A Class of Functional Methods for Error-Contaminated Survival Data Under Additive Hazards Models with Replicate Measurements

ABSTRACT Covariate measurement error has attracted extensive interest in survival analysis. Since Prentice, a large number of inference methods have been developed to handle error-prone data that are modulated with proportional hazards models. In contrast to proportional hazards models, additive hazards models offer a flexible tool to delineate survival processes. However, there is little research on measurement error effects under additive hazards models. In this article, we systematically investigate this important problem. New insights into measurement error effects are revealed, as opposed to well-documented results for proportional hazards models. In particular, we explore asymptotic bias of ignoring measurement error in the analysis. To correct for the induced bias, we develop a class of functional correction methods for measurement error effects. The validity of the proposed methods is carefully examined, and we investigate issues of model checking and model misspecification. Theoretical results are established, and are complemented with numerical assessments. Supplementary materials for this article are available online.


Introduction
Covariate measurement error has long been a concern in survival analysis, and has attracted extensive research interest. Since Prentice (1982), a large number of inference methods have been developed to handle error-prone data (e.g., Nakamura 1992;Buzas 1998;Hu, Tsiatis, and Davidian 1998;Huang and Wang 2000;Li and Lin 2003;Hu and Lin 2004;Song and Huang 2005;Yi and Lawless 2007;Zucker and Spiegelman 2008). Although discussion on survival data with measurement error is not restricted to a single type of model, proportional hazards models have been the center of existing research. The impact of covariate error is well understood for such models.
Proportional hazards models (Cox 1972) specify that covariates have multiplicative effects on the hazard ratio; a most appeal of such models is that baseline hazard functions can be left unspecified when conducting inference about covariate effects based on partial likelihood functions. In contrast to proportional hazards models, additive hazards models offer a flexible tool to delineate survival data (Breslow and Day 1980;Cox and Oakes 1984). Lin and Ying (1994) developed an inference method for covariate effects based on pseudo-score functions, and a key catch of this method is that baseline hazard functions are left unmodeled. Furthermore, this method allows a closed form of the estimator of regression parameters.
Relative to a large body of literature on proportional hazards models with covariate measurement error, there is little research on measurement error effects under additive hazards models, although several authors investigated this problem. Sun, Zhang, and Sun (2006) considered additive hazards models for the case with replicates of mismeasured covariates, and justified asymptotic results using empirical process theory.  Kulich and Lin (2000) proposed an unbiased corrected pseudoscore approach for the case where a validation sample is available. However, a number of important questions remain unexplored. For instance, as indicated by the work for proportional hazards models, many correction methods can be developed to account for error effects. Are there any intrinsic connections among those methods? How do we assess the validity of the proposed methods that essentially rely on correct model specification? Does measurement error in covariates have the same effects on additive hazards models as those on proportional hazards models? Can we reveal new insights by exemplifying the unique features of additive hazards models?
In this article, we examine these important questions. In particular, we explore asymptotic bias induced from the naive analysis with measurement error ignored. To correct for the induced bias, we develop a class of correction methods to exemplify the unique features of additive hazards models. The validity of the proposed methods is carefully examined, and we investigate issues of model checking and model misspecification. Theoretical results are rigorously established, and are complemented with various numerical assessments. In addition, different from the most work that assumes classical additive error models with error distributions specified, in this article, we relax the requirement of specifying a full distributional assumption for error terms. With availability of replicated measurements, we consider a flexible model for measurement error processes, which assumes only an additive structure. Moreover, we employ the socalled "functional modeling" approach for which the distribution of the true covariates is left unmodeled (Carroll et al. 2006). Therefore, our proposed methods are robust to misspecification of the distribution of the true covariates.
The remainder is organized as follows. In Section 2, we introduce the basic model setup and estimation in the absence of measurement error. In Section 3, we conduct a bias study for the naive estimator that ignores covariate measurement error. In Section 4, we propose an approach based on pseudo-score functions to deal with survival data with replicates of mismeasured covariates. Asymptotic results are established. In Section 5, we propose an estimating equation-based method. In Section 6, numerical studies for the estimators are provided. In Section 7, we study the impact of model misspecification and propose a goodness-of-fit test statistic. In Section 8, a real data example is provided. Concluding discussion is provided in the last section.

