Using Controlled Feeding Study for Biomarker Development in Regression Calibration for Disease Association Estimation

Abstract

Correction for systematic measurement error in self-reported data is an important challenge in association studies of dietary intake and chronic disease risk. The regression calibration method has been used for this purpose when an objectively measured biomarker is available. However, a big limitation of the regression calibration method is that biomarkers have only been developed for a few dietary components. We propose new methods to use controlled feeding studies to develop valid biomarkers for many more dietary components and to estimate the diet disease associations. Asymptotic distribution theory for the proposed estimators is derived. Extensive simulation is performed to study the finite sample performance of the proposed estimators. We applied our method to examine the associations between the sodium/potassium intake ratio and cardiovascular disease incidence using the Women’s Health Initiative cohort data. We discovered positive associations between sodium/potassium ratio and the risks of coronary heart disease, nonfatal myocardial infarction, coronary death, ischemic stroke, and total cardiovascular disease.

1. Introduction

There is an urgent need to obtain reliable information on dietary patterns that can reduce the risks of various chronic diseases, such as cancer, cardiovascular disease (CVD) and diabetes. Although a positive association between obesity and cancer risk is well established (Adams et al., 2006), epidemiological studies have not shown convincing evidence that key energy balance factors, including total energy intake, are risk factors for major chronic diseases (WCRF/AICR, 2007). A likely cause of this apparent discrepancy is bias in dietary assessment, which is known to be challenging to deal with, see e.g. (Paeratakul et al., 1998). There is strong evidence (Prentice et al., 2011) that the misreporting of dietary energy intake is related to individual characteristics (for example, body mass index (BMI)). Such systematic measurement error leads to bias that cannot be automatically corrected (Carroll et al., 2006). The challenge of having measurement error and its impact in nutritional study have been reviewed by Freedman et al. (2011). Methods to deal with dietary measurement error in nutritional studies has been studied for years and various approaches have been proposed (Bartlett & Keogh, 2018, Hu & Lin, 2002, Huang & Wang, 2000, Li & Ryan, 2006, Song & Huang, 2005, Wang et al., 1997, Yan & Yi, 2015, Zucker, 2005). Among these methods, we will focus on regression calibration method which allows us to handle the covariate dependent measurement error. Regression calibration has been proposed to correct measurement error in covariates (Prentice, 1982, Rosner et al., 1990) and has the advantage of being easy to implement. Some previous methodology, and their applications to the Women’s Health Initiative (WHI) (Prentice, 1982, Prentice et al., 2011, Shaw & Prentice, 2012, Zheng et al., 2014), have shown the validity and value of using (joint) regression calibration approaches to tackle this issue when objective measurements are available to be used as biomarkers of dietary intakes. These biomarkers are used to build calibration equations for self-reported measurements of the exposures of interest. Calibrated intake estimates can then be used to estimate associations between these dietary exposures and the risks of various diseases.

There remains an important research gap in building satisfactory biomarkers for many nutritional (and physical activity) variables using only univariate objective measurements. Therefore, regression models have been used to build biomarkers with multiple predictor variables. For example in the WHI Nutrition and Physical Activity Assessment Study (NPAAS), to correct systematic measurement error in the self-reported food frequency questionnaire (FFQ) data from the full cohort (with 161,808 subjects), blood and urine measurements were collected for a subgroup (450 subjects) of the cohort (Prentice et al., 2011). Besides, a feeding study was performed on another smaller subgroup (153 subjects) where both blood and urine measurements and the provided dietary intake information were collected (Lampe et al., 2017). This controlled feeding study used a novel design. Rather than feed all women the same standard diets, each woman was provided food that mimicked her habitual diet as described by her 4-day food record (4FDR) with adjustment based on individual discussion with the study dietitian (Lampe et al., 2017).

One might consider a natural strategy of developing calibration equations using biomarkers developed from the feeding study: First, with the 153 subjects from the feeding study, one builds a model to predict the provided dietary intakes using the blood and urine measurements along with some personal characteristics of the subjects (Lampe et al., 2017). Second, estimation of the ‘true’ dietary intake is made for the 450 subjects using their blood and urine measurements and the prediction model from the first step; and a calibration equation for the self-reported dietary intakes can be established. Previously developed regression calibration methods (e.g. Zheng et al. (2014)) used an externally developed biomarker, which plausibly satisfies the classical measurement error assumption. The classical measurement error assumption requires the measurement error to be independent of the true value. This assumption, however, will generally be violated by this feeding study based biomarker development procedure because the residual of the regression model is indpendent of the predicted value rather than the true value. Ignoring this violation of the classical measurement error assumption causes bias in the subsequent estimates of the calibrated dietary intake and the diet-disease association. To tackle this issue, in this project, we aim to develop new calibration methods that can allow for Berkson-type errors (i.e, an error that is independent of the error-prone variable rather than the error-free variable (Carroll et al., 2006)) in the biomarkers. Using our new methods, we establish consistent estimators for diet-disease associations; we incorporate variation in the biomarker construction step and the calibration step when estimating asymptotic distributions. From these, one can build valid confidence intervals for disease association parameters.

To evaluate the performance of the new methods and demonstrate their real data application, we applied the new methods to evaluate the association between the dietary sodium to potassium ratio and CVD risk. This association has been evaluated in a recent WHI study using a regression calibration method where the nutrient intake was calibrated by a single measurement biomarker (from a single 24-hour urine collection) (Huang et al., 2013, Prentice et al., 2017). In a different cohort, the large international Prospective Urban Rural Epidemiology (PURE) Study measured sodium and potassium intakes using fasting morning spot urine (O’Donnell et al., 2014). These previous studies suggested positive associations between CVD and sodium to potassium ratio. However, the single measure biomarker is suboptimal for the performance of the regression calibration approach, as it has a lower than desirable correlation with the true dietary intakes in the WHI context. The data from NPAAS provide a golden opportunity to build potentially stronger biomarkers for sodium and potassium intakes into disease association studies using novel regression methods.

2. Framework and Notation

We are interested in studying the association between a specific dietary intake $Z\in \mathbb{R}$ (for example the (log-transformed) dietary sodium-potassium ratio) and the time, $T$, to development of a certain chronic disease. We further consider some potential confounding variables, which we call personal characteristics $V\in {\mathbb{R}}^q$ where $q$ is the number of covariates. We use a Cox model to model the hazard of the response:

$$\lambda (t\mid Z,V,Q)=\lambda (t\mid Z,V)={\lambda }_0\left(t\right)\exp \left(\left(Z,V^{\mathrm{\top }}\right)\theta \right),$$ (1)

where $\theta ={\left({\theta }_z,{\theta }_v^{\mathrm{\top }}\right)}^{\mathrm{\top }}\in {\mathbb{R}}^{(1+q)},{\theta }_z$ is the parameter of interest, and ${\lambda }_0\left(t\right)$ is a ‘baseline’ hazard function. However, instead of observing $Z$, we only collect data on the self-reported dietary intake $Q\in \mathbb{R}$, which may be biased from $Z$ in a manner that depends on personal characteristics:

$$Q=\left(1,Z,V^{\mathrm{\top }}\right)a+{\epsilon }_q,$$ (2)

where $a\in {\mathbb{R}}^{(2+q)}$ is a unknown parameter vector and ${\epsilon }_q$ is a mean 0 random error that is independent of $Z$ and $V$.

The main purpose of the regression calibration equation development is to discover the relationship between $Q$ and $Z$. In the WHI feeding study, we provide the subjects’ diets using standardized food, in a manner that mimics the subjects’ usual diet, while using dietary components having well-characterised nutrient content (Lampe et al., 2017). We denote the true unobservable dietary intake within this two-week feeding period as $X=Z+{\epsilon }_x$, where ${\epsilon }_x\sim N\left(0,{\sigma }_x^2\right)$ is independent of $Z$ and $V$. A subtle issue with the feeding study relates to measurement error induced by food packaging. For example, a bag of chips that is labeled with 100 calories in energy could actually have 101 calories. Therefore, the observed short-term dietary intake $\tilde{X}$ in the feeding study can be modeled as $\tilde{X}=X+{\tilde{\epsilon }}_x$, where ${\tilde{\epsilon }}_x\sim N\left(0,{\tilde{\sigma }}_x^2\right)$ is independent of ${\epsilon }_x,$ $Z$ and $V$. Figure 1 displays the study design, including the feeding study (Sample 1) for biomarker development, the biomarker substudy (Sample 2) for calibration equation development, and the full cohort (Sample 3) for disease association analyses. When a self-reported intake $Q$ is also available for the feeding study samples, the bias of self-reported dietary intake can be calibrated directly (see section 3.4). However, the concurrent self-reported dietary intake $Q$ is typically unavailable in Sample 1. To obtain that one would need to have a long-term feeding study where individuals report their provided dietary intakes over preceding months (e.g. 3 months). In addition, the $Q$ value obtained just prior to the feeding period in NPAAS-FS is not collected at the same time as biomarker $W$ and it might be inappropriately strongly correlated with $\tilde{X}$. Therefore, in practice, our best choice is to use the baseline $Q$ collected at a different time (at baseline for Sample 3 in our example) for Sample 1. This baseline $Q$ has been successfully used in studies for multiple dietary components (e.g. protein and carbohydrate) (Prentice et al., 2021a,b). However, there is a time interval between the data collection of this baseline $Q$ and measurement time for $(\bar{X},V,W,Z)$ in Sample 1. Therefore, there is a concern that the conditional distribution $(Q\mid \tilde{X},V,W,Z)$ in Sample 1 might be different from Samples 2 and 3 for some dietary components. In this case, we treat $Q$ as unavailable in Sample 1. Even when $Q$ is available, the sample size of a feeding study is typically quite limited, possibly leading to non-satisfactory efficiency for disease association estimates. Alternatively, a biomarker $W\in {\mathbb{R}}^P$, which is composed of $p$ objectively measured blood and urine measurements can be used to bridge between $\tilde{X}$ in the feeding study sample and $Q$ from another larger sample. We consider the blood and urine measurements $W$ are affected by the short term diet $X$ while the self-reported questionnaire data are directly affected by the long term diet $Z$. We assume $W$ to follow a parametric model:

$$W={\left[\left(1,X,V^{\mathrm{\top }}\right)B\right]}^{\mathrm{\top }}+{\epsilon }_w,$$

where $B\in {\mathbb{R}}^{(2+q)\times p}$ is an unknown parameter and ${\epsilon }_w\sim N\left(0,{\sigma }_w^2{I}_p\right)$ is independent of ${\epsilon }_x,{\tilde{\epsilon }}_x,{\epsilon }_q,Z,V$ and $B$.

Fig. 1:

a. The flow chart of the whole process from biomarker construction to association estimation; b. Direct Acyclic Graph for the variables; c. Measurements between datasets. In the plots, we use $Z$ to denote the (unobservable) true long-term dietary intake, $V$ the personal characteristics, $W$ the biomarker from objectively blood and urine measurements, $Q$ the self-reported dietary intake, $X$ the (unobservable) true short-term dietary intake, $\tilde{X}$ the observed short-term dietary intake, $\left(T^{\star },\mathrm{\Delta }\right)$ the time to event/censoring and indicator of event/censoring.

Estimation of the association between $Z$ and $T$ is composed of 3 stages with non-overlapping samples from the same underlying population: namely, 1. the biomarker construction stage, 2. the calibration stage, and 3. the association assessment stage (Figure 1(b)). For each stage, a different sample is used. We denote the sample size of stage $k$ as $n_k$. For stage 1, we have $n_1$ samples and for every individual $i$, we have data $\left({\tilde{X}}_i,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }}\right)$ and possibly $Q_i$ available; for stage 2, we have $n_2$ samples and for every individual $i$, we have $\left(Q_i,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }}\right)$; for stage 3, we have $n_3$ samples and for every individual $i$, we have $\left(Q_i,V_i^{\mathrm{\top }}\right)$ and the composite outcome $(T_i^{\mathrm{*}}=T_i\wedge C_i,{\Delta }_i=I(T_i\le C_i))$ where $T_i$ is the time to disease occurrence and $C_i$ is a potential censoring time.

The estimation procedure is as follows. In stage 1, we use data from the biomarker construction stage to establish the biomarker. This model can be built by regressing the observed short term dietary intakes $\tilde{X}$ on one of the following:

1. blood/urine measurements $W$ and personal characteristics $V$;

2. blood/urine measurements $W$, self-reported dietary intake $Q$, and personal characteristics $V$;

3. self-reported dietary intake $Q$ and personal characteristics $V$.

Remark 1

As mentioned above, self-reported dietary intake $Q$ might be considered to be unavailable in stage 1. In this case, we treat $Q$ in stage 1 as unavailable and adopt option (i) in stage 1. When $Q$ is available in stage 1, option (ii) potentially improves the estimation of $\tilde{X}$ in stage 1. When the biomarker $W$ is unavailable, option (iii) directly model $\tilde{X}$ based on $Q$ and $V$; the performance will potentially be affected by the limited sample size $n_1$.

In Stage 2, using data from the calibration stage, we build a calibration equation using the self-reported log-transformed dietary intake $Q$ and the personal characteristics $V$ to predict the true dietary intake $X$ if (i) or (ii) is used in Stage 1. If (iii) is used in Stage 1, then the calibration equation was already established in Stage 1 and thus Stage 2 is omitted. One caveat is that developing biomarker using (i) would lead to Berkson-type error and have impact on the regression calibration, as will be shown later. To see this, asymptotically $\widehat{X}$ converges to :

$$\mathbb{E}[X\mid W,V]=X-{\epsilon }_x=X+{\epsilon }_x^{\prime },$$

where, under a normality assumption for $X,{\epsilon }_x^{\prime }=-{\epsilon }_x$ is independent of $\mathbb{E}[X\mid W,V]$ but not independent of $X$. Therefore ${\epsilon }_x^{\prime }$ is a Berkson-type error instead of a classical measurement error. Therefore new methods are needed to account for this property.

In Stage 3, we only have information on the self-reported dietary intake $Q$, the personal characteristics $V$, and the composite survival outcome $(T^{\mathrm{*}},\Delta )$. We use the calibration equation developed in Stage 2 to calibrate the self-reported dietary intake for the large cohort and perform disease association analyses.

Figure 1(c) shows the variable availabilities among the three samples. When $Q$ is not available in the stage 1 sample, option (i) can be considered for this stage.

Remark 2

In Stage 1, we consider both scenarios where $Q$ is available in Sample 1 and $Q$ is not available in Sample 1 (in the sense that the joint distribution of $(Q,\tilde{X},V,W)$ are different between Stage 1 and the other two stages). When $Q$ is not available in Sample 1, although the joint distribution of $(Q,\tilde{X},V,W)$ is not nonparametrically identifiable, under our model assumption the first order moment $\mathbb{E}(\tilde{X}\mid Q,V,W)$ is identifiable, which enables us to obtain the consistent estimator of $\tilde{X}$ when using regression calibration approach under the rare event assumption. More details are discussed in Section 7.

3. Method

In this section, we first introduce the traditional regression calibration method (Method 1) and then our three newly proposed methods (Method 2–4). We have $n=n_1+n_2+n_3$ subjects $S_1,\cdots ,S_n$, with $S_{1:n_1}$ being the biomarker construction sample; $S_{n_1+1:n_1+n_2}$ the calibration sample; and $S_{n_1+n_2+1:n}$ the association assessment sample.

3.1. Method 1: The naïve three-step approach

We first consider the naïve three-step approach.

