CA 94720, USA
* To whom correspondence should be addressed.
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SUMMARY |
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 TOP SUMMARY 1. INTRODUCTION2. ATTRIBUTABLE RISK FUNCTIONS3. ESTIMATION AND INFERENCES4. NUMERICAL STUDIES5. DISCUSSIONAPPENDIXREFERENCES |
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Time-to-event endpoints are often used in clinical and epidemiological studies to evaluate disease association with hazardous exposures. In the statistical literature of time-to-event analysis, such association is usually measured by the hazard ratio in the proportional hazards model. In public health, it is also of important interest to assess the excess risk attributable to an exposure in a given population. In this article, we extend the notion of ‘population attributable fraction’ for the binary outcomes to the attributable risk function for the event times in prospective studies. A simple estimator of the time-varying attributable risk function is proposed under the proportional hazards model. Its inference procedures are established. Monte-Carlo simulation studies are conducted to evaluate its validity and performance. The proposed methodology is motivated and demonstrated by the data collected in a multicenter acquired immunodeficiency syndrome (AIDS) cohort study to estimate the attributable risk of human immunodeficiency virus type 1 (HIV-1) infections due to several potential risk factors.
Keywords: Attributable fraction; Epidemiologic methods; HIV/AIDS prevention; Population etiologic fraction; Risk assessment
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1. INTRODUCTION |
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TOPSUMMARY1. INTRODUCTION2. ATTRIBUTABLE RISK FUNCTIONS3. ESTIMATION AND INFERENCES4. NUMERICAL STUDIES5. DISCUSSIONAPPENDIXREFERENCES |
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The Multicenter AIDS Cohort Study (MACS) is an ongoing prospective cohort study of the natural history of human immunodeficiency virus type 1 (HIV-1) infection among homosexual and bisexual men to identify their associated risk factors (Kaslow et al., 1987
). In the MACS and many other prospective studies, various time-to-event endpoints are frequently collected to assess their association with potential risk factors. In particular, the time-to-event outcomes can be the times to HIV seroconversion since enrolment for the MACS HIV uninfected participants. It is thus of important research and public health interest to investigate how the time-to-HIV-seroconversion outcomes are associated with the risk factors, such as needle sharing or having sex with an acquired immunodeficiency syndrome (AIDS) partner. In Figure 1, the Kaplan–Meier estimates of time-to-HIV-seroconversion are plotted for these two risk factors. Apparently, both risk factors are associated with unusually high risk of HIV seroconversion during the observation period.
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To measure the association between time-to-HIV-seroconversion and risk factors, the proportional hazards model (Cox, 1972
),
 | (1.1) |
is often used in the statistical and medical literatures, where Z is the covariate for the risk factors and 
is the regression parameter. Here, 
is an unspecified baseline hazard function and 
is the hazard function for Z. When Z is a risk factor indicator, 
is the hazards ratio. The parameter hence measures the relative risk in the proportional hazards model. Specifically, for the time-to-HIV-seroconversion in the MACS between 1984 and 1999, we estimate that 
(
) for the risk factor of needle sharing (
) against no sharing (
), which means that needle-sharing practice is associated with an almost doubled hazard of HIV seroconversion. Similarly, we estimate that 
(
) for the risk factor of having sex with an AIDS partner, implying a 59.9% increase in the hazard of HIV seroconversion for those having sex with an AIDS partner (
).
The measure of association itself, however, may be insufficient to determine what public health implication the risk factors would have on the priority of community-level prevention, since it does not take into account the prevalence of risk factors in a given population. In the MACS cohort, although engaging in the needle-sharing practice is associated with a seemingly greater risk of HIV seroconversion than having sex with an AIDS partner, the proportion of this practice in the MACS initially HIV uninfected cohort is much smaller. It is merely 1.3% compared with that of 58.6% for having sex with an AIDS partner. Without taking into account the discrepancy in the prevalence of risk factors, a prevention program that is solely based on the strength of association may not deliver its full potential on the community risk reduction. In fact, since 1980s, there has been growing public health interest in the disease risk attributable to a risk factor, or potentially preventable if the risk factor is eliminated from a given population, as reviewed in Uter and Pfahlberg (2001). An attributable risk would take into account both the strength of association and the prevalence of the risk factors in the population. For example, Silverberg et al. (2004) recently studied the attributable risk of the AIDS cases among 525 HIV-1 seroconverted participants due to six host restriction gene variants in the MACS cohort.
