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

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).

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 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 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 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

$$\mathrm{\lambda }(t;\mathbf{\xi })={\mathrm{\lambda }}_0(t)\text{exp}\{\mathit{\beta }\prime \mathbf{\xi }\},t\ge 0,$$ (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 {C_i, W_i, T_i, D_i}, i = 1, … , n₁ in the main study, and {c_i, C_i, W_i, T_i}, i = n₁ + 1, … , n in the validation study. Because data on the outcome, D_i 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 $\mathrm{\lambda }(t|\mathbf{c},\mathbf{W})={\mathrm{\lambda }}_0(t)\exp ({\mathit{\beta }}_1^{\prime }\mathbf{c}+{\beta }_2^{\prime }\mathbf{W})$, i.e. if the proportional hazards 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

$$\begin{array}{cc}\mathrm{\lambda }(t|\mathbf{C},\mathbf{W})\hfill & ={\mathrm{\lambda }}_0(t)\mathrm{E}(\text{exp}({\mathit{\beta }}_1^{\prime }\mathbf{c}+{\mathit{\beta }}_2^{\prime }\mathbf{W})|\mathbf{C},\mathbf{W},T\ge t)\hfill \\ \hfill & ={\mathrm{\lambda }}_0(t)\text{exp}({\mathit{\beta }}_2^{\prime }\mathbf{W})\mathrm{E}(\text{exp}({\mathit{\beta }}_1^{\prime }\mathbf{c})|\mathbf{C},\mathbf{W},T\ge t)\hfill \\ \hfill & \approx {\mathrm{\lambda }}_0(t)\text{exp}({\mathit{\beta }}_2^{\prime }\mathbf{W})\mathrm{E}(\text{exp}({\mathit{\beta }}_1^{\prime }\mathbf{c})|\mathbf{C},\mathbf{W}),\hfill \end{array}$$ (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 $\mathrm{E}(\text{exp}({\mathit{\beta }}_1^{\prime }\mathbf{c})|\mathbf{C},\mathbf{W})$. There are two basic ways of 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 μ_c(C, W) = E(c|C, W), and uses the first-order approximation $\mathrm{E}(\exp ({\mathit{\beta }}_1^{\prime }c)|\mathbf{C},\mathit{W})\approx \text{exp}({\mathit{\beta }}_1^{\prime }{\mathbf{\mu }}_{\mathbf{c}})$. This approach leads naturally to imputing μ_c(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 μ_c(C, W) = E(c|C, W) and the conditional variance Σ_c(C, W) = Cov(c|C, W). The approximation is given by

$$\mathrm{\lambda }(t|\mathbf{C},\mathbf{W})\approx {\mathrm{\lambda }}_0(t)\text{exp}({\mathit{\beta }}_2^{\prime }\mathbf{W})\text{exp}({\mathit{\beta }}_1^{\prime }{\mathit{\mu }}_{\mathbf{c}}+\frac{1}{2}{\mathit{\beta }}_1^{\prime }{\mathrm{\Sigma }}_{\mathbf{c}}{\mathit{\beta }}_1),$$ (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 ${\widehat{\beta }}_{\mathit{\text{ORC}}}={\widehat{\beta }}_{\mathit{\text{naive}}}/{\widehat{\alpha }}_1\text{and}\widehat{\text{Var}}({\widehat{\beta }}_{\mathit{\text{ORC}}})=\frac{1}{{\widehat{\alpha }}_1^2}\widehat{\text{Var}}({\widehat{\beta }}_{\mathit{\text{naive}}})+\frac{{\widehat{\beta }}_{\mathit{\text{naive}}}^2}{{\alpha }_1^4}\widehat{\text{Var}}({\widehat{\alpha }}_1)$ (Spiegelman et al., 1997), where β̂_naive 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.

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

$$\begin{array}{cc}\mathrm{\lambda }(t|\mathbf{C},\mathbf{W})\hfill & ={\mathrm{\lambda }}_0(t)\mathrm{E}(\text{exp}({\mathit{\beta }}_1^{\prime }\mathbf{c}+{\mathit{\beta }}_2^{\prime }\mathbf{W})|\mathbf{C},\mathbf{W},T\ge t)\hfill \\ \hfill & \approx {\mathrm{\lambda }}_0(t)\text{exp}({\mathit{\beta }}_2^{\prime }\mathbf{W})\text{exp}({\mathit{\beta }}_1^{\prime }\mathrm{E}(\mathbf{c}|\mathbf{C},\mathbf{W},T\ge t)).\hfill \end{array}$$ (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 {C_i(t), W_i(t), T_i, D_i}, i = 1, …, n₁ in the main study, and {c_i(t), C_i(t), W_i(t), T_i}, i = n₁ + 1, … , n in the validation study, where the time-varying surrogate exposure C_i(t) is measured at a discrete grid of time points and the true exposure c_i(t) is also measured on certain occasions in the validation study. The gold standard c_i(t) is usually measured much less frequently than the surrogate C_i(t).