Step 1: With the stage 1 sample, regress the consumed diet $\left(\tilde{X}\right)$ on the blood and urine measurements $\left(W\right)$ and the personal characteristics $\left(V\right)$ to obtain

$${\widehat{\beta }}_1={\left\{\sum _{i=1}^{n_1} {\left(1,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}\left(1,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }}\right)\right\}}^{-1}\left\{\sum _{i=1}^{n_1} {\left(1,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}{\tilde{X}}_i\right\}.$$

Step 2: For the stage 2 sample, compute ${\widehat{X}}_{1i}=\left(1,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }}\right){\widehat{\beta }}_1$ for $i=n_1+1,\cdots ,n_1+n_2$ as a predictor for $X$. Then regress ${\widehat{X}}_1$ on $Q$ and $V$ to obtain calibration equation parameter estimate

$${\widehat{\gamma }}_1={\left\{\sum _{i=n_1+1}^{n_1+n_2} {\left(1,Q_i,V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}\left(1,Q_i,V_i^{\mathrm{\top }}\right)\right\}}^{-1}\left\{\sum _{i=n_1+1}^{n_1+n_2} {\left(1,Q_i,V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}{\widehat{X}}_{1i}\right\}.$$

Step 3: We first estimate $Z$ in the stage 3 sample with ${\widehat{Z}}_{1i}=\left(1,Q_i,V_i^{\mathrm{\top }}\right){\widehat{\gamma }}_1$ for $i=n_1+n_2+1,\cdots ,n_1+n_2+n_3$. Then we estimate the association of $Z$ with the time-to-event endpoint $(T^{\mathrm{*}},\Delta )$ by solving the score equation for Cox model:

$$0=\sum _{i=n_1+n_2+1}^{n_1+n_2+n_3} {\int }_0^{\tau } \left[\left(\begin{array}{c}{\widehat{Z}}_{1i}\\ V_i\end{array}\right)-\sum _j \frac{Y_j(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{({\widehat{Z}}_{1j},V_j^{\mathrm{\top }})\theta \right\}}{\sum _k Y_k(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{({\widehat{Z}}_{1k},V_k^{\mathrm{\top }})\theta \right\}}\left(\begin{array}{c}{\widehat{Z}}_{1j}\\ V_j\end{array}\right)\right]d{N}_i(t).$$

where $\tau$ is a pre-specified large number and we assume $P(C>\tau )>0,N_i(t)=I({\mathrm{\Delta }}_i=1,T_i^{\mathrm{*}}\le t)$ and $Y_i(t)=I(T_i^{\mathrm{*}}\ge t)$. In application, $\tau$ is typically defined to be the largest follow-up time in the (Step 3) sample.

Remark 3

We show in the Appendix A.1 that

$$\mathbb{E}({\widehat{Z}}_1\mid Q,V)=\mathrm{B}\mathrm{F}\times \mathbb{E}(Z\mid Q,V)+(1-\mathrm{B}\mathrm{F})\times \mathbb{E}(Z\mid V)\ne \mathbb{E}(Z\mid Q,V),$$

where the bias factor (BF) is given by

$$\mathrm{B}\mathrm{F}=1-\frac{\mathrm{Var}\left(X∣V,W\right)}{\mathrm{Var}\left(X∣V\right)}=R_{1\mid V}^2.$$

Such BF will lead to bias in the estimation of association parameter $\theta$. Upon further assuming $\mathbb{E}(X\mid V)=\left(1,V^{\mathrm{\top }}\right)\delta$ where $\delta \in {\mathbb{R}}^{1+q}$ is unknown parameter, we show in Appendix A.1 and A.2 that the estimator ${\widehat{\theta }}_1\to {\theta }_1^{\mathrm{*}}$ as $n\to \mathrm{\infty }$, with ${\theta }_{1z}^{\mathrm{*}}\approx {\mathrm{B}\mathrm{F}}^{-1}\times {\theta }_z$ and ${\theta }_{1v}^{\mathrm{*}}\approx {\theta }_v-\frac{1-\mathrm{B}\mathrm{F}}{\mathrm{B}\mathrm{F}}{\theta }_{1z}$. The approximation error is ignorable only under rare disease assumption, i.e., $P(T<t\mid Z,V)\to 0$ when $n\to \mathrm{\infty }$ for all levels of $Z$ and $V$ and $t\in [0,\tau ]$. Here we assume $\mathrm{B}\mathrm{F}>0$. When $\mathrm{B}\mathrm{F}=0$, the variance of ${\hat{\theta }}_1$ will be infinity because ${\theta }_z$ is not identifiable. By definition of BF, when $\mathrm{B}\mathrm{F}=0$, the biomarker $W$ contains no information of $X$ given $V$, therefore we are not able to separate the effect of $X$ and $V$.

3.2. Method 2: Three-step with Bias Correction

As shown for Method 1, we have bias in ${\widehat{Z}}_1$ when using ${\widehat{X}}_1$ in Step 2. Therefore, we propose a bias-corrected estimator ${\widehat{X}}_{2i}={\widehat{\mathrm{B}\mathrm{F}}}^{-1}{\widehat{X}}_{1i}$ where

$$\widehat{\mathrm{B}\mathrm{F}}={\widehat{R}}_{1\mid V}^2=1-\frac{\widehat{Var}(\tilde{X}\mid V,W)-{\tilde{\sigma }}_x^2}{\widehat{Var}(\tilde{X}\mid V)-{\tilde{\sigma }}_x^2}$$

is the estimated BF. Here, we assume that ${\tilde{\sigma }}_x^2$ is known. In many real applications, ${\tilde{\sigma }}_x^2$ can be estimated with a separate experiment, as we discuss in Section 7. When ${\tilde{\sigma }}_x^2$ is not available, we may vary this parameter to perform sensitivity analysis.

Then we have

$$\widehat{{\gamma }_2}=\sum _{i=n_1+1}^{n_1+n_2} {\left\{{\left(1,Q_i,V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}\left(1,Q_i,V_i^{\mathrm{\top }}\right)\right\}}^{-1}\sum _{i=n_1+1}^{n_1+n_2} \left\{{\left(1,Q_i,V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}{\hat{X}}_{2i}\right\}.$$

In step 3, we predict the exposure by ${\widehat{Z}}_{2i}=\left(1,Q_i,V_i^{\mathrm{\top }}\right)\widehat{{\gamma }_2}$ for $i=n_1+n_2+1,\cdots ,n_1+n_2+n_3$ and obtain estimator ${\widehat{\theta }}_2$ by solving the following estimating equations:

$$0=\sum _{i=n_1+n_2+1}^{n_1+n_2+n_3} {\int }_0^{\tau } \left[\left(\begin{array}{c}{\widehat{Z}}_{2i}\\ V_i\end{array}\right)-\sum _j \frac{Y_j(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{({\widehat{Z}}_{2j},V_j^{\mathrm{\top }})\theta \right\}}{\sum _k Y_k(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{({\widehat{Z}}_{2k},V_k^{\mathrm{\top }})\theta \right\}}\left(\begin{array}{c}{\widehat{Z}}_{2j}\\ V_j\end{array}\right)\right]d{N}_i(t).$$

Alternatively, instead of correcting $\widehat{X}$ which allows estimation of both ${\hat{\theta }}_{2z}$ and ${\hat{\theta }}_v$, we can also compute ${\widehat{\theta }}_{2z}$ as ${\widehat{\theta }}_{1z}\times \widehat{\mathrm{B}\mathrm{F}}$.

Remark 4

This method does not require the self-reported dietary intake data $\left(Q\right)$ to be collected in the feeding study. As a remark, even if the self-reported data is available in the feeding study (Stage 1), the association between the self-reported and the actual dietary intake in the feeding study might be different from that association from the cohort for reasons such as the modification of the dietary pattern during the controlled feeding study, or the potential change in dietary preference in the period of the feeding study. Method 2 is robust to such an association difference since we have not directly included $Q$ in Stage 1 for biomarker construction. However, when $Q$ is available among Stage 1 samples and its association with $Z$ remains unchanged among all stages, we can use its information to improve efficiency. We will next propose two methods that require the availability of the self-reported dietary intake in the feeding study and assume the association between the self-reported dietary intake and the actual dietary intake to be the same among all three samples.

3.3. Method 3: Three-step with $Q$ available in stage 1

When the self-reported dietary intake $Q$ is available from the feeding study and the distribution of $\left(Q\right|Z,V)$ are the same between controlled feeding study and the cohort, then the bias in the naïve estimator can be corrected simply by including $Q$ in the biomarker development equation because the inclusion of $Q$ guarantees that

$$E[\widehat{Z}\mid Q,V]=E[E[Z\mid Q,V,W]\mid Q,V]=E[Z\mid Q,V].$$

The three steps of the first method remain the same, but in the first step regression model, the self-reported dietary intake $\left(Q\right)$ is added. That is, for the first step, we regress $\tilde{X}$ on $W,V$ and $Q$ to build the biomarker, and then use $W,V$ and $Q$ to predict $Z$ in the second step. Specifically, we have

$${\widehat{\beta }}_3={\left\{\sum _{i=1}^{n_1} {\left(1,W_i^{\mathrm{\top }},Q_i,V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}\left(1,W_i^{\mathrm{\top }},Q_i,V_i^{\mathrm{\top }}\right)\right\}}^{-1}\left\{{\left(1,W_i^{\mathrm{\top }},Q_i,V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}{\tilde{X}}_i\right\},$$

${\widehat{X}}_{3i}=\left(1,W_i^{\mathrm{\top }},Q_i,V_i^{\mathrm{\top }}\right){\widehat{\beta }}_3$ for $i=n_1+1,\dots ,n_1+n_2$, and then

$${\widehat{\gamma }}_3={\left\{\sum _{i=n_1+1}^{n_1+n_2} {\left(1,Q_i,V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}\left(1,Q_i,V_i^{\mathrm{\top }}\right)\right\}}^{-1}\left\{{\left(1,Q_i,V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}{\widehat{X}}_{3i}\right\},$$

and ${\widehat{Z}}_{3i}=\left(1,Q_i,V_i^{\mathrm{\top }}\right){\widehat{\gamma }}_3$ for $i=n_1+n_2+1,\cdots ,n_1+n_2+n_3$. We obtain ${\widehat{\theta }}_3$ by solving the following estimating equations:

$$0=\sum _{i=n_1+n_2+1}^{n_1+n_2+n_3} {\int }_0^{\tau } \left[\left(\begin{array}{c}{\widehat{Z}}_{3i}\\ V_i\end{array}\right)-\sum _j \frac{Y_j(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{({\widehat{Z}}_{3j},V_j^{\mathrm{\top }})\theta \right\}}{\sum _k Y_k(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{({\widehat{Z}}_{3k},V_k^{\mathrm{\top }})\theta \right\}}\left(\begin{array}{c}{\widehat{Z}}_{3j}\\ V_j\end{array}\right)\right]d{N}_i(t)$$

3.4. Method 4: Direct Estimation

When $Q$ is available from the feeding study, another possibility is to ignore the second dataset (or not require a second dataset to be available) and directly build the estimating equation by regressing $\tilde{X}$ on $Q$ and $V$ in the first step and directly apply it in the third step. All other steps remain the same as the third method, except that we ignore the $n_2$ calibration samples and directly build the calibration equation using the feeding study by regressing $\tilde{X}$ on $V$ and $Q$ and use the calibration equation to predict $Z$ and perform a regression of $Y$ on $Z$ and $V$ in the full cohort to estimate the association parameter. In other words, we have

$${\widehat{\gamma }}_4={\left\{\sum _{i=1}^{n_1} {\left(1,Q_i,V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}\left(1,Q_i,V_i^{\mathrm{\top }}\right)\right\}}^{-1}\left\{{\left(1,Q_i,V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}{\tilde{X}}_i\right\},$$

with ${\widehat{Z}}_{4i}=\left(1,Q_i,V_i^{\mathrm{\top }}\right){\widehat{\gamma }}_4$ for $i=n_1+n_2+1,\cdots ,n_1+n_2+n_3$ and ${\widehat{\theta }}_4$ by solving the following estimating equations:

$$0=\sum _{i=n_1+n_2+1}^{n_1+n_2+n_3} {\int }_0^{\tau } \left[\left(\begin{array}{c}{\widehat{Z}}_{4i}\\ V_i\end{array}\right)-\sum _j \frac{Y_j(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{({\widehat{Z}}_{4j},V_j^{\mathrm{\top }})\theta \right\}}{\sum _k Y_k(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{({\widehat{Z}}_{4k},V_k^{\mathrm{\top }})\theta \right\}}\left(\begin{array}{c}{\widehat{Z}}_{4j}\\ V_j\end{array}\right)\right]d{N}_i(t).$$

4. Asymptotics

In this section, we derive the asymptotic distributions of the four estimators and show that Method 1 tends to give biased result while the other three methods provide consistent estimators under rare disease assumption $\left(P\right(T<t\mid Z,V)\to 0$ when $n\to \mathrm{\infty }$ for $t\in [0,\tau ])$.

In practice, the violation of rare disease assumption will lead to estimation bias in Methods 2–4 (i.e., ${\theta }_2^{\mathrm{*}},{\theta }_3^{\mathrm{*}},{\theta }_4^{\mathrm{*}}$ as shown in theorems below can be different from $\theta$), but the extent of the biases from Methods 2–4 are usually much smaller than that of Method 1 according to our numerical study results. Here we summarize the asymptotic results in the following theorem with the proofs given in Appendix A.2.

Theorem 1:

With $\frac{n_3}{n_2}\to C_2$ and $\frac{n_2}{n_1}\to C_1$, for $k=\mathrm{1,2},\mathrm{3,4}$, we have

$$\sqrt{n_3}({\widehat{\theta }}_k-{\theta }_k^{\mathrm{*}})\to N\left(0,{\Sigma }_{\theta k}\right)$$

where ${\Sigma }_{\theta k}=I_{\theta k}^{-1}\left(I_{\theta k}+C_2{I}_{\gamma k}{\Sigma }_{\gamma k}{I}_{\gamma k}^{\mathrm{\top }}\right)I_{\theta k}^{-\mathrm{\top }}$ can be consistently estimated by ${\widehat{\Sigma }}_{\theta k}={\widehat{I}}_{\theta k}^{-1}({\widehat{I}}_{\theta k}+\frac{n_3}{n_2}{\widehat{I}}_{\gamma k}{\widehat{\Sigma }}_{\gamma k}{\widehat{I}}_{\gamma k}^{\mathrm{\top }}){\widehat{I}}_{\theta k}^{-\mathrm{\top }}$ with the detailed expressions defined in the proofs in Appendix A. Here the $I_{\theta k}$ component is the same for $k=\mathrm{2,3},4$, though its corresponding estimate ${\hat{I}}_{\theta k}$ can be different.

Intuitively, method 4 should be less efficient compared with method 2 and 3 as it ignores the information contained in $W$. Heuristically, to compare the efficiency of the estimators from method 3 and method 4, we can show that

$$E[Var({\hat{Z}}_3\mid Q,V)]/E[Var({\hat{Z}}_4\mid Q,V)]=\frac{n_1}{n_2}+(1-\frac{n_1}{n_2})(1-R_{(X,W)\mid Q,V}^2),$$ (3)

which suggests that the key quantity to characterise the usefulness of biomarker $W$ is the partial coefficient of multiple determination between $X$ and $W$ conditional on $Q$ and $V(R_{(X,W)\mid Q,V}^2)$. The closer the value of $R_{(X,W)|Q,V}^2$ is towards 0, the ‘weaker’ the biomarker is; the closer it is towards 1, the ‘stronger’ the biomarker is. Consider two extreme examples: (1) when $R_{(x,W)|Q,V}^2=0$, the relative efficiency is 1; method 3 does not have an efficiency gain compared with method 4; the biomarker is completely useless. (2) when $R_{(X,W)\mid QV}^2=1$, the relative efficiency is $\frac{n_1}{n_2}$. This is as if we have observed all $X$ information in the NPAAS dataset and the efficiency gain is proportional to the sample size gain. The asymptotic variance comparing ${\theta }_3$ and ${\widehat{\theta }}_4$ is always less than or equal to 1, which indicates a potential loss of efficiency by ignoring the second dataset. Also, method 2 should be less efficient than method 3 when $Q$ is available in the first stage sample as it ignores the information on $Q$ in the first sample.