In public health sciences, the parameter that characterizes the attributable risk is often referred to as the population attributable fraction or the population etiologic fraction (Benichou, 2000, p. 51). When the event outcomes are binary, it is usually defined as (Levin, 1953)
 | (1.2) |
where D is a binary event indicator and Z is the binary risk factor indicator. When the HIV seroconversion is considered binary, according to (1.2), the attributable fraction for the HIV seroconversions is calculated as 0.23% for engaging in the needle-sharing practice, while it is 24.1% for having sex with an AIDS partner. That is, in the MACS cohort, a fraction of 0.23% of the HIV seroconversion incidences would be attributed to the needle-sharing practice, but it becomes 24.1% that would be attributed to having sex with an AIDS partner. More discussion on the implication, estimation, and application of the attributable risk can be found in Walter (1976), Greenland and Robins (1988), and Gefeller (1992).
For the attributable risk characterized by the population attributable fractions, statistical methods have been mainly focused on its estimation and inference in various epidemiological study samplings, such as case-control (Drescher and Schill, 1991), cross-sectional (Basu and Landis, 1993), and cohort designs (Benichou, 2001). The recent work of Graubard and Fears (2005) further developed more general methodologies to estimate the population attributable fraction across a variety of sample designs. Nevertheless, most of these methods are developed for the binary event outcomes. Only a few of them such as those of Greenland (2001) and Silverberg et al. (2004) may be potentially extended to the outcomes of censored time-to-event, which are frequently encountered in the studies such as the MACS.
In the sections to follow, we propose some attributable risk measures for the time-to-event outcomes. These measures are functions of time. They allow the attributable risk to be time varying. Under the widely used proportional hazards model, we develop a simple estimator for the hazard-based attributable risk function. Simulation studies are conducted to evaluate its validity and performance. We apply the proposed methodology to the publicly available portion of the MACS data from the study inception in 1984 to 1999. Some technical details are included in the Appendix.
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2. ATTRIBUTABLE RISK FUNCTIONS |
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TOPSUMMARY1. INTRODUCTION2. ATTRIBUTABLE RISK FUNCTIONS3. ESTIMATION AND INFERENCES4. NUMERICAL STUDIES5. DISCUSSIONAPPENDIXREFERENCES |
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Let T be the nonnegative random variable of the time-to-event. A natural extension of
for T is, for some 
,
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where 
values are the absolute risk functions, i.e. cumulative distribution functions of T. Thus, the attributable fraction of disease risk due to an exposure can be time varying. When t is the end of the follow-up period for a cohort study, 
, say, 
is the attributable fraction in (1.2). For rare diseases, when values are usually approximated by their respective cumulative hazard functions of 
, 
can also be expressed in 
. Within an infinitesimal neighborhood of t, an alternative measure of the attributable risk function for T is thus
 | (2.1) |
which is called attributable hazard function, given the nature of its definition. An extended measure is the average attributable hazard function, i.e. 
on 
. In particular, 
is a useful summary measure of 
. In addition, the range of is 
. Under the proportional hazards model (1.1), 
for all 
if and only if 
. Since 
in (1.1), 
.
