Joint nonparametric correction estimator for excess relative risk regression in survival analysis with exposure measurement error

SUMMARY

Observational epidemiological studies often confront the problem of estimating exposure-disease relationships when the exposure is not measured exactly. In the paper, we investigate exposure measurement error in excess relative risk regression, which is a widely used model in radiation exposure effect research. In the study cohort, a surrogate variable is available for the true unobserved exposure variable. The surrogate variable satisfies a generalized version of the classical additive measurement error model, but it may or may not have repeated measurements. In addition, an instrumental variable is available for individuals in a subset of the whole cohort. We develop a nonparametric correction (NPC) estimator using data from the subcohort, and further propose a joint nonparametric correction (JNPC) estimator using all observed data to adjust for exposure measurement error. An optimal linear combination estimator of JNPC and NPC is further developed. The proposed estimators are nonparametric, which are consistent without imposing a covariate or error distribution, and are robust to heteroscedastic errors. Finite sample performance is examined via a simulation study. We apply the developed methods to data from the Radiation Effects Research Foundation, in which chromosome aberration is used to adjust for the effects of radiation dose measurement error on the estimation of radiation dose responses.

1 Introduction

Estimating exposure-disease relationships in epidemiological studies may encounter the challenge of exposure measurement error. This is especially common when the exposure is quantitative and must be measured or estimated from characteristics of the individual and/or circumstances of exposure. Some of the most important examples of this problem arise in the field of radiation epidemiology. In nearly all settings in which humans are exposed to ionizing radiation, the doses to target organs can only be estimated, and in some settings of great public health significance (e.g., environmental exposures, unmonitored occupational accidents), the dose estimates can be quite uncertain. It is widely recognized that errors or uncertainties in exposure variables can introduce bias into estimates of exposure-disease relationships, however the most appropriate methods for counteracting such bias remain to be identified.

An important study of radiation exposure and cancer is from the Radiation Effects Research Foundation (RERF). Since 1947, the RERF and its predecessor the Atomic Bomb Casualty Commission have conducted health follow-up studies on survivors of the 1945 atomic bombings of Hiroshima and Nagasaki. The RERF has examined the links between radiation exposure and disease outcomes, cell and genetic damage, and other factors, in a population of about 200,000 survivors and their children. The Life Span Study (LSS) is the center of the RERF’s research programs, and is widely acknowledged to be the most important single source of information regarding risk of late effects due to radiation exposure. The Adult Health Study is a subset of nearly 20,000 LSS members who have been offered clinical examinations at RERF every 2 years since 1958 as part of the study’s follow-up. Because of the large size and long followup of the RERF cohorts, they have been the most important sources of information for the estimation of radiation health effects and for setting standards for radiation protection.

The linear excess relative risk (ERR) model is well-applied for exposure effect estimation in environmental and occupational epidemiology (Thomas, 1981). An important reason for this model is that in many applications, such as radiation effects, the exposure effect is more linear than exponential. Analyses of exposure-disease relationships in the RERF cohorts rely heavily on individual estimates of radiation doses from the atomic bombings. Those doses were of course not measured, but must be estimated from the available data, which consists of information about each survivor’s location and shielding at the time of exposure, obtained through past interviews, and the results of separately performed physical calculations of radiation source terms and transport through air and shielding materials. The current dosimetry system, designated DS02, was implemented in 2002, replacing the previous version (DS86), and was able to estimate doses for about 86,000 LSS members among about 94,000 individuals who were in the city survivors. Pierce, Stram and Vaeth (1990), Pierce, et al. (1992), and Pierce, Vaeth and Cologne (2008) discussed error correction techniques in the LSS data. The atomic-bomb survivor dose estimates used in most analyses have been corrected for random dosimetry errors under an error model in which the random error was assumed to have a standard deviation equal to 35% of the true dose. The primary cause of random error was assumed to be errors in the reported location and shielding situations of the survivors at the time of the bombings. The aforementioned correction methods for measurement error were based primarily on the so called regression calibration (RC) method, in which the true unobserved radiation dose is replaced by an estimate of its conditional expectation given the observed covariates (Carroll et al., 2006, chapter 4). A parametric RC estimator was applied in a measurement error modeling by Sposto, et al. (1991), in which they estimated the true unobserved radiation dose by the conditional expectation of the DS86 dose and epilation group indicator (severity of epilation being correlated with exposure). In their parametric model, they assumed that the distribution of the logarithm of DS86 dose was normally distributed with mean logarithm of the unobserved true dose, and with a known standard deviation (dose error parameter).

DS02, like DS86, provides only a single point estimate of dose for each person, not a set of replicate estimates. However, an additional variable that is associated with the unobserved true dose may provide information about the error in the DS02 estimate if the additional variable is an instrumental variable (IV). Roughly speaking, a variable is an IV if it is correlated with the unobserved exposure, independent of the measurement error of the surrogate variable for the true exposure, and independent of the outcome variable given covariates. Stram et al. (1993) presented analyses of the effect of radiation exposure, age, age at exposure, sex, time of assay and city of exposure on the proportion of T-lymphocytes exhibiting stable chromosome aberrations based on data of 1703 subjects collected between 1968 and 1985 by the RERF. Kodama et al. (2001) later examined chromosome aberrations based on data of 3042 subjects. Both studies found significant radiation dose-response relationships for percent stable chromosome aberration (%SCA) in the A bomb survivors, providing compelling evidence that %SCA is a useful, long-term marker of radiation exposure. Figure 1 shows the association between %SCA and DS02 (similar to data in Kodama et al., 2001). We used a quadratic-linear function and a lowess curve to model the relation between the proportion of aberrant cells and gamma marrow dose for both Hiroshima and Nagasaki, respectively. We will later discuss the use of %SCA as an IV for radiation exposure in the data section.

Figure 1.

Percent of lymphocytes with stable chromosome aberrations versus DS02 gamma marrow dose by city. The solid curve was from lowess smoothing and the dashed line was from fitting a quadratic function.

In this paper, we investigate ERR hazard regression when the covariates may be measured with errors. In general, radiation exposure may contain a mixture of additive and Berkson errors; as described in Pierce, et al. (2008). However, for the development of our methods there will be many technical challenges involved when the measurement error is from a mixture of classical and Berkson errors. Hence, in this paper we will focus on additive errors in the methodology development. Methods for ERR when the radiation exposure may contain a mixture of errors may be an interesting topic for future research. We assume both unbiased surrogates and IVs are available in a subset of the study cohort. The subset is called a calibration sample. In the non-calibration sample, an unbiased surrogate variable is available, but not an IV. There are 2 major contributions from our paper. First, we propose a new estimating equation for the ERR model in the absence of measurement error. Second, we propose a new method for covariate measurement error in ERR models. The proposed estimator contains a new idea of pseudo-parameters so that the 3-step estimation procedure can be implemented by a set of joint estimating equations. Our new method is different from the best linear estimator of Wang (2012) in non-censored outcome regression, and the generalized method of moments of Song and Wang (2014) in Cox regression. Section 2 describes the regression models for our problem. In Section 3 we investigate a semiparametric RC estimator that is different from the parametric RC approach (Pierce, Stram and Vaeth, 1990; Sposto, Stram and Awa, 1991). In Section 4, we develop a nonparametric correction (NPC) estimator using the calibration sample. We propose a joint NPC (JNPC) estimator using all observed data in Section 5. An optimal linear combination (OLC) estimator of JNPC and NPC is further developed. Finite sample performance of the proposed estimators is investigated in Section 6. The semiparametric RC, NPC, JNPC, and OLC estimators are robust to heteroscedastic measurement errors. We apply the methods to the RERF data in Section 7 to study the association between radiation exposure and solid cancer when %SCA is used as an IV. Some concluding remarks are given in Section 8. Regularity conditions are deferred to the Appendix, and sketched proofs are given in the supplementary materials (Web Section 7).