Additive Hazards Model
For i = 1, . . . , n, let T i be the failure time, C i be the censoring time, and Z i (t ) = (X T i , V T i (t )) T be a vector of covariates, where X i is a p × 1 vector of time-independent and error-prone covariates, and V i (t ) is a q × 1 vector of covariates that are precisely measured and possibly time-dependent. As common in practice, the V i (t ) are assumed to be external covariates (Kalbfleisch and Prentice 2002, p. 197). We consider that the hazard function of T i is related to Z i (·) through the additive hazards model where λ 0 (·) is an unspecified baseline hazard function, and β = (β T x , β T v ) T is a vector of unknown regression parameters. Let 0 (t ) = t 0 λ 0 (u)du be the baseline cumulative hazard function. T i and C i are assumed to be conditionally independent given the covariates.
Suppose individuals are observed over a common time inter-

Estimation in The Absence of Measurement Error
If X i were precisely measured, then estimation of β can be carried out using the pseudo-score functions proposed by Lin and Ying (1994): Solving U (β ) = 0 gives an estimator of β, which has an explicit form given bŷ where a ⊗ 2 = aa T for a column vector a.
This estimator is consistent, provided certain regularity conditions hold. Indeed, U (β ) can be equivalently written as . . , n} be the σ -field generated by the event, covariates, and observation histories prior to time t for all subjects. Then M i (t; β, 0 ) is an F t -adapted martingale (e.g., Kalbfleisch and Prentice 2002, sec. 5.3). Consequently, E{U (β )} = 0, that is, U (β ) are unbiased estimating functions of β. By estimating function theory, under regularity conditions, solving U (β ) = 0 leads to a consistent estimator of β (e.g., Yanagimoto and Yamamoto 1991).
Noting that E{M i (t; β, 0 )} = 0 by the martingale property of M i (t; β, 0 ), we estimate the baseline cumulative hazard function by solving

Measurement Error Model
Suppose X i is repeatedly measured n i times, resulting in the surrogates W ir , r = 1, . . . , n i . We assume an additive measurement error model: where the ir are independent and identically distributed with mean 0 and a positive-definite variance matrix 0 , i = 1, . . . , n; r = 1, . . . , n i . We assume that the ir are independent of X i , V i (t ), T i , and C i . This assumption implies that given the true covariates Z i (t ), the surrogate measurements W i j are independent of T i and C i . This assumption is analogous to the usual nondifferential error mechanism for uncensored data (Carroll et al. 2006, p. 36).
, and 0 q is the q × q matrix of elements 0. With the replicates W ir , we estimate the covariance matrix 0 bŷ

Asymptotic Bias Analysis
We investigate measurement error effects on the structure of the hazard function. We derive the hazard function based on the observed covariates (W T i· , V T i (t )) T , and let λ * (t;W i· , V i (t )) denote this hazard function. With the assumption made on the measurement error process, The expression (5) indicates that the hazard function for the observed covariates may retain the additive structure while the risk difference has a more complicated form than (1). Since the conditional expectation E{X i |T i ≥ t,W i· , V i (t )} generally differs fromW i· , (5) suggests that the naive analysis withW i· replacing X i would lead to biased results. We now quantify the asymptotic bias resulted from the naive analysis.
That is, U nv (β ) is the naive pseudo-score function that is obtained by replacing (5), and then applying the pseudo-score function form (2) to the observed data. Letβ nv be the solution of U nv (β ) = 0.
Following the discussion of Yi and Reid (2010), we can show that under certain regularity conditions,β nv converges in probability to a limit, say β * nv , as n → ∞. We can further characterize the relationship between β and β * nv , given by The details are presented in Appendix A2 of the supplementary material. It is immediate that from (7), if β = 0, then β * nv = 0, where · is the Euclidean norm. If Z i (t ) contains only a univariate X i , then |β * nv | < |β|, suggesting an attenuated measurement error effect. If X i and V i (t ) are univariates and are independent, and either V i (t ) or X i are independent of the at risk process Y i (t ), then |β * nv,x | < |β x | and β * nv,v = β v , where β x and β v (or β * nv,x and β * nv,v ) are components of β (or β * nv ) corresponding to the covariates X i and V i (t ), respectively. The justifications are provided in Appendix A2 of the supplementary material.