The exposure variables of interest in these studies are, as noted in the introduction, generally some function of the time-varying point exposures. Denote the function of the time-varying true exposure c_i(t) as x_i(t), and the function of the time-varying surrogate exposure C_i(t) as X_i(t). Time-varying error-free covariates, and functions of these error-free covariates, are denoted by Z_i(t). Then, the main/external validation study data take the form {X_i(t), Z_i(t), T_i, D_i}, i = 1, … , n₁ for the main study, and {x_i(t), X_i(t), Z_i(t), T_i}, i = n₁ + 1, … , n for the validation study, and (4) becomes

$$\mathrm{\lambda }(t|\mathbf{X}(t),\mathbf{Z}(t))\approx {\mathrm{\lambda }}_0(t)\text{exp}({\mathit{\beta }}_1^{\prime }\mathrm{E}(\mathbf{x}(t)|\mathbf{X}(t),\mathbf{Z}(t),T\ge t)+{\mathit{\beta }}_2^{\prime }\mathbf{Z}(t)).$$

The basic idea here is that the measurement error model

$$\mathrm{E}({\mathbf{x}}_i(t)|{\mathbf{X}}_i(t),{\mathbf{Z}}_i(t))={\mathit{\alpha }}_0(t)+{\mathit{\alpha }}_1(t){\mathbf{X}}_i(t)+{\mathit{\alpha }}_2(t){\mathbf{Z}}_i(t)$$ (5)

is re-estimated from the validation study at each main study failure time, and the true exposure is re-estimated for everyone at risk. Then, β̂_RRC will solve

$${\displaystyle \sum _{i=1}^{n_1}{\displaystyle {\int }_0^{t*}\left\{\left(\begin{array}{c}{\widehat{\mathbf{x}}}_i(\widehat{\mathit{\psi }},t)\hfill \\ {\mathbf{Z}}_i(t)\hfill \end{array}\right)-\frac{{\mathbf{S}}^{(1)}(\mathit{\beta },\widehat{\mathit{\psi }},t)}{S^{(0)}(\mathit{\beta },\widehat{\mathit{\psi }},t)}\right\}}}{N}_i(\mathit{\text{dt}})=0$$ (6)

where

$$S^{(0)}(\mathit{\beta },\widehat{\mathit{\psi }},t)={n_1}^{-1}{\displaystyle \sum _{i=1}^{n_1}{Y}_m(i,t)\text{exp}\{{\mathit{\beta }}_1^{\prime }{\widehat{\mathbf{x}}}_i(\widehat{\mathit{\psi }},t)+{\mathit{\beta }}_2^{\prime }{\mathit{Z}}_i(t)\},}$$

and

$${\mathbf{S}}^{(1)}(\mathit{\beta },\widehat{\mathit{\psi }},t)={n_1}^{-1}{\displaystyle \sum _{i=1}^{n_1}{Y}_m(i,t)\left(\begin{array}{c}{\widehat{\mathbf{x}}}_i(\widehat{\mathit{\psi }},t)\hfill \\ {\mathbf{Z}}_i(t)\hfill \end{array}\right)\text{exp}\{{\mathit{\beta }}_1^{\prime }{\widehat{\mathbf{x}}}_i(\widehat{\mathit{\psi }},t)+{\mathit{\beta }}_2^{\prime }{\mathbf{Z}}_i(t)\}.}$$

Here, $\beta =({\beta }_1^{\prime },{\beta }_2^{\prime })$, N_i(t) = I(T_i ≤ t, D_i = 1) is the counting process corresponding to T_i. N_i(dt) = 1 if the subject i fails at the failure time t, N_i(dt) = 0 otherwise. Y_m(i, t_l) = I(T_i ≥ t_l) is the risk process indicator in the main study, the subscript ‘m’ indicates the main study, the subscript ‘v’ indicates the validation study. dim(α₀(t_l)) = p₁, dim(α₁(t_l)) = p₁ × p₁, dim(α₂(t_l)) = p₁ × p₂, ψ(t_l) = (α₀(t_l), α₁(t_l), α₂(t_l)), l = 1, … , r, and r is the number of unique failure times in the main study. Next, we explain how to obtain estimates of α₀(t_l), α₁(t_l), α₂(t_l):

1. Order the unique failure times that occur in the main study as t₁ < t₂ < … < t_r. Find the r risk sets R_m(t_l), l = 1, 2, … , r, where R_m(t_l) is the set of individuals in the main study who are alive and uncensored at a time just prior to t_l. Generate the risk process indicator Y_m(i, t_l) so that Y_m(i, t_l) = 1 if i ∈ R_m(t_l), and Y_m(i, t_l) = 0 otherwise.