2 Statistical Models

In our problem of interest, survival, surrogate variable and IV data are involved. Assume that the study cohort consists of n subjects. For i = 1, …, n, let X_i be the primary exposure variable that may be associated with a disease outcome. Let Z_i be a vector of covariates measured without an error, such as age, gender and body mass index. We assume that a surrogate variable is available that follows the classical additive measurement error model

$$W_i=X_i+U_i,$$ (1)

where E(U_i|X_i, Z_i) = 0, and var(U_i|X_i, Z_i) may not be a constant. The error even may directly depend on X, as in the additional simulations in Web Section 5, as long as the conditional expectation is zero. This more general association would allow more general applications of the methods to be developed. We assume that X_i is a scalar variable for notational simplicity. In the RERF example, X_i is the unobserved true radiation exposure, W_i is the DS02 exposure estimate for subject i. Here we assume there is only one W_i, but the methods can be easily applied to the situation when replicates are available. In the RERF example, only one DS02 estimate is available for each individual. Let $T_i^0$ be the survival time of the ith subject, and C_i be the censoring time. The response consists of observed variables $T_i\equiv min(T_i^0,C_i)$ and ${\delta }_i\equiv I(T_i^0\le C_i)$, where I(·) is the indicator function. Of interest is the relationship between survival time $T_i^0$ and covariates X_i and Z_i, but $T_i^0$ is subject to censoring and thus is not fully observed. We assume that $T_i^0$ is independent of C_i given X_i and Z_i, and the hazard function λ(·) of $T_i^0$ given (X_i, Z_i) follows the ERR hazard model

$$\lambda (t|X_i,{\mathit{Z}}_i)={\lambda }_0(t)r(\mathit{\beta },X_i,{\mathit{Z}}_i),\text{where}r(\mathit{\beta },X_i,{\mathit{Z}}_i)=e^{{{\mathit{\beta }}^{\prime }}_2{\mathit{Z}}_i}(1+{\beta }_1{X}_i{e}^{{{\mathit{\beta }}^{\prime }}_3{\mathit{Z}}_i}),$$ (2)

and $({\beta }_1,{{\mathit{\beta }}^{\prime }}_2,{{\mathit{\beta }}^{\prime }}_3)\prime \equiv \mathit{\beta }$ is the vector of parameters of interest and λ₀(·) is an unspecified baseline hazard function. In the ERR model, β₂ is to model the background disease rate as a function of covariate Z, β₃ is for effect modification, and β₁ is ERR per unit dose for the effect modification at baseline (Z = 0). The correctly measured covariates Z may be time-dependent. In the RERF example, we consider %SCA as an IV. Denote the IV by Q_i, which satisfies

$$Q_i=h\left(X_i,{\mathit{Z}}_i\right)+V_i,\text{where}\mathrm{E}\left(V_i|X_i,{\mathit{Z}}_i\right)=0,$$ (3)

for any function h(X, Z), V_i is independent of the outcome (T_i, δ_i), and var(V_i|X_i, Z_i) may not be a constant. For example, $h(X_i,{\mathit{Z}}_i)={\alpha }_0+{\alpha }_1{X}_i+{\alpha }_2{X}_i^2+{{\alpha }^{\prime }}_3{\mathit{Z}}_i+{{\alpha }^{\prime }}_4{X}_i{\mathit{Z}}_i+{\alpha }_5{X}_i^2{\mathit{Z}}_i$; a reasonable linear-quadratic relation between %SCA and gamma marrow dose. In the paper, we assume there is only one observation of a single IV for each subject in the calibration sample, but the methods to be developed later can be applied to the situation when repeated Q_i are available. If the ith subject is in the calibration sample, then it is denoted by η_i = 1. That is, if η_i = 1 then {T_i, δ_i, W_i, Q_i, Z_i} are available. If η_i = 0, then the ith subject is in the non-calibration sample, in which {T_i, δ_i, W_i, Z_i} are available, but Q_i is not available. Generally, because X_i is not observable, equation (3) itself is not identifiable (even if there are repeated Q_i measurements). But, the parameters in (1) – (3) can be identified when the observed data involved in the 3 models are used in the analysis.

Many parametric and nonparametric methods have been developed for proportional hazards regression. However, to our knowledge, ERR regression with covariate measurement errors is less developed. The main contribution of the paper is to propose a consistent estimator for the ERR regression parameters under a general IV model (3). The parameters are identifiable given that a subsample is available which contains data in (1) – (3). The nonparametric estimation method to be developed will make use of Q under (3) without the need to have replicates in W.

3 Semiparametric Regression Calibration Method

A naive method is to use W_i to replace the unobserved X_i in the analysis. However, it is well-known that the naive estimator may cause very serious bias. In our problem, we may calculate the expected value of the hazard function given the observed data, namely the induced hazard function. It can be seen that the induced hazard function can be expressed as λ(t |Q_i, Z_i) = λ₀(t)r(β, E(X_i|Q_i, Z_i, T_i ≥ t), Z_i). The above equation indicates that an approximation based on the idea of replacing the unobserved X_i by E(X_i|W_i, Q_i, Z_i) may be valid. This replacement approach is often called the regression calibration (RC) estimator (Carroll, et al., 2006, Chapter 4) in the literature of measurement error. In Cox regression, Prentice (1982) proposed the RC estimator by noticing if disease is rare then the condition T ≥ t may be neglected.

A common implementation for RC is to assume that the joint distribution of X, W, Q and Z is multivariate-normal, or the joint distribution of X, W, Q given Z is multivariate-normal. When Z_i is discrete, E(X_i|W_i, Q_i, Z_i) serves as a replacement for X_i within the calibration sample, and E(X_i|W_i, Z_i) serves as a replacement for X_i within the non-calibration sample. If given Z_i = z, (X_i, W_i, Q_i) is multivariate normal, then the calibration function E(X_i|W_i, Q_i, Z_i) can be calculated by

$${\mu }_{X|Z}+({\sum }_{XW|Z,}{\sum }_{XQ|Z}){\left(\begin{array}{cc}{\sigma }_{W|Z}^2& {\sum }_{WQ|Z}\\ {\sum }_{WQ|Z}& {\sigma }_{Q|Z}^2\end{array}\right)}^{-1}\left(\begin{array}{c}{W}_i-{\mu }_{X|Z}\\ Q_i-{\mu }_{Q|Z}\end{array}\right),$$

where µ_X|Z is the conditional expectation of X_i given Z_i = z, Σ_XQ|Z = cov(X_i, Q_i|Z_i = z) and ${\sigma }_{Q|Z}^2=\text{var}(Q_i|{\mathit{Z}}_i=z)$ for any X, Q and Z.

The RC estimator may replace X by E(X|W, Q, Z) by using the above formula, or by E(X|W, Z) by a similar formula. But, the calculation would involve many nuisance parameters, such as the regression coefficients and the first 2 moments in the modeling of Q given X (not observed) and Z. The nuisance parameters can be estimated by the method of moments; a special example can be seen in the first paragraph of Web Section 3 of the supplementary materials. However, instead of the above method, the RC can be implemented by a more robust approach to calculate E(W |Q, Z) for E(X|Q, Z), by noticing that W given X follows a classical additive error model. In the calibration sample, we could consider a regression model for W given (Q, Z). For example, W may be modeled as linear, such as $\mathrm{E}(W|Q,\mathit{Z})={\alpha }_0+{\alpha }_{1}Q+{{\alpha }^{\prime }}_2\mathit{Z}$ if the association is appropriate. The RC estimator for this example is to use the calibration sample such that X_i is replaced by ${\widehat{\alpha }}_0+{\widehat{\alpha }}_{1}Q+{\widehat{\alpha }}_2{\mathit{Z}}_i$, where ${\widehat{\alpha }}_0$, ${\widehat{\alpha }}_1$ and ${\widehat{\mathit{\alpha }}}_2$ consistent for are estimates α₀, α₁ and α₂. RC using the non-calibration sample is not straight-forward since Q is not available. This RC estimator is semiparametric in the sense that we only assume the regression model of W given (Q, Z), but not a covariate distribution. This semiparametric RC estimator is different from the parametric RC estimator of Sposto, Stram and Awa (1991) and Neriishi et al. (1991) in which they assumed a log-normal distribution for W given X. They also assumed that the measurement error variance is known. The semiparametric RC is approximately consistent since the induced hazard function given (Q_i, Z_i) is λ(t | Q_i, Z_i) = λ₀(t)r{β, E(X_i|Q_i, Z_i, T_i ≥ t), Z_i}. The approximation error was from the replacement of E(X_i|Q_i, Z_i, T_i ≥ t) by E(X_i|Q_i, Z_i) when applying the semiparametric RC estimator. From our simulation results, the approximation error in general is very minimal, and the estimator performs similarly to other consistent estimators. In the supplementary materials, Web Section 2 provides details on a risk set RC estimator which uses E(X|Q, Z, T ≥ t) as a replacement for X. In addition, since the estimation of the calibration function E(X|Q, Z) is based on a linear regression model, the semiparametric RC would not cause bias even when the measurement errors are heteroscedastic.