Corrected Pseudo-Score Approach
As shown in Section 3, the naive analysis with measurement error ignored yields biased estimation of β. We now develop an inference method for β with measurement error effects taken into account. The idea is to find sensible estimating functions of β, which satisfy two key conditions: (1) estimating functions must be computable in the sense of being expressed in terms of the observed data and parameters, and (2) estimating functions are unbiased. By estimating function theory, solving the resulting estimating equations leads to a consistent estimator of β if suitable regularity conditions hold.
Using the pseudo-score functions (2) with X i replaced bȳ W i· gives us computable estimating functions, U nv (β ), of β. But as implied by the discussion in Section 3, these estimating functions U nv (β ) are not unbiased. As suggested by Yi and Reid (2010), a quick remedy to fixing this is to modify U nv (β ) by subtracting their expectation E{U nv (β )} so that the resulting estimating functions, U nv (β ) − E{U nv (β )}, are unbiased. However, evaluation of E{U nv (β )} is generally complicated due to the involvement of the joint distribution of the survival, censoring, and covariate processes, thus making the modified estimating functions U nv (β ) − E{U nv (β )} unappealing. To get around this problem, we alternatively evaluate the conditional expectation of U nv (β ), given F τ . As shown in Appendix A3 of the supplementary material, This identity motivates us to consider corrected pseudo-score functions: By that E{U (β )} = 0, we obtain that E{Ũ c (β )} = 0, implying thatŨ c (β ) are unbiased estimating functions.
To use the corrected pseudo-score functionsŨ c (β ) to estimate β, we need to replace 1 with its consistent estimateˆ 1 , and let U c (β ) denote the resultant estimating functions. One might expect that the substitution ofˆ 1 for 1 would break down the unbiasedness ofŨ c (β ), but this is not the case here.
Therefore, U c (β ) are unbiased estimating functions due to that E{Ũ c (β )} = 0. Letβ c be the solution to the equations U c (β ) = 0. It is seen thatβ We comment that numerically,β c performs stably. This can be explained by that the inverse matrix (scaled by n −1 ) inβ c converges almost surely to a positive definite matrix under mild regularity conditions, thus singularity does not occur in the asymptotic sense. Details on this point are included in Lemma A.1 of Appendix A4 in the supplementary material.
Next, we discuss estimation of the baseline cumulative hazard function (Lin and Ying 1994) leads to an estimator of 0 (t ), sayˆ 0 (t;β c ), given bŷ To ensure monotonicity, we propose to use˜ 0 (t;β c ) = max 0≤s≤tˆ 0 (s;β c ) to estimate 0 (t ) as in Lin and Ying (1994). Asymptotic properties ofβ c andˆ 0 (t;β c ) are summarized in the following theorems, whose proofs are included in Appendices A4 and A5 of the supplementary material. Let Theorem 1. Under regularity conditions R1-R8 listed in Appendix A1 of the supplementary material, we have Theorem 2. Under regularity conditions R1-R8 listed in Appendix A1 of the supplementary material, we have where means weak convergence, l ∞ [0, τ ] is the space of all bounded functions on [0, τ ] (van der Vaart and Wellner 1996), for time points s and t, and We comment that, as seen from the proofs of Theorems 1 and 2, the term 2 β − ρ 0 E(S i ) 1 β in c and i (t ) can be interpreted as the substitution effect of replacing 1 with its consistent estimateˆ 1 . If there are no replicate measurements, that is, n i = 1, i = 1, . . . , n, and 1 is known, then the asymptotic results ofβ c andˆ 0 (t;β c ) are given by Theorems 1 and 2 with the term 2 β − ρ 0 E(S i ) 1 β removed from c and i (t ). More details are included in Appendix A6 of the supplementary material.
Theorem 2 implies that Pr{sup 0≤t≤τ n 1/2 |ˆ 0 (t;β c ) − 0 (t )| ≤ x} → Pr{sup 0≤t≤τ |G(t )| ≤ x} as n → ∞ for any x ≥ 0. It is difficult to use this result to conduct inference about 0 (t ) because the Gaussian process G(t ) does not have the independent increment property and has a complex form.
To get around this difficulty, we suggest using resampling techniques to construct confidence bands for survival curves.
The proof of Theorem 3 is deferred to Appendix A7 of the supplementary material. This theorem suggests that we can legitimately use the distribution of sup 0≤t≤τ |Ŵ n (t )| to approximate that of sup 0≤t≤τ |G(t )|, and thus that of . . , n} independently from the standard normal distribution for a large number of times, say 1000, and calculate sup 0≤t≤τ |Ŵ n (t )| each time. Thus, we obtain an empirical quantileq α , and to obtain an approximate (1 − α) confidence band of 0 (t ) as