5. Simulation Studies

We performed simulations to study the finite sample behavior of our proposed estimators. We generate exposure and covariates from the following models:

$$\begin{array}{l}\left(Z,V\right)\sim N\left(\mathbf{0},\left(\begin{array}{cc}1-{\sigma }_x^2& \rho \\ \rho & 1\end{array}\right)\right),\\ W=b_0+b_{1}X+b_{2}V+{\epsilon }_w,\\ X=Z+{\epsilon }_x,\\ \hat{X}=X+{\epsilon }_x,\\ Q=a_0+a_{1}Z+a_{2}V+{\epsilon }_q,\end{array}$$

where ${\epsilon }_w$, ${\epsilon }_x$, ${\tilde{\epsilon }}_x$ and ${\epsilon }_q$ are independently sampled from normal distributions with mean zero and with respective standard deviations ${\sigma }_w$, ${\sigma }_x$, ${\tilde{\sigma }}_x$ and ${\sigma }_q$. The event time is generated from Cox regression model:

$$\lambda \left(t∣Z,X,V,W,Q\right)=\lambda \left(t∣Z,V\right)={\lambda }_0\left(t\right)\exp \left({\theta }_{z}Z+{\theta }_{v}V\right).$$

Three different censoring distributions are considered: (1) Mixed censoring: the censoring time is sampled from a mixture of Unif(0,10) and a point mass at 10 with equal probability; (2) Uniform censoring: the censoring time is sampled from Unif(0,10); (3) Exponential censoring: the censoring time is sampled from exponential distribution with mean 10. We simulated with three sample sizes, $\left(n_1,n_2,n_3\right)=(150,300,5150)$, $\left(n_1,n_2,n_3\right)=(300,600,10300)$ and $\left(n_1,n_2,n_3\right)=(600,1200,20600)$. We set $b_0=5,b_2=1,$ ${\sigma }_x=0.2,$ ${\tilde{\sigma }}_x=0.5,$ ${\theta }_z=0.4,$ ${\theta }_v=0.6,$ ${\sigma }_w=1$ and ${\lambda }_0(t)=0.002,$ ${\lambda }_0(t)=0.004$ and ${\lambda }_0(t)=0.004t$. Then we changed the values of $b_1,$ $\rho ,$ $a_1,$ $a_2$ and ${\sigma }_q$ to vary the values of the coefficients of multiple determinations $\left(R^2\mathrm{s}\right)$, and partial $R^2\mathrm{s}$ conditional on $V$, that quantify the strength of $W$ and $Q$ in predicting $Z$. The detailed mathematical definitions and values of these $R^2\mathrm{s}$ under 6 different settings can be found in the Appendix B. The $R^2\mathrm{s}$ for regressions in step 1 and 2 in all settings are also listed in the Appendix B. For settings 1–3, we have weak $Q$ effects $(R_{ZQ\mid V}^2=0.13)$, with the effect of $W,R_{ZW\mid V}^2$, decreasing from 0.49 to 0.28. For settings 4–6, we have a stronger $Q$ effect $(R_{ZQ\mid V}^2=0.2)$ with the effect of $W,R_{ZW\mid V}^2$, decreasing from 0.53 to 0.19. The regression parameters and baseline hazards are chosen to illustrate the performance under the event rate of our real data application; the event rate ranges from 2% to 24% as listed in Appendix B.

Tables 1, 2 and 3 summarizes the simulation results for all six settings. The bias, mean estimated standard error (SE), empirical standard deviation (SD), and coverage rate (CR) of 95% nominal confidence interval for ${\theta }_z$ for all 4 methods with the three sample sizes from 1000 simulations are listed. The results show that method 1 tends to be biased even when the R² is high; while our proposed methods 2–4 performed uniformly better across all settings, with much smaller bias and reduced SE. When the sample size is small, i.e., $\left(n_1,n_2,n_3\right)=(150,300,5150)$, the CR for Method 1 seems to be acceptable, but it is because of the SE for Method 1 is also large under this small sample size; when the sample size is doubled in the setting where $\left(n_1,n_2,n_3\right)=(300,600,10300)$, the SE becomes smaller and the bias dominates the error. Among the three newly proposed methods, Method 3 performs better than Method 4 under larger sample sizes ($\left(\left(n_1,n_2,n_3\right)=(300,600,10300)\right.$ and $\left(n_1,n_2,n_3\right)=(600,1200,20600)$); observation is in accordance with our theoretical result for the asymptotic distributions. Method 2 has the best performance when $W$ has strong effect and $Q$ has weak effect (Setting 1). When the biomarker has a weak effect (Settings 3 and 6) and the sample size for the feeding study $\left(n_1\right)$ is small, Method 2 may have relatively large bias and variance because the estimated $BF$ is close to 0. The performance of Method 3 and Method 4 are less sensitive to the strength of $W$ and for small sample size, $\left(\left(n_1,n_2,n_3\right)=(150,300,5150)\right)$, the performance of Method 4 is good. Specifically, when the effect of $Q$ is weak (Setting 1–3), Method 4 outperforms Method 3 with smaller variance.

Table 1:

Simulation results comparing different methods’ performance in terms of bias, mean estimated standard error (SE), empirical standard deviation (SD) and coverage rate (CR) for 95% confidence interval when ${\lambda }_0\left(t\right)=0.002$.

Censoring	Setting	Method	$(n_1,n_2,n_3)=(150,300,5150)$				$(n_1,n_2,n_3)=(300,600,10300)$				$(n_1,n_2,n_3)=(600,1200,20600)$
			Bias	SE	SD	CR	Bias	SE	SD	CR	Bias	SE	SD	CR
Mixed	1	1	0.48	0.56	0.56	0.97	0.40	0.36	0.37	0.87	0.39	0.34	0.33	0.82
		2	0.05	0.29	0.29	0.97	0.01	0.19	0.19	0.96	0.01	0.18	0.17	0.96
		3	0.07	0.47	0.65	0.97	0.01	0.19	0.19	0.97	0.01	0.17	0.17	0.96
		4	0.05	0.30	0.30	0.98	0.02	0.19	0.20	0.97	0.01	0.18	0.18	0.96
	2	1	0.67	0.72	0.73	0.97	0.56	0.45	0.46	0.84	0.55	0.41	0.40	0.77
		2	0.06	0.31	0.31	0.97	0.01	0.19	0.20	0.97	0.01	0.18	0.18	0.97
		3	0.07	0.63	0.84	0.97	0.01	0.19	0.19	0.97	0.01	0.18	0.17	0.96
		4	0.05	0.30	0.30	0.98	0.02	0.19	0.20	0.97	0.01	0.18	0.18	0.96
	3	1	1.38	1.82	1.97	0.98	1.09	0.74	0.76	0.79	1.05	0.64	0.64	0.67
		2	0.10	0.50	0.55	0.96	0.02	0.21	0.21	0.96	0.02	0.19	0.18	0.96
		3	0.07	0.55	0.73	0.97	0.01	0.19	0.20	0.97	0.01	0.18	0.17	0.96
		4	0.05	0.30	0.30	0.98	0.02	0.19	0.20	0.97	0.01	0.18	0.18	0.96
	4	1	0.38	0.35	0.34	0.91	0.35	0.24	0.24	0.75	0.35	0.22	0.22	0.67
		2	0.03	0.19	0.19	0.96	0.01	0.13	0.13	0.96	0.01	0.12	0.12	0.96
		3	0.02	0.19	0.19	0.97	0.00	0.13	0.13	0.95	0.01	0.12	0.12	0.95
		4	0.02	0.19	0.19	0.97	0.01	0.13	0.14	0.95	0.01	0.12	0.12	0.96
	5	1	0.73	0.55	0.55	0.87	0.66	0.36	0.37	0.58	0.66	0.32	0.32	0.46
		2	0.03	0.21	0.22	0.96	0.01	0.14	0.14	0.96	0.01	0.13	0.13	0.96
		3	0.02	0.19	0.20	0.97	0.00	0.13	0.13	0.96	0.01	0.12	0.12	0.95
		4	0.02	0.19	0.19	0.97	0.01	0.13	0.14	0.95	0.01	0.12	0.12	0.96
	6	1	2.17	2.31	2.87	0.91	1.77	0.86	0.90	0.42	1.71	0.71	0.71	0.24
		2	0.09	0.45	0.60	0.94	0.02	0.17	0.17	0.96	0.02	0.14	0.14	0.96
		3	0.02	0.19	0.20	0.97	0.01	0.13	0.13	0.96	0.01	0.12	0.12	0.95
		4	0.02	0.19	0.19	0.97	0.01	0.13	0.14	0.95	0.01	0.12	0.12	0.96
Uniform	1	1	0.46	0.66	0.67	0.97	0.41	0.43	0.45	0.89	0.41	0.41	0.40	0.85
		2	0.04	0.34	0.34	0.97	0.02	0.22	0.23	0.96	0.02	0.21	0.21	0.97
		3	0.07	0.59	0.87	0.98	0.02	0.22	0.23	0.96	0.02	0.21	0.21	0.95
		4	0.04	0.35	0.36	0.97	0.02	0.23	0.24	0.96	0.02	0.21	0.21	0.95
	2	1	0.65	0.84	0.86	0.97	0.58	0.53	0.55	0.87	0.57	0.49	0.49	0.82
		2	0.05	0.36	0.37	0.97	0.02	0.23	0.23	0.96	0.02	0.21	0.21	0.96
		3	0.08	0.80	1.14	0.97	0.02	0.22	0.23	0.96	0.02	0.21	0.21	0.95
		4	0.04	0.35	0.36	0.97	0.02	0.23	0.24	0.96	0.02	0.21	0.21	0.95
	3	1	1.32	1.95	2.10	0.98	1.11	0.86	0.90	0.84	1.07	0.76	0.76	0.76
		2	0.08	0.55	0.60	0.96	0.03	0.24	0.25	0.96	0.02	0.22	0.22	0.97
		3	0.07	0.69	0.99	0.98	0.02	0.22	0.23	0.95	0.02	0.21	0.21	0.95
		4	0.04	0.35	0.36	0.97	0.02	0.23	0.24	0.96	0.02	0.21	0.21	0.95
	4	1	0.37	0.42	0.40	0.94	0.35	0.28	0.28	0.82	0.35	0.27	0.27	0.77
		2	0.02	0.23	0.22	0.97	0.01	0.15	0.15	0.97	0.01	0.15	0.15	0.96
		3	0.02	0.22	0.23	0.97	0.00	0.15	0.15	0.96	0.01	0.14	0.14	0.95
		4	0.02	0.22	0.22	0.97	0.01	0.15	0.16	0.96	0.01	0.15	0.15	0.96
	5	1	0.72	0.63	0.63	0.92	0.66	0.42	0.42	0.70	0.66	0.38	0.38	0.60
		2	0.03	0.24	0.24	0.96	0.01	0.16	0.16	0.97	0.01	0.15	0.15	0.96
		3	0.02	0.22	0.23	0.97	0.00	0.15	0.15	0.96	0.01	0.14	0.14	0.96
		4	0.02	0.22	0.22	0.97	0.01	0.15	0.16	0.96	0.01	0.15	0.15	0.96
	6	1	2.12	2.34	2.66	0.94	1.77	0.97	1.00	0.59	1.71	0.82	0.82	0.43
		2	0.08	0.46	0.54	0.94	0.02	0.19	0.18	0.96	0.02	0.16	0.16	0.97
		3	0.02	0.23	0.23	0.97	0.01	0.15	0.15	0.96	0.01	0.15	0.15	0.96
		4	0.02	0.22	0.22	0.97	0.01	0.15	0.16	0.96	0.01	0.15	0.15	0.96
Exponential	1	1	0.44	0.51	0.52	0.95	0.41	0.33	0.35	0.83	0.39	0.30	0.30	0.79
		2	0.03	0.26	0.27	0.97	0.01	0.17	0.18	0.96	0.01	0.16	0.16	0.96
		3	0.05	0.44	0.63	0.96	0.01	0.17	0.18	0.96	0.01	0.16	0.16	0.97
		4	0.03	0.27	0.27	0.96	0.02	0.18	0.19	0.95	0.01	0.16	0.16	0.96
	2	1	0.63	0.66	0.68	0.95	0.57	0.41	0.44	0.80	0.54	0.37	0.37	0.72
		2	0.04	0.28	0.29	0.96	0.02	0.18	0.18	0.96	0.01	0.16	0.16	0.97
		3	0.06	0.60	0.82	0.96	0.01	0.17	0.18	0.96	0.01	0.16	0.16	0.96
		4	0.03	0.27	0.27	0.96	0.02	0.18	0.19	0.95	0.01	0.16	0.16	0.96
	3	1	1.29	1.66	1.79	0.97	1.10	0.69	0.74	0.73	1.03	0.59	0.59	0.61
		2	0.07	0.46	0.51	0.95	0.02	0.19	0.20	0.95	0.01	0.17	0.17	0.97
		3	0.05	0.52	0.71	0.96	0.02	0.17	0.18	0.95	0.01	0.16	0.16	0.96
		4	0.03	0.27	0.27	0.96	0.02	0.18	0.19	0.95	0.01	0.16	0.16	0.96
	4	1	0.37	0.32	0.31	0.91	0.34	0.22	0.23	0.69	0.34	0.20	0.20	0.63
		2	0.02	0.18	0.17	0.97	0.00	0.12	0.12	0.96	0.00	0.11	0.11	0.96
		3	0.02	0.17	0.17	0.97	0.00	0.11	0.12	0.96	0.00	0.11	0.11	0.95
		4	0.02	0.17	0.17	0.97	0.01	0.12	0.12	0.95	0.00	0.11	0.11	0.95
	5	1	0.72	0.51	0.51	0.85	0.66	0.33	0.35	0.49	0.64	0.29	0.29	0.39
		2	0.03	0.19	0.20	0.97	0.01	0.13	0.13	0.96	0.00	0.11	0.11	0.96
		3	0.02	0.17	0.18	0.97	0.00	0.12	0.12	0.95	0.00	0.11	0.11	0.95
		4	0.02	0.17	0.17	0.97	0.01	0.12	0.12	0.95	0.00	0.11	0.11	0.95
	6	1	2.11	2.05	2.38	0.88	1.76	0.81	0.87	0.32	1.67	0.65	0.65	0.15
		2	0.08	0.40	0.49	0.93	0.02	0.16	0.16	0.94	0.01	0.13	0.13	0.96
		3	0.02	0.17	0.18	0.96	0.00	0.12	0.12	0.95	0.00	0.11	0.11	0.95
		4	0.02	0.17	0.17	0.97	0.01	0.12	0.12	0.95	0.00	0.11	0.11	0.95

Table 2:

Simulation results comparing different methods’ performance in terms of bias, mean estimated standard error (SE), empirical standard deviation (SD) and coverage rate (CR) for 95% confidence interval when ${\lambda }_0\left(t\right)=0.004$.

