03706251170BiometricsBiometricsBiometrics0006-341X1541-042030525191655569410.1111/biom.13012NIHMS1015626ArticleA Modified Partial Likelihood Score Method for Cox Regression with Covariate Error Under the Internal Validation DesignZuckerDavid M.1*ZhouXin2LiaoXiaomei3LiYi4SpiegelmanDonna5Department of Statistics and Data Science, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, ISRAELDepartment of Biostatistics, Harvard T.H. Chan School of Public Health, 677 Huntington Avenue, Boston, MA 02115, U.S.A.,Department of Biostatistics, Harvard T.H. Chan School of Public Health, 677 Huntington Avenue, Boston, MA 02115, U.S.A. Currently employed at AbbVie Incorporated, North Chicago, IL, U.S.A.Department of Biostatistics, University of Michigan School of Public Health, 1415 Washington Heights, Ann Arbor, MI 48109-2029, U.S.A.,Department of Biostatistics, Yale School of Public Health and Department of Statistics, Yale University, New Haven, CT 06520 U.S.A, and Departments of Epidemiology, Biostatistics, Nutrition and Global Health, Harvard T.H. Chan School of Public Health, Boston, MA 02115, U.S.A

Current address: Center for Methods in Implementation and Prevention Science (CMIPS) and Department of Biostatistics, Yale School of Public Health, 60 College Street, New Haven, CT 06520, U.S.A

david.zucker@mail.huji.ac.il6320191342019620192382019752414427Summary:

We develop a new method for covariate error correction in the Cox survival regression model, given a modest sample of internal validation data. Unlike most previous methods for this setting, our method can handle covariate error of arbitrary form. Asymptotic properties of the estimator are derived. In a simulation study, the method was found to perform very well in terms of bias reduction and confidence interval coverage. The method is applied to data from Health Professionals Follow-Up Study (HPFS) on the effect of diet on incidence of Type II diabetes.

Cox modelMeasurement errormodified scoreIntroduction

In the Cox (1972) regression model for survival data, the hazard function λ(t|x) for an individual with covariate vector x ∈ IRp is modeled semiparametrically as λ(t|x)=λ0(t)exp(βTx), where β ∈ IRp is a vector of regression coefficients and λ0(t) is an unspecified baseline hazard function λ0(t). Cox proposed drawing inference on β based on the notion of partial likelihood, which was subsequently justified rigorously by Tsiatis (1981), who used classical limit theory, and by Andersen and Gill (1982), who used a martingale theory approach.

In many applications, however, the covariate X is not measured exactly, but is subject to measurement error of some degree, often substantial. Thus, instead of observing X, we observe a surrogate measure W. Starting from Prentice (1982), a considerable literature has been developed on inference for the Cox regression model with covariate error in various contexts; see Zucker (2005) for a brief review.

The existing methods generally involve some model assumptions on the joint distribution of the true covariate and the surrogate. Many of the methods make use of specific parametric forms for this joint distribution. Other methods, such as those of Huang and Wang (2000) and Kong and Gu (1999), avoid use of a specific parametric form but still rely on an assumption that the covariate error is of independent additive structure. Some papers, such as Zhou and Pepe (1995), Zhou and Wang (2000), and Chen (2002), present methods without this additive error assumption for the internal validation design in which there is a subsample of individuals with a measurement on both the true covariate and the surrogate. These methods, however, have challenges as well. The approach taken by Zhou and Pepe (1995) and by Zhou and Wang (2000) involves stratification or smoothing in the covariate space; when the number of covariates is moderate to large, this approach breaks down due to the “curse of dimensionality.” Chen (2002) assumes that it is possible to form a satisfactory initial estimate of the regression coefficient vector based on the validation sample alone. This is not the case, however, for studies where the event rate is low to moderate, the main study sample size is in the thousands to hundreds of thousands, and the validation study sample size is, as in all applications we know of, only a few hundred. Under these circumstances, the number of events in the validation study is very small, so that a satisfactory initial estimate of the regression coefficient vector based on the validation sample alone cannot be obtained. Thus, in such situations, which often arise in practice, Chen’s approach is problematic.

This paper presents a new method for the Cox model with covariate error, which overcomes the limitations of previously proposed methods. The method involves a modified version of the classical Cox partial likelihood score function, with the internal validation data incorporated in a suitable way. Our approach is very simple in concept. It is in the spirit of Lin and Ying’s (1993) work on Cox regression with incomplete covariate data. There is also some resemblance to Huang and Wang’s (2000) method for Cox regression with covariate error, and to work of Kulich and Lin (2000, 2004). The method requires no assumptions on the form of the covariate error. It is especially designed for the internal validation design with a relatively small validation sample and a moderate to large number of covariates, which, as indicated above, is a challenging situation that often arises in epidemiological studies. The method is easy to implement, and its practical utility is backed by large-sample theory and small-sample simulations.

The outline of the remainder of the paper is as follows. Section 2 presents the proposed method and its asymptotic properties, Section 3 a simulation study, Section 4 an application to data from the Health Professionals Follow-Up Study (HPFS), and Section 5 a brief summary. The Web Appendix provides theoretical details.

The Proposed Method and Its Asymptotic PropertiesThe Proposed Method

We assume a classical survival data setup. We have i.i.d. observations on n individuals. Associated with each individual i is a set of random variables (Ti°, Ci, Xi, Wi), with Ti° representing the time to event, Ci representing the time to censoring, Xi representing a p-vector of true covariate values, and Wi representing a p-vector of surrogate covariate values. We assume that the covariates are arranged so that the first p1 covariates are the error-prone covariates and the remaining p2 = p − p1 covariates are error-free. For the error-free covariates, the relevant component of Wi is identical to the corresponding component of Xi. We denote the maximum follow-up time by τ. The available data on all individuals consist of (Ti, δi, Wi), where Ti=min(Ti°,Ci) is the follow-up time and δi=I(Ti°⩽Ci), with I(·) being the indicator function, is the event indicator. In addition, within the main study we have a random internal validation sample of size m of individuals with both Xi and Wi observed. We take m = ceil(πn), where π is a specified number in (0, 1) and ceil(u) denotes the smallest integer greater than or equal to u. We define ωi to be equal to 1 if individual i is in the internal validation sample and 0 otherwise. Thus, the random vector (ω1, …, ωn) has a uniform distribution over the finite set O(n,m) of vectors with m ones and n − m zeros (i.e., O(n,m) expresses the various ways of selecting m elements from a set of n elements). We write π^=m/n. Note that π^ is not an estimate, but rather is fixed by design. Also, as usual, we define Yi(t) = I(Ti ⩾ t) and Ni(t) = δiI(Ti ⩽ t). Left truncation is handled by setting Yi(t) to zero until the time at which individual i comes under observation.