2. Find the r risk sets R_v(t_l), l = 1, 2, … , r, in the validation study. Generate the risk process indicator Y_v(i, t_l) = I(T_i ≥ t_l), so that Y_v(i, t_l) = 1 if i ∈ R_v(t_l), and Y_v(i, t_l) = 0 otherwise. For any given t, let ${\mathbf{X}}_i^{*}(t),{\mathbf{x}}_i^{*}(t)\text{and}{\mathbf{Z}}_i^{*}(t)$ be the most recent observed values of X_i(t), x_i(t) and Z_i(t) prior to t. We then run, for each failure time t_l, a regression of ${\mathbf{x}}_i^{*}(t_l)\text{on}{\mathbf{X}}_i^{*}(t_l)\text{and}{\mathbf{Z}}_i^{*}(t_l)$ on the subjects in R_v(t_l), obtaining α̂₀(t_l), α̂₁(t_l), α̂₂(t_l).

3. Estimate ${\widehat{\mathbf{x}}}_i(t_l)=[{\widehat{\mathit{\alpha }}}_0(t_l)+{\widehat{\mathit{\alpha }}}_1(t_l){\mathbf{X}}_i^{*}(t_l)+{\widehat{\mathit{\alpha }}}_2(t_l){\mathbf{Z}}_i^{*}(t_l)]\cdot Y_m(i,t_l)$ for each subject i in the risk set R_m(t_l) in the main study, l = 1, 2, … , r.

4. Fit the “naive” Cox model on (x̂_i(t_l), Z_i(t_l), Y_m(i, t_l), T_i, D_i) to get the risk set regression calibration estimator β̂_RRC.

Non-differential measurement error and random censorship are also required here to ensure validity of estimation and inference, as in the original regression calibration estimate discussed in Section 3. In addition, we assume that the measurement error model (5) that holds in the validation study is applicable to the main study as well. This assumption is the transportability assumption discussed by Carroll et al. (2006).

5. Var(β̂_RRC) for the main/ external validation study

We write the score equation (6) as ${\displaystyle {\sum }_{i=1}^{n_1}{\mathbf{U}}_{{\mathit{\beta }}_i}}(\mathit{\beta }|\widehat{\mathit{\psi }})=\mathbf{0}$, where

$${\mathbf{U}}_{\mathit{\beta }i}(\mathit{\beta }|\widehat{\mathit{\psi }})=D_i\left\{\left(\begin{array}{c}{\widehat{\mathbf{x}}}_i(T_i)\hfill \\ {\mathbf{Z}}_i(T_i)\hfill \end{array}\right)-\frac{{\mathbf{S}}^{(1)}(\mathit{\beta },\widehat{\mathit{\psi }},T_i)}{S^{(0)}(\mathit{\beta },\widehat{\mathit{\psi }},T_i)}\right\},$$ (7)

with ψ̂ = (ψ̂(t₁), ψ̂(t₂), … , ψ̂(t_r)), dim(ψ̂) = p₁ × (p₁+p₂+1)r and dim(U_βi) = (p₁+p₂) × 1. Define ${\widehat{\mathbf{U}}}^{*}(\widehat{\mathit{\beta }},\widehat{\psi })={\displaystyle {\sum }_{i=1}^{n_1}\frac{\partial {\mathbf{U}}_{\mathit{\beta }i}(\mathit{\beta }|\mathit{\psi })}{\partial \mathit{\psi }}}{|}_{(\mathit{\beta }|\mathit{\psi })=({\widehat{\mathit{\beta }}}_{\mathit{\text{RRC}}},\widehat{\mathit{\psi }})}$, then dim(Û*) = (p₁ + p₂) × p₁(p₁ + p₂+1)r. Denote the covariance matrix of ψ̂ as V_ψ̂, so dim(V_ψ̂) = p₁(p₁ + p₂ + 1)r × p₁(p₁ + p₂ + 1)r.

In the validation study, ψ̂ solves

$${\displaystyle \mathbf{\sum }_{i=1}^{n_2}{\mathbf{U}}_{\mathit{\psi }i,j}(\mathit{\psi })=\mathbf{0},}$$ (8)

where ${\mathbf{U}}_{\mathit{\psi }i,j}(\mathit{\psi })=({\mathbf{U}}_{{\mathit{\alpha }}_{0}i,j}^{\prime }(\mathit{\psi }),{\mathbf{U}}_{{\mathit{\alpha }}_{1}i,j}^{\prime }(\mathit{\psi }),{\mathbf{U}}_{{\mathit{\alpha }}_{2}i,j}^{\prime }(\mathit{\psi }))$ is defined as follows

