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Survival analysis with error-prone time-varying covariates: a risk

set calibration approach

Xiaomei Liao1,2,*, David M. Zucker3, Yi Li1, and Donna Spiegelman1,2,**

1Department of Biostatistics, Harvard School of Public Health, Boston, MA, 02115,U.S.A.

2Department of Epidemiology, Harvard School of Public Health, Boston, MA, 02115,U.S.A.

3Department of Statistics, Hebrew University, Mt. Scopus, 91905 Jerusalem, Israel.

Summary

Occupational, environmental, and nutritional epidemiologists are often interested in estimating the

prospective effect of time-varying exposure variables such as cumulative exposure or cumulative

updated average exposure, in relation to chronic disease endpoints such as cancer incidence and

mortality. From exposure validation studies, it is apparent that many of the variables of interest are

measured with moderate to substantial error. Although the ordinary regression calibration

approach is approximately valid and efficient for measurement error correction of relative risk

estimates from the Cox model with time-independent point exposures when the disease is rare, it

is not adaptable for use with time-varying exposures. By re-calibrating the measurement error

model within each risk set, a risk set regression calibration method is proposed for this setting. An

algorithm for a bias-corrected point estimate of the relative risk using an RRC approach is

presented, followed by the derivation of an estimate of its variance, resulting in a sandwich

estimator. Emphasis is on methods applicable to the main study/external validation study design,

which arises in important applications. Simulation studies under several assumptions about the

error model were carried out, which demonstrated the validity and efficiency of the method in

finite samples. The method was applied to a study of diet and cancer from Harvard’s Health

Professionals Follow-up Study (HPFS).

Keywords

Cox proportional hazards model; Measurement error; Risk set regression calibration; Time-

varying covariates

1. Introduction

Many epidemiological studies involve survival data with covariates measured with error: the

true covariate value c, as defined by some “gold standard”, is represented approximately by

a surrogate measure C. Often, interest centers on cumulatively updated total or cumulatively

updated average levels of a time-varying exposure, which are computed from a series of

error-prone point exposure measurements. For example, in prospective studies of diet and

*stxia@channing.harvard.edu. **stdls@channing.harvard.edu.

Supplementary Material

Web Appendix A, referenced in Section 6 is available under the Paper Information link at the Biometrics website http:

www.tibs.org/biometrics.

Web Appendix B, referenced in Section 5 is available under the Paper Information link at the Biometrics website

http:www.tibs.org/biometrics.

NIH Public Access

Author Manuscript

Biometrics. Author manuscript; available in PMC 2011 March 17.

Published in final edited form as:

Biometrics. 2011 March ; 67(1): 50–58. doi:10.1111/j.1541-0420.2010.01423.x.

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health such as the Nurses’ Health Study (Hunter et al., 1996), the primary exposure variable

is the cumulatively updated average dietary intake of a given nutrient. Similarly, in

prospective studies of the effects of air pollution on health, there is often interest in the

effect of cumulative exposure to specific pollutants (Zanobetti et al., 2000). Typically the

surrogate point exposures are measured once at each point of a specified grid, and are

validated at timepoints in a coarser grid (e.g., Willett et al., 1985; Brauer et al., 2003). There

is a practical need for statistical methods suited specifically for such applications.

Covariate measurement error in the Cox survival regression model was first addressed by

Prentice (1982), in the setting of a time-invariant exposure. Under certain assumptions, with

a linear Gaussian model for c given C, the regression calibration estimator emerged. In the

Cox model, the regression calibration estimator is not consistent, but it is a good

approximation under certain conditions. In later papers by many authors, more accurate

methods were developed for various settings; see Zucker (2005) for a review. We note in

particular the risk set regression calibration estimator, which Xie et al. (2001) developed in

the setting of a time-invariant exposure under a main/reliability study design. Xie et al.

(2001) assumed the classical homoscedastic measurement error model C = c+ϵ, with E(ϵ) =

0 and Var(ϵ) constant.

Time-varying covariates are more challenging to handle. A number of papers have dealt

with measurement error in time-varying covariates. Huang and Wang (2000) presented an

elegant solution for the setting where the classical homoscedastic error model applies and

replicate measurements of the surrogate covariate are available on all (or a sizeable sample

of) study individuals at all times at which events occur.

In practice, as noted above, measurements on the surrogate are usually available only on an

intermittent basis. A common strategy is to use the last available covariate measurements,

although this strategy can lead to some bias (Raboud et al., 1993). A number of authors have

developed more sophisticated methods, based on the joint modeling paradigm. A joint

model consists of a model for the covariate process (often a mixed linear model) and a

model for the hazard of the relevant event given the covariate (typically a Cox model).

Considerable work along this line has been published; Tsiatis and Davidian (2004) provide a

recent review.

The joint modeling approach has a number of features that impede its use in applications.