We assume, as usual, that Ti° and Ci are conditionally independent given Xi. We assume further that the measurement error is noninformative in the sense that Wi is conditionally independent of (Ti°, Ci) given Xi. We make no assumptions about the form of the measurement error. Finally, we assume that the survival time Ti° follows the Cox model (1). We denote the true value of β by β*. We present our development for the case of the classical Cox relative risk function eβTXi, but it is straightforward to extend the development to more general relative risk functions, as in Thomas (1981) and in Breslow and Day, 1993, Sec.5.1(c).

We construct our procedure as follows. Let E^ denote empirical expectation, so that, for example, E^{Y(t)exp(βTX)}=1n∑j=1nYj(t)exp(βTXj), E^{Y(t)Xexp(βTX)}=1n∑j=1nYj(t)Xjexp(βTXj).

In the absence of measurement error, the Cox partial likelihood score function is given by UCOX(β)=1n∑i=1nδi(Xi−[E^{Y(t)Xexp(βTX)}E^{Y(t)exp(βTX)}]t=Ti).

When X is measured only for a sample of the individuals and only W is available for the others, a naive Cox analysis involves simply substituting W in place of X for the individuals without a measurement of X. In other words, defining Wi°=ωiXi+(1−ωi)Wi, the naive Cox analysis is based on the score function UNAI(β)=1n∑i=1nδi(Wi°−[E^{Y(t)W°exp(βTW°)}E^{Y(t)exp(βTW°)}]t=Ti), with E^{Y(t)exp(βTW°)} = S0°(t,β)=1n∑j=1nYj(t)exp(βTWj°), E^{Y(t)W°exp(βTW°)}=S1°(t,β)=1n∑j=1nYj(t)Wj°exp(βTWj°)

We denote the corresponding estimator by β^NAI. The terms in UNAI(β) are of “observed – expected” form, but the “expected” term is incorrect. Consequently, the naive score function does not have zero asymptotic expectation under β*, and therefore β^NAI is biased.

An improved estimator can be obtained using regression calibration, which is an established technique for measurement error problems; see, for example, Carroll et al. (2006, Chapter 4). In regression calibration, we redefine Wi° to be Wi°=ωiXi+(1−ωi)X^i, with X^ir (r = 1, …, p1) defined as X^ir=α^r0+∑s=1pα^rsWis where α^r0,…,α^rp are the ordinary least squares estimates of the regression of Xir on Wi based on the internal validation sample. Having redefined Wi°, we redefine S0°(t,β) and S1°(t,β) correspondingly. We denote the resulting estimator by β^RC. In (4), for the sake of generality, we have included all of the components of Wi in the regression, but in typical applications of regression calibration the regression model for Xir includes only Wir and perhaps one or two additional components of Wi. Substantial improvement is often achieved with regression calibration approach, but the “expected” term is still not exactly correct, and therefore the resulting estimator is not exactly consistent. The regression calibration approximation is good when the degree of measurement error is small or the regression coefficients of the error-prone covariates are small, but otherwise the approximation can be unsatisfactory (Spiegelman, Rosner, and Logan, 2000).

We present an estimator that builds on the regression calibration estimator but is exactly consistent. As in regression calibration, we use the regression model (4). However, we use this model only as a working model, and it is not necessary for the model to be correct for our estimator to be consistent. As with standard regression calibration, it is possible in principle, as written in (4), to include all the components of Wi in the model, but in practice we recommend using only Wir and perhaps one or two additional components.

The idea of our approach is to replace the incorrect “expected” term with a correct one. Let α(r) be the column vector with components αr0, αr1, …, αrp, let α denote the vector formed by stacking the vectors α(r) one on top of the other, and let α* denote the true value of α. To emphasize the dependence of X^ir on α, we denote the vector of Xir’s by X^i(α).

Define S0a(t,β)=1n∑j=1nωjYj(t)exp(βTXj) S0b(t,β,α)=1n∑j=1n(1−ωj)Yj(t)exp{βTX^j(α)} S0c(t,β,α)=1n∑j=1nωjYj(t)exp{βTX^j(α)} S1a(t,β)=1n∑j=1nωjYj(t)Xjexp(βTXj) S1b(t,β,α)=1n∑j=1n(1−ωj)Yj(t)X^j(α)exp{βTX^j(α)} S1c(t,β,α)=1n∑j=1nωjYj(t)X^j(α)exp{βTX^j(α)} S˜1a(t,β,α)=1n∑j=1nωjYj(t)X^j(α)exp(βTXj) S0(t,β,α)=S0a(t,β)+{S0a(t,β)S0c(t,β,α)}S0b(t,β,α) ϕ˜(t,β,α)=S0b(t,β,α)/S0c(t,β,α) S1(t,β,α)=S1a(t,β)+S1b(t,β,α)+ ϕ˜(t,β,α){ S˜1a(t,β,α)−S1c(t,β,α)}

We then take the score function to be UMS(β,α)=1n∑i=1nδi{Wi°−S1(Ti,β,α)S0(Ti,β,α)}.

The estimator β^MS is defined to be the solution to the score equation UMS(β,α^)=0. We could have used ϕ^=(1−π^)/π^=(n−m)/m in place of ϕ˜(t,β,α), but we found that better finite-sample performance is achieved with ϕ˜(t,β,α).

The motivation behind UMS(β, α) is as follows. The regression calibration function URC(β) can be written in counting process notation as URC(β,α)=1n∑i=1n∫0τ{Wi°−E(t,β,α)}dNi(t) with E(t,β,α)=S1°(t,β,α)S0°(t,β,α).