Censoring	Setting	Method	$(n_1,n_2,n_3)=(150,300,5150)$				$(n_1,n_2,n_3)=(300,600,10300)$				$(n_1,n_2,n_3)=(600,1200,20600)$
			Bias	SE	SD	CR	Bias	SE	SD	CR	Bias	SE	SD	CR
Mixed	1	1	0.44	0.41	0.43	0.97	0.40	0.26	0.29	0.74	0.38	0.24	0.24	0.66
		2	0.03	0.21	0.22	0.95	0.01	0.14	0.15	0.94	0.01	0.12	0.12	0.96
		3	0.04	0.37	0.54	0.95	0.01	0.14	0.14	0.94	0.01	0.12	0.12	0.97
		4	0.03	0.22	0.23	0.94	0.01	0.14	0.16	0.94	0.01	0.12	0.13	0.96
	2	1	0.62	0.54	0.59	0.93	0.56	0.33	0.36	0.68	0.38	0.24	0.24	0.66
		2	0.04	0.23	0.25	0.95	0.01	0.14	0.15	0.94	0.01	0.12	0.12	0.96
		3	0.05	0.50	0.71	0.95	0.01	0.14	0.15	0.94	0.01	0.12	0.12	0.97
		4	0.03	0.22	0.23	0.94	0.01	0.14	0.16	0.94	0.01	0.12	0.13	0.96
	3	1	1.29	1.64	2.00	0.94	1.08	0.57	0.62	0.55	1.03	0.47	0.47	0.36
		2	0.07	0.44	0.51	0.94	0.02	0.16	0.17	0.94	0.01	0.14	0.14	0.96
		3	0.05	0.43	0.62	0.95	0.01	0.14	0.15	0.94	0.01	0.12	0.12	0.97
		4	0.03	0.22	0.23	0.94	0.01	0.14	0.16	0.94	0.01	0.12	0.13	0.96
	4	1	0.37	0.26	0.27	0.82	0.35	0.17	0.19	0.52	0.34	0.15	0.15	0.39
		2	0.02	0.14	0.15	0.95	0.00	0.10	0.10	0.95	0.00	0.08	0.09	0.95
		3	0.01	0.14	0.14	0.95	0.00	0.09	0.10	0.95	0.00	0.08	0.08	0.95
		4	0.01	0.14	0.14	0.95	0.01	0.10	0.10	0.94	0.00	0.08	0.09	0.95
	5	1	0.70	0.42	0.45	0.71	0.66	0.27	0.30	0.23	0.64	0.23	0.23	0.14
		2	0.02	0.16	0.18	0.94	0.01	0.10	0.11	0.95	0.00	0.09	0.09	0.95
		3	0.01	0.14	0.14	0.95	0.00	0.09	0.10	0.95	0.00	0.08	0.08	0.95
		4	0.01	0.14	0.14	0.95	0.01	0.10	0.10	0.94	0.00	0.08	0.09	0.95
	6	1	2.09	1.91	2.33	0.75	1.77	0.72	0.79	0.07	1.67	0.54	0.54	0.02
		2	0.08	0.38	0.49	0.92	0.02	0.14	0.14	0.94	0.01	0.11	0.11	0.95
		3	0.01	0.14	0.15	0.95	0.00	0.09	0.10	0.95	0.00	0.08	0.09	0.95
		4	0.01	0.14	0.14	0.95	0.01	0.10	0.10	0.94	0.00	0.08	0.09	0.95
Uniform	1	1	0.47	0.51	0.51	0.96	0.41	0.33	0.35	0.83	0.39	0.30	0.29	0.77
		2	0.05	0.26	0.26	0.97	0.02	0.17	0.18	0.96	0.01	0.16	0.15	0.96
		3	0.06	0.44	0.64	0.97	0.02	0.17	0.18	0.95	0.01	0.16	0.15	0.96
		4	0.05	0.27	0.27	0.97	0.02	0.18	0.19	0.96	0.01	0.16	0.16	0.96
	2	1	0.66	0.66	0.68	0.95	0.57	0.40	0.43	0.79	0.55	0.36	0.36	0.69
		2	0.05	0.28	0.29	0.97	0.02	0.17	0.18	0.96	0.01	0.16	0.16	0.96
		3	0.07	0.60	0.83	0.97	0.02	0.17	0.18	0.95	0.01	0.16	0.15	0.96
		4	0.05	0.27	0.27	0.97	0.02	0.18	0.19	0.96	0.01	0.16	0.16	0.96
	3	1	1.36	1.87	2.17	0.96	1.10	0.68	0.72	0.72	1.04	0.58	0.57	0.57
		2	0.09	0.51	0.59	0.95	0.02	0.19	0.20	0.96	0.02	0.17	0.17	0.96
		3	0.07	0.52	0.72	0.97	0.02	0.17	0.18	0.96	0.01	0.16	0.15	0.96
		4	0.05	0.27	0.27	0.97	0.02	0.18	0.19	0.96	0.01	0.16	0.16	0.96
	4	1	0.37	0.32	0.31	0.91	0.35	0.21	0.22	0.68	0.35	0.20	0.20	0.59
		2	0.02	0.17	0.17	0.96	0.01	0.12	0.12	0.95	0.01	0.11	0.11	0.96
		3	0.01	0.17	0.17	0.97	0.00	0.11	0.12	0.95	0.01	0.11	0.11	0.96
		4	0.01	0.17	0.17	0.97	0.01	0.12	0.12	0.95	0.01	0.11	0.11	0.96
	5	1	0.71	0.50	0.51	0.85	0.66	0.33	0.34	0.46	0.66	0.29	0.29	0.36
		2	0.02	0.19	0.20	0.96	0.01	0.13	0.13	0.95	0.01	0.11	0.11	0.96
		3	0.01	0.17	0.18	0.97	0.01	0.11	0.12	0.95	0.01	0.11	0.11	0.96
		4	0.01	0.17	0.17	0.97	0.01	0.12	0.12	0.95	0.01	0.11	0.11	0.96
	6	1	2.11	2.18	2.75	0.89	1.77	0.80	0.85	0.29	1.70	0.64	0.64	0.15
		2	0.08	0.43	0.57	0.94	0.02	0.15	0.15	0.95	0.02	0.13	0.13	0.96
		3	0.01	0.17	0.18	0.97	0.01	0.12	0.12	0.95	0.01	0.11	0.11	0.96
		4	0.01	0.17	0.17	0.97	0.01	0.12	0.12	0.95	0.01	0.11	0.11	0.96
Exponential	1	1	0.44	0.41	0.43	0.94	0.40	0.26	0.29	0.74	0.38	0.24	0.24	0.66
		2	0.03	0.21	0.22	0.95	0.01	0.14	0.15	0.94	0.01	0.12	0.12	0.96
		3	0.04	0.37	0.54	0.95	0.01	0.14	0.14	0.94	0.01	0.12	0.12	0.97
		4	0.03	0.22	0.23	0.94	0.01	0.14	0.16	0.94	0.01	0.12	0.13	0.96
	2	1	0.62	0.54	0.59	0.93	0.56	0.33	0.36	0.68	0.54	0.29	0.29	0.55
		2	0.04	0.23	0.25	0.95	0.01	0.14	0.15	0.94	0.01	0.13	0.13	0.96
		3	0.05	0.50	0.71	0.95	0.01	0.14	0.15	0.94	0.01	0.12	0.12	0.97
		4	0.03	0.22	0.23	0.94	0.01	0.14	0.16	0.94	0.01	0.12	0.13	0.96
	3	1	1.29	1.64	2.00	0.94	1.08	0.57	0.62	0.55	1.03	0.47	0.47	0.36
		2	0.07	0.44	0.51	0.94	0.02	0.16	0.17	0.94	0.01	0.14	0.14	0.96
		3	0.05	0.43	0.62	0.95	0.01	0.14	0.15	0.94	0.01	0.12	0.12	0.97
		4	0.03	0.22	0.23	0.94	0.01	0.14	0.16	0.94	0.01	0.12	0.13	0.96
	4	1	0.37	0.26	0.27	0.82	0.35	0.17	0.19	0.52	0.34	0.15	0.15	0.39
		2	0.02	0.14	0.15	0.95	0.00	0.10	0.10	0.95	0.00	0.08	0.09	0.95
		3	0.01	0.14	0.14	0.95	0.00	0.09	0.10	0.95	0.00	0.08	0.08	0.95
		4	0.01	0.14	0.14	0.95	0.01	0.10	0.10	0.94	0.00	0.08	0.09	0.95
	5	1	0.70	0.42	0.45	0.71	0.66	0.27	0.30	0.23	0.64	0.23	0.23	0.14
		2	0.02	0.16	0.18	0.94	0.01	0.10	0.11	0.95	0.00	0.09	0.09	0.95
		3	0.01	0.14	0.14	0.95	0.00	0.09	0.10	0.95	0.00	0.08	0.08	0.95
		4	0.01	0.14	0.14	0.95	0.01	0.10	0.10	0.94	0.00	0.08	0.09	0.95
	6	1	2.09	1.91	2.33	0.75	1.77	0.72	0.79	0.07	1.67	0.54	0.54	0.02
		2	0.08	0.38	0.49	0.92	0.02	0.14	0.14	0.94	0.01	0.11	0.11	0.95
		3	0.01	0.14	0.15	0.95	0.00	0.09	0.10	0.95	0.00	0.08	0.09	0.95
		4	0.01	0.14	0.14	0.95	0.01	0.10	0.10	0.94	0.00	0.08	0.09	0.95

Table 3:

Simulation results comparing different methods’ performance in terms of bias, mean estimated standard error (SE), empirical standard deviation (SD) and coverage rate (CR) for 95% confidence interval when ${\lambda }_0\left(t\right)=0.0004t$.

Censoring	Setting	Method	$(n_1,n_2,n_3)=(150,300,5150)$				$(n_1,n_2,n_3)=(300,600,10300)$				$(n_1,n_2,n_3)=(600,1200,20600)$
			Bias	SE	SD	CR	Bias	SE	SD	CR	Bias	SE	SD	CR
Mixed	1	1	0.44	0.51	0.52	0.95	0.41	0.33	0.35	0.83	0.38	0.23	0.23	0.63
		2	0.03	0.26	0.27	0.97	0.01	0.17	0.18	0.96	0.00	0.12	0.12	0.95
		3	0.05	0.44	0.63	0.96	0.01	0.17	0.18	0.96	0.00	0.12	0.12	0.95
		4	0.03	0.27	0.27	0.96	0.02	0.19	0.20	0.97	0.01	0.18	0.18	0.95
	2	1	0.63	0.66	0.68	0.95	0.57	0.41	0.44	0.80	0.53	0.28	0.28	0.51
		2	0.04	0.28	0.29	0.96	0.02	0.18	0.18	0.96	0.01	0.12	0.12	0.95
		3	0.06	0.60	0.82	0.96	0.01	0.17	0.18	0.96	0.00	0.12	0.12	0.95
		4	0.03	0.27	0.27	0.96	0.02	0.18	0.19	0.95	0.00	0.12	0.12	0.95
	3	1	1.29	1.66	1.79	0.97	1.10	0.69	0.74	0.73	1.02	0.45	0.46	0.32
		2	0.07	0.46	0.51	0.95	0.02	0.19	0.20	0.95	0.01	0.13	0.13	0.96
		3	0.06	0.52	0.71	0.96	0.02	0.17	0.18	0.95	0.00	0.12	0.12	0.95
		4	0.03	0.27	0.27	0.96	0.02	0.18	0.19	0.95	0.00	0.12	0.12	0.95
	4	1	0.37	0.32	0.31	0.91	0.34	0.22	0.23	0.69	0.33	0.15	0.15	0.39
		2	0.02	0.18	0.17	0.97	0.00	0.12	0.12	0.96	0.00	0.08	0.08	0.95
		3	0.02	0.17	0.17	0.97	0.00	0.11	0.12	0.96	0.00	0.08	0.08	0.95
		4	0.02	0.17	0.17	0.97	0.01	0.12	0.12	0.95	0.00	0.08	0.08	0.95
	5	1	0.72	0.51	0.51	0.85	0.66	0.33	0.35	0.49	0.63	0.22	0.23	0.14
		2	0.03	0.19	0.20	0.97	0.01	0.13	0.13	0.96	0.00	0.09	0.09	0.95
		3	0.02	0.17	0.18	0.97	0.00	0.12	0.12	0.95	0.00	0.08	0.08	0.95
		4	0.02	0.17	0.17	0.97	0.01	0.12	0.12	0.95	0.00	0.08	0.08	0.95
	6	1	2.11	2.05	2.38	0.88	1.76	0.81	0.87	0.32	1.65	0.53	0.54	0.02
		2	0.08	0.40	0.49	0.93	0.02	0.16	0.16	0.94	0.00	0.10	0.10	0.96
		3	0.02	0.18	0.18	0.96	0.00	0.12	0.12	0.95	0.00	0.08	0.08	0.95
		4	0.02	0.17	0.17	0.97	0.01	0.12	0.12	0.95	0.00	0.08	0.08	0.95
Uniform	1	1	0.42	0.33	0.36	0.92	0.38	0.20	0.21	0.59	0.37	0.17	0.17	0.42
		2	0.02	0.17	0.19	0.93	0.00	0.11	0.11	0.95	0.00	0.09	0.09	0.93
		3	0.04	0.36	0.65	0.94	0.09	0.10	0.11	0.95	0.00	0.09	0.09	0.94
		4	0.02	0.18	0.18	0.94	0.01	0.11	0.12	0.93	0.00	0.09	0.09	0.94
	2	1	0.60	0.44	0.52	0.90	0.54	0.26	0.26	0.43	0.52	0.21	0.22	0.25
		2	0.03	0.19	0.23	0.93	0.00	0.11	0.11	0.95	0.00	0.09	0.09	0.93
		3	0.05	0.53	0.87	0.94	0.00	0.11	0.11	0.95	0.00	0.09	0.09	0.94
		4	0.02	0.18	0.18	0.94	0.01	0.11	0.12	0.93	0.00	0.09	0.09	0.94
	3	1	1.24	1.41	1.74	0.90	1.04	0.47	0.49	0.23	1.00	0.36	0.37	0.10
		2	0.06	0.39	0.50	0.92	0.01	0.13	0.13	0.93	0.00	0.10	0.11	0.93
		3	0.04	0.44	0.74	0.94	0.00	0.11	0.11	0.94	0.00	0.09	0.09	0.94
		4	0.02	0.18	0.18	0.94	0.01	0.11	0.12	0.93	0.00	0.09	0.09	0.94
	4	1	0.34	0.20	0.21	0.73	0.31	0.14	0.14	0.31	0.31	0.11	0.11	0.14
		2	0.00	0.11	0.12	0.93	−0.01	0.07	0.07	0.91	−0.01	0.06	0.06	0.93
		3	0.00	0.10	0.12	0.94	−0.01	0.07	0.07	0.92	−0.01	0.06	0.06	0.93
		4	0.00	0.11	0.11	0.92	−0.01	0.07	0.07	0.93	−0.01	0.06	0.06	0.93
	5	1	0.67	0.35	0.37	0.52	0.62	0.22	0.23	0.04	0.61	0.17	0.18	0.01
		2	0.01	0.13	0.15	0.93	−0.01	0.08	0.08	0.91	−0.01	0.07	0.07	0.93
		3	0.00	0.11	0.12	0.94	−0.01	0.07	0.07	0.92	−0.01	0.06	0.06	0.93
		4	0.00	0.11	0.11	0.92	−0.01	0.07	0.07	0.93	−0.01	0.06	0.06	0.93
	6	1	1.99	1.70	2.09	0.55	1.67	0.62	0.66	0.01	1.60	0.44	0.45	0.00
		2	0.06	0.34	0.45	0.92	0.00	0.12	0.12	0.91	0.00	0.09	0.09	0.93
		3	0.00	0.11	0.12	0.93	−0.01	0.07	0.07	0.92	−0.01	0.06	0.06	0.93
		4	0.00	0.11	0.11	0.92	−0.01	0.07	0.07	0.93	−0.01	0.06	0.06	0.93
Exponential	1	1	0.43	0.44	0.48	0.95	0.38	0.28	0.28	0.79	0.37	0.19	0.19	0.49
		2	0.03	0.22	0.25	0.95	0.00	0.14	0.15	0.95	0.00	0.10	0.10	0.95
		3	0.05	0.45	0.78	0.95	0.00	0.14	0.15	0.95	0.00	0.10	0.10	0.93
		4	0.02	0.23	0.24	0.95	0.01	0.15	0.16	0.94	0.00	0.10	0.10	0.95
	2	1	0.62	0.57	0.65	0.93	0.54	0.34	0.35	0.72	0.52	0.23	0.24	0.36
		2	0.04	0.25	0.28	0.95	0.01	0.15	0.15	0.94	0.00	0.10	0.10	0.95
		3	0.06	0.65	1.04	0.95	0.00	0.14	0.15	0.95	0.00	0.10	0.10	0.93
		4	0.02	0.23	0.24	0.95	0.01	0.15	0.16	0.94	0.00	0.10	0.10	0.95
	3	1	1.29	1.62	2.00	0.95	1.05	0.59	0.59	0.61	1.01	0.39	0.40	0.16
		2	0.07	0.45	0.57	0.93	0.01	0.17	0.17	0.94	0.00	0.11	0.12	0.95
		3	0.05	0.54	0.88	0.95	0.00	0.14	0.15	0.95	0.00	0.10	0.10	0.94
		4	0.03	0.23	0.24	0.95	0.01	0.15	0.16	0.94	0.00	0.10	0.10	0.95
	4	1	0.36	0.27	0.28	0.86	0.33	0.18	0.19	0.59	0.32	0.13	0.13	0.22
		2	0.01	0.15	0.15	0.95	0.00	0.10	0.10	0.95	−0.01	0.07	0.07	0.93
		3	0.01	0.14	0.16	0.95	0.00	0.10	0.10	0.95	−0.01	0.07	0.07	0.94
		4	0.01	0.15	0.15	0.95	0.00	0.10	0.10	0.95	−0.01	0.07	0.97	0.94
	5	1	0.69	0.44	0.46	0.76	0.64	0.28	0.29	0.29	0.62	0.19	0.20	0.04
		2	0.02	0.17	0.18	0.94	0.00	0.11	0.11	0.94	0.00	0.07	0.08	0.94
		3	0.01	0.15	0.16	0.95	0.00	0.10	0.10	0.95	−0.01	0.07	0.07	0.95
		4	0.01	0.15	0.15	0.95	0.00	0.10	0.10	0.95	−0.01	0.07	0.07	0.94
	6	1	2.05	1.92	2.33	0.80	1.73	0.73	0.76	0.12	1.63	0.47	0.49	0.00
		2	0.07	0.38	0.51	0.92	0.01	0.14	0.14	0.94	0.00	0.09	0.10	0.94
		3	0.01	0.15	0.16	0.95	0.00	0.10	0.10	0.95	−0.01	0.07	0.07	0.95
		4	0.01	0.15	0.15	0.95	0.00	0.10	0.10	0.95	−0.01	0.07	0.07	0.94