$$\begin{array}{cc}{\mathbf{U}}_{{\mathit{\alpha }}_{0}i,j}(\mathit{\psi })\hfill & =Y_{\nu }(i,t_j)[{\mathbf{x}}_i-{\mathit{\alpha }}_0(t_j)-{\mathit{\alpha }}_1(t_j){\mathbf{X}}_i-{\mathit{\alpha }}_2(t_j){\mathbf{Z}}_i]\prime \hfill \\ {\mathbf{U}}_{{\mathit{\alpha }}_{1}i,j}(\mathit{\psi })\hfill & =Y_{\nu }(i,t_j){\mathbf{X}}_i[{\mathbf{x}}_i-{\mathit{\alpha }}_0(t_j)-{\mathit{\alpha }}_1(t_j){\mathbf{X}}_i-{\mathit{\alpha }}_2(t_j){\mathbf{Z}}_i]\prime \hfill \\ {\mathbf{U}}_{\mathit{\alpha }2i,j}(\mathit{\psi })\hfill & =Y_{\nu }(i,t_j){\mathbf{Z}}_i[{\mathbf{x}}_i-{\mathit{\alpha }}_0(t_j)-{\mathit{\alpha }}_1(t_j){\mathbf{X}}_i-{\mathit{\alpha }}_2(t_j){\mathbf{Z}}_i]\prime \hfill \end{array}$$ (9)

for j = 1, … , r, i = 1, … , n₂, dim(U_{α0i, j}) = 1 × p₁, dim(U_{α1i, j}) = p₁ × p₁, dim(U_{α2i, j}) = p₂ × p₁, dim(U_{ψi, j}(ψ)) = p₁ × (p₁ + p₂ + 1), U_ψi = (U_{ψi, 1}, U_{ψi, 2}, … , U_{ψi, r}) and dim(U_ψi) = p₁ × (p₁ + p₂ + 1)r.

Then, as shown in Web Appendix B, Cov(ψ̂) can be estimated by $\frac{1}{n_2}{\widehat{\mathbf{V}}}_{\widehat{\mathit{\psi }}}$, with V̂_ψ̂ constructed as

$${\widehat{\mathbf{V}}}_{\widehat{\mathit{\psi }}}={\left[\frac{1}{n_2}{\displaystyle \sum _{i=1}^{n_2}\frac{\partial {\mathbf{U}}_{\mathit{\psi }i}(\mathit{\psi })}{\partial \mathit{\psi }}}\right]}^{-1}\left[\frac{1}{n_2}{\displaystyle \sum _{i=1}^{n_2}{\mathbf{U}}_{\mathit{\psi }i}(\mathit{\psi })\otimes {\mathbf{U}}_{\mathit{\psi }i}^{\prime }(\mathit{\psi })}\right]{\left[\frac{1}{n_2}{\displaystyle \sum _{i=1}^{n_2}\frac{\partial {\mathbf{U}}_{\mathit{\psi }i}(\mathit{\psi })}{\partial \mathit{\psi }}}\right]}^{-1}{|}_{\mathit{\psi }=\widehat{\mathit{\psi }}}.$$ (10)

Following the derivation in Web Appendix B, we have

$$\widehat{\text{Var}}({\widehat{\mathit{\beta }}}_{\mathit{\text{RRC}}})=\frac{1}{n_1}{\widehat{\mathbf{I}}}_{\mathit{\beta }}^{-1}{\widehat{\mathbf{H}}}_{\mathit{\beta },\mathit{\psi }}{\widehat{\mathbf{I}}}_{\mathit{\beta }}^{-1}$$ (11)

where

$${\widehat{\mathbf{I}}}_{\mathit{\beta }}^{-1}={\left[\frac{1}{n_1}{\displaystyle \sum _{i=1}^{n_1}\frac{\partial {\mathbf{U}}_{\mathit{\beta }i}(\mathit{\beta }|\widehat{\psi })}{\partial \mathit{\beta }}}\right]}^{-1}{|}_{\mathit{\beta }={\widehat{\mathit{\beta }}}_{\mathit{\text{RRC}}}},$$ (12)

$${\widehat{\mathbf{H}}}_{\mathit{\beta },\mathit{\psi }}={\widehat{\mathbf{H}}}_{\mathit{\beta }}+\frac{1}{n_1{n}_2}{\widehat{\mathbf{U}}}^{*}(\mathit{\beta },\mathit{\psi }){\widehat{\mathbf{V}}}_{\mathit{\psi }}{\widehat{\mathbf{U}}}^{*}{(\mathit{\beta },\mathit{\psi })}^{\prime },$$