Let us now define dMi(t)=dNi(t)−Yi(t)eβ*TXiλ0(t)dt. We can then write URC(β,α)=1n∑i=1n∫0τ{Wi°−E(t,β,α)}Yi(t)eβ*TXiλ0(t)dt+1n∑i=1n∫0τ{Wi°−E(t,β,α)}dMi(t)=∫0τ{S1°°(t,β*,α)−E(t,β,α)S0d(t,β*)}λ0(t)dt+1n∑i=1n∫0τ{Wi°−E(t,β,α)}dMi(t). where S0d(t,β)=1n∑j=1nYj(t)eβTXj S1oo(t,β,α)=1n∑j=1nYj(t)Wj°eβTXj=S1a(β,α)+1n∑j=1n(1−ωj)Yj(t)X^j(α)eβTXj

Using counting process theory (Gill, 1984), it can be seen that the second term of (16) has expectation zero. In the absence of measurement error, the value at β* of the quantity in brackets in the first term of (16) is zero, so that the score function is unbiased. In the presence of measurement error, the value at β* of this quantity is in general nonzero. We need to redefine E(t, β, α) so that the limiting value of this quantity at β*, α* is zero. Define EX(t,β)=E{Y(t)exp(βTX)}, EXX(t,β)=E{Y(t)Xexp(βTX)}, EWX(t,β,α)=E{Y(t)X^(α)exp(βTX)}.

The limiting value of S0d(t, β) is then EX(t,β,α) and the limiting value of S1°°(t,β,α) is s1(t,β,α)=πEXX(t,β,α)+(1−π)EWX(t,β,α). We thus have to redefine E(t, β, α) so that its limiting value is equal to s1(t,β,α)/EX(t,β,α). Taking E(t, β, α) = S1(t, β, α)/S0(t, β, α) achieves this objective. At the same time, our estimator reduces to the usual Cox estimator under zero measurement error. We regard this reducibility property to be important for measurement error correction methods.

We reiterate that our method makes no assumptions about the form of the covariate error, and that the model (4) is only a working model, with our estimator still being consistent even if the working model is misspecified. In addition, our method requires only estimation of unconditional means involving Y, W, and X, and therefore does not require use of smoothing methods. For this reason, a modestly-sized internal validation sample is sufficient. By contrast, the approaches taken by Zhou and Pepe (1995) and by Zhou and Wang (2000) require consistent estimates of conditional means, which involve stratification or smoothing in the covariate space, and thus require a larger validation sample. In addition, since our method is based on separate empirical averages for each risk set, a rare disease approximation is not needed.

We have worked in the setting of time-independent covariates, but it is possible to consider extension to the case of time-dependent covariates. When the covariate processes are measured on an approximately continuous basis (W(t) for the full cohort and X(t) for the internal validation sample), the method and its asymptotic theory carries over with notational changes only. Since the method is based on separate empirical averages for each risk set, changes over time in the measurement error distribution are handled automatically. The method and the asymptotic theory also carry over to the case where the covariate processes are measured only intermittently, as commonly occurs in practice, but the processes vary slowly, so that carrying forward the last observed covariate value is a reasonable approximation. In the case where the the covariate processes are measured only intermittently and vary more rapidly, the extension to the case of time-dependent covariates is more complex and is beyond the scope of this paper.

Asymptotic Properties

The asymptotic properties of the estimator are presented in the following theorem.

Theorem 1: Under the regularity conditions stated in the Web Appendix, β^MS converges almost surely to β*, and n(β^MS−β*) is asymptotically mean-zero multivariate normal with covariance matrix that can be estimated consistently by the sandwich-type estimator described below.

We present here a sketch of the proof of this result. The details are presented in the Web Appendix.

The consistency proof hinges on the fact that, as explained above, UMS(β, α,) is constructed so that it converges to a limit u(β, α) for which u(β*, α*) = 0. We can then appeal to arguments of Foutz (1977) to obtain the consistency result.

The asymptotic normality proof is based on estimating equations theory, and uses an argument along the lines of Lin and Wei (1989). Setting θ = (β, α), we can define the estimator θ^ of θ to be the solution θ^ to U(θ)=0 with U(θ)=(U(1),U(2)), where U(1)(θ) is the UMS(β, α) defined in (15) and U(2)(θ) is given by stacking the vectors Ur(2)(α)=1n∑i=1nωi(Xir−αr0−∑s=1pαrsWis)[1Wi] where we include Ur(2) only for covariates that are subject to measurement error. We can write U(2)(θ)=1n∑i=1nωiZi(12)(θ) with Zi(12)(θ)=(xi⊗w¯i)−{Ip1⊗(w¯iw¯iT)}α where xi consists of Xi1, …, Xip1, w¯i consists of a 1 followed by the components of Wi, ⊗ denotes the Kronecker product, and Ib denotes the b × b identity matrix. The vector U(2)(θ) is of length (p + 1)p1. When the model for a given Xir includes only some of the Wis’s, we delete the superfluous elements of α and Zi(12)(θ).

In the Web Appendix we show that U(1)(θ*) is asymptotically equivalent to the quantity U(1)*(θ*)=π^{1m∑i=1nωiZi(11)(θ*)}+(1−π^){1n−m∑i=1n(1−ωi)Zi(21)(θ*)} where {Zi(11)(θ):ωi=1} and {Zi(21)(θ):ωi=0} are each sets of i.i.d. vectors with mean zero under θ = θ*, the expressions for which are presented in the Web Appendix. Thus, the solution to U(θ*)=0 is asymptotically equivalent to the solution to U*(θ*)=0, with U*=(U(1)*,U(2)). Let Zi(1) denote the stacked vector formed by Zi(11) and Zi(12) and let Zi(2) denote the stacked vector formed by Zi(21) and the zero vector of length (p + 1)p1. We can then write U*(θ)=π^{1m∑i=1nωiZi(1)(θ*)}+(1−π^){1n−m∑i=1n(1−ωi)Zi(2)(θ*)}.

Define C1= Cov (Zi(1)), C2= Cov (Zi(2)), and C=π^C1+(1−π^)C2. We see that the asymptotic distribution of nU*(θ*) is mean-zero normal with covariance matrix C. Consequently n(β^MS−β*) is asymptotically mean-zero normal with covariance matrix V=RCRT, where R is the matrix consisting of the first p rows of d(θ)−1, where d(θ) is the limiting value of the matrix D(θ) given by −1 times the Jacobian of U(θ). In principle, we can estimate V by V^=R^C^R^, where R^ consists of the first p rows of D(θ^)−1 and C^=π^C^1+(1−π^)C^2, where C^s is the sample covariance of Zi(s)(θ), i.e. C^s=1n∑i=1nZi(s)(θ)Zi(s)(θ)T.

In actuality, the terms of U(1)* involve additional unknown quantities, so we compute C^s using the sample covariance of the vectors Z^i(s)(θ^) defined by replacing these quantities with consistent estimates. The detailed derivations of the expressions for Zi(11)(θ), Zi(21)(θ), and and D(θ) are presented in the Web Appendix.