To evaluate the relationships of bias versus $R_{\tilde{X}WV}^2\left(R_1^2\right)$ and bias versus $R_{\tilde{X}W\mid V}^2\left(R_{1v}^2\right)$, we vary the parameter $\rho$ from 0 to 0.6 in setting 1 and setting ${\sigma }_w=1$ and ${\sigma }_w=1.7$, respectively, keeping all other parameters unchanged. Figure 2 shows the estimated bias for method 1 with respect to the squared multiple correlation coefficient from the first step $(R_{\tilde{X}WV}^2)$ and squared multiple partial correlation coefficient given covariate $V$ from the first step $(R_{\tilde{X}W\mid V}^2)$. Based on the results, we can see that the bias is not a monotonic decreasing function of $R_{\bar{X}WV}^2$, but is a decreasing function of $R_{\tilde{X}W\mid V}^2$, which is consistent with our theoretical derivation. Figure 2 suggests that the requirement $R^2>0.36$ (Lampe et al., 2017) is insufficient as a single criterion to decide whether a biomarker is useful or not. In particular, the partial $R^2$ after adjusting for the personal characteristic effect is an important factor influencing the bias of the current biomarker-based regression calibration for the association study. Figure 3 shows the relationship between SD of the BF corrected estimator of the association parameter and the $R_{\tilde{X}WV}^2$ and $R_{\tilde{X}W\mid V}^2$. Based on Figure 3, we can see that the SD is a decreasing function of $R_{\tilde{X}W\mid V}^2$ rather than $R_{\tilde{X}WV}^2$, which indicates that the partial $R^2$, instead of $R^2$, is an essential factor affecting SD. Therefore, to evaluate whether the calibration equation is useful, we should consider both the partial $R^2$ and the $R^2$. Similar patterns for the other two estimators with the use of self-reported dietary data from feeding study (method 3 and method 4) are shown in Figure 4.

Fig. 2:

Bias from method 1 in relation with $R_1^2$ and $R_{1v}^2$

Fig. 3:

SD of method 1 and method 2 in relation with $R_1^2$ and $R_{1v}^2$

Fig. 4:

SD of method 3 and method 4 in relation with $R_1^2$ and $R_{1v}^2$

Additional simulation in Appendix B indicates that method 2 is robust to the assumption that the relationships between the self-reported dietary intake and the short-term dietary intake are the same between sample 1 and sample 2. Also, it showed that with the bias correction, one may not need to be too stringent on the threshold of partial $R^2{(R}_{1v}^2)$.

6. Real Data Example

We illustrate our methods with the WHI NPAAS feeding study $(n=153)$, NPAAS biomarker study $(n=450)$ and the full WHI cohort data. We treat log-transformed self-reported sodium and potassium intake from FFQ as $Q$. Variables including age, BMI, race/ethnicity, education level, self-reported physical activity, and smoking status are used as $V$; the 24 hour urine sodium and potassium measurements are used as $W$ and the disease outcomes are several CVD categories, including total coronary heart disease (CHD) and its myocardial infarction (MI) and coronary death components, total stroke and its hemorrhagic and ischemic components, total CVD comprised of CHD and stroke, coronary artery bypass grafting (CABG) and percutaneous coronary intervention (PCI), and total CVD that also includes CABG and PCI, and heart failure. The prevalence of CVD events range from 1–10% (Prentice et al., 2017), which suggests the rare disease assumption is not severely violated. Follow-up times began with the time of FFQ measurement (year-1 visit in DM-C and at enrollment in OS) and continued until the earliest of the specific CVD outcomes under analysis, death, loss to follow-up, or September 30, 2010, whichever occurred first. In our analysis, the hazard rates was modeled as implicitly conditioning on the continued survival of the study subject. This means that death is not a source of censoring in our formulation. Rather death simply limits the follow-up period over which hazard rate information is obtained for thestudy subject. This is distinct from regarding death as censoring non-fatal outcomes, which a competing risk formulation would do. Using the log-transformed self-reported sodium-potassium ratio as a single predictor for the ‘true’ log-transformed dietary sodium-potassium ratio, we obtained $R^2=0.36$ and an increased $R^2=0.38$ can be obtained by using the self-reported log sodium and log potassium as separate predictors. Therefore for the analysis, we used these two self-reported measurements as predictors for the ‘true’ log-transformed dietary sodium-potassium ratio. The further inclusion of personal characteristics increased the $R^2$ to 0.45 with a partial $R^2$ conditional on personal characteristics equal to 0.37. We tested the normality of log-transformed self-reported sodium and potassium intake $\left(Q\right)$, log-transformed 24 hour urine sodium and potassium measurements $\left(W\right)$, and the log-transformed assessed sodium/potassium ratio $\left(\tilde{X}\right)$. There is no evidence that any of the normality assumptions are violated $(p>0.1)$.

For the feeding study, there were moderate measurement errors in the assessed consumed dietary data, so we differentiated the short term dietary intake X and the observed consumed dietary intake $\tilde{X}$ and adjusted the bias factor estimate by $\widehat{BF}=1-\frac{\widehat{\mathrm{V}\mathrm{a}\mathrm{r}}(\tilde{X}\mid W,V)-{\hat{\tilde{\sigma }}}_x^2}{\widehat{\mathrm{V}\mathrm{a}\mathrm{r}}(\tilde{X}\mid V)-{\hat{\tilde{\sigma }}}_x^2}$ where ${\hat{\tilde{\sigma }}}_x^2$ was treated as a sensitivity parameter since we do not have lab replication data for nutrition evaluation to estimate the food packaging error and nutrient composition database error accurately. We set ${\hat{\tilde{\sigma }}}_x^2$ at several different levels to illustrate the potential bias.

The estimated hazard ratio (HR) for a 20% increase in dietary intake is shown in Table 4. Under the assumption that there was no measurement error in the consumed dietary data in the controlled feeding study, we estimate the bias factor will be about 0.37. This will lead to the most conservative (bias toward the null) HR estimate. The HR estimated from method 3 (three-step with FFQ approach) is about the same as the estimate using method 2 with a BF around 0.78, which is equivalent to the case where approximately 50% of the variation in the estimated consumed diet data is from ‘noise’. This noise level is about the same across all disease outcomes and is consistent with our estimation of total energy expenditure (Zheng et al., 2014).

Table 4:

Association between 20% increase in sodium/potassium ratio with various cardiovascular diseases.

Outcome	Method 1		Method 3		Method 4
	HR	95% CI	HR	95% CI	HR	95% CI
CHD	1.20	(1.06, 1.36)	1.16	(1.04, 1.28)	1.21	(1.03,1.43)
Nonfatal MI	1.22	(1.04, 1.42)	1.16	(1.04, 1.30)	1.19	(1.01,1.41)
Coronary death	1.21	(1.07, 1.38)	1.17	(1.03, 1.33)	1.27	(0.98,1.64)
Stroke	1.11	(1.02, 1.21)	1.10	(1.00, 1.20)	1.17	(1.02,1.34)
Ischemic Stroke	1.18	(1.06, 1.31)	1.15	(1.02, 1.29)	1.24	(1.05,1.46)
Hemorrhagic Stroke	0.86	(0.67, 1.10)	0.90	(0.73, 1.11)	0.92	(0.63,1.35)
Total CVD	1.15	(1.08, 1.22)	1.12	(1.03, 1.21)	1.17	(1.06,1.29)
Revascularization	1.21	(1.06, 1.37)	1.15	(1.04, 1.28)	1.18	(1.01,1.39)
Non-Revascularization	1.15	(1.05, 1.26)	1.12	(1.03, 1.22)	1.18	(1.04,1.34)
Heart Failure	1.06	(0.94, 1.19)	1.03	(0.94, 1.14)	1.00	(0.84,1.18)
Outcome	Method 2 (no Error)		Method 2 (40% Error)		Method 2 (50% Error)
	HR	95% CI	HR	95% CI	HR	95% CI
CHD	1.07	(1.02, 1.12)	1.13	(1.04, 1.23)	1.16	(1.04,1.30)
Nonfatal MI	1.08	(1.03, 1.13)	1.14	(1.03, 1.25)	1.17	(1.03,1.34)
Coronary death	1.07	(1.02, 1.14)	1.14	(1.02, 1.26)	1.17	(1.03,1.34)
Stroke	1.04	(1.00, 1.08)	1.07	(1.00, 1.15)	1.09	(1.00,1.18)
Ischemic Stroke	1.06	(1.01, 1.11)	1.11	(1.02, 1.21)	1.14	(1.03,1.27)
Hemorrhagic Stroke	0.94	(0.86, 1.03)	0.90	(0.77, 1.06)	0.88	(0.72,1.08)
Total CVD	1.05	(1.02, 1.09)	1.10	(1.03, 1.16)	1.12	(1.04,1.21)
Revascularization	1.07	(1.02, 1.12)	1.13	(1.03, 1.25)	1.17	(1.03,1.32)
Non-Revascularization	1.05	(1.02, 1.09)	1.10	(1.03, 1.17)	1.12	(1.03,1.22)
Heart Failure	1.02	(0.98, 1.07)	1.04	(0.96, 1.12)	1.05	(0.94,1.16)

Compared with the results from (Prentice et al., 2017), we have the same findings that sodium/potassium ratio is positively associated with the risk of CHD, nonfatal myocardial infarction, coronary death, ischemic stroke, total CVD, coronary revascularization, non-revascularization and negatively associated with hemorrhagic stroke, but the conservative lower bounds of such effects are much smaller than previously presented. Method 3 gives a slightly wider confidence interval compared with (Prentice et al., 2017), but the difference for point estimate is modest. Method 2 with a 50% error assumption gives a point estimate and a confidence interval that are similar to those from method 3, which is consistent with what we have observed from simulations. For method 2 with no error assumption, since the BF is over-estimated, the confidence interval is narrower than method 3 because of the shrinkage effect. However, since the assumption of no error is implausible, it leads to biased results towards the null and the comparison of efficiency is not of much interest. Method 4 has a wider confidence interval compared with method 3, indicating that the urine biomarker is ‘strong’ and provides useful independent information beyond the self-reported data.

Estimation procedures for non-linear hazard ratio have not been fully developed when using the regression calibration procedure (Prentice & Huang, 2018). However, a test for non-linearity is carried out by including a quadratic term in calibrated intake in the hazard ratio model and testing for a zero coefficient does not show significant nonlinear effects (Prentice et al., 2017).

7. Discussion

In this work, we showed both theoretically and numerically that the näive regression calibration method (Method 1) leads to biased estimation of the association between the exposure and the outcome, even when the $R^2$ from the first step regression is large. In addition, $R^2$ itself is not a sufficient criterion for the evaluation of the biomarkers in general; the partial $R^2$ also needs be taken into consideration. Also the choice of $V$ (to condition on) should not include potential mediators.

We proposed three new regression calibration methods which can provide consistent estimation of the association under the rare disease assumption. The newly proposed methods has significantly improved performance over Method 1. We performed extensive simulations to compare the performance of the different estimators to provide guidance on future study designs.

In practice, Method 1 should not be used independently due to its potential large bias, unless the biomarker is very strong. The other three methods have their advantages/disadvantages. Method 4 is the simplest one in design and requires fewest assumptions. However, it depends on the availability of a strong dietary instrument and a large number of subjects in the feeding study to accurately characterize the association between the dietary instrument and the true dietary intake. Method 3 is another simple three-step approach. From both the theoretical result and the numerical studies, Method 3 uniformly outperforms Method 4 with large sample size. It allows the use of biomarker information efficiently and is robust to the measurement error in the assessed diet from the controlled feeding study. Method 3 works well under the assumption that the association between the dietary instrument and the true dietary intake are the same between the cohort and the controlled feeding study subgroup. In this case, we give the preference to Method 3 for studies on dietary components whose intakes do not change dramatically over time. Method 3 has been successfully used in several recent studies (Prentice et al., 2021a,b).

When such dietary instruments are not available, we need to consider Method 2. Method 2 does not use dietary instrument information in the biomarker development stage, it depends only on the objective measurements. Therefore, as long as the model is correctly specified, the same biomarker can be used to build calibration equations for the dietary instruments that are available for the cohort but are not necessarily measured in the controlled feeding study. To compare the performance of Method 2 and Method 3, note that when we have relatively strong biomarker and weak dietary instruments, Method 2 has better performance than Method 3; whereas when we have relatively weak biomarker and strong dietary instrument, Method 3 has better performance.