4 Nonparametric Correction Estimation Using Calibration Sample

In this section, we develop a consistent estimation method for measurement error by correcting the observed data estimating score using the calibration sample. Let N_i(t) = I[T_i ≤ t, δ_i = 1] be the counting process, and Y_i(t) = I[T_i ≥ t] be the at risk process. Generally speaking, if $\widehat{\mathbf{\Psi }}(T,\delta ,X,\mathit{Z})$ is a full data score such that $\mathrm{E}\{\widehat{\mathbf{\Psi }}(T,\delta ,X,\mathit{Z})\}=0$ when evaluated at the true parameters, then ${\widehat{\mathbf{\Psi }}}_c(T,\delta ,X,\mathit{Z})$ is a corrected score if it has the same limit as $\widehat{\mathbf{\Psi }}(T,\delta ,X,\mathit{Z})$ does when n → ∞. As discussed in Thomas (1981) and Prentice and Mason (1986), the partial likelihood score for ERR model may encounter finite sample challenges since the relative risk function is involved in the denominator of the estimating equation. To avoid this issue, we propose to use the following estimating equation (when there is no measurement error),

$$n^{-1}{\displaystyle \sum _{i=1}^n{\displaystyle {\int }_0^{\tau }\left\{\left(\begin{array}{c}{X}_i\\ {\mathit{Z}}_i\\ X_i\mathit{Z}\end{array}\right)-\frac{{\displaystyle {\sum }_{j=1}^n{Y}_j(t)(X_j,{{\mathit{Z}}^{\prime }}_j,X_j{{\mathit{Z}}^{\prime }}_j)\prime r(\mathit{\beta },X_j,{\mathit{Z}}_j)}}{{\displaystyle {\sum }_{j=1}^n{Y}_j(t)r(\mathit{\beta },X_j,{\mathit{Z}}_j)}}\right\}d{N}_i(t)}}=\mathbf{0}.$$ (4)

Similar to Cox regression, $M_i(t)=N_i(t)-{\displaystyle {\int }_0^t{Y}_i(t)r(\mathit{\beta },X_i,{\mathit{Z}}_i){\lambda }_0(t)dt}$ can be shown to be a martingale. As in Web Section 1 of the supplementary materials, (4) can be written as a martingale representation and hence (4) can be called a martingale-based estimating equation (MEE). In addition to the finite sample advantage discussed above, the second advantage of MEE over partial likelihood is from the viewpoint of constructing an unbiased estimating equation with the presence of covariate measurement errors. As one referee pointed out, Buzas (1998) showed that there does not exist a corrected score for a linear partial likelihood risk function. Hence, the MEE is an important tool in the development of corrected score estimation for covariate measurement error in the ERR model. In the absence of measurement error, the MEE estimator is better than partial likelihood when the sample size is small (such as n = 100). We also examined the asymptotic efficiency, the MEE estimator is highly efficient for small exposure effects, and it still has good efficiency for large exposure effects (Web Table 2).

In Cox regression with additive measurement errors, Huang and Wang (2000) proposed a nonparametric correction estimator. It was nonparametric in the sense that there was no need to assume the distribution of the covariates or measurement errors. In our problem, the main regression is the ERR hazard regression, and hence a new methodology for consistent estimation is needed. Recall that Q given (X, Z) has mean h(X, Z). We note that by some calculations, the following score can be shown to be unbiased:

$$\mathbf{\Phi }\left(\mathit{\beta },\tau \right)=\mathrm{E}\left[{\displaystyle {\int }_0^{\tau }\left\{\left(\begin{array}{c}h\left(X,\mathit{Z}\right)\\ \mathit{Z}\\ h\left(X,\mathit{Z}\right)\mathit{Z}\end{array}\right)-\frac{\mathrm{E}\{Y\left(t\right)(h\left(X,\mathit{Z}\right),{\mathit{Z}}^{\prime },h\left(X,\mathit{Z}\right){\mathit{Z}}^{\prime })\prime r\left(\mathit{\beta },X,\mathit{Z}\right)}{\mathrm{E}\left\{Y\left(t\right)r\left(\mathit{\beta },X,\mathit{Z}\right)\right\}}\right\}dN\left(t\right)}\right].$$

Hence, we develop a nonparametric correction score estimator using the calibration sample solving the following estimating equation:

$${\displaystyle \sum _{i=1}^n{\eta }_i\varphi \left(\mathit{\beta },W_i,Q_i,{\mathit{Z}}_i\right)}\equiv {\displaystyle \sum _{i=1}^n{\eta }_i{\displaystyle {\int }_0^{\tau }\left\{\left(\begin{array}{c}{Q}_i\\ {\mathit{Z}}_i\\ Q_i{\mathit{Z}}_i\end{array}\right)-\frac{{\displaystyle {\sum }_{j=1}^n{\eta }_j{Y}_j(t)(Q_j,{{\mathit{Z}}^{\prime }}_j,Q_j{\mathit{Z}}_j)\prime r(\mathit{\beta },W_j,{\mathit{Z}}_j)}}{{\displaystyle {\sum }_{j=1}^n{\eta }_j{Y}_j(t)r(\mathit{\beta },W_j,{\mathit{Z}}_j)}}\right\}d{N}_i(t)}}=0.$$ (5)

The NPC estimator is nonparametric since the estimating equation is unbiased without using a parametric assumption on either the covariates or the measurement errors. We denote the NPC estimator by ${\widehat{\mathit{\beta }}}_{\mathit{npc}}^c$, and define G(β) = −E{∂ηϕ(W, Q, Z, β)/∂β}. Let

$${\varphi }^{\ast }\left(\mathit{\beta },W_i,Q_i,{\mathit{Z}}_i\right)={\eta }_i{\displaystyle {\int }_0^{\tau }\left[\left(\begin{array}{c}{Q}_i\\ {\mathit{Z}}_i\\ Q_i{\mathit{Z}}_i\end{array}\right)-\frac{\mathrm{E}\{Y(t)(h\left(X,\mathit{Z}\right),{\mathit{Z}}^{\prime },h\left(X,\mathit{Z}\right){\mathit{Z}}^{\prime })\prime r\left(\mathit{\beta },X,\mathit{Z}\right)\}}{\mathrm{E}\{Y(t)r\left(\mathit{\beta },X,\mathit{Z}\right)\}}\right]d{N}_i\left(t\right)}-{\displaystyle {\int }_0^{\tau }\frac{Y_i\left(t\right)r\left(\mathit{\beta },X_i,{\mathit{Z}}_i\right)}{\mathrm{E}\left\{Y\left(t\right)r\left(\mathit{\beta },X,\mathit{Z}\right)\right\}}\left[\left(\begin{array}{c}{Q}_i\\ {\mathit{Z}}_i\\ Q_i{\mathit{Z}}_i\end{array}\right)-\frac{\mathrm{E}\{Y\left(t\right)(h\left(X,\mathit{Z}\right),{\mathit{Z}}^{\prime },h\left(X,\mathit{Z}\right){\mathit{Z}}^{\prime })\prime r\left(\mathit{\beta },X,\mathit{Z}\right)\}}{\mathrm{E}\left\{Y\left(t\right)r\left(\mathit{\beta },X,\mathit{Z}\right)\right\}}\right]d\overline{N}\left(t\right)},$$ (6)

where $\overline{N}\left(t\right)=\mathrm{E}\left\{N\left(t\right)\right\}$. The asymptotic distribution of ${\widehat{\mathit{\beta }}}_{\mathit{npc}}^c$ is given below:

Proposition 1

Under Regularity Conditions (A1) – (A8) given in the Appendix, ${\widehat{\mathit{\beta }}}_{\mathit{npc}}^c$ converges to β in probability, and $n_c^{1/2}\left({\widehat{\mathit{\beta }}}_{\mathit{npc}}^c-\mathit{\beta }\right)$ is asymptotically normal with mean 0 and variance

$${\left\{\mathit{G}\left(\mathit{\beta }\right)\right\}}^{-1}\mathrm{E}[\eta {\varphi }^{\ast }\left(\mathit{\beta },W,Q,\mathit{Z}\right){\left\{{\varphi }^{\ast }\left(\mathit{\beta },W,Q,\mathit{Z}\right)\right\}}^{\prime }]{\{{\mathit{G}}^{-1}\left(\mathit{\beta }\right)\}}^{\prime }.$$

We note that Proposition 1 holds even if var(U_i|X_i, Z_i) or var(V_i|X_i, Z_i) is not a constant. A sketch of the proof of Proposition 1 is given in the Web Appendix. Estimation of the standard error of ${\widehat{\mathit{\beta }}}_{\mathit{npc}}$ can be obtained by a sandwich estimator and noting that ϕ^∗(β, W_i, Z_i, Q_i), i = 1, …, n are independent. It is easier to estimate G(β) by numerical derivatives, but this can be done by analytic calculation too.