Simulation Study

We examined the performance of the proposed method in a simulation study. We constructed the simulation setup so as to be representative of a typical epidemiological cohort study. We considered a setup where the time metameter is age, with age at entry to the study being uniformly distributed over the interval 30 to 50 years. The study horizon was 12 years. We took the censoring distribution to be exponential with a rate of 1% per year. We took the baseline survival function to be Weibull with shape parameter 5, as in Zucker and Spiegelman (2004, 2008). In terms of the sample size and the event rate (determined by the Weibull scale parameter), we considered two scenarios: a rare event scenario with n = 10, 000 and a cumulative event rate of about 5% (so that the number of events is about 500), and a common event scenario with n = 500 and a cumulative event rate of about 25% (so that the number of events is about 125). The internal validation sample size was 200. Thus, in the rare event case, the internal validation sample size included a mere handful of events, which may hamper the use of Chen’s (2002) approach.

We carried out two sets of simulations. In the first set, we worked with a single covariate X, generated from a standard normal distribution. We considered two measurement error models, as follows:

Independent Measurement Error Model: W = X + ϵ with ϵ ~ N(0, a) independently of X

Dependent Measurement Error Model: W = X + ϵ with ϵ |X ~ N(0, a(1 + |X|))

We chose a range of a values corresponding to the following range of values for the correlation between X and W: 0.9, 0.7, 0.5. Finally, we took eβ = 1.5, 2.5, or 4. We compared our proposed estimator (MS) against Chen’s (2002) estimator (CH), the regression calibration estimator obtained by replacing X by X^ in the Cox score function (RC), the “complete case” (CC) estimator based only on the data with a measurement of X, In the second set of simulations, we worked with five covariates X1, …, X5, with X1 error-prone and the other covariates error-free. We took the five covariates to be N(0, 1) random variables, either independent or equally-correlated with a correlation of 0.2. We took the hazard function to be λ(t) = λ0(t) exp(β1×1 + β2×2 + β3×3 + β4×4 + β5×5) with β2 = β3 = β4 = β5 = log(1.5), where, as before, we took λ0(t) to be Weibull with shape parameter 5 and eβ = 1.5, 2.5, or 4. The other settings were as in the the first set of simulations. The simulation results were based on 10,000 replications. If the zero-finding procedure with our method failed to converge, we used the RC estimate. In the univariate simulations this usually occurred in less than 1% of the replications, and the worst instance it occurred in 6% of the replications. In the multivariate simulations, convergence failure usually occurred in less than 5% of the replications, and in the worst instance it occurred in 10% of the replications. In both the univariate and multivariate simulation, the worst instance was with highest value of β1 and highest degree of measurement error. The results for the rare event scenario are presented in Tables 1–6. The corresponding results for the common event scenario are presented in the Supplementary Web Materials in Tables S1–S6.

The naive estimator was seriously biased in all cases studied, often dramatically. In the single covariate setup, the MS method exhibited low bias across the board, while the RC method often exhibited appreciable bias, especially under the dependent error model, with the bias increasing as the true β increases and as the degree of measurement error increases. In the rare disease case, as expected, the CC method had very high variance, while the variance of the MS method was usually considerably lower. In the common disease case, the MS method had lower variance than the CC method in most configurations, although there are some configurations in which the CC method had lower variance. As expected, Chen’s method performed very well in the common disease setup, where the MS method and Chen’s method are comparable in terms of bias, variance and coverage probability. In the rare disease setup, Chen’s estimator had low bias is some cases and considerable bias in other cases. In addition, the standard deviation of Chen’s estimator was substantially greater than that of the MS estimator, in some cases around 3 times greater. Also, the estimate of the standard deviation tended to underestimate, leading to considerably lower than nominal confidence interval coverage rates.

In the multiple-covariate setup, MS method exhibited noticeable bias in some configurations, but the bias with the MS method was typically lower than with the RC method, often considerably so. The patterns were similar across the disease incidence levels (common/rare) and the measurement error models (independent/dependent). The performance of the MS method with dependent covariates was similar to that with independent covariates, and no systematic trends emerged between the dependent covariate case and the independent covariate case in the relative performance of the MS method as compared with the other methods. Chen’s method had a noticeably lower standard deviation than the MS method in the multivariate common disease setting with for large β and moderate correlation between the surrogate and the true exposure (Tables S3–S6 in the Web Appendix, bottom panel). To explore the relative performance of the two methods further, we conducted additional simulations with eβ = 4 under an “intermediate event rate” scenario with n = 500, validation sample size of 200, and a cumulative event rate of about 15% (Table S7 in the Web Appendix) In these simulations, Chen’s method again had a noticeably lower standard deviation than the MS method; at the same time, Chen’s estimate of the standard deviation of the estimate was noticeably lower than the empirical standard deviation. As a rough practical guide, we suggest that the MS estimator is to be preferred when the number of events in the validation study is very low, while Chen’s estimator is to be preferred when the number of events in the validation study is 30 or more, with some caution needed with Chen’s estimate of the standard deviation of the estimator.

In both the single-covariate and the multiple-covariate setups, the empirical coverage rate of the asymptotic confidence interval based on the MS method is generally close to the nominal level of 95%, while for the RC method the coverage rate tended to be considerably below nominal for eβ = 4.

For the multiple-covariate setup, we conducted additional simulations to examine the bias of the MS method for larger sample sizes. These results are reported in the Supplementary Web Materials in Tables S8–S9. When the sample size is increased, the bias decreases, eventually to a very small level.

Example

We illustrate the method on data from the Health Professionals Follow-Up Study (HPFS), a prospective cohort study of 51,529 middle-aged (age 40–75 years at baseline) male health professionals. Participants were recruited in 1986 and were mailed questionnaires every other year to assess health status and lifestyle. Here, we analyze the relationship between onset of Type 2 diabetes (T2D) and a diet score relating to intake of carbohydrates, protein, and fat (de Koning et al., 2011). The diet score ranged from 0 to 30, with the score increasing under a decrease in carbohydrate intake or an increase in protein or fat intake. The analysis included the 41,616 study participants who were free of T2D, cardiovascular disease, or cancer at baseline, among whom there were 2,790 cases of incident T2D during follow-up. Diet was assessed with a 131-item semiquantitative food frequency questionnaire (FFQ), an instrument which is subject to substantial measurement error. In a subsample of 105 participants, another diet assessment was carried out using a more accurate diet record (DR). The analysis was stratified by age and adjusted for body mass index (BMI). We analyzed the data using the naive Cox method, the RC method, the complete case method, Chen’s method, and our proposed MS method. There were only 6 events among the 105 individuals in the validation sample, which puts Chen’s method and the complete case method at a very severe disadvantage. Table 5 presents the results for the various methods. For the regression coefficient for the diet score, the RC estimate is considerably larger than the naive estimate, and the MS and complete case estimates are noticeably larger than the RC estimate. The estimate with Chen’s method was lower than that with the naive method. The standard error with Chen’s method was a bit over 1.5 times the standard error with the MS method. For the regression coefficient for BMI, the estimates were similar across all methods, and the standard error with Chen’s method was 2.7 times that of the standard error with the MS method.