To gain some concrete sense of the proposed attributable risk functions of 
and , we assume that the proportional hazards model (1.1) holds for the exponential baseline hazard functions of 1 and 
, representing relatively frequent and rare diseases, respectively. Let 
for the exposed 
against the unexposed 
. Three proportions of exposure are considered: 25%, 50%, and 75%, respectively. As shown in Figure 2, the attributable risk function defined by either or is not necessarily constant over time, even when the baseline hazard function itself and the exposure prevalence are constant. When the baseline hazard function is relatively large, the attributable risk functions change more rapidly over time, less otherwise. That is, when the disease is more (less) frequent among the unexposed subjects, the disease risk attributable to the exposure tends to change more (less) rapidly over time. In addition, by comparing with , we find that better approximates for the less frequent disease and the smaller proportion of exposure.
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with constant 
. Solid lines are 
. Dashed lines are 
.Given the popularity of the proportional hazards model in the literature, it is foreseeable that
may be easier to be adapted in model-based estimation than , since 
is expressed directly in hazard functions. Note that 
in (2.1) is, however, the hazard function of the marginal distribution 
, by ignoring the heterogeneity among the subjects in the given population. It usually does not equal 
where 
is the distribution function of 
By Bayes Theorem, we have
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where 
, 
, and 
is the conditional distribution function of Z given 
. As a result,
 | (2.2) |
When the actual timing of the events is ignored, i.e. the event outcomes are binary, under the logistic regression model
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where 
and are the regression parameters, Drescher and Becher (1997) discovered that the attributable fraction could be expressed as
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for the rare diseases. Compared with in (2.2), this would be exactly 
if the actual occurrences of the events were scaled up to the maximum follow-up time of . Thus, is considered as a natural extension of for the logistic regression model to the time-to-event outcomes for the proportional hazards model.
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3. ESTIMATION AND INFERENCES |
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TOPSUMMARY1. INTRODUCTION2. ATTRIBUTABLE RISK FUNCTIONS3. ESTIMATION AND INFERENCES4. NUMERICAL STUDIES5. DISCUSSIONAPPENDIXREFERENCES |
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We follow standard notation and assumptions to establish the estimation and inference procedures for the proposed
under the proportional hazards model, when the time-to-event outcomes are subject to censoring. Suppose that there are n subjects recruited in the cohort study. Let 
and 
be the time-to-event and the censoring time, respectively, 
. The observed data consist of n independent identically distributed (iid) 
, , where 
and 
. Denote the at-risk indicator 
. Consider 
and 
, 
.
Assume that 
is the true value of the regression parameter in the semiparametric proportional hazards model (1.1). The baseline hazard function is unspecified. The maximum partial likelihood estimator, 
, can then be obtained by solving the partial score equations
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where 
and 
. Let 
. Standard martingale theory of counting processes in Andersen and Gill (1982) shows that is consistent and 
is asymptotically equivalent to
 | (3.1) |
where 
are martingales with respect to the filtration of 
Moreover, the baseline hazard function 
can be estimated by the Breslow estimator of 
. Considering the Nelson–Aalen estimator for the marginal hazard function of , i.e. 
, a natural estimator of the attributable fraction function in (2.2) is thus
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Alternatively, denote 
. As derived in Xu and O'Quigley (2000), when is independent of and 
, 
values are the conditional probabilities of the subjects observed to fail at t given that one of the at-risk subjects would fail at the same time. Therefore, the conditional distribution function of Z given , , can be consistently estimated by 
This fact leads to the same estimator of 
To make inferences based on the proposed 
, we develop its asymptotic properties as shown in the Appendix. Under the specified regularity conditions, is uniformly consistent for for 
, i.e. 
In addition, 
converges weakly to a zero-mean Gaussian process. Its covariance function of 
, 
, is consistently estimated by 
, where 
is
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and 
, respectively. As a result, the variance of 
is approximately 
, and the pointwise 
confidence intervals for 
can be constructed as
 |
where 
is the 
th percentile of the standard normal distribution.