The approach is computationally intensive. In addition, it puts an undesirable focus on

modeling the exposure process, which requires significant effort but is of no intrinsic

interest to the investigators. Moreover, model checking is problematic, because the covariate

measurement times are typically too few and too sparse for effective model checking.

As we stated at the outset, we are specifically interested in epidemiological applications

involving cumulatively updated total or average exposure. The Willett’s classic textbook on

nutritional epidemiology cites hundreds of papers which use the cumulatively updated

average exposure variable in survival data models, and, similarly, the environmental

epidemiology textbooks by Thomas and by Savitz and Steenland cite hundreds of papers

using the cumulative exposure and distributed lagged exposure variable in survival data

models (Willett, 1998; Thomas, 2009; Steenland and Savitz, 1997). Commonly, in these

studies, the point exposures are subject to considerable measurement error, while the error

induced by carrying forward the most recent cumulative exposure value is less serious. This

motivates an effort to develop methods for analyzing the effect of cumulative exposure in

the presence of measurement error in the point exposures, without invoking the complex

joint modeling approach. Methodology of this sort would have immediate applicability in a

wide range of large-scale epidemiological studies, including in our own work Harvard’s

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Nurses’ Health Studies, the Harvard Professionals Follow-Up Study, the Harvard Six Cities

Study, and many others. It would allow cumulative exposure effects to be assessed in a

practical way that meets the needs of the applied reality.

The purpose of this paper is to develop such a method. Our approach is to extend the risk set

regression calibration method of Xie et al. (2001) from the setting of a time-invariant

covariate with a classical measurement model to the setting of cumulative exposure with

respect to a time-varying covariate. We work with a measurement error model that is

substantially more general than the classical model, and our method is appropriately

designed to handle this more complex error structure. Instead of the replicate measures

setting, we work under the main study/validation study design, which is suitable for studies

in nutritional and environmental epidemiology, where a gold standard measure of exposure,

or an unbiased measure thereof, often exists.

Section 2 provides a motivating example. Section 3 presents notation and background.

Section 4 presents the method. In Section 5, we derive the variance of the proposed

estimator for the case of the main study / external validation design. Section 6 presents

simulation studies of the method for a range of scenarios motivated by practical

applications, including time-varying exposures with different correlation structures, and rare

and common disease settings. In Section 7, we illustrate the method in the setting of the

example presented in Section 2. Section 8 provides a discussion.

2. Motivating example

We present here an example of the type of problem that motivated our work. The Health

Professionals’ Follow-Up Study (HPFS) is an ongoing prospective cohort study of cancer

and heart disease among 51529 U.S. male health professionals who responded to a mailed

baseline questionnaire in 1986, asking about demographics, family history of disease, diet,

smoking, physical activity, medications and other lifestyle factors. Every two years, study

participants receive questionnaires to update health status information and potential risk

factors.

In this paper, we consider data from the HPFS concerning the relationship between total

calcium intake and risk of fatal prostate cancer (Giovannucci et al., 2006). Total vitamin E

intake was the only important confounder with respect to this relationship. Nutrient intake,

including calcium and vitamin E, was assessed via a food frequency questionnaire (FFQ) in

1986, 1990, 1994, 1998 and 2002. After excluding men with a history of cancer at baseline,

or who did not adequately complete the 1986 dietary questionnaire, there were 390703

person-year observed in the main study with 357 fatal prostate cancer cases among 47760

subjects between 1986 to 2004. In our analysis, we used age as the time scale, as is more

suitable for epidemiologic studies of chronic disease (Korn, Graubard, and Midthune, 1997),

hence a left-truncated analysis is implied here.

The FFQ measures dietary intake with some degree of error and more reliable information

can be obtained from a diet record (DR) (Willett, 1998). In the HPFS validation study, 2

weeks of weighed diet records (DR) were observed in 127 person-years among 127 study

participants. To increase the validation study sample size, we incorporated another dietary

validation study, the Eating at America’s Table Study (EATS) in our analysis (Subar et al.,

2001). EATS is a study that was designed to validate the Diet History Questionnaire (DHQ),

a food frequency questionnaire (FFQ) similar to the one used in HPFS. In EATS, the

exposures was validated by four telephone-administered 24-hour dietary recalls. We discuss

later the transportability of the EATS validation study to the HPFS cohort. The goal of the

analysis is to estimate the relationship between dietary calcium intake and prostate cancer

based on the HPFS main cohort data, using the validation data to correct for the

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measurement error associated with the dietary intake variables. Note that, in the context of

this study, calcium intake is a time-dependent covariate. The variable of interest is

cumulatively updated average calcium intake, which is the average of all reported calcium

intake up to the present time.