Summary and Discussion

We have developed a new method for covariate error correction in the Cox survival regression model, given internal validation data. The method can handle covariate error of arbitrary form, not just independent additive measurement error. Only a modestly-sized internal validation sample is required. The method can handle the case where the number of covariates in moderate to large. In a simulation study, the method was found to perform very well in terms of bias reduction and confidence interval coverage.

We have worked in the setting of time-independent covariates, but it is possible to consider extension to the case of time-dependent covariates. When the covariate processes are measured on an approximately continuous basis (W(t) for the full cohort and X(t) for the internal validation sample), the method and its asymptotic theory carries over with notational changes only. The same is true in the case where the covariate processes are measured only intermittently, as commonly occurs in practice, but the processes vary slowly, so that carrying forward the last observed covariate value is a reasonable approximation.

If the association between W and X is very weak, the proposed estimate will remain consistent and asymptotically normal, but the variance will be very high. If there is no association at all between W and X, then W is not a suitable surrogate for X and no correction method will help. If the relationship between W and X is highly nonlinear, the working model (4) can be modified to include nonlinear W terms. A plot of Xir versus Wir for the individuals in the internal validation sample can be used to examine whether nonlinear W terms are needed in the working model for Xir.

Supplementary MaterialAcknowledgements

We thank Yi-Hau Chen for sharing with us the code for his method. In addition, we thank the editor, associate editor, and referees for helpful comments that led to substantial improvements in the paper.

Supplementary Materials

The Web Appendix, referenced in Sections 2 and 3, is available with this paper at the Biometrics website on Wiley Online Library, as is the code we used to implement the various method.

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Simulation results for the single-covariate rare disease case with independent measurement error. β* is the true value of β. Bias(%) is the relative bias, i.e. Bias (%)=100×(β^−β*)/β*. IQR is 0.74 times the interquartile range of the β^ values. SE is the mean of the estimated standard error of β^. SD is the empirical standard deviation of the β^ values. CR is the empirical coverage rate of the asymptotic 95% confidence interval. Methods considered: MS = modified score, CH = Chen, RC = regression calibration, CC = complete case, NA = naive.

Corr(X, W)exp(β*)β*MethodMeanMedianIQRSESDCR
β^Bias(%)β^Bias(%)
0.901.50.4055MS0.4050−0.10.4011−1.10.05770.05250.05170.965
GH0.42304.30.43136.40.16210.13730.17430.879
RG0.4036−0.50.4032−0.60.05620.05110.05060.957
GG0.41883.30.41632.70.31200.33840.36840.945
NA0.3287−18.90.3309−18.40.04040.04020.03860.543
0.701.50.4055MS0.40880.80.4040−0.40.07370.07380.07530.945
GH0.42775.50.44038.60.25450.21260.29790.855
RG0.4029−0.60.4032−0.60.07040.06880.06900.965
GG0.41883.30.41632.70.31200.33840.36840.945
NA0.1993−50.90.2011−50.40.03460.03130.03060.000
0.501.50.4055MS0.41291.80.41031.20.10810.10990.11860.938
GH0.43266.70.445910.00.30060.25180.35580.859
RG0.4030−0.60.4004−1.30.10520.09760.10110.938
GG0.41883.30.41632.70.31200.33840.36840.945
NA0.1022−74.80.1036−74.40.02370.02240.02230.000
0.902.50.9163MS0.92791.30.92210.60.07500.07200.07210.949
GH0.94012.60.92891.40.18570.15990.21200.875
RG0.9098−0.70.9045−1.30.06190.05840.05750.973
GG0.93802.40.92591.00.34010.35900.41090.941
NA0.7412−19.10.7406−19.20.03930.04090.03950.004
0.702.50.9163MS0.94493.10.93762.30.12690.12790.13240.957
GH0.95454.20.95113.80.27560.23940.34770.867
RG0.8944−2.40.8910−2.80.09040.08780.08450.949
GG0.93802.40.92591.00.34010.35900.41090.941
NA0.4434−51.60.4438−51.60.02720.03150.03070.000
0.502.50.9163MS0.96205.00.94603.20.20690.22630.24010.957
GH0.95774.50.96014.80.31520.27610.40260.855
RG0.8785−4.10.8766−4.30.13450.12630.12580.914
GG0.93802.40.92591.00.34010.35900.41090.941
NA0.2254−75.40.2270−75.20.01910.02240.02230.000
0.904.01.3863MS1.42142.51.40801.60.11660.11620.11960.930
GH1.43593.61.41592.10.23520.20040.26250.875
RG1.3464−2.91.3476−2.80.07180.06870.06510.906
GG1.44604.31.41342.00.38810.40630.45900.957
NA1.0967−20.91.099 7−20.70.04490.04260.04220.000
0.704.01.3863MS1.48627.21.458 75.20.22710.24050.25900.957
GH1.46545.71.41962.40.32810.28370.39860.856
RG1.2863−7.21.2901−6.90.11120.10790.10200.781
GG1.44604.31.41342.00.38810.40630.45900.957
NA0.6384−53.90.6388−5 3.90.03370.03190.03120.000
0.504.01.3863MS1.49928.11.44464.20.33840.36980.37860.944
GH1.47526.41.41552.10.36360.31680.44970.863
RG1.2358−10.91.2302−11.30.16500.14960.14900.7 39
GG1.44604.31.41342.00.38810.40630.45900.957
NA0.3206−76.90.3228−76.70.02270.02250.02210.000

Simulation results for the single-covariate rare disease case with dependent measurement error. β* is the true value of β. Bias(%) is the relative bias, i.e. Bias (%)=100×(β^−β*)/β*. IQR is 0.74 times the interquartile range of the β^ values. SE is the mean of the estimated standard error of β^. SD is the empirical standard deviation of the β^ values. CR is the empirical coverage rate of the asymptotic 95% confidence interval. Methods considered: MS = modified score, CH = Chen, RC = regression calibration, CC = complete case, NA = naive.