5 Joint Nonparametric Correction Estimation

The NPC estimator described in the previous section uses the calibration sample only, but not all the observed data. In this section, our goal is to further make use of the non-calibration sample as well to improve efficiency. We first note that a naive estimator using all surrogate variables W_i, i = 1, …, n is to solve the following estimating equation:

$${\displaystyle \sum _{i=1}^n{\displaystyle {\int }_0^t\left\{\left(\begin{array}{c}{W}_i\\ {\mathit{Z}}_i\\ W_i\mathit{Z}\end{array}\right)-\frac{{\displaystyle {\sum }_{j=1}^n{Y}_j(t){(W_j,{{\mathit{Z}}^{\prime }}_j,W_j{{\mathit{Z}}^{\prime }}_j)}^{\prime }r(\mathit{\beta },W_j,{\mathit{Z}}_j)}}{{\displaystyle {\sum }_{j=1}^n{Y}_j(t)r(\mathit{\beta },W_j,{\mathit{Z}}_j)}}\right\}d{N}_i(t)}}=0.$$

The above estimating equation is not unbiased. By direct calculation of $\mathrm{E}\{Y(t){(W_i,{{\mathit{Z}}^{\prime }}_j,W_j{{\mathit{Z}}^{\prime }}_j)}^{\prime }r(\mathit{\beta },W,\mathit{Z})\}$ and E{Y (t)r(β, W, Z)}, a corrected score estimating equation can be written as

$${\displaystyle \sum _{i=1}^n{\displaystyle {\int }_0^{\tau }\left\{\left(\begin{array}{c}{W}_i\\ {\mathit{Z}}_i\\ W_i{\mathit{Z}}_i\end{array}\right)-\frac{{\displaystyle {\sum }_{j=1}^n{Y}_j(t){(W_j,{{\mathit{Z}}^{\prime }}_j,W_j{{\mathit{Z}}^{\prime }}_j)}^{\prime }r(\mathit{\beta },W_j,{\mathit{Z}}_j)}}{{\displaystyle {\sum }_{j=1}^n{Y}_j(t)r(\mathit{\beta },W_j,{\mathit{Z}}_j)}}-D(\mathit{\beta },t)\right\}d{N}_i(t)}}=0.$$ (7)

where $D(\mathit{\beta },t)=\mathrm{E}\{Y(t)\exp ({{\mathit{\beta }}^{\prime }}_3\mathit{Z}+{{\mathit{\beta }}^{\prime }}_2\mathit{Z}){({\beta }_1{U}^2,0,{\beta }_1{\mathit{Z}}^{\prime }{U}^2)}^{\prime }\}/\mathrm{E}\{Y(t)r(\mathit{\beta },W,\mathit{Z})\}$. Therefore, if the distribution of the measurement error (U) is known, then a parametric corrected score estimation can be obtained. However, in Web Section 3, we show that in general parametric corrected score estimation has poor performance. Because in general the measurement error is unknown, our goal is to develop a nonparametric approach to estimating the bias correction term D(β, t) without assuming any additional distributional assumptions on either the covariates or the measurement errors.

Our idea is to use the calibration sample to estimate the bias term D(β, t) in (7). However, estimating D(β, t) directly from the calibration data seems rather challenging. Hence, we consider an approach to nonparametrically estimate it using the calibration data indirectly. By applying (7) to the calibration sample, the bias term is the limit of the following empirical average:

$$n_c^{-1}{\displaystyle \sum _{i=1}^n{\eta }_i{\displaystyle {\int }_0^{\tau }\left\{\left(\begin{array}{c}{W}_i\\ {\mathit{Z}}_i\\ W_i{\mathit{Z}}_i\end{array}\right)-\frac{{\displaystyle {\sum }_{j=1}^n{\eta }_j{Y}_j\left(t\right){\left(W_j,{{\mathit{Z}}^{\prime }}_j,W_j{{\mathit{Z}}^{\prime }}_j\right)}^{\prime }r\left(\mathit{\beta },W_j,{\mathit{Z}}_j\right)}}{{\displaystyle {\sum }_{j=1}^n{\eta }_j{Y}_j\left(t\right)r\left(\mathit{\beta },W_j,{\mathit{Z}}_j\right)}}\right\}d{N}_i\left(t\right)}}\equiv \widehat{\mathit{B}}\left(\mathit{\beta }\right).$$

Let the limit of $\widehat{\mathit{B}}\left(\mathit{\beta }\right)$ be denoted by B(β). We now apply the consistent estimator ${\widehat{\mathit{\beta }}}_{\mathit{npc}}^c$ developed in the last section to the above equation. The bias term B(β) can be estimated by $\widehat{\mathit{B}}({\widehat{\mathit{\beta }}}_{\mathit{npc}}^c)$. Therefore, if we use γ to denote ${\widehat{\mathit{\beta }}}_{\mathit{npc}}^c$, then an unbiased estimating equation for the bias term can be expressed as

$$n_c^{-1}{\displaystyle \sum _{i=1}^n{\eta }_i{\displaystyle {\int }_0^{\tau }\left\{\left(\begin{array}{c}{W}_i\\ {\mathit{Z}}_i\\ W_i{\mathit{Z}}_i\end{array}\right)-\frac{{\displaystyle {\sum }_{j=1}^n{\eta }_j{Y}_j(t){(W_j,{{\mathit{Z}}^{\prime }}_j,W_j{{\mathit{Z}}^{\prime }}_j)}^{\prime }r(\mathit{\gamma },W_j,{\mathit{Z}}_j)}}{{\displaystyle {\sum }_{j=1}^n{\eta }_j{Y}_j(t)r(\mathit{\gamma },W_j,{\mathit{Z}}_j)}}-\mathit{\xi }\right\}d{N}_i(t)}}=0,$$ (8)

where ξ = ξ(γ) is the same as $\widehat{\mathit{B}}\left(\mathit{\gamma }\right)/\{n_c^{-1}{\displaystyle {\sum }_{i=1}^n{\eta }_i{\displaystyle {\int }_0^{\tau }d{N}_i\left(t\right)}}\}$. From the arguments given above, the new bias-correction estimator can be obtained by the following procedure. (i) We first obtain ${\widehat{\mathit{\beta }}}_{\mathit{npc}}^c$ using the calibration sample, and let it be denoted by γ. (ii) Then we estimate ξ by equation (8) using the calibration sample in which γ is from step (i). (iii) Finally we estimate β using the observed data from the whole cohort based on the following estimating equation:

$${\displaystyle \sum _{i=1}^n{\displaystyle {\int }_0^{\tau }\left\{\left(\begin{array}{c}{W}_i\\ {\mathit{Z}}_i\\ W_i{\mathit{Z}}_i\end{array}\right)-\frac{{\displaystyle {\sum }_{j=1}^n{Y}_j(t){(W_j,{{\mathit{Z}}^{\prime }}_j,W_j{{\mathit{Z}}^{\prime }}_j)}^{\prime }r(\mathit{\beta },W_j,{\mathit{Z}}_j)}}{{\displaystyle {\sum }_{j=1}^n{Y}_j(t)r(\mathit{\beta },W_j,{\mathit{Z}}_j)}}-\widehat{\mathit{\xi }}\right\}d{N}_i(t)}}=0.$$ (9)

The root for (9) is our proposed joint nonparametric correction (JNPC) estimator which jointly uses the calibration sample to estimate the bias correction term and all of the observed data in the bias-correction estimating equation. The basic reasoning for the consistency of the JNPC estimator is that we use the calibration sample to estimate the bias correction term ξ consistently, and hence solving (9) will lead to a consistent estimator using all the observed data.

In the arguments given above, we intentionally used γ, rather than β, for the regression coefficients when using the calibration sample to estimate the bias correction term. An important reason for doing so is to have a better look at the estimating equations involved. So, $\widehat{\mathit{\gamma }}$ is actually the NPC estimator using the calibration sample when β is denoted by γ. Hence γ itself can be considered as a pseudo-parameter vector. With this concept, we would treat the problem as joint estimation of (β, ξ, γ). Let θ = (β′, ξ′, γ′)′. Then we could express the estimating procedure as the following joint estimating equations for θ:

$$n^{-1}{\displaystyle \sum _{i=1}^n\left\{\begin{array}{c}{\displaystyle {\int }_0^{\tau }\left\{\left(\begin{array}{c}{W}_i\\ {\mathit{Z}}_i\\ W_i{\mathit{Z}}_i\end{array}\right)-\frac{{\displaystyle {\sum }_{j=1}^n{Y}_j(t){(W_j,{{\mathit{Z}}^{\prime }}_j,W_j{{\mathit{Z}}^{\prime }}_j)}^{\prime }r(\mathit{\beta },W_j,{\mathit{Z}}_j)}}{{\displaystyle {\sum }_{j=1}^n{Y}_j(t)r(\mathit{\beta },W_j,{\mathit{Z}}_j)}}-\mathit{\xi }\right\}d{N}_i(t)}\\ {\eta }_i{\displaystyle {\int }_0^{\tau }\left\{\left(\begin{array}{c}{W}_i\\ {\mathit{Z}}_i\\ W_i{\mathit{Z}}_i\end{array}\right)-\frac{{\displaystyle {\sum }_{j=1}^n{\eta }_j{Y}_j(t){(W_j,{{\mathit{Z}}^{\prime }}_j,W_j{{\mathit{Z}}^{\prime }}_j)}^{\prime }r(\mathit{\gamma },W_j,{\mathit{Z}}_j)}}{{\displaystyle {\sum }_{j=1}^n{\eta }_j{Y}_j(t)r(\mathit{\gamma },W_j,{\mathit{Z}}_j)}}-\mathit{\xi }\right\}d{N}_i(t)}\\ {\eta }_i{\displaystyle {\int }_0^{\tau }\left\{\left(\begin{array}{c}{Q}_i\\ {\mathit{Z}}_i\\ Q_i{\mathit{Z}}_i\end{array}\right)-\frac{{\displaystyle {\sum }_{j=1}^n{\eta }_j{Y}_j(t){(Q_j,{{\mathit{Z}}^{\prime }}_j,Q_j{{\mathit{Z}}^{\prime }}_j)}^{\prime }r(\mathit{\gamma },W_j,{\mathit{Z}}_j)}}{{\displaystyle {\sum }_{j=1}^n{\eta }_j{Y}_j(t)r(\mathit{\gamma },W_j,{\mathit{Z}}_j)}}\right\}d{N}_i(t)}\end{array}\right\}}=0.$$ (10)