Estimating Equation Approach
Instead of focusing on the pseudo-score functions U (β ) alone as in the previous section, we now jointly look at unbiased estimating equations for β and 0 (·). Our starting point is the fact These results suggest that dM i (t; β, 0 ) and Z i (t )dM i (t; β, 0 ) can be used to construct unbiased estimating functions for 0 (·) and β if X i were error-free. As . By the error model (4), it is easily seen that this replacement does not change the property (11), but it breaks down (12). That is, Hence, we construct two sets of unbiased estimating equations: Now we investigate whether (13) and (14) are adequate for estimating β (a finite-dimensional parameter) and 0 (t ) (a function). Since the function 0 (t ) can be regarded as an infinite-dimensional parameter, the usual estimating equation theory does not guarantee that solving (13) and (14) simultaneously leads to appropriate estimators. For example, given an arbitrary estimator of 0 (t ), sayˆ 0 (t ), which satisfies both (13) and (14),ˆ 0 (t ) + C would also satisfy (13) and (14) for any constant C, yielding an unidentifiability issue. To resolve this problem, we adopt an ad hoc procedure, which shares the same spirit as that of Lin and Ying (1994), and thus identifiability can be achieved.
Note that (13) (14), we obtain as an estimator of β, where we set t = τ to fully use the whole dataset. Pluggingβ e back intoˆ 0 (t; β), we obtain an estimator for the baseline cumulative hazard function It is interesting to note thatβ e differs fromβ c by a factor {1 − 1/ n j=1 Y j (t )}, andˆ 0 (t;β e ) andˆ 0 (t;β c ) assume the same form but with a different estimatorβ e orβ c . In the following corollary, we show that asymptoticallyβ e behaves the same aŝ β c , andˆ 0 (t;β e ) behaves the same asˆ 0 (t;β c ). The proof is sketched in Appendix A8 of the supplementary material.
Corollary 1. Under regularity conditions R1-R8 listed in Appendix A1 of the supplementary material, we have where G(t ) is the Gaussian process defined in Theorem 2.

Empirical Studies
We conduct various simulation studies to evaluate the finite sample performance of the proposed estimators. In particular, we contrast our proposed estimatorsβ c andβ e to the naive estimatorβ nv , the regression calibration estimatorβ rc (Prentice 1982), and the estimatorβ szs by Sun, Zhang, and Sun (2006).