Corr(X, W)exp(β*)β*MethodMeanMedianIQRSESDCR
β^Bias(%)β^Bias(%)
0.901.50.4055MS0.4043−0.30.4012−1.10.05580.05250.05160.957
GH0.42554.90.43146.40.16280.13450.17290.871
RG0.4005−1.20.4009−1.10.05170.04990.04970.949
GG0.41883.30.41632.70.31200.33840.36840.945
NA0.3299−18.60.3315−18.30.03920.04000.03820.531
0.701.50.4055MS0.40710.40.40550.00.07390.07300.07290.953
GH0.42725.40.44068.70.26220.21260.30120.867
RG0.3992−1.50.3980−1.80.06940.06690.06670.957
GG0.41883.30.41632.70.31200.33840.36840.945
NA0.1974−51.30.1985−51.00.03220.03080.03010.000
0.501.50.4055MS0.41562.50.41111.40.10200.11220.11750.953
GH0.42675.20.42875.70.30360.25090.35610.855
RG0.4016−1.00.3995−1.50.10070.09740.09950.949
GG0.41883.30.41632.70.31200.33840.36840.945
NA0.1023−74.80.1042−74.30.02240.02230.02240.000
0.902.50.9163MS0.92260.70.9091−0.80.07630.08400.08010.949
GH0.93862.40.92831.30.18350.16230.22100.859
RG0.8798−4.00.8778−4.20.05790.05500.05520.887
GG0.93802.40.92591.00.34010.35900.41090.941
NA0.7247−20.90.7229−21.10.03540.03920.03760.000
0.702.50.9163MS0.94152.80.92210.60.13410.14330.13460.961
GH0.94923.60.95063.70.27320.24310.36130.867
RG0.8631−5.80.8669−5.40.08610.08070.07790.879
GG0.93802.40.92591.00.34010.35900.41090.941
NA0.4268−53.40.4264−5 3.50.02610.02970.02920.000
0.502.50.9163MS0.96164.90.93652.20.21440.28610.23270.953
GH0.93672.20.93672.20.30460.27770.39130.871
RG0.8675−5.30.8698−5.10.13600.12570.12460.902
GG0.93802.40.92591.00.34010.35900.41090.941
NA0.2230−75.70.2246−75.50.02020.02190.02260.000
0.904.01.3863MS1.40561.41.3595−1.90.10870.22760.22030.928
GH1.42803.01.404 71.30.24890.20890.29070.883
RG1.2652−8.71.2619−9.00.06590.06350.06080.508
GG1.44604.31.41342.00.38810.40630.45900.957
NA1.0416−24.91.0429−24.80.04170.03920.03860.000
0.704.01.3863MS1.45595.01.40361.20.23590.30210.29200.939
GH1.45174.71.404 31.30.34660.28880.41620.859
RG1.2086−12.81.2094−12.80.09630.09620.09060.535
GG1.44604.31.41342.00.38810.40630.45900.957
NA0.5969−56.90.5988−56.80.03040.02880.02840.000
0.504.01.3863MS1.46635.81.40391.30.31860.38210.37930.934
GH1.45645.11.39220.40.37050.31920.45300.856
RG1.2105−12.71.1978−13.60.15720.14900.14780.696
GG1.44604.31.41342.00.38810.40630.45900.957
NA0.3135−77.40.3149−77.30.02390.02140.02210.000

Simulation results for the multiple-covariate rare disease case with independent covariates and independent measurement error. β* is the true value of β. Bias(%) is the relative bias, i.e. Bias (%)=100×(β^−β*)/β*. IQR is 0.74 times the interquartile range of the β^ values. SE is the mean of the estimated standard error of β^. SD is the empirical standard deviation of the β^ values. CR is the empirical coverage rate of the asymptotic 95% confidence interval. Methods considered: MS = modified score, CH = Chen, RC = regression calibration, CC = complete case, NA = naive.

Corr(X, W)exp(β*)β*MethodMeanMedianIQRSESDCR
β^Bias(%)β^Bias(%)
0.901.50.4055MS0.41181.60.41011.10.05120.06840.05730.949
GH0.41362.00.41061.30.19620.13520.24320.772
RG0.40740.50.40630.20.04860.05230.05070.945
GG0.471716.30.453311.80.36260.37820.45380.961
NA0.3321−18.10.3337−17.70.04200.04100.04080.567
0.701.50.4055MS0.41753.00.40670.30.07610.08600.08140.957
GH0.42925.80.451811.40.32210.20260.40500.749
RG0.40660.30.40740.50.07000.07090.06840.949
GG0.471716.30.453311.80.36260.37820.45380.961
NA0.1997−50.80.2017−50.20.03330.03190.03200.000
0.501.50.4055MS0.43006.10.41482.30.11050.14020.13400.952
GH0.42745.40.454212.00.35180.23580.48520.749
RG0.40810.60.40650.30.10500.10160.09860.957
GG0.471716.30.453311.80.36260.37820.45380.961
NA0.1013−75.00.1021−74.80.02480.02280.02280.000
0.902.50.9163MS0.94292.90.92891.40.10760.12720.12300.937
GH0.97576.50.94803.50.21100.15960.26070.785
RG0.9109−0.60.9133−0.30.05360.05950.05690.961
GG1.075717.41.030512.50.37160.41510.46760.945
NA0.7424−19.00.7429−18.90.04410.04150.04180.016
0.702.50.9163MS0.96575.40.93982.60.19010.20150.21460.955
GH1.021111.40.97786.70.31170.22870.41240.754
RG0.8962−2.20.8896−2.90.08830.09010.08650.930
GG1.075717.41.030512.50.37160.41510.46760.945
NA0.4408−51.90.4428−51.70.03110.03180.03330.000
0.502.50.9163MS1.026412.00.93562.10.30040.31310.32270.938
GH1.033012.71.010010.20.34530.26020.47510.762
RG0.8868−3.20.8738−4.60.12350.13080.13080.930
GG1.075717.41.030512.50.37160.41510.46760.945
NA0.2228−75.70.2244−75.50.02210.02260.02360.000
0.904.01.3863MS1.43953.81.41752.30.12450.13330.14900.943
GH1.50518.61.45915.20.31510.20870.46700.769
RG1.3467−2.91.3496−2.60.07320.07050.06830.902
GG1.656319.51.584 714.30.48730.50550.59710.945
NA1.0974−20.81.0969−20.90.04400.04380.04530.000
0.704.01.3863MS1.525310.01.44103.90.31800.33290.34860.936
GH1.559212.51.50458.50.47890.28220.58240.741
RG1.2853−7.31.2782−7.80.10740.11100.10910.805
GG1.656319.51.584 714.30.48730.50550.59710.945
NA0.6327−54.40.6349−54.20.03540.03260.03470.000
0.504.01.3863MS1.540711.11.44724.40.43630.45110.44040.925
GH1.590314.71.525510.00.52490.31380.62860.706
RG1.2423−10.41.2252−11.60.14810.15460.15460.789
GG1.656319.51.584 714.30.48730.50550.59710.945
NA0.3156−77.20.3164−77.20.02380.02290.02360.000