In addition to the pointwise confidence intervals, it is also of practical interest to consider simultaneous 
th percentile confidence bands, 
and 
, say, such that 
Due to the fact that there is no independent increment structure in the limiting process of 
, it is not straightforward to be transformed into the standard Brownian bridge in direct confidence bands calculation. To find appropriate confidence bands, however, the simulation approach in Lin et al. (1994) can be adapted for ease of implementation. Specifically, consider n iid standard normal deviates 
in
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For any set of finite number of time points 
, 
, the conditional limiting distribution of 
given the observed 
is the same as the unconditional distribution of 
. As a result, 
and have the same limiting distribution by the tightness of 
(Lin et al., 1994). Therefore, th percentile simultaneous confidence bands can be constructed as 
, where 
is computed such that
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4. NUMERICAL STUDIES |
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TOPSUMMARY1. INTRODUCTION2. ATTRIBUTABLE RISK FUNCTIONS3. ESTIMATION AND INFERENCES4. NUMERICAL STUDIES5. DISCUSSIONAPPENDIXREFERENCES |
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Simulations are conducted to evaluate the validity and performance of the proposed estimator of in Section 3. In addition to assuming that the baseline hazard functions are constant of 0.01 and 1.00, respectively, time-to-events are generated according to the proportional hazards model (1.1) with 
and 
, respectively. Sample sizes are selected to be 200 and 500, respectively. Each subject's binary exposure indicator is generated according to the Bernoulli trial with the exposure probability of 25% and 50%, respectively. Censoring times are generated to yield about 30% and 10% of censored observations. The estimators and their associated variances are calculated at the 75 percentile and median of the marginal survival distribution, 
and 
, respectively.
Simulation results are listed in Table 1. For each entry in the table, 1000 simulated data sets are generated to calculate the bias and 95% nominal coverage probability. Here the bias is the difference between the average of the 1000 estimates and the true attributable fraction, and the 95% nominal coverage probability is the percentage of 1000 95% confidence intervals containing the true attributable fraction. As shown in the table, the proposed estimators are virtually unbiased and their confidence intervals maintain the desired coverage probabilities. In addition, the sample standard errors of each 1000 and the average of 1000 
are calculated, respectively. It is shown that they are close to each other, which suggests the accuracy of the calculated variance.
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We apply the proposed attributable risk functions to the publicly released portion of the MACS data set (http://www.statepi.jhsph.edu/macs/pdt.html). The MACS is an ongoing prospective study of the natural and treatment histories of HIV-1 infection in homosexual and bisexual men in four U.S. cities of Baltimore, Maryland; Chicago, Illinois; Pittsburgh, Pennsylvania and Los Angeles, California since 1984 (Kaslow et al., 1987). The full MACS cohort consists of 5622 HIV-1 seropositive and seronegative participants recruited between 1984 and 1985. They are followed up at six-month intervals. In our application, we use a subset of 3341 participants of the original cohort who were in definitive HIV-1 infection-free status at the initial enrolment and followed through 1999. The mean age of the selected participants is 33.5 years (SE=7.7). Among them, 88.0% are white, 10.7% are black, and 1.3% are other races, with about 85% of the participants having more than high school education. By the end of 1999, a total of 508 cases of seroconversions are identified in the data set.
The primary time-to-event outcome of interest is the time to HIV-1 seroconversion since the study enrolment. Due to the follow-up scheme of six-month visit intervals, the exact dates of seroconversion are often unknown. We thus followed the convention in Kingsley et al. (1991), Detels et al. (1998), and Silverberg et al. (2004) to calculate the proxy seroconversion date at one-third of the time between the last HIV-1 seronegative study visit and the first HIV-1 seropositive study visit. Censored observations are mainly due to loss-to-followup, HIV-1-unrelated death or the data inclusion cutoff of December 31, 1999.