$${\widehat{\mathbf{H}}}_{\mathit{\beta }}=\frac{1}{n_1}{\displaystyle \sum _{i=1}^{n_1}{\tilde{\mathbf{U}}}_{\mathit{\beta }i}(\mathit{\beta },\widehat{\mathit{\psi }}){\tilde{\mathbf{U}}}_{\mathit{\beta }i}^{\prime }(\mathit{\beta },\widehat{\mathit{\psi }})}{|}_{\mathit{\beta }={\widehat{\mathit{\beta }}}_{\mathit{\text{RRC}}}},$$ (13)

with

$$\begin{array}{cc}{\tilde{\mathbf{U}}}_{\mathit{\beta }i}(\mathit{\beta },\widehat{\mathit{\psi }})=\hfill & {\mathbf{U}}_{\mathit{\beta }i}(\mathit{\beta }|\widehat{\mathit{\psi }})-{\displaystyle \sum _{i=1}^{n_1}\frac{D_j{Y}_m(i,T_j)\text{exp(}{\mathit{\beta }}_1^{\prime }{\widehat{\mathbf{x}}}_i(T_j)+{\mathit{\beta }}_2^{\prime }{\mathbf{Z}}_i(T_j))}{n_1{S}^{(0)}(\mathit{\beta },\widehat{\mathit{\psi }},T_j)}}\hfill \\ \hfill & \cdot \left\{\left(\begin{array}{c}{\widehat{\mathbf{x}}}_i(\widehat{\mathit{\psi }},T_j)\hfill \\ {\mathbf{Z}}_i(T_j)\hfill \end{array}\right)-\frac{{\mathbf{S}}^{(1)}(\mathit{\beta },\widehat{\mathit{\psi }},T_j)}{S^{(0)}(\mathit{\beta },\widehat{\mathit{\psi }},T_j)}\right\}.\hfill \end{array}$$

The asymptotic distribution theory proven in Web Appendix B follows arguments in Andersen and Gill (1982) and Lin and Wei (1989); the reader is referred to Web Appendix B for further details.

6. A simulation study under the main study/external validation study

We report finite-sample simulation results for our proposed method under the main study / external validation study design, following the algorithm in Sections 4 and 5. We consider a single error-prone covariate c with surrogate C. All simulation results are based on 1000 replications.

We consider two event rate scenarios: rare disease and common disease. Motivated by our real data set, the Health Professional Follow-up Study, we set the number of events to be around 500. For the rare disease situation, we set the cumulative event rate at 1%, and thus the main study sample size is set at n₁ = 50; 000. For the common disease situation, we set the cumulative event rate at 50%, and thus the main study sample size is set at n₁=1000. The cases we considered for the validation study size were n₂ = 150 and n₂ = 500. A validation sample size of n₂ = 150 is very common in nutritional epidemiology studies. The case of n₂ = 500 is less common, but does arise in some applications, particularly when two or more validation studies can be combined, as in the example in Section 2.

We take the baseline hazard function to be of the Weibull form λ₀(t) = θν(νt)^θ−1, with θ = 6, which is typical of many epithelial cancers (Armitage and Doll (1961); Breslow and Day (1993)). Censoring is assumed exponential with a rate of 1% per year. The maximum follow-up time is taken to be t^* =50 years. The parameter ν is set to achieve the specified cumulative event rate.

In preliminary simulations, we examined the case where the covariate is time-invariant, and compared the ORC method to the RRC method. These results are summarized in Web Table 1 of Web Appendix A.1. The parameter which describes the extent of measurement error, ρ = Corr(c, C) was varied across the values 0.3, 0.6, 0.9. The key parameter in the simulation comparisons is η = β²Var(c|C) = (1 − ρ₂)β²σ², where β is the regression coefficient and σ² is the variance of the true exposure. When η is small, the ORC method was adequate, but when η is large, the RRC method was clearly superior. These findings are consistent with those of Xie et al. (2001).

We turn now to the time-varying exposure situation. In line with the motivating example in Section 2, we worked with the cumulatively updated average exposure (Hu et al., 1999). The true and surrogate exposures, x(t) and X(t), were defined as follows:

$$\mathbf{x}(t)=\frac{1}{t_k-t_0}{\displaystyle \sum _{m=1}^k\mathbf{c}(t_{m-1})(t_m-t_{m-1}),\mathbf{X}(t)}\frac{1}{t_k-t_0}{\displaystyle \sum _{m=1}^k\mathbf{C}(t_{m-1})(t_m-t_{m-1}),}$$ (14)

for t ∈ [t_k, t_k+1) with 1 ≤ k ≤ p, where the set {t₀, t₁, t₂, …, t_p} are the times at which c_i(t) is measured. C_i(t) can be measured on the same time scale, or, as is typically the case in applications, C_i(t) is measured on a much finer time scale. In the simulation study below, we used the same time scale for c_i(t) and C_i(t).