The creation of the pseudo-parameter vector γ in (10) may be non-standard, but it could be understood by the reasoning of the 3-step procedure given above. Let (10) be denoted by Ψ(θ, W, Q, Z) = 0. If the true value of β is β₀, then the true value of γ is also β₀. The third part of Ψ(θ, W, Q, Z) is used to obtain the NPC estimate (using the calibration sample only) of β, denoted by γ, so that it can be used for the first 2 parts of Ψ(θ, W, Q, Z). The second part of Ψ(θ, W, Q, Z) is to obtain a bias correction estimate (ξ) using the calibration sample, which will then be applied to the first part of Ψ(θ, W, Q, Z) for the primary parameter estimation for β using all of the observed data. The main advantage of expressing the JNPC estimator as stacked estimating equations is that the large sample distribution can be seen more easily. Although the equations contain empirical averages (such as $n^{-1}{\displaystyle {\sum }_{j=1}^n{\mathit{Y}}_j\left(t\right)r(\mathit{\beta },{\mathit{W}}_j,{\mathit{Z}}_j)}$), they could be expressed as independent terms much more easily than treating the estimator as being obtained via an estimating equation but with estimated parameters ( $\widehat{\mathit{\xi }}$ and ${\widehat{\mathit{\beta }}}_{\mathit{npc}}^c$) as in (9).

Write Ψ_n(θ, W, Q, Z) as ${\displaystyle {\sum }_{i=1}^n\psi (\mathit{\theta },W_i,Q_i,{\mathit{Z}}_i)}$, which is not a sum of independent scores. Then by the functional delta method, it can be shown that $n^{-1/2}{\mathbf{\Psi }}_n(\mathit{\theta },W,Q\mathit{Z})=n^{-1/2}{\displaystyle {\sum }_{i=1}^n{\psi }^{\ast }(\mathit{\theta },W_i,Q_i,{\mathit{Z}}_i)+o_p\left(1\right)},$ where

$${\psi }^{\ast }\left(\mathit{\theta },W_i,Q_i,{\mathit{Z}}_i\right)=\{\begin{array}{c}{\displaystyle {\int }_0^{\tau }\left\{\left(\begin{array}{c}{W}_i\\ {\mathit{Z}}_i\\ W_i{\mathit{Z}}_i\end{array}\right)-\frac{\mathrm{E}\left\{Y\left(t\right){\left(W,{\mathit{Z}}^{\prime },W{\mathit{Z}}^{\prime }\right)}^{\prime }r\left(\mathit{\beta },W,\mathit{Z}\right)\right\}}{\mathrm{E}\left\{Y\left(t\right)r\left(\mathit{\beta },W,\mathit{Z}\right)\right\}}-\mathit{\xi }\right\}d{N}_i\left(t\right)}-{\displaystyle {\int }_0^{\tau }\frac{Y_i\left(t\right)r\left(\mathit{\beta },W_i,{\mathit{Z}}_i\right)\}}{\mathrm{E}\left\{Y\left(t\right)r\left(\mathit{\beta },W,\mathit{Z}\right)\right\}}\left\{\left(\begin{array}{c}{W}_i\\ {\mathit{Z}}_i\\ W_i{\mathit{Z}}_i\end{array}\right)-\frac{\mathrm{E}\left\{Y\left(t\right){\left(W,{\mathit{Z}}^{\prime },W{\mathit{Z}}^{\prime }\right)}^{\prime }r\left(\mathit{\beta },W,\mathit{Z}\right)\right\}}{\mathrm{E}\left\{Y\left(t\right)r\left(\mathit{\beta },W,\mathit{Z}\right)\right\}}\right\}d{\overline{N}}_i\left(t\right)},\\ {\eta }_i{\displaystyle {\int }_0^{\tau }\left\{\left(\begin{array}{c}{W}_i\\ {\mathit{Z}}_i\\ W_i{\mathit{Z}}_i\end{array}\right)-\frac{\mathrm{E}\left\{Y\left(t\right){\left(W,{\mathit{Z}}^{\prime },W{\mathit{Z}}^{\prime }\right)}^{\prime }r\left(\mathit{\gamma },W,\mathit{Z}\right)\right\}}{\mathrm{E}\left\{Y\left(t\right)r\left(\mathit{\gamma },W,\mathit{Z}\right)\right\}}-\mathit{\xi }\right\}d{N}_i\left(t\right)}-{\displaystyle {\int }_0^{\tau }\frac{Y_i\left(t\right)r\left(\mathit{\gamma },W_i,{\mathit{Z}}_i\right)}{\mathrm{E}\left\{Y\left(t\right)r\left(\mathit{\gamma },W,\mathit{Z}\right)\right\}}\left\{\left(\begin{array}{c}{W}_i\\ {\mathit{Z}}_i\\ W_i{\mathit{Z}}_i\end{array}\right)-\frac{\mathrm{E}\left\{Y\left(t\right){\left(W,{\mathit{Z}}^{\prime },W{\mathit{Z}}^{\prime }\right)}^{\prime }r\left(\mathit{\gamma },W,\mathit{Z}\right)\right\}}{\mathrm{E}\left\{Y\left(t\right)r\left(\mathit{\gamma },W,\mathit{Z}\right)\right\}}\right\}d{\overline{N}}_i\left(t\right),}\\ {\eta }_i{\displaystyle {\int }_0^{\tau }\left\{\left(\begin{array}{c}{Q}_i\\ {\mathit{Z}}_i\\ Q_i{\mathit{Z}}_i\end{array}\right)-\frac{\mathrm{E}\left\{Y\left(t\right){\left(Q,{\mathit{Z}}^{\prime },Q\mathit{Z}\right)}^{\prime }r\left(\mathit{\gamma },W,\mathit{Z}\right)\right\}}{\mathrm{E}\left\{Y\left(t\right)r\left(\mathit{\gamma },W,\mathit{Z}\right)\right\}}\right\}d{N}_i\left(t\right)}-{\displaystyle {\int }_0^{\tau }\frac{Y_i\left(t\right)r\left(\mathit{\gamma },W_i,{\mathit{Z}}_i\right)}{\mathrm{E}\left\{Y\left(t\right)r\left(\mathit{\gamma },W,\mathit{Z}\right)\right\}}\left\{\left(\begin{array}{c}{Q}_i\\ {\mathit{Z}}_i\\ Q_i{\mathit{Z}}_i\end{array}\right)-\frac{\mathrm{E}\left\{Y\left(t\right){\left(Q,{\mathit{Z}}^{\prime },Q{\mathit{Z}}^{\prime }\right)}^{\prime }r\left(\mathit{\gamma },W,\mathit{Z}\right)\right\}}{\mathrm{E}\left\{Y\left(t\right)r\left(\mathit{\gamma },W,\mathit{Z}\right)\right\}}\right\}d{\overline{N}}_i\left(t\right)}.\end{array}$$

Let $\mathcal{G}(\mathit{\theta })=-\left(\partial /\partial \mathit{\theta }\right)\mathrm{E}\left\{{\mathbf{\Psi }}_n\left(\mathit{\theta },W,Q,\mathit{Z}\right)\right\}=-\left(\partial /\partial \mathit{\theta }\right)\mathrm{E}\left\{\mathit{\psi }\left(\mathit{\theta },W,Q,\mathit{Z}\right)\right\}$. Let the JNPC estimator for θ be denoted by ${\widehat{\mathit{\theta }}}_{\mathit{jnpc}}$, and its corresponding component for β be denoted by ${\widehat{\mathit{\beta }}}_{\mathit{jnpc}}$. The asymptotic distribution of ${\widehat{\mathit{\theta }}}_{\mathit{jnpc}}$ is given below:

Proposition 2

Under Regularity Conditions (A1) – (A8) given in the Appendix ${\widehat{\mathit{\theta }}}_{\mathit{jnpc}}$ converges to θ in probability, and $n_c^{1/2}({\widehat{\mathit{\theta }}}_{\mathit{jnpc}}-\mathit{\theta })$ is asymptotically normal with mean 0 and variance

$${\{\mathcal{G}(\mathit{\theta })\}}^{-1}\mathrm{E}[{\psi }^{\ast }(\mathit{\theta },W,Q,\mathit{Z})\{{\psi }^{\ast }{(\mathit{\theta },W,Q,\mathit{Z})}^{\prime }\}]{\{{\mathcal{G}}^{-1}(\mathit{\theta })\}}^{\prime }.$$

A sketch of the proof of Proposition 2 is given in the Web Appendix. Estimation of the standard error of ${\widehat{\mathit{\theta }}}_{\mathit{jnpc}}$ can be obtained by the sandwich estimator

$${\{\widehat{\mathcal{G}}({\widehat{\mathit{\theta }}}_{\mathit{jnpc}})\}}^{-1}{n}^{-1}\left[{\displaystyle \sum _{i=1}^n{\widehat{\psi }}^{\ast }({\widehat{\mathit{\theta }}}_{\mathit{jnpc}},W_i,Q_i,{\mathit{Z}}_i){\{{\widehat{\psi }}^{\ast }({\widehat{\mathit{\theta }}}_{\mathit{jnpc}},W_i,Q_i,{\mathit{Z}}_i)\}}^{\prime }}\right]{\{{\widehat{\mathcal{G}}}^{-1}({\widehat{\mathit{\theta }}}_{\mathit{jnpc}})\}}^{\prime }.$$

In the formula above, $\widehat{\mathcal{G}}(\mathit{\theta })=-\left(\partial /\partial \mathit{\theta }\right)\left\{{\mathbf{\Psi }}_n\left(\mathit{\theta },W,Q,\mathit{Z}\right)\right\}$, which is easier to obtain by numerical derivatives than analytic calculation. In addition, ${\widehat{\psi }}^{\ast }(\mathit{\theta },W_i,{\mathit{Z}}_i,Q_i)$ is the same as ${\widehat{\psi }}^{\ast }(\mathit{\theta },W_i,{\mathit{Z}}_i,Q_i)$ except that terms such as E{Y (t)(W, Z′, W Z)′r(β, W, Z)} would be replaced by their corresponding sample means. We note that Proposition 2 holds even if var(U_i|X_i, Z_i) or var(V_i|X_i, Z_i) is not a constant.

An anonymous referee kindly suggested an OLC estimator that is the most efficient among the linear combination of ${\widehat{\mathit{\beta }}}_{\mathit{npc}}^c$ and ${\widehat{\mathit{\beta }}}_{\mathit{jnpc}}$. Define $\widehat{\mathit{\beta }}\left(\mathbf{\Delta }\right)\equiv \mathbf{\Delta }{\widehat{\mathit{\beta }}}_{\mathit{npc}}^c+\left(I-\mathbf{\Delta }\right){\widehat{\mathit{\theta }}}_{\mathit{jnpc}}$, for any $\mathbf{\Delta }$ of dimension p×p, where p is the dimension of β, and I is the identity matrix of dimension p × p. An optimal Δ, denoted by Δ*, would minimize the asymptotic variance of $\widehat{\mathit{\beta }}\left(\mathbf{\Delta }\right)$_. By direct calculation, ∆^∗ satisfies:

$${\mathbf{\Delta }}^{\ast }={\left[\mathrm{var}\{n^{1/2}({\widehat{\mathit{\beta }}}_{\mathit{npc}}^c-{\widehat{\mathit{\beta }}}_{\mathit{jnpc}})\}\right]}^{-1}\left[\mathrm{var}\{n^{1/2}({\widehat{\mathit{\beta }}}_{\mathit{jnpc}}-\mathit{\beta })-\mathrm{cov}\{n_c^{1/2}({\widehat{\mathit{\beta }}}_{\mathit{jnpc}}-\mathit{\beta }),n_c^{1/2}({\widehat{\mathit{\beta }}}_{\mathit{npc}}^c-\mathit{\beta })\}\right].$$

Additional details of the OLC estimator, denoted by ${\widehat{\mathit{\beta }}}_{\mathit{olc}}\equiv \widehat{\mathit{\beta }}\left({\mathbf{\Delta }}^{\ast }\right)$, are provided in Web Section 4.

6 Simulation Study

We conducted a simulation study to compare the proposed estimators with the other methods discussed above. The naive estimator was to use W to replace X. We considered the RC estimator that replaced X by E(W|Z, Q). The NPC estimator was to solve (5), the JNPC estimator was to solve (10), and the OLC estimator was described above. In Table 1, covariates X_i, i = 1, …, n, were generated from a uniform distribution with mean µ_x = 2 and variance ${\sigma }_x^2=1$. Surrogates W_i, i = 1, …, n, were generated by W_i = X_i + U_i, where U_i was normal with mean 0 and standard deviation σ_u = 0.5. The IV Q_i was generated by $Q_i={\gamma }_0+{\gamma }_1{X}_i+{\gamma }_2{X}_i^2+V_i$, where γ₀ = 0.2, γ₁ = 0.55, γ₂ = 0.1, and σ_v = 0.707. The sample size of the whole cohort in the simulation was n = 1000 or n = 2000. The sample size of the calibration sample was given in the tables. The failure times were generated by the ERR hazard model with a baseline hazard function λ₀(t) = 1. In general, the performance of the estimators was not affected by the baseline hazard function (such as λ₀(t) = 2t, for a Weibull failure time distribution). In Table 1, we took the 50th percentile of the failure time of the cohort as the censoring time for each subject (censoring rate = 50%). All the parameters are given in the tables. The standard errors of the RC estimates were obtained by a sandwich variance estimator where the vector of the estimating equations was obtained by stacking the estimating equations for β and the estimating equations for regression W given Q discussed in Section 3.

Table 1.

Simulation study with various proportions of the calibration sample

		Naive	RC	NPC	JNPC	Naive	RC	NPC	JNPC
		n = 1000				n = 2000
β1 = 0.2
nc/n = 0.30	Bias	−0.036	0.048	0.045	0.026	−0.041	0.038	0.037	0.023
	SD	0.140	0.247	0.235	0.170	0.088	0.172	0.170	0.134
	ASE	0.134	0.255	0.248	0.182	0.091	0.166	0.164	0.125
	CP	0.876	0.920	0.918	0.922	0.866	0.948	0.954	0.936
nc/n = 0.50	Bias	−0.041	0.032	0.031	0.018	−0.044	0.024	0.023	0.019
	SD	0.094	0.190	0.186	0.151	0.071	0.126	0.124	0.099
	ASE	0.100	0.182	0.179	0.146	0.069	0.121	0.120	0.100
	CP	0.876	0.942	0.944	0.928	0.808	0.952	0.948	0.962
nc/n = 0.70	Bias	−0.044	0.017	0.017	0.015	−0.044	0.016	0.015	0.010
	SD	0.083	0.147	0.146	0.138	0.059	0.097	0.093	0.085
	ASE	0.084	0.143	0.142	0.133	0.059	0.099	0.098	0.088
	CP	0.846	0.930	0.932	0.942	0.816	0.956	0.954	0.950
β1 = 0.5
nc/n = 0.30	Bias	−0.123	0.179	0.152	0.101	−0.138	0.099	0.090	0.068
	SD	0.241	0.767	0.683	0.490	0.149	0.399	0.386	0.319
	ASE	0.229	0.800	0.718	0.496	0.152	0.386	0.373	0.283
	CP	0.740	0.896	0.900	0.926	0.730	0.924	0.926	0.928
nc/n = 0.50	Bias	−0.135	0.124	0.102	0.075	−0.147	0.061	0.051	0.036
	SD	0.172	0.430	0.418	0.343	0.105	0.283	0.267	0.221
	ASE	0.168	0.439	0.426	0.340	0.114	0.279	0.252	0.211
	CP	0.728	0.920	0.920	0.924	0.664	0.930	0.926	0.928
nc/n = 0.70	Bias	−0.143	0.098	0.086	0.054	−0.153	0.051	0.043	0.033
	SD	0.141	0.374	0.343	0.313	0.094	0.220	0.205	0.194
	ASE	0.139	0.344	0.344	0.291	0.095	0.213	0.207	0.188
	CP	0.686	0.920	0.920	0.926	0.564	0.962	0.958	0.940

Note: The naive estimator replaced the unobserved X by W using calibration sample, the RC estimator replaced X by E(W|Q). The NPC estimator solved (5), and the JNPC estimator solved (10). Covariates were from a uniform distribution with µ_x = 2, σ_x = 1. The measurement error was normal with mean 0 and σ_u = .5. In addition, $Q_i={\gamma }_0+{\gamma }_1{X}_i+{\gamma }_2{X}_i^2+V_i$, where γ₀ = 0.2, γ₁ = 0.55, γ₂ = 0.1, and σ_v = 0.707. The censoring percentage was 50%. The correlation coefficient between W and Q was about 0.715. The “bias” was calculated by taking the average of $\widehat{\mathit{\beta }}-\mathit{\beta }$ from 500 replicates, “SD” was the sample standard deviation of the estimators, “ASE” was the average of the estimated standard errors of the estimators, and “CP” was the 95% Wald-type confidence interval coverage probability.