In addition to the aforementioned two risk factors of needle-sharing practice and having sex with an AIDS partner, several other prominent risk factors are also examined, i.e. unprotected receptive/insertive anal sex and using cocaine. In Figure 3, survival functions of time-to-HIV-seroconversion are plotted for each additional risk factor, respectively. For the same risk factor of anal sex, the plot shows different patterns between being receptive or being insertive: the receptive participants are associated with higher risk of HIV-seroconversion, while the insertive participants do not show strong association. In fact, the estimates of regression parameter in the proportional hazards model show that being unprotected receptive is significantly associated with 72.8% more of hazard but being unprotected insertive is not significantly associated with higher hazard. For the risk factor of using cocaine, the hazard is more than doubled for those using cocaine, which also shows higher risk in the plot of its survival functions.
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To assess the attributable risk of HIV-seroconversion due to these risk factors, we first calculate their usual
as defined in (1.2). The results are shown in Table 2, along with other summary statistics and the proportional hazards model estimates. Among the five selected risk factors, using cocaine attributes the most to the population risk by 28.1% given its greatest magnitude of association and high prevalence. Having sex with an AIDS partner and unprotected receptive anal sex attributes similarly to the population risk by 24.1% and 23.5%, respectively, given their similar magnitude of association and prevalence. Although unprotected insertive anal sex is the most prevalent risk factor, its attribution to the population risk is by a small fraction of 1.1%. The attribution is even smaller for needle sharing, regardless of its high association, due to merely 43 (out of 3293) participants using shared needles. We also examine the potential interaction between needle sharing and having sex with an AIDS partner. For those engaged in both of the risk factors, their incidence rate is 31.6%. Because of the low prevalence in needle sharing, however, they consist only 1.2% of the total MACS cohort and the associated attributable risk is about 1.3%. According to Miettinen (1974), this may mean a negative interaction, since 
. More discussion on judging the interaction can be found in Walter (1976). Nevertheless, in a group similar to the MACS cohort, when an HIV-prevention program is planned to reduce the preventable HIV-transmission, top priorities should be given to the effort of reducing cocaine use, having sex with an AIDS partner and unprotected receptive anal sex.
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in MACSWe further calculate and plot the proposed attributable risk function for each of the five risk factors in Figure 4, respectively. As shown in the figure, all the attributable risk functions do not appear constant over time. For having sex with an AIDS partner, unprotected receptive anal sex and using cocaine, their attributable risk functions change from 26%, 28.3%, and 32.3% to 25.4%, 27.5%, and 30.5%, respectively, over a 15-year period. Although they are generally decreasing, their attributions to the population risk are little changed. They remain the highly attributable risk factors for the HIV seroconversion. An effective prevention program targeting these prominent attributable risk factors should be designed and implemented to reduce the population risk. For needle sharing, although its attribution to the population risk changed about 10% over time, its attribution may not be always decreasing over time. Unprotected insertive anal sex also does not show consistent pattern of attribution to the population risk. Although unprotected insertive anal sex starts relative low attribution, it may arise as time progresses.
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for the MACS by risk factors (a) 
: needle sharing; (b) 
: having sex with an AIDS partner; (c) 
: unprotected anal sex (receptive); (d) 
: unprotected anal sex (insertive); (e) 
: using cocaine.For these model-based attributable risk functions, we use the risk factors measured at the baseline and assume constant regression coefficients in the proportional hazards models. Therefore, according to (2.2), the overall time-varying pattern in the attributable risk functions should be determined by
, i.e. the time-varying proportion of the risk factor Z in the MACS cohort. In fact, as the time progresses, those already seroconverted prior to t shall not be considered in . The cohort thus tends to be ‘healthier’, which may explain the generally downward pattern of the attributable risk functions. |
5. DISCUSSION |
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TOPSUMMARY1. INTRODUCTION2. ATTRIBUTABLE RISK FUNCTIONS3. ESTIMATION AND INFERENCES4. NUMERICAL STUDIES5. DISCUSSIONAPPENDIXREFERENCES |
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The attributable risk function is introduced in this article to capture the time-varying contribution of both the relative risk and the prevalence of risk factors to a disease progression. It provides a prospective profile of the population attributable risk in time. It can be also used in planning and prioritizing a prevention program. For example, in promoting a community vaccination program, the attributable risk function can be calculated to measure the timely impact on the preventable infections due to the vaccine.