The true and surrogate cumulatively updated average exposures were generated as follows:

1. The true exposure c_i ~ MV N(μ_c, Σ_c), where c_i is a p–vector with p as the number of the observation time points, μ_c is the mean vector and Σ_c is a covariance matrix. Without loss of generality, we consider a simple case with μ_c = 0 and Σ_c such that Σ_c(j, k) = 1 if j = k and ${\mathrm{\Sigma }}_c(j,k)={\rho }_I^{{\left|j-k\right|}^{\tau }}$ if j ≠ k, with τ = 0 or 1. When τ = 0, a compound symmetry covariance structure is obtained, and the intra-class correlation ρ_ICS was set at 0:3, 0:6, 0:9. When τ = 1, an AR(1) structure is obtained. To put these two covariance scenarios on an equal footing, we set the average correlation of ρ_IAR over [0; t^*] equal to ρ_ICS, i.e.,

   $$\frac{1}{t*}{\displaystyle {\int }_0^{t*}{({\rho }_{I_{\text{AR}}})}^t\mathit{\text{dt}}}={\rho }_{I_{\text{CS}}}.$$ (15)

   Solving this equation, we obtained the corresponding values of ρ_IAR as 0.938, 0.978, 0.996.

2. C_ij = c_ij+e_ij, e_ij ~ N(0, Δ), with the measurement error variance Δ given by $\mathrm{\Delta }=\frac{1}{{\rho }^2}-1$, where ρ is the correlation between C_ij and c_ij and varied as 0:3, 0:6, 0:9.

3. A cumulative average exposure x(t) and X(t) was then generated by

   $$x_i(t_k)=\frac{1}{t_k-t_0}{\displaystyle \sum _{m=1}^k{c}_i(t_{m-1})(t_m-t_{m-1}),X_i(t_k)=\frac{1}{t_k-t_0}}{\displaystyle \sum _{m=1}^k{C}_i(t_{m-1})(t_m-t_{m-1})},$$

   for 1 ≤ k ≤ p, over the time points {t₀, t₁, t₂, … , t_p}. In applications, time could be expressed in terms of participant’s age, time on the study, calendar year, or in some other appropriate manner. In our simulations, for simplicity, we set t_j = 5 * j, for j = 0, 1, … , 10. Thus, there were 10 exposure measurements in total over the study period of t^* = 50 years. This scheme was patterned after the measurement schedule in our motivating example, HPFS, and the other Harvard cohort studies.

The survival data were generated according to the hazard model λ(t|x(t)) = λ₀(t) exp(βx(t)), where x(t) is as defined in (14). Web Appendix A.2 presents details of the data generation. Since survival is usually measured on a finer time scale than that defined by the exposure measurement schedule, we generated the survival time on a finer time scale, based on t_j = j, for j = 0, 1, … , 50.

Table 1 presents results for the compound symmetry structure with validation sample size n₂ = 150. More results with different validation sample sizes are presented in Web Table 2 of Web Appendix A.3. The results for the AR(1) structure, which were similar, are presented in Web Table 3 of Web Appendix A.3. Overall, the RRC estimator performed very well in this time-varying cumulatively updated average exposure setting, with the good performance being robust across different levels of the autocorrelation in the true exposure process and the different correlation structures. The bias was very small for all scenarios and became even smaller as the autocorrelation became higher. An increased bias was seen with the AR(1) correlation structure when the autocorrelation was low, consistent with the fact that within-person variability increases more quickly with time for the AR(1) structure than the CS structure. However, the worst bias seen was only 5%. The coverage probability was nearly accurate in all cases considered. Increasing the frequency of exposure measurements improved the results even more (data not shown), although the bias was already minimal with the exposure frequencies investigated here.

Table 1.

Results for simulation of cumulatively updated average exposure with a compound symmetry covariance structure, for different intra-class correlation ρ_ICS.