One limitation of Method 2 is that it requires the knowledge of the variation of noise in the assessed diet, ${\tilde{\sigma }}_x^2$. The major portion of this variation might be from the inaccuracy of the records from the nutritional database (a bag of chips labeled with 100 cal might be 90 cal or 110 cal). Ideally, if this variation information can be added to the nutritional database, then the problem will be solved. We recommend when producing the standardized food, analyzing multiple samples for each type of food used in the feeding study during the nutritional analysis stage and reporting the standard error for each food type. When we do not have ${\tilde{\sigma }}_x^2$ available, a sensitivity analysis can be done. Alternatively, when there are more than one self-reported measurements exist (denoted as $Q_1$ and $Q_2$ for example) and only one of them (e.g. $Q_1$) is available in the first stage sample, we can estimate ${\tilde{\sigma }}_x$ by comparing the estimator ${\hat{\theta }}_3$ and ${\hat{\theta }}_1$ using $Q_1$ only and then apply Method 2 with $Q_2$ or both $Q_1$ and $Q_2$ as shown in Appendix B.

As usual in Cox regression models without measurement errors in the predictors, if $Z$ and $V$ are highly correlated, then we will face variance inflation and will not be able to estimate the coefficient before $Z$ precisely without extremely large sample size. The potential collinearity between $W$ and $V$ will not cause issue itself since we are only interested in predicting $E[Z\mid W,V]$. However, if $E[Z\mid W,V]$ and $V$ are highly correlated (i.e, the biomarker cannot provide enough additional information about the true diet given the adjusted confounding variables), then we will face the variance inflation issue.

Here we emphasize that our proposed approach is different from simple imputation (SI), and SI is not suitable for our measurement error correction problem. First, for Steps 1 and 2 of the biomarker-calibrated approach, we do not simply use SI to generate one or more observations from the conditional distribution of the unobserved dietary variable given available data, instead, we make a good approximation to the expectation of the unobserved ‘true’ intake given available data. Also, when calculating the asymptotic variance, unlike SI, which only uses the variance estimated from the last step, our proposed regression calibration method also incorporates the variation in the regression models for Steps 1 and 2. Hence, to attain similar efficiency of the regression calibration approach, with SI, a very large number of imputations are needed, which is not practical. Under the classical measurement error assumption and the rare event assumption, it has been shown in previous literature (Prentice, 1982) that a two-step regression calibration approach leads to a consistent estimator. By carefully selecting the set of variables that can be included in the regression at each step, we extended the existing literature and showed that our proposed three-step estimator is also consistent under similar assumptions.

Multiple imputation is not suitable for the measurement measure error correction problem in general since the true dietary intakes $Z$ and $X$ are never observed. The measurement error contained in $\tilde{X}$ leads to biased estimates using multiple imputation. In addition, due to the fact that $n_3$ is much larger than $n_2$ and $n_1$, the multiple imputation algorithm is quite inefficient both statistically and computationally for our application.

To evaluate the generalizability of the developed biomarker and calibration equation, it will be of interest to validate and/or apply the biomarker and calibration equation developed in this study to other nutritional study such as Observing Protein and Energy Nutrition (OPEN) (Subar et al., 2003) and Nurses’s Health Study (NHS) (Belanger et al., 1978).

Table 7:

Simulation results comparing different methods’ performance when the correlation structures between the short term dietary intake and true dietary intakes of the controlled feeding study and the full cohort are different when ${\lambda }_0\left(t\right)=0.004$.

Censor	Setting	Diff	Method	$(n_1,n_2,n_3)=(150,300,5150)$				$(n_1,n_2,n_3)=(300,600,10300)$				$(n_1,n_2,n_3)=(600,1200,20600)$
				Bias	SE	SD	CR	Bias	SE	SD	CR	Bias	SE	SD	CR
Mix	1	0.1	1	0.44	0.41	0.43	0.94	0.40	0.26	0.29	0.74	0.38	0.24	0.24	0.66
		0.1	2	0.03	0.21	0.22	0.95	0.01	0.14	0.15	0.94	0.01	0.12	0.12	0.96
		0.1	3	0.02	0.81	1.03	0.96	0.03	0.14	0.16	0.95	0.02	0.13	0.13	0.97
		0.1	4	0.08	0.27	0.28	0.96	0.06	0.17	0.20	0.96	0.05	0.14	0.14	0.97
		0.3	1	0.44	0.41	0.43	0.94	0.40	0.26	0.29	0.74	0.38	0.24	0.24	0.66
		0.3	2	0.03	0.21	0.22	0.95	0.01	0.14	0.15	0.94	0.01	0.12	0.12	0.96
		0.3	3	0.01	6.7	2.74	0.97	0.07	0.17	0.18	0.98	0.07	0.15	0.14	0.97
		0.3	4	0.23	0.54	0.56	0.98	0.17	0.26	0.32	0.99	0.16	0.19	0.20	0.97
		0.5	1	0.44	0.41	0.43	0.94	0.40	0.26	0.29	0.74	0.38	0.24	0.24	0.66
		0.5	2	0.03	0.21	0.29	0.95	0.01	0.14	0.15	0.94	0.01	0.12	0.12	0.96
		0.5	3	0.22	2.27	1.38	0.98	0.14	0.21	0.24	0.99	0.13	0.17	0.17	0.96
		0.5	4	0.56	10.97	3.62	0.99	0.39	13.57	5.49	1.00	0.40	0.36	0.41	0.96
	4	0.1	1	0.37	0.26	0.27	0.82	0.35	0.17	0.19	0.52	0.34	0.15	0.15	0.39
		0.1	2	0.02	0.14	0.15	0.95	0.00	0.10	0.10	0.95	0.00	0.08	0.09	0.95
		0.1	3	0.03	0.14	0.15	0.96	0.02	0.10	0.10	0.96	0.02	0.09	0.09	0.95
		0.1	4	0.04	0.16	0.17	0.96	0.04	0.11	0.11	0.96	0.03	0.09	0.10	0.96
		0.3	1	0.37	0.26	0.27	0.82	0.35	0.17	0.19	0.52	0.34	0.15	0.15	0.39
		0.3	2	0.02	0.14	0.15	0.95	0.00	0.10	0.10	0.95	0.00	0.08	0.09	0.95
		0.3	3	0.07	0.17	0.19	0.97	0.05	0.11	0.12	0.96	0.05	0.10	0.10	0.95
		0.3	4	0.16	0.28	0.37	0.99	0.13	0.15	0.17	0.98	0.13	0.12	0.13	0.91
		0.5	1	0.37	0.26	0.27	0.82	0.35	0.17	0.19	0.52	0.34	0.15	0.15	0.39
		0.5	2	0.02	0.14	0.15	0.95	0.00	0.10	0.10	0.96	0.00	0.08	0.09	0.95
		0.5	3	0.14	0.26	0.39	0.98	0.11	0.13	0.14	0.97	0.10	0.11	0.11	0.92
		0.5	4	0.46	1.43	1.43	1.00	−0.20	6.72	21.5	0.98	0.31	0.20	0.21	0.79
Unif	1	0.1	1	0.47	0.51	0.51	0.96	0.41	0.33	0.35	0.83	0.39	0.30	0.29	0.77
		0.1	2	0.05	0.26	0.26	0.97	0.02	0.17	0.18	0.96	0.01	0.16	0.15	0.96
		0.1	3	0.03	0.96	1.20	0.98	0.03	0.18	0.19	0.96	0.03	0.16	0.16	0.96
		0.1	4	0.09	0.32	0.33	0.98	0.06	0.20	0.22	0.96	0.05	0.17	0.17	0.97
		0.3	1	0.47	0.51	0.51	0.96	0.41	0.33	0.35	0.83	0.39	0.30	0.29	0.77
		0.3	2	0.05	0.26	0.26	0.97	0.02	0.17	0.18	0.96	0.01	0.16	0.15	0.96
		0.3	3	0.03	7.56	3.08	0.98	0.08	0.21	0.22	0.97	0.07	0.18	0.18	0.97
		0.3	4	0.25	0.62	0.65	0.99	0.19	0.30	0.34	0.98	0.16	0.23	0.23	0.97
		0.5	1	0.78	0.51	0.51	0.94	0.41	0.33	0.35	0.83	0.39	0.30	0.29	0.77
		0.5	2	0.05	0.26	0.26	0.97	0.02	0.17	0.18	0.96	0.01	0.16	0.15	0.96
		0.5	3	0.29	3.73	2.37	0.99	0.15	0.25	0.27	0.98	0.13	0.21	0.21	0.96
		0.5	4	0.57	12.47	4.62	0.99	0.40	14.07	5.68	0.99	0.40	0.40	0.43	0.97
	4	0.1	1	0.37	0.32	0.31	0.91	0.35	0.21	0.22	0.68	0.35	0.20	0.20	0.59
		0.1	2	0.02	0.17	0.17	0.96	0.01	0.12	0.12	0.95	0.01	0.11	0.11	0.96
		0.1	3	0.03	0.18	0.19	0.97	0.02	0.12	0.12	0.95	0.02	0.11	0.11	0.96
		0.1	4	0.05	0.19	0.20	0.98	0.04	0.13	0.14	0.96	0.04	0.12	0.12	0.96
		0.3	1	0.37	0.32	0.31	0.91	0.35	0.21	0.22	0.68	0.35	0.20	0.20	0.59
		0.3	2	0.02	0.17	0.17	0.96	0.01	0.12	0.12	0.95	0.01	0.11	0.11	0.96
		0.3	3	0.07	0.21	0.24	0.98	0.06	0.13	0.14	0.96	0.06	0.12	0.12	0.95
		0.3	4	0.17	0.34	0.46	0.99	0.14	0.18	0.19	0.97	0.13	0.15	0.15	0.92
		0.5	1	0.37	0.32	0.31	0.91	0.35	0.21	0.22	0.68	0.35	0.20	0.20	0.59
		0.5	2	0.02	0.17	0.17	0.96	0.01	0.12	0.12	0.95	0.01	0.11	0.11	0.96
		0.5	3	0.14	0.31	0.49	0.99	0.11	0.16	0.17	0.96	0.11	0.14	0.14	0.93
		0.5	4	0.45	1.39	1.34	1.00	−0.09	4.27	16.53	0.98	0.32	0.24	0.24	0.85
Exp	1	0.1	1	0.44	0.41	0.43	0.94	0.40	0.26	0.29	0.74	0.38	0.24	0.24	0.66
		0.1	2	0.03	0.21	0.22	0.95	0.01	0.14	0.15	0.94	0.01	0.12	0.12	0.96
		0.1	3	0.02	0.81	1.03	0.96	0.03	0.14	0.16	0.95	0.02	0.13	0.13	0.97
		0.1	4	0.08	0.27	0.28	0.96	0.06	0.17	0.20	0.96	0.05	0.14	0.14	0.97
		0.3	1	0.44	0.41	0.43	0.94	0.40	0.26	0.29	0.74	0.38	0.24	0.24	0.66
		0.3	2	0.03	0.21	0.22	0.95	0.01	0.14	0.15	0.94	0.01	0.12	0.12	0.96
		0.3	3	0.01	6.7	2.74	0.97	0.07	0.17	0.18	0.98	0.07	0.15	0.14	0.97
		0.3	4	0.23	0.54	0.56	0.98	0.17	0.26	0.32	0.99	0.16	0.19	0.20	0.97
		0.5	1	0.44	0.41	0.43	0.94	0.40	0.26	0.29	0.74	0.38	0.24	0.24	0.66
		0.5	2	0.03	0.21	0.22	0.95	0.01	0.14	0.15	0.94	0.01	0.12	0.12	0.96
		0.5	3	0.22	2.27	1.38	0.98	0.14	0.21	0.24	0.99	0.13	0.17	0.17	0.96
		0.5	4	0.56	10.97	3.62	0.99	0.39	13.57	5.49	1.00	0.40	0.36	0.41	0.96
	4	0.1	1	0.37	0.26	0.27	0.82	0.35	0.17	0.19	0.52	0.34	0.15	0.15	0.39
		0.1	2	0.02	0.14	0.15	0.95	0.00	0.10	0.10	0.95	0.00	0.08	0.09	0.95
		0.1	3	0.03	0.14	0.15	0.96	0.02	0.10	0.10	0.96	0.02	0.09	0.09	0.95
		0.1	4	0.04	0.16	0.17	0.96	0.04	0.11	0.11	0.96	0.03	0.09	0.10	0.96
		0.3	1	0.37	0.26	0.27	0.82	0.35	0.17	0.19	0.52	0.34	0.15	0.15	0.39
		0.3	2	0.02	0.14	0.15	0.95	0.00	0.10	0.10	0.95	0.00	0.08	0.09	0.95
		0.3	3	0.07	0.17	0.19	0.97	0.06	0.11	0.12	0.96	0.05	0.10	0.10	0.95
		0.3	4	0.16	0.28	0.37	0.99	0.14	0.15	0.17	0.98	0.13	0.12	0.13	0.91
		0.5	1	0.37	0.26	0.27	0.82	0.35	0.17	0.19	0.52	0.34	0.15	0.15	0.39
		0.5	2	0.02	0.14	0.15	0.95	0.00	0.10	0.10	0.95	0.00	0.08	0.09	0.95
		0.5	3	0.14	0.26	0.39	0.98	0.11	0.13	0.14	0.97	0.10	0.11	0.11	0.92
		0.5	4	0.46	1.43	1.43	1.00	−0.20	6.72	21.5	0.98	0.31	0.20	0.21	0.79

Appendix A: Technical Details

A.1: Bias Factor Derivation

In this subsection, we derived the asymptotic bias of $\theta$ for method 1 under rare disease assumption by checking how $E(\hat{X}\mid Q,V)$ is biased away from $E(X\mid Q,V)$. For the first step regression model of $\tilde{X}$ on $(W,V)$, we have

$$\hat{X}=E(\tilde{X}\mid W,V)=E(X\mid W,V)=E(X\mid V)+\{W-E(W\mid V){\}}^{\mathrm{\top }}{\Sigma }_{WW\mid V}^{-1}{\Sigma }_{XW\mid V}^{\mathrm{\top }}.$$

where ${\Sigma }_{WW\mid V}=\mathrm{V}\mathrm{a}\mathrm{r}(W\mid V)$ and ${\Sigma }_{XW\mid V}=\mathrm{C}\mathrm{o}\mathrm{v}(X,W\mid V)$ are conditional variance covariance matrices. Plugging this into the second step regression model, we have

$$\begin{array}{l}E(\hat{X}\mid Q,V)=E\{E(\hat{X}\mid X,Q,V)\mid Q,V\}=E\{E(\hat{X}\mid X,V)\mid Q,V\}\\ =E\left[E\left[E(X\mid V)+\{W-E(W\mid V){\}}^{\mathrm{\top }}{\Sigma }_{WW\mid V}^{-1}{\Sigma }_{XW\mid V}^{\mathrm{\top }}\mid X,V\right]\mid Q,V\right]\\ =E\left[E(X\mid V)+\{E(W\mid X,V)-E(W\mid V){\}}^{\mathrm{\top }}{\Sigma }_{WW\mid V}^{-1}{\Sigma }_{XW\mid V}^{\mathrm{\top }}\mid Q,V\right]\\ =E\left[E(X\mid V)+\{X-E(X\mid V)\}{\Sigma }_{XX\mid V}^{-1}{\Sigma }_{XW\mid V}{\Sigma }_{WW\mid V}^{-1}{\Sigma }_{XW\mid V}^{\mathrm{\top }}\mid Q,V\right]\\ =E\left[E(X\mid V)+\{X-E(X\mid V)\}{\rho }_V\mid Q,V\right]\\ ={\rho }_{V}E(X\mid Q,V)+\left(1-{\rho }_V\right)E(X\mid V).\end{array}$$

When the bias factor ${\rho }_V$ is a constant over $V$, we simply denote it as $\rho$ and we have $\rho {\theta }_z^{\mathrm{*}}={\theta }_z$, or ${\theta }_z^{\mathrm{*}}={\rho }^{-1}{\theta }_z$ with appropriate adjustment for $V$. Explicitly, we have

$$\begin{array}{l}\rho =1-{\mathrm{\Sigma }}_{XX\mid V}^{-1}\left({\mathrm{\Sigma }}_{XX\mid V}-{\mathrm{\Sigma }}_{XW\mid V}{\mathrm{\Sigma }}_{WW\mid V^{-1}}^{{\mathrm{\Sigma }}_{XW\mid V}^{\mathrm{\top }}}\right)\\ =1-\frac{\mathrm{V}\mathrm{a}\mathrm{r}(X\mid W,V)}{\mathrm{V}\mathrm{a}\mathrm{r}(X\mid V)}=R_{X,W\mid V}^2.\end{array}$$

If we further have $E(X\mid V)=V\delta$ is a linear function of $V$, then $(1-\rho )\delta {\theta }_z^{\mathrm{*}}+{\theta }_v^{\mathrm{*}}={\theta }_v$, or ${\theta }_v^{\mathrm{*}}={\theta }_v-\frac{(1-\rho )\delta }{\rho }{\theta }_z$.