Simulation results for the multiple-covariate rare disease case with independent covariates and dependent measurement error. β* is the true value of β. Bias(%) is the relative bias, i.e. Bias (%)=100×(β^−β*)/β*. IQR is 0.74 times the interquartile range of theβ^ values. SE is the mean of the estimated standard error of β^. SD is the empirical standard deviation of the β^ values. CR is the empirical coverage rate of the asymptotic 95% confidence interval. Methods considered: MS = modified score, CH = Chen, RC = regression calibration, CC = complete case, NA = naive.

Corr(X, W)exp(β*)β*MethodMeanMedianIQRSESDCR
β^Bias(%)β^Bias(%)
0.901.50.4055MS0.40981.10.40570.10.05580.05640.05460.948
GH0.41572.50.41131.40.21280.13340.24100.760
RG0.4045−0.20.4017−0.90.04820.05110.05040.949
GG0.471716.30.453311.80.36260.37820.45380.961
NA0.3339−17.60.3367−16.90.04170.04080.04060.571
0.701.50.4055MS0.41412.10.40620.20.07350.08330.08000.957
GH0.42504.80.447210.30.30610.20260.40280.733
RG0.4036−0.50.4043−0.30.07100.06900.06780.953
GG0.471716.30.453311.80.36260.37820.45380.961
NA0.1984−51.10.1994−50.80.03310.03150.03200.000
0.501.50.4055MS0.43126.40.41231.70.11640.14600.14530.957
GH0.42996.00.43467.20. 36860.23460.47250.753
RG0.40840.70.40700.40.10480.10170.10040.953
GG0.471716.30.453311.80.36260.37820.45380.961
NA0.1019−74.90.1031−74.60.02600.02270.02330.000
0.902.50.9163MS0.94142.70.91830.20.08390.10500.11480.956
GH0.97416.30.94373.00.22530.16170.27000.782
RG0.8828−3.70.8821−3.70.04920.05620.05480.891
GG1.075717.41.030512.50.37160.41510.46760.945
NA0.7284−20.50.7276−20.60.04110.03990.04000.004
0.702.50.9163MS0.95043.70.92230.70.12750.13660.14290.935
GH1.009610.20.97756.70.31640.22950.41970.754
RG0.8686−5.20.8685−5.20.07950.08330.08030.883
GG1.075717.41.030512.50.37160.41510.46760.945
NA0.4271−53.40.4276−5 3.30.02980.03020.03180.000
0.502.50.9163MS1.00239.40.92521.00.22440.23820.23520.928
GH1.017311.00.98487.50.34870.25940.47390.750
RG0.8800−4.00.8685−5.20.13070.13110.13230.906
GG1.075717.41.030512.50.37160.41510.46760.945
NA0.2219−75.80.2228−75.70.02350.02210.02380.000
0.904.01.3863MS1.41552.11.3822−0.30.15490.17200.18810.942
GH1.48256.91.43243.30.32990.21260.44560.764
RG1.2701−8.41.2703−8.40.06450.06540.06390.571
GG1.656319.51.584 714.30.48730.50550.59710.945
NA1.0474−24.41.0466−24.50.04050.04050.04240.000
0.704.01.3863MS1.43393.41.3776−0.60.33080.35630.36660.930
GH1.529710.31.47116.10.46860.28370.56520.732
RG1.2156−12.31.2144−12.40.10540.09970.09800.567
GG1.656319.51.584 714.30.48730.50550.59710.945
NA0.5969−56.90.6004−56.70.03200.02970.03200.000
0.504.01.3863MS1.49858.11.39040.30.45070.47400.46780.926
GH1.567313.11.50508.60.53240.31300.63220.710
RG1.2240−11.71.2001−13.40.15600.15490.15610.758
GG1.656319.51.584 714.30.48730.50550.59710.945
NA0.3112−77.60.3120−77.50.02490.02200.02340.000

Simulation results for the multiple-covariate rare disease case with dependent covariates and independent measurement error. β* is the true value of β. Bias(%) is the relative bias, i.e. Bias (%)=100×(β^−β*)/β*. IQR is 0.74 times the interquartile range of the β^ values. SE is the mean of the estimated standard error of β^. SD is the empirical standard deviation of the β^ values. CR is the empirical coverage rate of the asymptotic 95% confidence interval. Methods considered: MS = modified score, CH = Chen, RC = regression calibration, CC = complete case, NA = naive.