		Estimated β̂(SÊ[β̂])		Percent Bias(%)		95% CI Coverage(%)
ρICS	ρ	Naive	RRC	Naive	RRC	Naive	RRC
		n1 = 50000,	n2 = 150,	Rare disease
0.3	0.3	0.117(0.036)	0.502(0.179)	−76.6	0.4	0.0	95.6
	0.6	0.318(0.058)	0.500(0.098)	−36.3	0.1	12.8	94.1
	0.9	0.464(0.070)	0.499(0.077)	−7.3	−0.3	91.2	95.6
0.6	0.3	0.172(0.032)	0.509(0.118)	−65.6	1.7	0.0	95.0
	0.6	0.373(0.048)	0.498(0.069)	−25.4	−0.4	24.5	95.6
	0.9	0.474(0.054)	0.495(0.057)	−5.1	−0.9	92.6	94.9
0.9	0.3	0.212(0.030)	0.502(0.090)	−57.5	0.3	0.0	94.2
	0.6	0.405(0.041)	0.503(0.056)	−19.1	0.6	37.4	95.0
	0.9	0.486(0.045)	0.501(0.047)	−2.9	0.2	92.8	94.5
		n1 = 1000,	n2 = 150,	Common disease
0.3	0.3	0.105(0.035)	0.492(0.193)	−79.1	−1.5	0.0	94.5
	0.6	0.293(0.058)	0.490(0.103)	−41.3	−2.1	4.7	94.4
	0.9	0.438(0.071)	0.490(0.078)	−12.5	−2.0	87.3	95.6
0.6	0.3	0.153(0.033)	0.509(0.135)	−69.4	1.8	0.0	94.1
	0.6	0.352(0.049)	0.503(0.076)	−29.6	0.7	14.3	94.8
	0.9	0.457(0.057)	0.498(0.061)	−8.5	−0.5	88.2	95.1
0.9	0.3	0.185(0.031)	0.502(0.107)	−62.9	0.3	0.0	94.3
	0.6	0.380(0.044)	0.504(0.063)	−24.1	0.8	22.0	94.1
	0.9	0.466(0.049)	0.501(0.052)	−6.8	0.1	89.8	94.6

True β = 0.5, the study duration t^* = 50, the number of simulation replications B = 1000.

In the rare disease situation, the cumulative incidence is about 1% with n₁ = 50000.

In the common disease situation, the cumulative incidence is about 50% with n₁ = 1000.

7. Application of the method

We illustrate our method in an analysis of the HPFS study described in Section 2. Recall that the goal is to estimate the relationship between dietary calcium intake and prostate cancer, adjusting for the confounding effect of vitamin E intake. Calcium and vitamin E intake were assessed in the full cohort every four years using a food frequency questionnaire (FFQ). The FFQ is known to be error-prone, so we use data from two validation studies where the nutrients were measured using both the FFQ and the more accurate diet record (DR): the HPFS validation study and the EATS validation study.

As a preliminary step, we checked the transportability of the EATS data to the HPFS population. To this end, we ran a regression analysis of DR on FFQ for each study, adjusting for age in 5-year age groups. The slope for total calcium intake was 0.445 in EATS and 0.371 in HPFS. For total vitamin E, the slope was 0.818 in EATS and 0.762 in HPFS. Given the similarity of these estimates, the transportability assumption appeared reasonable. The rightmost major column of Table 2 shows the basic characteristics of the HPFS main study, so they can be compared to the characteristics of the two validation studies.

Table 2.

Basic characteristics of the study population.

Variable	Exposure type	Validation study						Main study
		HPFS (n21 = 127, Age:(39 – 75))			EATS (n22 = 446, Age:(21 – 76))			HPFS(n1 = 47760)
		Mean(s.d.)	Range	Correlation	Mean(s.d.)	Range	Correlation	Mean(s.d.)	Range
Total calciuma	DR	0.43(0.12)	(0.24, 0.87)		0.46(0.16)	(0.18, 1.02)		-	-
(1838 mg/day)	FFQ	0.47(0.17)	(0.25, 1.17)	0.51	0.53(0.22)	(0.14, 1.54)	0.61	0.50(0.21)	(0.07,3.96)
Total vitamin Eb	DR	1.32(0.42)	(0.85, 2.72)		1.36(0.56)	(0.57, 3.03)		-	-
(mg/day)	FFQ	1.20(0.49)	(0.71, 2.97)	0.89	1.33(0.53)	(0.33, 2.87)	0.79	1.42(0.54)	(0.48,3.02)

We then carried out a further empirical check of the transportability assumption. We defined a binary indicator Study_ind with value 1 if the study was HPFS and 0 if the study was EATS, and then ran the regression model

$${\mathit{\text{Calc}}}_{\mathit{\text{dr}}}={\gamma }_0+{\gamma }_1{\mathit{\text{Calc}}}_{\mathit{\text{ffq}}}+{\gamma }_2{\mathit{\text{Calc}}}_{\mathit{\text{ffq}}}*{\mathit{\text{Study}}}_{\mathit{\text{ind}}},$$ (16)

and similarly for vitamin E. To test the hypothesis that there is no between-studies variation in the slope, i.e. γ₂ = 0 by age group, we fit regression models by 5-year age groups to the combined validation study, using model (16). We found that the null hypothesis was accepted for most age groups, except in age group 56 – 60, the p–value for the total calcium intake by study interaction was less than 0.001, and in age group older than 71, the p–value for the total vitamin E intake by study interaction was less than 0.03. We tailored our analyses accordingly: in analyses where both total calcium intake and total vitamin E intake were regarded as error-prone exposures, we excluded the EATS study participants from the risk sets in those ages; in analyses where only the total calcium intake was regarding as being measured with error, we kept the EATS study participants in the risk set of age 71 and older, but we excluded the EATS study subjects from the age of 56 – 60 risk sets.