A.2: Proof of Theorem 1

Before proof of main theorem 1, we first proof the lemma below which give the asymptotics for the third step regression given the $\hat{\gamma }$ from the second step.

Lemma 1

Assume $n_3/n_2\to C_2<\mathrm{\infty }$ and ${\theta }^{\mathrm{*}}$ is the unique solution to $EU\left(\theta ,{\gamma }^{\mathrm{*}}\right)=0$ for an estimating function $U(\theta ,\gamma )$. If $\hat{\theta }$ solve the estimating equation $0=n_3^{-1}{\sum }_{i=1}^{n_3} U_i(\theta ,\hat{\gamma })$ where $\sqrt{n_2}\left(\hat{\gamma }-{\gamma }^{\mathrm{*}}\right)\to N\left(0,{\mathrm{\Sigma }}_{\gamma }\right)$ and $\hat{\gamma }$ is independent of $U_i(\theta ,\gamma )$, then we have $\sqrt{n_3}(\hat{\theta }-{\theta }^{\mathrm{*}})\to N\left(0,{\Sigma }_{\theta }\right)$ where

$${\Sigma }_{\theta }=I_{\theta }^{-1}\left(J_{\theta }+C_2{I}_{\gamma }{\mathrm{\Sigma }}_{\gamma }{I}_{\gamma }^{\mathrm{\top }}\right)I_{\theta }^{-\mathrm{\top }},$$

where $I_{\theta }=E\left(-{\frac{\partial U(\theta ,\gamma )}{\partial \theta }|}_{{\theta }^{\mathrm{*}},{\gamma }^{\mathrm{*}}}\right),$ $I_{\gamma }=E\left(-{\frac{\partial U(\theta ,\gamma )}{\partial \gamma }|}_{{\theta }^{\mathrm{*}},{\gamma }^{\mathrm{*}}}\right)$ and $J_{\theta }=Var(U({\theta }^{\mathrm{*}},{\gamma }^{\mathrm{*}}))$. This variance can be consistently estimated by

$${\stackrel{\mathrm{ˆ}}{\Sigma }}_{\theta }={\hat{I}}_{\theta }^{-1}\left({\hat{J}}_{\theta }+\frac{n_3}{n_2}{\hat{I}}_{\gamma }{\stackrel{\mathrm{ˆ}}{\mathrm{\Sigma }}}_{\gamma }{\hat{I}}_{\gamma }^{\mathrm{\top }}\right){\hat{I}}_{\theta }^{-\mathrm{\top }},$$

where ${\hat{I}}_{\theta }=n_3^{-1}{\sum }_{i=1}^{n_3} \left(-{\frac{\partial U_i(\theta ,\gamma )}{\partial \theta }|}_{\hat{\theta },\hat{\gamma }}\right),{\hat{I}}_{\gamma }=n_3^{-1}{\sum }_{i=1}^{n_3} \left(-{\frac{\partial U_i(\theta ,\gamma )}{\partial \gamma }|}_{\hat{\theta },\hat{\gamma }}\right),{\hat{J}}_{\theta }=n_3^{-1}{\sum }_{i=1}^{n_3} U_i(\hat{\theta },\hat{\gamma })U_i^{\mathrm{\top }}(\hat{\theta },\hat{\gamma })$ and ${\hat{\Sigma }}_{\gamma }$ is a consistent estimator of ${\Sigma }_{\gamma }$.

Proof

We can derive the asymptotic for $\hat{\theta }$ as below:

$$\begin{array}{l}0=\sum _{i=1}^{n_3} U_i(\hat{\theta },\hat{\gamma })\\ =\sum _{i=1}^{n_3} \left\{U_i({\theta }^{\mathrm{*}},{\gamma }^{\mathrm{*}})+{\frac{\partial U_i(\theta ,\gamma )}{\partial \theta }|}_{{\theta }^{\mathrm{*}},\gamma }(\hat{\theta }-{\theta }^{\mathrm{*}})+{\frac{\partial U_i(\theta ,\gamma )}{\partial \gamma }|}_{{\theta }^{\mathrm{*}},\gamma }(\hat{\gamma }-{\gamma }^{\mathrm{*}})+o(\Vert \hat{\theta }-{\theta }^{\mathrm{*}}\Vert ,\Vert \hat{\gamma }-{\gamma }^{\mathrm{*}}\Vert )\right\}.\end{array}$$

With Uniform Law of Large Number (ULLN), we have

$${n_3^{-1}\sum _{i=1}^{n_3} \frac{\partial U_i(\theta ,\gamma )}{\partial \theta }|}_{{\theta }^{\mathrm{*}},{\gamma }^{\gamma }}=E\left({\frac{\partial U(\theta ,\gamma )}{\partial \theta }|}_{{\theta }^{\mathrm{*}},{\gamma }^{\mathrm{*}}}\right)+o_p(1)=-I_{\theta }+o_p(1).$$

Similarly, we have

$${n_3^{-1}\sum _{i=1}^{n_3} \frac{\partial U_i(\theta ,\gamma )}{\partial \gamma }|}_{{\theta }^{\mathrm{*}},{\gamma }^{\mathrm{*}}}=-I_{\gamma }+o_p\left(1\right).$$

Plug in the Taylor expansion, notice that $EU({\theta }^{\mathrm{*}},{\gamma }^{\mathrm{*}})=0$, we have

$$\begin{array}{l}0=n_3^{-1}\sum _{i=1}^{n_3} \{U_i({\theta }^{\mathrm{*}},{\gamma }^{\mathrm{*}})-EU({\theta }^{\mathrm{*}},{\gamma }^{\mathrm{*}})\}-I_{\theta }(\hat{\theta }-{\theta }^{\mathrm{*}})-I_{\gamma }(\hat{\gamma }-{\gamma }^{\mathrm{*}})\\ +o(\Vert \hat{\theta }-{\theta }^{\mathrm{*}}\Vert ,\Vert \hat{\gamma }-{\gamma }^{\mathrm{*}}\Vert )+o_p(1)\mathrm{*}∥\hat{\gamma }-{\gamma }^{\mathrm{*}}∥+o_p(1)\mathrm{*}\Vert \hat{\theta }-{\theta }^{\mathrm{*}}\Vert .\end{array}$$

So we have

$$\sqrt{n_3}(\hat{\theta }-{\theta }^{\mathrm{*}})=I_{\theta }^{-1}\left\{n_3^{-1/2}\sum _{i=1}^{n_3} U_i({\theta }^{\mathrm{*}},{\gamma }^{\mathrm{*}})-I_{\gamma }\sqrt{n_3}\left(\hat{\gamma }-{\gamma }^{\mathrm{*}}\right)\right\}+o_p(1).$$

By Central Limit Theorem (CLT), we have

$$n_3^{-1/2}\sum _{i=1}^{n_3} \{U_i({\theta }^{\mathrm{*}},{\gamma }^{\mathrm{*}})-EU({\theta }^{\mathrm{*}},{\gamma }^{\mathrm{*}})\}{\Rightarrow }_{d}N\left(0,J_{\theta }\right)$$

So we have

$${\mathrm{\Sigma }}_{\theta }=I_{\theta }^{-1}\left\{J_{\theta }+C_2{I}_{\gamma }{\mathrm{\Sigma }}_{\gamma }{I}_{\gamma }^{\mathrm{\top }}\right\}{I}_{\theta }^{-\mathrm{\top }}.$$

By ULLN and continuous mapping theorem, we have ${\hat{I}}_{\theta }=I_{\theta }+o_p(1),$ ${\hat{I}}_{\gamma }=I_{\gamma }+o_p(1)$ and ${\hat{J}}_{\theta }=J_{\theta }+o_p(1)$ By assumption ${\hat{\Sigma }}_{\gamma }={\Sigma }_{\gamma }+o_p(1)$ and $n_3/n_2\to C_2<\mathrm{\infty }$, using continuous mapping theorem, we have ${\hat{\Sigma }}_{\theta }{\Sigma }_{\theta }+o_p(1)$.

Now we provide the detail forms for the quantities involved in a Cox regression model.

Lemma 2

Assume $n_3/n_2\to C_2<\mathrm{\infty }$ and ${\theta }^{\mathrm{*}}$ is the unique solution to

$$0=U\left(\theta ,{\gamma }^{\mathrm{*}}\right)=E{\int }_0^{\tau } \left[\left(\begin{array}{c}{Z}^{\mathrm{*}}\\ V\end{array}\right)-\frac{E\left[\left(\begin{array}{c}{Z}^{\mathrm{*}}\\ V\end{array}\right)Y(t)\mathrm{e}\mathrm{x}\mathrm{p}\{(Z^{\mathrm{*}},V^{\mathrm{\top }})\theta \}\right]}{E\left[Y(t)\mathrm{e}\mathrm{x}\mathrm{p}\{(Z^{\mathrm{*}},V^{\mathrm{\top }})\theta \}\right]}\right]dN(t).$$

where $Z^{\mathrm{*}}=\mathbb{X}{\gamma }^{\mathrm{*}}$. If $\hat{\theta }$ solve the estimating equation

$$0=n_3^{-1}\sum _{i=1}^{n_3} {\int }_0^{\tau } \left[\left(\begin{array}{c}{\hat{Z}}_i\\ V_i\end{array}\right)-\sum _{j=1}^{n_3} \frac{Y_j(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{({\hat{Z}}_j,V_j^{\mathrm{\top }})\theta \right\}}{{\sum }_{k=1}^{n_3} Y_k(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{({\hat{Z}}_k,V_k^{\mathrm{\top }})\theta \right\}}\left(\begin{array}{c}{\hat{Z}}_j\\ V_j\end{array}\right)\right]d{N}_i(t),$$

where ${\hat{Z}}_i={\mathbb{X}}_i\hat{\gamma }$ and $\sqrt{n_2}\left(\hat{\gamma }-{\gamma }^{\mathrm{*}}\right)\to N\left(0,{\Sigma }_{\gamma }\right)$, then we have $\sqrt{n_3}(\hat{\theta }-{\theta }^{\mathrm{*}})\to N\left(0,{\Sigma }_{\theta }\right)$ where

$$\begin{array}{c}{\Sigma }_{\theta }=I_{\theta }^{-1}\left(J_{\theta }+C_2{I}_{\gamma }{\Sigma }_{\gamma }{I}_{\gamma }^{\mathrm{\top }}\right)I_{\theta }^{-\mathrm{\top }},\\ I_{\theta }=E\left[Y_i(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{(Z_i^{\mathrm{*}},V_i^{\mathrm{\top }})\theta \right\}{\left(\begin{array}{c}{Z}_i^{\mathrm{*}}\\ V_i\end{array}\right)}^{\otimes 2}\right]\\ J_{\theta }=E{\int }_0^{\tau } {\left[\left(\begin{array}{c}{Z}_i^{\mathrm{*}}\\ V_i\end{array}\right)-\frac{s^{(1)}(\theta ,t)}{s^{\left(0\right)}(\theta ,t)}\right]}^{\otimes 2}d{N}_i(t)\\ I_{\gamma }=-E\left[{\int }_0^{\tau } \left\{\left(\begin{array}{c}{\mathbb{X}}_i\\ 0\end{array}\right)-\frac{A(\theta ,t)}{s^{\left(0\right)}(\theta ,t)}+\frac{s^{(1)}(\theta ,t)G(\theta ,t)}{s^{\left(0\right)}(\theta ,t{)}^2}\right\}d{N}_i(t)\right].\end{array}$$

where $a^{\otimes 2}=a{a}^T$ and

$$\begin{array}{ll}A(\theta ,t)& =E\left[Y_i(t)\left(\begin{array}{c}1+{\theta }_Z{Z}_i^{\mathrm{*}}\\ {\theta }_Z{V}_i\end{array}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left\{(Z_i^{\mathrm{*}},V_i^{\mathrm{\top }})\theta \right\}{\mathbb{X}}_i\right],\\ G(\theta ,t)& =E\left[Y_i(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{(Z_i^{\mathrm{*}},V_i^{\mathrm{\top }})\theta \right\}{\theta }_Z{\mathbb{X}}_i\right],\\ s^{\left(0\right)}(\theta ,t)& =E\left[Y_i(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{(Z_i^{\mathrm{*}},V_i^{\mathrm{\top }})\theta \right\}\right],\\ s^{(1)}(\theta ,t)& =E\left[Y_i(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{(Z_i^{\mathrm{*}},V_i^{\mathrm{\top }})\theta \right\}\left(\begin{array}{c}{Z}_i^{\mathrm{*}}\\ V_i\end{array}\right)\right].\end{array}$$

This variance can be consistently estimated by

$${\hat{\Sigma }}_{\theta }={\hat{I}}_{\theta }^{-1}\left({\hat{J}}_{\theta }+\frac{n_3}{n_2}{\hat{I}}_{\gamma }{\stackrel{\mathrm{ˆ}}{\Sigma }}_{\gamma }{\hat{I}}_{\gamma }^{\mathrm{\top }}\right){\hat{I}}_{\theta }^{-\mathrm{\top }},$$

where

$$\begin{array}{c}{\hat{I}}_{\theta }=n_3^{-1}\sum _{i=1}^{n_3} \left[Y_i(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{({\hat{Z}}_i,V_i^{\mathrm{\top }})\hat{\theta }\right\}{\left(\begin{array}{l}{\hat{Z}}_i\\ V_i\end{array}\right)}^{\otimes 2}\right]\\ {{\hat{J}}_{\theta }=n_3^{-1}\sum _{i=1}^{n_3} {\Delta }_i\left[\left(\begin{array}{c}{\hat{Z}}_i\\ V_i\end{array}\right)-\sum _j \frac{Y_j\left(T_i\right)\mathrm{e}\mathrm{x}\mathrm{p}\left\{({\hat{Z}}_j,V_j^{\mathrm{\top }})\hat{\theta }\right\}}{\sum _k Y_k\left(T_i\mathrm{e}\mathrm{x}\mathrm{p}\left\{({\hat{Z}}_k,V_k^{\mathrm{\top }})\hat{\theta }\right\}\right.}\left(\begin{array}{c}{\hat{Z}}_j\\ V_j\end{array}\right)\right]}^{\otimes 2}\\ {\hat{I}}_{\gamma }=n_3^{-1}\sum _{i=1}^{n_3} {\Delta }_i\left\{\left(\begin{array}{c}{\mathbb{X}}_i\\ 0\end{array}\right)-\frac{\hat{A}(\hat{\theta },T_i)}{s^{\left(0\right)}(\hat{\theta },T_i)}+\frac{{\hat{s}}^{(1)}(\hat{\theta },T_i)\hat{G}(\hat{\theta },T_i)}{{\hat{s}}^{\left(0\right)}{(\hat{\theta },T_i)}^2}\right\}\\ \hat{A}(\theta ,t)=n_3^{-1}\sum _{i=1}^{n_3} \left[Y_i\left(t\right)\left(\begin{array}{c}1+{\theta }_Z{\hat{Z}}_i\\ {\theta }_Z{V}_i\end{array}\right)\exp \left\{({\hat{Z}}_i,V_i^{\mathrm{\top }})\theta \right\}{\mathbb{X}}_i\right],\\ \hat{G}(\theta ,t)=n_3^{-1}\sum _{i=1}^{n_3} \left[Y_i(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{({\hat{Z}}_i,V_i^{\mathrm{\top }})\theta \right\}{\theta }_Z{\mathbb{X}}_i\right],\end{array}$$

${\hat{\Sigma }}_{\gamma }$ is a consistent estimator of ${\Sigma }_{\gamma }$ and $n_3/n_2\to C_2$.