Corr(X, W)exp(β*)β*MethodMeanMedianIQRSESDCR
β^Bias(%)β^Bias(%)
0.901.50.4055MS0.40951.00.4045−0.20.05260.05700.05890.957
GH0.41993.60.41492.30.17930.13640.21200.825
RG0.3954−2.50.3916−3.40.05030.04820.05100.941
GG0.43737.80.42083.80.37440.35290.38910.961
NA0.3221−20.60.3215−20.70.03930.03780.03980.3 75
0.701.50.4055MS0.41392.10.40630.20.07870.09020.08830.949
GH0.43557.40.43637.60.29040.20440.34140.7 74
RG0.3768−7.10.3721−8. 20.05960.06380.06570.910
GG0.43737.80.42083.80.37440.35290.38910.961
NA0.1861−54.10.1840−54.60.03040.02880.03050.000
0.501.50.4055MS0.43206.50.3959−2.40.12140.15330.14290.941
GH0.43918.30.44148.90.35020.23810.39770.7 70
RG0.3603−11.10.3601−11.20.08240.08850.08700.906
GG0.43737.80.42083.80.37440.35290.38910.961
NA0.0916−77.40.0905−77.70.02040.02030.02120.000
0.902.50.9163MS0.93752.30.92010.40.08370.11900.11090.937
GH0.95073.80.92190.60.19210.15690.23240.840
RG0.8767−4.30.8719−4.80.05410.05430.05810.875
GG0.98867.90.98037.00.32730.37070.41150.961
NA0.7140−22.10.7141−22.10.04530.03730.03950.000
0.702.50.9163MS0.97386.30.94032.60.18250.21510.20350.934
GH0.98867.90.96705.50.29720.22960.35360.809
RG0.8191−10.60.8130−11.30.08620.08010.08270.699
GG0.98867.90.98037.00.32730.37070.41150.961
NA0.4042−55.90.4042−55.90.03060.02800.02970.000
0.502.50.9163MS1.011210.40.91720.10.29530.33480.32500.928
GH0.99658.80.96044.80.34750.26070.41620.805
RG0.7736−15.60.7737−15.60.11500.11180.10990.683
GG0.98867.90.98037.00.32730.37070.41150.961
NA0.1975−78.40.1969−78.50.02160.01960.02060.000
0.904.01.3863MS1.43613.61.39800.80.15370.23730.19280.938
GH1.46385.61.399 71.00.26680.21050.37590.824
RG1.2871−7.21.2805−7.60.06750.06480.06650.606
GG1.619116.81.51459.20.45350.48860.72400.969
NA1.0479−24.41.0419−24.80.04250.03920.04210.000
0.704.01.3863MS1.50638.71.41752.30.29520.34540.30200.920
GH1.551511.91.46045.30.39850. 28870.50800.807
RG1.1605−16.31.1516−16.90.10260.09860.09940.383
GG1.619116.81.51459.20.45350.48860.72400.969
NA0.5725−58.70.5714−58.80.03480.02850.03100.000
0.504.01.3863MS1.49317.71.40661.50.37960.44360.41720.918
GH1.545511.51.44214.00.45680.31850.55940.792
RG1.0735−22.61.0742−22.50.13300.13340.12960.367
GG1.619116.81.51459.20.45350.48860.72400.969
NA0.2753−80.10.2737−80.30.02560.01970.02090.000

Simulation results for the multiple-covariate rare disease case with with dependent covariates and dependent measurement error. β* is the true value of β. Bias(%) is the relative bias, i.e.Bias (%)=100×(β^−β*)/β*. IQR is 0.74 times the interquartile range of the β^ values. SE is the mean of the estimated standard error of β^. SD is the empirical standard deviation of the β^ values. CR is the empirical coverage rate of the asymptotic 95% confidence interval. Methods considered: MS = modified score, CH = Chen, RC = regression calibration, CC = complete case, NA = naive.

Corr(X, W)exp(β*)β*MethodMeanMedianIQRSESDCR
β^Bias(%)β^Bias(%)
0.901.50.4055MS0.41321.90.4034−0.50.05350.06380.07170.949
GH0.42565.00.41843.20.17160.15100.21170.840
RG0.3882−4.30.3845−5.20.04750.04670.04940.930
GG0.43737.80.42083.80.37440.35290.38910.961
NA0.3201−21.10.3190−21.30.03850.03730.03890.363
0.701.50.4055MS0.40971.00.3965−2.20.07410.09370.08750.953
GH0.42474.70.3979−1.90.31050.24020.39890.832
RG0.3656−9.80.3627−10.50.05520.06120.06210.875
GG0.43737.80.42083.80.37440.35290.38910.961
NA0.1804−55.50.1782−56.00.03080.02800.02940.000
0.501.50.4055MS0.42494.80.3938−2.90.12260.16630.15910.936
GH0.3966−2.20.3846−5.20.36260.28140.48330.816
RG0.3519−13.20.3498−13.70.08480.08760.08420.890
GG0.43737.80.42083.80.37440.35290.38910.961
NA0.0897−77.90.0885−78.20.02260.02000.02090.000
0.902.50.9163MS0.93301.80.9018−1.60.08990.11340.12290.929
GH0.94252.90.9136−0.30. 21880.18300.26000.871
RG0.8400−8.30.8350−8.90.04950.05120.05410.625
GG0.98867.90.98037.00.32730.37070.41150.961
NA0.6922−24.50.6923−24.40.03860.03560.03700.000
0.702.50.9163MS0.94993.70.9150−0.10.15120.18000.18430.921
GH0.97706.60.93712.30.31830.27310.43400.855
RG0.7789−15.00.7735−15.60.07990.07330.07420.484
GG0.98867.90.98037.00.32730.37070.41150.961
NA0.3834−58.20.3838−58.10.02970.02620.02730.000
0.502.50.9163MS0.97636.50.8980−2.00.26040.30320.28470.896
GH0.96595.40.92851.30.32320.31160.49710.863
RG0.7554−17.60.7568−17.40.11470.11100.10630.641
GG0.98867.90.98037.00.32730.37070.41150.961
NA0.1929−79.00.1929−78.90.02140.01890.02000.000
0.904.01.3863MS1.42132.51.3515−2.50.15230.18130.17130.870
GH1.46685.81.40181.10.28390.23500.41040.832
RG1.2054−13.11.1996−13.50.05610.06020.06090.160
GG1.619116.81.51459.20.45350.48860.72400.969
NA0.9925−28.40.9893−28.60.03870.03620.03860.000
0.704.01.3863MS1.43043.21.3619−1.80.27690.31310.30110.870
GH1.51999.61.42612.90.35730.32320.54170.848
RG1.0864−21.61.0744−22.50.09610.08830.08860.129
GG1.619116.81.51459.20.45350.48860.72400.969
NA0.5337−61.50.5336−61.50.03090.02590.02810.000
0.504.01.3863MS1.44894.51.3289−4.10.36430.45920.43740.871
GH1.52049.71.44374.10.36620.35630.56150.848
RG1.0487−24.41.049 3−24.30.13920.13320.12770.3 24
GG1.619116.81.51459.20.45350.48860.72400.969
NA0.2686−80.60.2676−80.70.02300.01880.02040.000

HPFS Results. SE = standard error of estimate. SE Ratio = Ratio between the standard error of the estimate and the standard error of the modified score estimate. Methods considered: MS = modified score, CH = Chen, RC = regression calibration, CC = complete case, NA = naive.

Diet Score CoefficientBMI Coefficient
MethodEstimateSESE RatioEstimateSESE Ratio
Naive0.02160.00270.11070.08670.00190.2346
CC0.07880.07383.02460.09130.133516.4815
RC0.04850.00960.39340.08670.00780.9630
CH0.01360.03831.56970.08000.02202.7160
MS0.07120.02441.00000.08650.00811.0000