It needs to point out that the attributable risk function in this article is estimated under the proportional hazards model. We conveniently chose for the main development in this article. To rely less on the assumptions of the proportional hazards model, we shall use expressed in s and their nonparametric estimates, which are also more meaningful in terms of the absolute risks. Since 
by Taylor expansion, the high-order terms of and have additional influence on the exact difference of 
. Technical investigation can be conducted to evaluate this difference more generally. In addition, we adopted the stronger version of independence assumption in Xu and O'Quigley (2000) to estimate 
. When the risk factors are mostly categorical with a finite number of categories, such as binary in this article, this assumption can be replaced by the usual weaker version of the conditional independence assumption, as seen in Murray and Tsiatis (1996).
Nevertheless, in this article, we establish a framework of the attributable risk function for the time-to-event outcomes with mostly single binary risk factors. It needs to be further extended to the risk factors of the more general types, either multilevel categorical or continuous. When the possible confounding variables, such as age, gender, and socioeconomic status, are collected with the risk factors, an adjusted attributable risk function is also desirable, as argued in Benichou (2001). To avoid unnecessary technical presentation, we refer to a University of Washington Biostatistics Technical Report (http://www.bepress.com/uwbiostat/paper254/) for more details on some of the proposed extensions.
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APPENDIX |
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TOPSUMMARY1. INTRODUCTION2. ATTRIBUTABLE RISK FUNCTIONS3. ESTIMATION AND INFERENCES4. NUMERICAL STUDIES5. DISCUSSIONAPPENDIXREFERENCES |
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Let 
and 
, . To establish the asymptotic properties of , we assume the necessary regularity conditions specified in Theorem 4.1 of Andersen and Gill (1982):
is continuous, nondecreasing, and 
;
- There exists a compact neighborhood
of such that 
;
is positive definite, where 
.
First, we decompose the estimator 
into
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By Taylor's theorem, 
equals
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where 
is on the line segment connecting and . Following Theorem 4.1 and Corollary III.2 of Andersen and Gill (1982), we know that 
and 
are continuous in 
. In addition, is bounded away from zero on 
. Since 
and 
for , respectively,
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uniformly for . Due to the consistency of for , we know 
uniformly on 
. Also,
 | (A.1) |
It thus follows that 
uniformly for .
To prove the uniform consistency of 
, we only need to show 
. Under the proportional hazards model (1.1), equals
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Therefore, 
. On the other hand, 
equals
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Hence, 
. Under the assumption that C is independent of 
, 
. Therefore, the uniform consistency of holds.
To prove the asymptotic normality, 
can be written as
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By the expression of in (A.1) and the martingale representation of 
, it further equals
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where 
. Thus, 
. Here,
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Since 
and 
are both monotonic processes, converges weakly to a zero-mean Gaussian process with covariance function 
, as shown in the Example 2.11.16 of van der Vaart and Wellner (1996).
In order to prove that is consistently estimated by 
, it suffices to show by Cauchy–Schwarz inequality that
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These can be established by the consistencies of , the uniform consistency of 
, and 
, 
, respectively, following Lemma 1 of Lin et al. (2000).
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ACKNOWLEDGMENTS |
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The authors would like to thank Professor Scott Zeger, an Associate Editor and Dr. Ross Prentice for their constructive comments that lead to great improvement of the original manuscript. Conflict of Interest: None declared.
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TOPSUMMARY1. INTRODUCTION2. ATTRIBUTABLE RISK FUNCTIONS3. ESTIMATION AND INFERENCES4. NUMERICAL STUDIES5. DISCUSSIONAPPENDIXREFERENCES |
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Received August 25, 2005; revised December 5, 2005; revised February 2, 2006; accepted for publication February 9, 2006.
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