3. Definition and preliminary results

The Cox model (Cox, 1972) for censored survival data specifies the hazard rate λ(t) for the

survival time T of an individual with s-dimensional covariate vector ξ to have the form

(1)

where β is a s-vector of regression coefficients and λ0(t) is the underlying hazard function.

In the survival data setting with time-invariant covariates, a main/external validation study

design consists of data {Ci, Wi, Ti, Di}, i = 1, … , n1 in the main study, and {ci, Ci, Wi, Ti},

i = n1 + 1, … , n in the validation study. Because data on the outcome, Di is not available in

the validation study, we call this an external validation study. Here, c is the p1-vector of true

exposure which is subject to measurement error, and, in the main study, we observe a vector

of surrogate variables C instead. W is a p2-vector of error-free covariates . T is the follow-up

time, which is defined as the minimum of the potential failure time T0 and potential

censoring time V, i.e. T = min(T0, V, t*), where t* is the end of follow up; D is an indicator

for failure from the event of interest, n1 is the sample size of the main study, n2 is the sample

size of the validation study, and n = n1 + n2. Typically, c is expensive to measure, and hence

n1 >> n2. In what follows, we start by reviewing the ordinary regression calibration method

for several different error models.

Prentice (1982) shows that if

model holds in the perfectly measured covariates, if λ(t|c, C, W) = λ(t|c, W), i.e.

measurement error is non-differential and if λ(t|C, no censorship in [0, t)) = λ(t|C), i.e. if

there is random censorship conditional on the observed main study data, then

, i.e. if the proportional hazards

(2)

where β1 and β2 are respectively p1-vector and p2-vector of regression coefficients

corresponding to c and W, and following Prentice (1982), T ≥ t can be dropped out when the

event is rare.

From (2), we see that the critical quantity is

dealing with this quantity: exact evaluation or approximation. Exact evaluation requires

assuming a model for the full distribution of (c|C, W). Approximation can be carried out

using moment assumptions only. The simplest approximation involves modeling only the

conditional mean μc(C, W) = E(c|C, W), and uses the first-order approximation

. This approach leads naturally to imputing μc(C, W) for c

and running a standard Cox analysis. A more sophisticated approximation can be carried out

by introducing models for both the conditional mean μc(C, W) = E(c|C, W) and the

conditional variance Σc(C, W) = Cov(c|C, W). The approximation is given by

. There are two basic ways of

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(3)

which is obtained from a second-order Taylor approximation to the cumulant generating

function of (c|C, W). In the special case where (c|C, W) is multivariate normal, the second

order approximation is exact (Prentice (1982)); however, the approximation can be used

even in the non-normal case. The first-order approximation is the approach most commonly

taken.

Equation (3) allows for a semi-parametric error model (c|C, W), where only the conditional

mean and covariance of (c|C, W), rather than the whole distribution, needs to be specified.

For the ordinary regression calibration method, the multivariate results are similar to those

given for the logistic regression model in Rosner et al. (1990). For one-dimensional β

without any error-free covariates, when the disease is rare, or β is small, or if the

measurement error variance is small and constant, the ordinary regression calibration (ORC)

estimator is given by

et al., 1997), where β̂naive is the naive Cox regression estimate using the surrogate measure

C directly, and α̂1 is obtained in the validation study by fitting the linear regression model

given by E(c|C) = α0 + α1C and Var(c|C) = σ2.

(Spiegelman

4. Risk set regression calibration for time-varying exposures in a main

study/validation study design

The validity of the ordinary regression calibration method depends on the rare disease

assumption, i.e. when Pr(T ≥ t) ≈ 1, as shown in (2). Risk set regression calibration (RRC) is

an attempt to improve the estimator by recalibrating within each risk set (Xie et al., 2001).

Here, we consider the first order approximation of (2) as

(4)

Although T ≥ t is retained in (4), whenever Var(c|C, W, T ≥ t) and higher order moments

are not constants or are not independent of time, an asymptotic bias is expected to be

incorporated due to the effect of the higher order moments. Xie et al. (2001) explored

analytically the asymptotic bias of the RRC estimate and derived the sandwich variance

estimator for the main/reliability study design, in which one or more additional

measurements are obtained from a random subsample of study subjects, under the

assumption that the classic additive homoscedastic error model is suitable for the data at

hand. Since the assumption of classical additive error in a time-invariant exposure is rarely

suitable in nutritional and environmental epidemiology, it was necessary to extend the risk

set regression calibration method to time-varying exposures in the main study/external

validation study design, assuming a more general measurement error model.

With a time-varying point exposure, a main/external validation study design consists of data

{Ci(t), Wi(t), Ti, Di}, i = 1, …, n1 in the main study, and {ci(t), Ci(t), Wi(t), Ti}, i = n1 + 1,

… , n in the validation study, where the time-varying surrogate exposure Ci(t) is measured at

a discrete grid of time points and the true exposure ci(t) is also measured on certain

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