Design of Simulation
We consider n = 200 and generate 1000 simulations for each parameter configuration. We examine three scenarios for the bivariate time-independent covariates Z i = (X i , V i ) T . In Scenario 1, the covariates X i and V i are independently generated, where X i ∼ UNIF(−1, 1), and V i is a binary variable taking value 1 or 0 each with probability 0.5. Scenarios 2 and 3 correspond to that covariates X i and V i are correlated. In Scenario 2, (1) and Survival times are independently generated using the additive hazards model (1), where we take the baseline hazard function to be λ 0 (t ) = αγ t γ −1 , and we consider α = γ = 1 for Scenarios 1 and 2 and α = 0.5, γ = 2 for Scenario 3, respectively. The true values of β x and β v are set to be (β x , β v ) = (1, 0) for Scenario 1, and (0.5, 0.5) for Scenarios 2 and 3, respectively. Censoring times C i are generated from uniform distribution UNIF(0, C), where C is set as 4.6 for Scenario 1, 4.7 for Scenario 2, and 2.7 for Scenario 3, respectively. Roughly, 30% censoring percentages are produced for each scenario. The error model (4) is used to generate W ir where ir ∼ N(0, σ 2 ) for r = 1, . . . , n i , i = 1, . . . , n. We consider settings with σ = 0.25 or 0.75, and n i = 2. In Appendix A9 of the supplementary material, we illustrate how we generate survival times under the additive hazards model. In Appendix A10 of the supplementary material, we provide two additional simulation scenarios: (i) we study the impact of large coefficient β x on the performance of the regression calibration estimator and (ii) we generate covariates from normal distributions.

Performance of Estimators
In Table 1, we report the finite sample biases (Bias), the empirical variances (EVE), the average of the model-based variance estimates (MVE), the mean square errors (MSE), the coverage rate where var(β A ) is the estimated variances, and the subscript A refers to nv, szs, rc, c, and e accordingly. It is seen thatβ nv is always biased toward 0, with increasing magnitudes as measurement error becomes more substantial. These findings confirm the theoretical result revealed by the bias analysis in Section 3. The regression calibration estimator β rc only partially removes the bias induced from measurement error, and its variance estimate deviates from the empirical variance in some settings. The two proposed estimatesβ c andβ e have small finite sample biases. Their variance estimates agree reasonably well with the empirical variances, and the coverage rates agree well with the nominal level 95%. In contrast, when the measurement error is large, the variance ofβ szs is considerably larger than those ofβ c andβ e , and the model-based variance estimates ofβ szs tend to deviate from the empirical variance estimates with much larger magnitudes. Finally, we comment that the estimatorβ szs tends to behave less stably than the proposed estimatorsβ c andβ e , and the regression calibration estimator β rc . In our simulations, about 1% of divergence occurs for the estimatorβ szs when measurement error is large, whereas only 0.5% of divergence occurs forβ c ,β e , andβ rc .

Impact of The Number of Replicates
We now further evaluate the performance of the estimators for situations where some subjects may not have replicates W i j . Specifically, settings of different replicate numbers n i are considered for Scenario 3 described above. In Setting I, 150 out of n = 200 subjects are randomly selected to have two measurements, and the rest have a single measurement; whereas in Setting II, 100 out of n = 200 subjects are randomly selected to have two measurements, and the rest have a single measurement. We further consider two settings for which we use a probability mechanism to decide whether or not a subject has a single measurement. That is, we treat n i as a random variable taking value 1 or 2. Specifically, in Setting III, we assume that Pr(n i = 1) = 0.8 if T i ≤ median of all T i , and Pr(n i = 1) = 0.2 otherwise; in Setting IV, Pr(n i = 1) = 0.2 if T i ≤ median of all T i , and Pr(n i = 1) = 0.8 otherwise. Simulation results are summarized in Table  2. The primary finding is that the estimatorβ szs is not appropriate when the number of measurements depends on the underlying event failure time. The results show that when there is a portion of subjects that have a single measurement,β c andβ e have smaller variances thanβ szs . This demonstrates thatβ c and β e can effectively use information from subjects that have only a single measurement.  Finally, we consider Setting V where all subjects have only one single measurement, and the error variance is known to be 0.25 2 or 0.75 2 . The estimatorβ szs does not work for this setting as it is developed only for the case where all subjects must have replicated measurements for X i . However, our estimatorsβ c and β e can handle this scenario, and the simulation results show that they have satisfactory performance.