After completing this preliminary step, we proceeded to the main analysis. Table 3 gives the results of the RRC method for time-varying exposures developed in this paper compared to the naive Cox approach. The relative risks are given in units of 1838 mg/day for total calcium intake to facilitate comparison of the result reported in the original publication of these data (Giovannucci et al., 2006), where 1838 mg/day was the difference between the median of the top quintile and the median of bottom quintile. We performed the analysis using only the HPFS validation study and using both HPFS and EATS validation studies. Because the HPFS validation study had 127 person-year observations for 127 subjects, we grouped the risk sets by 4-year age interval to stabilize the estimates. This was not necessary when we included both validation studies. The results show that with more data in the validation study, the variance of the estimates is substantially lower. Moreover, with the added validation data, the de-attenuation effect is more pronounced, producing estimates in better accord with prior expectations. From the results including both HPFS and EATS validation studies, we can see that the under-estimation of the effect of calcium intake on fatal prostate cancer from the naive Cox approach was corrected by the RRC estimate. Total calcium intake had a significant positive association with fatal prostate cancer and the total vitamin E intake was weakly inversely associated, perhaps not associated at all.

Table 3.

Estimated coefficients and standard errors for HPFS of the relationship between total calcium intake and the fatal prostate cancer incidence.

		Total Calcium intakea			Total Vitamin E intakeb
Method	Validation study	Estimate(S.E.)	HRe(95% C.I.)	p-value	Estimate(S.E.)	HRe(95% C.I.)	p-value
Uncorrected	N/A	0.587(0.239)	1.80[1.13,2.87]	0.014	−0.274(0.103)	0.76[0.62,0.93]	0.008
RRCc	HPFS	0.623(0.653)	1.87[0.52,6.71]	0.340	−0.342(0.160)	0.71[0.52,0.97]	0.033
	HPFS + EATS	1.359(0.500)	3.89[1.46,10.37]	0.007	−0.374(0.120)	0.69[0.54,0.87]	0.002
RRCd	HPFS	0.150(0.766)	1.16[0.26,5.22]	0.845	−0.203(0.447)	0.82[0.34,1.96]	0.650
	HPFS + EATS	0.734(0.371)	2.08[1.01,4.31]	0.048	−0.370(0.201)	0.69[0.47,1.02]	0.065

^aTotal calcium intake is energy adjusted and the unit is 1838 mg/day.

^bTotal vitamin E intake is energy adjusted and log₁₀ transformed.

^cAssuming total calcium intake is error-prone, but total vitamin E intake is error-free.

^dAssuming both total calcium intake and total vitamin E intake are error-prone.

^eHR is the hazard ratio e^β.

8. Discussion

In this paper, we have considered the Cox survival regression model with time-dependent covariates subject to measurement error. We derived a bias-corrected point and interval estimate of the relative risk using a RRC approach. We emphasized the main study/external validation study design, which arises commonly in nutritional epidemiology, but has been given less attention in the literature than other designs, particularly in the setting of time-varying covariates. We focused on cumulatively updated total or average exposure, which is often of interest in nutritional and environmental epidemiology, but the approach could also be applied to analysis of time-varying point exposures. In addition, the approach extends in a straightforward way to cover left truncation. A FORTRAN program for the method is available at http://www.hsph.harvard.edu/faculty/spiegelman/rrc_timevarying_method.f. Simulation studies in the setting of cumulatively updated average exposure showed that the method performs very well.

Clearly, the effectiveness of the method depends on the size of the validation study. The analysis of the HPFS data in Section 7 illustrates this point. When we used only the data from the 127 participants in the HPFS validation study, and with vitamin E assumed to be measured without error, a very wide confidence interval for the effect of calcium intake was obtained, leading to results that were not very meaningful. When we included validation data data from the 573 men in the EATS study, the results were much more reasonable, and significant p-values for both calcium intake and vitamin E were obtained, even when both these nutrients were regarded as subject to with error.

To date, most validation studies available in nutritional epidemiology have only one measurement per subject. Only the Nurses Health Study has validation data repeatedly assessed more than once, and this is among a small number of subjects. The limited validation data makes it challenging to implement measurement error correction for time-varying exposures. However, we understand that new validation studies are underway, including one within our own group at the Nurses Health Study, involving a much larger number of subjects and repeated exposure assessments over time. Such expanded validation studies will provide a firmer basis of measurement error correction, and the method we have developed here provides a practical and reliable way for carrying out such correction.

In summary, measurement error in time-varying covariates, including those that are functions of a series of exposure measurements available up to a failure time, is an extremely common problem in nutritional and environmental epidemiology. The method developed in this paper provides a mechanism for handling this problem that is well-suited to these applications.