In Table 1, we considered two ERR values such that β₁ = 0.2 and β₁ = 0.5, respectively. These two β₁ parameters were in a possible range for the RERF data example. We considered various ratios of the calibration sample size to the whole cohort. It was seen that the naive estimator has large biases. The RC estimator performed reasonably well in terms of bias correction. The RC estimator and the NPC estimator using the calibration sample had similar performance, but the latter was slightly more efficient overall. JNPC estimator was better than RC and NPC in terms of bias and efficiency. In general, the efficiency gain from the JNPC estimator over the NPC estimator was higher when the calibration sample was smaller. While not provided in the table, the OLC estimator has a good efficiency gain over the JNPC estimator. For a small to moderate exposure effect with β₁ = 0.2, the efficiency gain of OLC over JNPC was about 20% when n = 1000, and was about 10% when n = 2000 (Web Section 4). In Table 2, we investigated the situation similar to Table 1 but the event rate was 0.3, 0.5 and 0.7, respectively. The calibration sample size was 50% of the whole cohort size. The findings of Table 2 were similar to those of Table 1. The coverage probabilities under some settings were not close to the nominal value of 0.95 when the total sample size n = 1, 000, but the situation was better when n = 2, 000.

Table 2.

Simulation study with various event rates

		Naive	RC	NPC	JNPC	Naive	RC	NPC	JNPC
		n = 1000				n = 2000
β1 = 0.2
event rate = 0.30	Bias	−0.070	0.068	0.066	0.055	−0.075	0.033	0.032	0.026
	SD	0.114	0.286	0.279	0.257	0.073	0.155	0.154	0.134
	ASE	0.110	0.278	0.274	0.231	0.075	0.163	0.162	0.140
	CP	0.790	0.910	0.912	0.914	0.728	0.934	0.934	0.940
event rate = 0.50	Bias	−0.074	0.033	0.031	0.029	−0.077	0.025	0.023	0.021
	SD	0.077	0.193	0.189	0.175	0.059	0.127	0.125	0.108
	ASE	0.083	0.184	0.181	0.158	0.057	0.122	0.121	0.106
	CP	0.758	0.942	0.940	0.932	0.628	0.952	0.948	0.946
event rate = 0.70	Bias	−0.075	0.016	0.015	0.010	−0.077	0.012	0.011	0.008
	SD	0.071	0.159	0.155	0.129	0.049	0.097	0.096	0.085
	ASE	0.069	0.144	0.143	0.124	0.048	0.097	0.097	0.082
	CP	0.710	0.902	0.906	0.930	0.566	0.940	0.940	0.942
β1 = 0.5
event rate = 0.30	Bias	−0.233	0.183	0.171	0.143	−0.228	0.167	0.156	0.101
	SD	0.148	0.791	0.731	0.680	0.114	0.562	0.533	0.395
	ASE	0.158	0.810	0.745	0.644	0.112	0.483	0.463	0.348
	CP	0.552	0.896	0.908	0.902	0.404	0.932	0.926	0.920
event rate = 0.50	Bias	−0.226	0.165	0.145	0.116	−0.240	0.065	0.055	0.048
	SD	0.128	0.496	0.462	0.416	0.084	0.287	0.278	0.251
	ASE	0.123	0.503	0.473	0.397	0.083	0.267	0.260	0.226
	CP	0.454	0.930	0.928	0.938	0.218	0.926	0.922	0.930
event rate = 0.70	Bias	−0.242	0.096	0.082	0.068	−0.242	0.0539	0.025	0.023
	SD	0.098	0.425	0.410	0.319	0.068	0.205	0.195	0.181
	ASE	0.099	0.364	0.351	0.293	0.069	0.209	0.201	0.178
	CP	0.322	0.932	0.926	0.940	0.122	0.946	0.940	0.946

Note: The naive estimator replaced the unobserved X by W using calibration sample, the RC estimator replaced X by E(W|Q). The NPC estimator solved (5), and the JNPC estimator solved (10). Covariates were from a uniform distribution with mean 2 and standard deviation 1. The measurement error was normal with mean 0 and σ_u = 0.707. In addition, $Q_i={\gamma }_0+{\gamma }_1{X}_i+{\gamma }_2{X}_i^2+V_i$, where γ₀ = 0.2, γ₁ = 0.55, γ₂ = 0.1, and σ_v = 0.707. The calibration sample size was 50% of the whole cohort size. The correlation coefficient between W and Q was about 0.65. The “bias” was calculated by taking the average of $\widehat{\mathit{\beta }}-\mathit{\beta }$ from 500 replicates, “SD” was the sample standard deviation of the estimators, “ASE” was the average of the estimated standard errors of the estimators, and “CP” was the 95% Wald-type confidence interval coverage probability.

Additional simulation results are provided in the supplementary materials. In Web Table 5, we examined the situation when X was from a mixture of normals, and measurement error U was heteroscedastic errors. The findings in Web Table 5 were rather similar to those in Tables 1 and 2. The OLC estimator was still the best estimator. The performance of RC, NPC, JNPC and OLC was not affected by the heteroscedastic errors. The reason is primarily because the hazard function is linear in X, which is similar to the phenomenon in linear or Poisson regression with non-censored data, that some corrected score estimators have been shown to be consistent under heteroscedastic errors. (Wang, 2012). We also conducted simulations (Web Section 5) with bivariate covariates (X, Z), and similarly the OLC estimator was better than RC, NPC and JNPC in terms of bias reduction and efficiency.

7 RERF Data Analysis

The RERF example was briefly described in the introduction. The analysis in this section will investigate an association between radiation exposure and solid cancer risk. The true radiation exposure in the risk assessment was defined as an unobserved true radiation dose, rather than an estimated exposure. In the application, DS02 bone marrow gamma dose was considered as an unbiased surrogate which follows an additive measurement error model. The %SCA variable is considered as an IV for the true radiation dose, which we will discuss further below. There were 3011 subjects who had both DS02 and %SCA data, and among them 420 subjects had solid cancer mortality by 2003. In our data demonstration, we did not use the rest of the LSS cohort as the non-calibration sample in the analysis due to the very large sample size and very skewed distribution in DS02. We randomly took 10% of individuals with DS02 in the range of 0.04 Gy and 0.8 Gy, and 50% with DS02 larger than 0.8 Gy. The stratified selection was to increase the relative proportion with high doses. With this sampling, we included 6102 subjects for the non-calibration sample. Hence, in our data analysis, 9113 individuals were included.

The association between DS02 bone marrow gamma dose and %SCA is shown in Figure 1. The data are mostly the same as those in Kodama et al. (2001). In the scatter plot, we also applied a quadratic-linear function and a smooth lowess curve to model the relation between the proportion of aberrant cells and the gamma marrow dose for both Hiroshima and Nagasaki, respectively. It was seen that the proportion of aberrant cells has a linear-quadratic relationship with estimated gamma marrow dose. When we calculated the semiparametric RC estimate, we assumed a linear-quadratic regression for estimated radiation dose given %SCA (for each city). We checked the conditional independence assumption of the IV. The partial correlation of %SCA and survival times adjusted for DS02 radiation was 0.076 (SE = 0.05, p-value = 0.13) among the non-censored survival times. This provides a supportive argument that %SCA is likely conditionally independent, or with mild dependence. In addition, we examined the residual plot of the residuals from regression of %SCA given DS02 radiation versus survival times. We first fit %SCA as a linear-quadratic regression model of DS02, and obtained the residual plot of the residuals versus the survival times among individuals with solid cancer mortality (Web Figure 1). The residuals in general showed no association with the survival times. Neriishi et al. (1991) and Sposto, Stram and Awa (1991) applied a parametric RC estimator by treating epilation as a covariate in their regression model (noncensored outcome regression with leukemia and %SCA outcomes, respectively). Their parametric RC estimator replaced the unobserved true radiation dose (X) by E(X|W, Z), where Z was epilation in the models. However, in their models, the early effect variable epilation served as the role of Z, not Q, since they wanted to evaluate the effect of epilation on the outcome variables. In order to be able to identify the parameters involved in their primary model and the measurement error model, they basically performed a type of sensitivity analysis by assuming the variance of the measurement errors. Of course, one strength of their methods was to evaluate the effect of an early effect variable (epilation) on the primary outcome, but their weakness was that they needed to assume the measurement error variance. In comparison, one strength of our methods was that we did not need to assume a known measurement error variance, but one limitation of our methods was that %SCA was assumed to be uncorrelated with solid cancer mortality given the true radiation dose and the covariates. Nevertheless, given the technical nature of the present paper, the data analysis in this section is primarily for demonstration of our new methods. We do not intend to interpret our findings as RERF results in radiation research since the analysis here has not adequately modeled the background rates. That is, the background rate of solid cancer mortality is assumed to be constant over age at time of bombing, age at risk, sex, among other potential factors.