Results on Cumulative Hazard Function
In Table 3, we use the procedure described in Section 4 to construct confidence bands of the baseline cumulative hazard function. Here we consider only Scenario 1. For each simulation run, we independently generate standard normal variables ξ i , i = 1, . . . , n, and repeat this procedure for 1000 times; we calculateŴ n (t ) each time and thus obtain the empirical upper 0.05-quantileq 0.05 . In the total number of 1000 simulation runs, we record the number of cases that sup t∈[0,τ ] √ n|ˆ 0 (t;β c ) − 0 (t )| is less thanq 0.05 , and produce the empirical coverage rate accordingly. We repeat the above procedure forˆ 0 (t;β c ), described in Section 4, and the naive cumulative hazard estimator based on Lin and Ying (1994), and further modify these two estimators by the procedure of Hall and Wellner (1980). Simulation results reveal that naively ignoring measurement error could result in low coverage rates, especially when measurement error is large. The corrected methods greatly outperform the naive method.

Model Misspecification
In the preceding sections, we explore various methods to correct for bias induced from measurement error. The validity of the proposed methods relies on the additive hazards model structure for survival data. An important concern therefore arises: what if the true hazard function λ(t; Z i (t )) is not of the additive hazards structure (1), but we incorrectly assume model form (1) to fit data. In this subsection, we investigate this problem. Suppose the true model is given by the Cox model but we incorrectly use the additive hazards model (1) to fit the data, where λ cox (t ) is the true baseline hazard function, and α represents the true covariate effects. Let β * c be the asymptotic limit ofβ c developed in Section 4. Then following Hattori (2006) and Yi and Reid (2010), we show that β * c is given by where E true represents the expectation taken under the true model (16) with cumulative hazard function (t; It is difficult to see how β * c differs from α based on (17). To gain an understanding of the relationship between β * c and α, we consider an approximation of (17) for the situation with small |α T Z i (t )|. Using the Taylor series expansion exp{α T Z i (t )} ≈ 1 + α T Z i (t ), we approximate the true hazard function (16) with an additive form: As a result, Expression (18) approximately quantifies the asymptotic bias of using the estimatorβ c under the misspecified model (1) to estimate the true covariate effects α. It is seen that the estimated covariate effects β * c approximately differ from the true covariate effects α by a product R of two nonnegative definite matrices. The factor R depends on both survival and censoring processes. Although the estimated covariate effects β * c and the true covariate effects α are different in general, they tend to have the same sign when the covariate is univariate. In a special situation where there is no covariate effect, the estimated effect β * c is close to zero.