We first applied the ERR model with DS02 as a single covariate. We also analyzed the data with DS02 and city in the model, but the city × DS02 interaction had no effect. Hence, in our adjusted analysis, we did not include the effect modification parameter; β₃ in (2). The analysis results are given in Table 3. The naive estimator under both models showed that bone marrow gamma dose had a significant association with solid cancer mortality. The unadjusted analysis from RC, NPC, JNPC and OLC all showed that the bone marrow gamma dose had an effect on solid cancer mortality, and the effects were stronger than the naive estimate. From the adjusted analysis, the RC, NPC, JNPC and OLC also showed a significant radiation effect. In the adjusted analysis, the effect of city was significant based on the JNPC estimator, but not the other estimators. This was due to the use of more individuals in JNPC, while the others used only the calibration sample. The efficiency gain of OLC or JNPC over NPC and RC was mainly due to the use of the non-calibration sample. The attenuation coefficient was assumed to be about 0.9 in the past (such as Pierce et al, 2008), but it could be about 0.55 based on the JNPC estimator in this demonstration of the methods investigated. The measurement error variance in Pierce et al. (2008) was assumed to be some known values in a sensitivity analysis; similar to Neriishi et al. (1991) and Sposto, et al. (1991). In comparison, our methods make use of %SCA as an IV without this additional assumption on the measurement error variance. Nevertheless, the result here is not based on a comprehensive analysis, and hence would not provide any epidemiologic interpretation to radiation research. Due to the complexity of the JNPC (or OLC) estimator, it would involve very significant efforts to apply the methods to a large data set and to adjust for important confounding factors in a comprehensive analysis.

Table 3.

RERF data analysis

	Naive	RC	NPC	JNPC	OLC
Gamma dose as covariate
${\widehat{\beta }}_1$	0.260	0.372	0.381	0.507	0.462
SE	0.103	0.166	0.169	0.159	0.156
	Gamma dose effect adjusted for city
${\widehat{\beta }}_1$	0.257	0.355	0.362	0.481	0.416
SE	0.103	0.165	0.166	0.157	0.152
${\widehat{\beta }}_2$	−0.056	−0.045	−0.050	−0.160	−0.157
SE	0.103	0.104	0.104	0.061	0.060

Note: The naive estimator replaced the unobserved X by W using calibration sample, the RC estimator replaced X by E(W|Q, Z). The NPC estimator solved (5), the JNPC estimator solved (10), and OLC was the optimal linear combination estimator of JNPC and NPC. When DS02 and city were in the model, the city × DS02 interaction had no effect. Hence, in the adjusted analysis, the hazard function was λ(t|X, Z) = exp(β₂Z)(1 + β₁X), in which Z was for city (binary).

8 Discussion

We have proposed the JNPC and OLC estimators for ERR regression in time to event analysis when covariates may be measured with additive errors. When there is no measurement error, we apply an unbiased estimating equation for the regression coefficient estimation, rather than from its partial likelihood due to the consideration of finite sample issues. The JNPC and OLC estimators are functional estimators in that they do not need to assume the covariate distribution. They also do not need to assume the distribution of the measurement errors. The methods will have important applications since in many studies the exposure variable and measurement error may not be from a normal or even any known parametric distribution. In addition to RERF, the methods can be modified and applied to nutritional biomarker data such as the Women’s Health Initiatives (Prentice et al., 2011).

We observed an important finding between the RC and NPC estimators; both using the calibration sample. Normally, in measurement error research the trade-off between bias and efficiency is often referred to the comparison between the naive estimator and a consistent estimator, but it is also applicable to a comparison between RC and NPC in non-linear regression. In logistic, Poisson (Wang, 2012), or Cox regression (Huang and Wang, 2000), RC in general has a bias concern but it is more efficient than a functional method (such as NPC). However, in our simulation study, the NPC estimator in general not only had smaller biases than the RC estimator but it also had smaller standard errors. This finding was somewhat different from the general phenomenon of bias-efficiency trade-off that was seen in measurement error literature. To further understand the RC estimator, we have added our study on risk set RC estimator in Web Section 2 of the supplementary materials. We examined an expression for E(X|Q, Z, T ≥ t) as a function of E(X|Q, Z) under a special case. We also examined a risk set RC estimator in our model. The risk set RC and NPC estimates were very close numerically in general (Web Table 2). The RC estimator was not as good as those, but the differences were in general small.

To simplify presentation, we have assumed that the calibration sample is a random sample of the study cohort. This can be extended to the situation when the calibration sample may not be a random sample. For example, the calibration sample may have a higher sampling weight for older people. Or, the calibration sample might also over-weight those with higher exposure. We suggest the use of inverse selection probability weighting to address this issue. Another future research is that measurement error of radiation exposure may be a mixture of Berkson and classical errors (as mentioned in the Introduction). If the errors in the doses are partly Berkson then this would lead to less attenuation of the dose response, than if they are entirely classical. Therefore, further research is warranted.

Our work has a few strengths. First, we propose novel statistical methodology for measurement error in covariates while the error-prone surrogate does not have replicates. The methods do not relay on parametric assumptions on either the covariates or measurement errors which are less developed in ERR regression with survival outcomes. Our methodology makes use of an IV variable to adjust for bias due to measurement error, but without assuming the measurement error variance under a sensitivity analysis (such as Pierce et al, 2008). However, our work also has a couple of limitations. First, the IV assumption may not hold in real applications. But, the value of the conditional independence of an IV is to add the strength to estimate the measurement error variance (which was assumed in Neriishi et al., 1991 and Sposto, Stram and Awa, 1991). Second, there are further computing efforts needed to implement the NPC, JNPC and OLC estimators so that they can be applied to general large data sets, also with more confounding variables to be included.

Appendix: Regularity Conditions

• (A1){N_i, Y_i, X_i, Z_i, W_i, η_i}, i = 1, …, n, are independent and identically distributed.
• (A2)The vector of regression parameters, β, belongs to a compact set $ℬ$.
• (A3)The baseline cumulative hazard function Λ₀(t) is continuous, Λ₀(τ) < ∞ and dΛ₀(0) > 0.
• (A4)For $\mathit{\beta }\in ℬ$, $1+{\beta }_{1}X\exp ({{\mathit{\beta }}^{\prime }}_2\mathit{Z})>0$ almost surely.
• (A5)The surrogate variable W satisfies that W = X + U, in which E(U|X, Z) = 0, E(U′U) < ∞, and U_i is independent of the outcome (T_i, δ_i).
• (A6)The instrumental variable Q satisfies (3) such that E(V|X, Z) = 0, E(V′V) < ∞, and V_i is independent of the outcome (T_i, δ_i).
• (A7)U, V, T⁰ and C are conditionally independent given (X, Z).
• (A8)E(X′X) < ∞, E(Z′Z) < ∞, E{h′(X, Z)h(X, Z)} < ∞, $E[{\sup }_{\mathit{\beta }}{X}^{\prime }X\exp \{2({{\beta }^{\prime }}_2+{{\mathbf{\beta }}^{\prime }}_2)\mathbf{Z}\}]<\infty$, $E[{\sup }_{\mathit{\beta }}{\mathit{Z}}^{\prime }Z\exp \{2({{\beta }^{\prime }}_2+{{\mathit{\beta }}^{\prime }}_3)\mathit{Z}\}]<\infty$, $\mathrm{E}[{\sup }_{\mathit{\beta }}h{(X,\mathit{Z})}^{\prime }h(X,\mathit{Z})\exp \{2({{\beta }^{\prime }}_2+{{\mathit{\beta }}^{\prime }}_3)\mathit{Z}\}]<\infty$.
• (A9)G(β) given in Section 4 and $\mathcal{G}(\mathit{\theta })$ given in Section 5 are both non-singular.
• (A10)The proportion of the calibration sample size n_c/n → ρ ∈ (0, 1).