Model Checking
In the previous subsection, it is seen that using the developed methods can yield biased estimates if the true covariate effects do not act additively on the hazard function. Thus, it is important to develop a model checking procedure for additive hazards models. Letˆ c be the empirical counterpart of c defined in Theorem 1, (ˆ −1 c ) j j be the jth diagonal element ofˆ −1 c , and 2 (t ) = diag(ρ 0 ρ −1 The following lemma describes the asymptotic behavior of n −1/2 U c (β c , t ). The proof is included in Appendix A11 of the supplementary material.
Lemma 1. Under regularity conditions R1-R8 listed in Appendix A1 of the supplementary material, we have where G 2 (t ) is a zero-mean Gaussian process with covariance function 2 (s, t ) = lim n→∞ n −1 n i=1 E[ 2,i (s) 2,i (t )] for time points s and t, and 2, Lemma 1 provides a basis for subsequent development of goodness-of-fit test. It says that if the additive hazards model and the additive error model are both correctly specified, n −1/2 U c (β c , t ) would fluctuate around zero randomly, provided that regularity conditions hold. This motivates us to propose an overall goodness-of-fit test statistic In the absence of measurement error, S c reduces to the overall goodness-of-fit test statistic by Kim, Song, and Lee (1998), which is a generalization of the test statistic for checking the Cox model assumption proposed by Lin, Wei, and Ying (1993).
The asymptotic distribution of S c is difficult to identify due to the complexity of the limit process G 2 (t ) associated with n −1/2 U c (β c , t ). However, an abnormally large value of S c can indicate that the additive hazards model and/or the error model are incorrectly specified. Now we describe an implementation procedure using the resampling techniques similar to those in Section 4. We generate {ξ i , i = 1, . . . , n} independently from the standard normal distribution, and calculatê , andÂ i (t ) andD c (t ) are the empirical versions of A i (t ) and D c (t ), respectively. Then S c can be used to assess goodness-of-fit because it mimics the behavior of S c asymptotically, as indicated below. The proof is sketched in Appendix A12 of the supplementary material.
Theorem 4. Assume regularity conditions R1-R8 listed in Appendix A1 of the supplementary material. Then conditional on the observed data is the Gaussian process defined in Lemma 1.
Theorem 4 also offers a justification to empirically evaluate the power of using S c for model checking. Specifically, we generate sets of iid standard normal variables {ξ i,k , i = 1, . . . , n} for N times, where N is a large number, say N = 1000. Then we calculate N copies ofŜ c , say {Ŝ c,k , k = 1, . . . , N}. Empirical quantiles of S c can then be obtained based on theŜ c,k . Now we numerically assess the performance of the proposed test statistic S c . First, we evaluate the empirical size of the test. We take the setting of Scenario 1 to generate the data, and consider two cases with no censoring or 30% censoring percentage.
The results for the empirical size of the corrected goodnessof-test statistic S c are summarized in Table 4, where the null hypothesis is that both of the additive hazards model and the additive error model are correctly specified. For comparison purposes, we also consider the naive goodness-of-test statistic S nv by naively applying the method of Kim, Song, and Lee (1998) with the difference between X i andW i· ignored, and the "true" goodness-of-test statistic, named S true , obtained by applying the method of Kim, Song, and Lee (1998) to the true covariate measurements.
It is observed that in the presence of censoring, the test size of S nv is close to the nominal level. However, when there is no censoring, the naive test statistic S nv yields test sizes that completely deviate from the nominal size 0.05. In contrast, the proposed statistic S c produces test sizes that are fairly close to the nominal level in all cases, and its performance is similar to the true goodness-of-test statistic S true .
Next, we evaluate the power of the proposed test statistic. We generate the survival times from the Cox model The covariates X i and V i are generated as in Scenario 1 in Section 6.1. The error model (4) is used to generate W ir where ir ∼ N(0, σ 2 ) for r = 1, . . . , n i , i = 1, . . . , n. We consider settings with σ = 0.25 or 0.75, and n i = 2. By taking C i to be ∞ or generating C i from UNIF(0, 4.6), we obtain two censoring scenarios: no censoring and 30% censoring, respectively. The results are summarized in Table 4. It is seen that the power of the proposed test statistic S c is fairly satisfactory, although the power would decrease when the degree of measurement error increases.

ACTG 175 Study
We apply the proposed methods to analyze the data arising from the AIDS Clinical Trials Group (ACTG) 175 study (Hammer et al. 1996). The following analysis of this dataset is merely an illustration of the proposed methods. The ACTG 175 study is a double-blind randomized clinical trial that evaluated the effects of the four types of HIV treatments: zidovudine only, zidovudine and didanosine, zidovudine and zalcitabine, and didanosine only. In this example, we are interested in evaluating how different treatments are associated with the survival time T i , which is defined to be the time to the occurrence of one of the events that CD4 counts decrease at least 50%, or disease progression to AIDS, or death, as in Hammer et al. (1996). Excluding the subjects who had missing values or unrecorded relevant information, we consider a subset of 2139 subjects out of 2467 subjects in the original dataset. About 75.6% of the outcome values are censored. Let V i be the treatment assignment indicator for subject i, where V i = 1 if subject i received the zidovudine only treatment, and 0 otherwise. In the ACTG 175 study, the baseline measurements on CD4 were collected before randomization, ranging from 200 to 500 per cubic millimeter. Let X i be the normalization version of the true baseline CD4 counts: log(CD4 counts + 1), which was not observed in the study. Two replicated baseline measurements of CD4 counts, denoted by W i1 and W i2 , after the same transformation as for X i , were observed for 2095 subjects, while the other 44 subjects were measured once for the CD4 counts at the baseline. An additive measurement error model is specified to link the underlying transformed CD4 counts with its surrogate measurements: where r = 1, 2 for i = 1, . . . , 2095, and r = 1 for i = 2096, . . . , 2139. Here, no specific distributional assumption is made for the errors ir except that the ir are assumed to be independent and identically distributed with mean zero and variance 0 . With the replicates, we estimate the error variance asˆ 0 = 0.035, and the variance of X i asˆ xx = 0.079. These estimates give the reliability ratioˆ xx /(ˆ xx +ˆ 0 ) = 69.3%, indicating a considerable degree of measurement error in this study.
We employ the additive hazards model to feature the dependence of T i on the covariates X i and V i : where λ 0 (t ) is the unspecified baseline hazard function, and β = (β x , β v ) T is the regression parameter.
We apply the methods considered in Section 6 to analyze the data: the data subsets with replicates and the entire dataset. The analysis results are shown in Table 5. The naive estimate of β x is smaller than those obtained from the other methods, while the naive estimate of β v is similar to those produced by the other methods. All the consistent methods and the regression calibration method produce similar results. Although estimates of β x and β v differ from method to method, all the results suggest that both CD4 counts and treatment are statistically significant.
We also apply the proposed test statistic S c to the ACTG 175 Study dataset. The p-value of the model test is 0.859, suggesting no evidence against the additive hazards model or the additive error model.

Extension and Discussion
In this article, we make a number of contributions on additive hazards models with measurement error. We propose several consistent and easily implemented estimators to correct for measurement error effects, and our methods are robust to possible misspecification for the distribution of the true covariates since they are functional modeling approaches (Carroll et al. 2006). Our development includes investigation of the impact of model misspecification of the survival process and construction of a test statistic for model checking. We rigorously establish asymptotic properties for the proposed estimators. Extensive numerical studies demonstrate satisfactory performance of our methods.
Our methods are explicitly developed for the additive error model (4). In fact, our methods can be modified to accommodate more general error models. For instance, consider a regression measurement error model where the error terms ir are iid with mean 0 and a positivedefinite variance matrix 0 , and are independent of N i (·), Y i (·), and Z i (·), i = 1, . . . , n; r = 1, . . . , n i . Here, γ x is a p × p matrix, and γ v is a p × q matrix. Model (19) accommodates a wide class of error models, including the classical additive model (4) if we set γ 0 = 0, and γ x = 1 with a zero vector γ v . In the following, we consider a special case that p = q = 1. We assume γ x = 0. LetX g,i = (W i· − γ v V i − γ 0 )/γ x , then replacing X i withX g,i in (2), we obtain a corrected pseudo-score function Consequently, solving U gc (β ) = 0 gives an estimator of β: Note that the derivation ofβ gc is similar to that ofβ c in Section 4. Similarly, we can construct other consistent estimator similar toβ e . Furthermore, we can construct estimators of 0 (t ) similar to previous sections, which, however, further requires that γ 0 is known or estimated by a validation subsample.
In this article, the covariates X i are assumed to be time-independent. The proposed methods can be potentially extended to the scenario where the covariates X i (t ) are timedependent. In this case, the development generally requires the surrogate W i (t ) to be observed for any time t, which is difficult if not impossible. One strategy to circumvent this problem is to impose additional model assumption. For instance, if we assume that the X i (t ) are piecewise constants over a sequence of time intervals, our proposed methods can be extended to accommodate this type of time-dependent covariates.
Finally, we note that our methods are developed for errorcontaminated survival data that are modulated by the additive hazards model (1). The additive hazards model (1) is a useful complement to the popularly used proportional hazards model. This model allows for a simple procedure for conducting inference on the model parameter β whose estimator can be explicitly expressed. However, to ensure a legitimate hazard function, the linear term β T Z i (t ) in model (1) must be constrained to be nonnegative (Aalen, Borgan, and Gjessing 2008, sec. 4.2). To avoid this nonnegativity constraint, one may consider alternative forms of model (1). For example, one may replace the linear term β T Z i (t ) by an exponential form exp{β T Z i (t )}. Alternative additive hazards models are discussed by Ying (1995, 1997). It would be interesting to modify our development here to other additive hazards models.