Proof Denote $U_i(\theta ,\gamma )={\int }_0^{\tau } \left[\left(\begin{array}{l}{Z}_i\\ V_i\end{array}\right)-\frac{E\left[Y_j(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{\left(Z_j,V_j^{\mathrm{\top }}\right)\theta \right\}\left(\begin{array}{l}{Z}_j\\ V_j\end{array}\right)\right]}{E\left[Y_k(t)\mathrm{e}\mathrm{x}\mathrm{p}\left\{\left(Z_k,V_k^{\mathrm{\top }}\right)\theta \right\}\right]}\right]d{N}_i(t)$ where $Z_i={\mathbb{X}}_i\gamma$. Then by definition of $I_{\theta },I_{\gamma },J_{\theta }$ in Lemma 1, we can compute the form of these terms as stated above. The convergence of $\hat{\theta }$ and $\hat{\gamma }$ as long as ULLN and continuous mapping ensure that we have

$$\begin{gathered} \hat{A}(\hat{\theta },t)=A({\theta }^{\mathrm{*}},t)+o_p(1) \\ \hat{G}(\hat{\theta },t)=G({\theta }^{\mathrm{*}},t)+o_p(1) \\ {\hat{s}}^{(0)}(\hat{\theta },t)=s^{(0)}({\theta }^{\mathrm{*}},t)+o_p(1) \\ {\hat{s}}^{(1)}(\hat{\theta },t)=s^{(1)}({\theta }^{\mathrm{*}},t)+o_p(1) \end{gathered}$$

which lead to the convergence ${\hat{I}}_{\theta }=I_{\theta }+o_p(1),{\hat{I}}_{\gamma }=I_{\gamma }+o_p(1)$ and ${\hat{J}}_{\theta }=J_{\theta }+o_p(1)$. Applying Lemma 1, we have the asymptotic for $\tilde{\theta }$ that solve the equation $0=n_3^{-1}{\sum }_{i=1}^{n_3} U_i(\theta ,\hat{\gamma })$. Now we just need to show $\tilde{\theta }$ and $\hat{\theta }$ is asymptotically equivalent, which is guaranteed by applying ULLN to get:

$$\begin{gathered} n_3^{-1}\sum _{i=1}^{n_3}{Y}_j(t)\text{exp}\left\{({\hat{Z}}_j,V_j^{\top })\theta \right\}=s^{\left(0\right)}\left(\theta ,t\right)+o_p(1) \\ {n}_3^{-1}\sum _{i=1}^{n_3}{Y}_j(t)\text{exp}\left\{({\hat{Z}}_j,V_j^{\top })\theta \right\}\left(\begin{array}{c}{\hat{Z}}_j\\ V_j\end{array}\right)=s^{(1)}\left(\theta ,t\right)+o_p(1) \end{gathered}$$

Here we would like to comment that for method 2–4, $Z^{\mathrm{*}}$ has the same expression $E(Z\mid Q,V)$ and thus the $I_{\theta }$ are the same though their estimated version ${\hat{I}}_{\theta }$ are different.

Now we handle the estimation of $\gamma$ in the following lemma.

Lemma 3

Assume $n_2/n_1\to C_1<\mathrm{\infty }$ and ${\gamma }^{\mathrm{*}}$ is the unique solution to $EU(\gamma ,{\beta }^{\mathrm{*}})=0$ for an estimating function $U(\gamma ,\beta )=[{\left(Q_i,V_i^{\mathrm{\top }}\right)}^T{\hat{X}}_i-{\left(Q_i,V_i^{\mathrm{\top }}\right)}^T\left(Q_i,V_i^{\mathrm{\top }}\right)\gamma ]$. If $\hat{\gamma }$ solve the estimating equation $0=n_3^{-1}{\sum }_{i=1}^{n_3} U_i(\gamma ,\hat{\beta })$ where $\sqrt{n_1}(\hat{\beta }-{\beta }^{\mathrm{*}})\to N\left(0,{\Sigma }_{\beta }\right)$, then we have $\sqrt{n_2}\left(\hat{\gamma }-{\gamma }^{\mathrm{*}}\right)\to N\left(0,{\Sigma }_{\gamma }\right)$ where

$${\Sigma }_{\gamma }=I_{\gamma }^{-1}\left(I_{\gamma }+C_1{I}_{\beta }{\mathrm{\Sigma }}_{\beta }{I}_{\beta }^{\mathrm{\top }}\right)I_{\gamma }^{-\mathrm{\top }},$$

where $I_{\gamma }=E\left(-{\frac{\partial U(\gamma ,\beta )}{\partial \gamma }|}_{\gamma ,{\beta }^{\mathrm{*}}}\right)=E[{\left(Q_i,V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}\left(Q_i,V_i^{\mathrm{\top }}\right)]$ and $I_{\beta }=E\left(-{\frac{\partial U(\gamma ,\beta )}{\partial \beta }|}_{{\gamma }^{\mathrm{*}},{\beta }^{\mathrm{*}}}\right)$. This variance can be consistently estimated by

$${\stackrel{\mathrm{ˆ}}{\Sigma }}_{\gamma }=I_{\gamma }^{-1}\left({\hat{I}}_{\gamma }+\frac{n_2}{n_1}{\hat{I}}_{\beta }{\stackrel{\mathrm{ˆ}}{\Sigma }}_{\beta }{I}_{\beta }^{\mathrm{\top }}\right)I_{\gamma }^{-\mathrm{\top }},$$

where ${\hat{Y}}_{\gamma }=n_2^{-1}{\sum }_{i=1}^{n_2} \left(-{\frac{\partial U_i(\gamma ,\beta )}{\partial \gamma }|}_{\gamma ,\hat{\beta }}\right),$ ${\hat{I}}_{\beta }=n_2^{-1}{\sum }_{i=1}^{n_2} \left(-{\frac{\partial U_i(\gamma ,\beta )}{\partial \beta }|}_{\hat{\beta },\hat{\beta }}\right)$ and ${\stackrel{\mathrm{ˆ}}{\Sigma }}_{\beta }$ is a consistent estimator of ${\Sigma }_{\beta }$.

Proof We can derive the asymptotic for $\hat{\gamma }$ as below:

$$\begin{array}{l}0=\sum _{i=1}^{n_2} U_i(\hat{\gamma },\hat{\beta })\\ =\sum _{i=1}^{n_2} \left\{U_i({\gamma }^{\mathrm{*}},{\beta }^{\mathrm{*}})+{\left.\frac{\partial U}{\partial \gamma }\right|}_{{\gamma }^{\mathrm{*}},{\beta }^{\mathrm{*}}}(\hat{\gamma }-{\gamma }^{\mathrm{*}})+{\frac{\partial U}{\partial \beta }|}_{{\gamma }^{\mathrm{*}},{\beta }^{\mathrm{*}}}(\hat{\beta }-{\beta }^{\mathrm{*}})+o(\Vert \hat{\gamma }-{\gamma }^{\mathrm{*}}\Vert ,\parallel \hat{\beta }-{\beta }^{\mathrm{*}}\Vert )\right\}.\end{array}$$

With ULLN,

$${n_2^{-1}\sum _{i=1}^{n_2} \frac{\partial U_i(\gamma ,\beta )}{\partial \gamma }|}_{\gamma ,\hat{\beta }}=E\left({\frac{\partial U}{\partial \gamma }|}_{\gamma ,\hat{\beta }}\right)+o_p(1)$$

and with continuous mapping theorem, we have

$$E\left({\frac{\partial U}{\partial \gamma }|}_{\gamma ,\hat{\beta }}\right)=E\left({\frac{\partial U}{\partial \gamma }|}_{\gamma ,{\beta }^{\mathrm{*}}}\right)+o_p(1)$$

So we have

$${n_2^{-1}\sum _{i=1}^{n_2} \frac{\partial U_i(\gamma ,\beta )}{\partial \gamma }|}_{\hat{\gamma },\hat{\beta }}=-I_{\gamma }+o_p\left(1\right).$$

Similarly, we have

$${n_2^{-1}\sum _{i=1}^{n_2} \frac{\partial U_i(\gamma ,\beta )}{\partial \beta }|}_{\hat{\gamma },\hat{\beta }}=-I_{\beta }+o_p(1).$$

Plug in the Taylor expansion, notice that $EU({\gamma }^{\mathrm{*}},{\beta }^{\mathrm{*}})=0$, we have

$$\begin{array}{l}0=n_2^{-1}\sum _{i=1}^{n_2} \{U_i({\gamma }^{\mathrm{*}},{\beta }^{\mathrm{*}})-EU({\gamma }^{\mathrm{*}},{\beta }^{\mathrm{*}})\}-I_{\gamma }\left(\hat{\gamma }-{\gamma }^{\mathrm{*}}\right)-I_{\beta }(\hat{\beta }-{\beta }^{\mathrm{*}})\\ +o(\Vert \hat{\gamma }-{\gamma }^{\mathrm{*}}\Vert ,\Vert \hat{\beta }-{\beta }^{\mathrm{*}}\Vert )+o_p(1)\mathrm{*}\Vert \hat{\beta }-{\beta }^{\mathrm{*}}\Vert +o_p(1)\mathrm{*}\Vert \hat{\gamma }-{\gamma }^{\mathrm{*}}\Vert .\end{array}$$

So we have

$$\sqrt{n_2}\left(\hat{\gamma }-{\gamma }^{\mathrm{*}}\right)=I_{\gamma }^{-1}\left\{n_2^{-1/2}\sum _{i=1}^{n_2} U_i({\gamma }^{\mathrm{*}},{\beta }^{\mathrm{*}})-I_{\beta }\sqrt{n_2}(\hat{\beta }-{\beta }^{\mathrm{*}})\right\}+o_p(1).$$

By CLT, we have

$$n_2^{-1/2}\sum _{i=1}^{n_2} \{U_i({\gamma }^{\mathrm{*}},{\beta }^{\mathrm{*}})-EU({\gamma }^{\mathrm{*}},{\beta }^{\mathrm{*}})\}{\Rightarrow }_{d}N(0,Var(U_i({\gamma }^{\mathrm{*}},{\beta }^{\mathrm{*}})))$$

So we have

$${\Sigma }_{\gamma }=I_{\gamma }^{-1}\left\{Var\left(U\right(\gamma ,\beta \left)\right)+C_2{I}_{\beta }{\mathrm{\Sigma }}_{\beta }{I}_{\beta }^{\mathrm{\top }}\right\}{I}_{\gamma }^{-\mathrm{\top }}.$$

As we have shown ${\hat{I}}_{\gamma }=I_{\gamma }+o_p(1)$, ${\hat{I}}_{\beta }=I_{\beta }+o_p(1)$ and by assumption ${\hat{\Sigma }}_{\beta }={\Sigma }_{\beta }+o_p(1)$ and $n_2/n_1\to C_1<\mathrm{\infty }$, using continuous mapping theorem, we have ${\stackrel{\mathrm{ˆ}}{\Sigma }}_{\gamma }={\Sigma }_{\gamma }+o_p(1)$.

Applying the general form to each specific regression method, we can obtain the asymptotic results we need.

Lemma 4

Assume $n_2/n_1\to C_1<\mathrm{\infty }$ and ${\gamma }_1^{\mathrm{*}}$ is the unique solution to $EU({\gamma }_1,{\beta }_1^{\mathrm{*}})=0$ for an estimating function $U\left({\gamma }_1,{\beta }_1\right)$. If ${\hat{\gamma }}_1$ solve the estimating equation $0=n_2^{-1}{\sum }_{i=1}^{n_2} U_i({\gamma }_1,{\hat{\beta }}_1)$ where $\sqrt{n_1}({\hat{\beta }}_1-{\beta }_1^{\mathrm{*}})\to N(0,{\Sigma }_{{\beta }_1})$, then we have $\sqrt{n_2}\left({\hat{\gamma }}_1-{\gamma }_1^{\mathrm{*}}\right)\to N\left(0,{\Sigma }_{{\gamma }_1}\right)$ where

$${\Sigma }_{{\gamma }_1}=I_{{\gamma }_1}^{-1}\left(I_{{\gamma }_1}+C_1{I}_{{\beta }_1}{\mathrm{\Sigma }}_{{\beta }_1}{I}_{{\beta }_1}^{\mathrm{\top }}\right)I_{{\gamma }_1}^{-\mathrm{\top }},$$

where $I_{{\gamma }_1}=E\left(-{\frac{\partial U\left({\gamma }_1,{\beta }_1\right)}{\partial {\gamma }_1}|}_{{\gamma }_1^{\mathrm{*}},{\beta }_1^{\mathrm{*}}}\right)$ and $I_{{\beta }_1}=E\left(-{\frac{\partial U\left({\gamma }_1,{\beta }_1\right)}{\partial {\beta }_1}|}_{{\gamma }_1^{\mathrm{*}},{\beta }_1^{\mathrm{*}}}\right)$. This variance can be consistently estimated by

$${\hat{\Sigma }}_{{\gamma }_1}={\hat{I}}_{{\gamma }_1}^{-1}\left({\hat{I}}_{{\gamma }_1}+\frac{n_2}{n_1}{\hat{I}}_{{\beta }_1}{\stackrel{\mathrm{ˆ}}{\mathrm{\Sigma }}}_{{\beta }_1}{\hat{I}}_{{\beta }_1}^{\mathrm{\top }}\right){\hat{I}}_{{\gamma }_1}^{-\mathrm{\top }},$$

where ${\hat{I}}_{{\gamma }_1}=n_2^{-1}{\sum }_{i=1}^{n_2} \left(-{\frac{\partial U_i\left({\eta }_1,{\beta }_1\right)}{\partial {\eta }_1}|}_{\hat{\gamma },{\hat{\beta }}_1}\right),{\hat{I}}_{{\beta }_1}=n_2^{-1}{\sum }_{i=1}^{n_2} \left(-{\frac{\partial U_i\left({\gamma }_1,{\beta }_1\right)}{\partial {\beta }_1}|}_{{\hat{\gamma }}_1,{\hat{\beta }}_1}\right)$ and ${\hat{\Sigma }}_{{\beta }_1}$ is a consistent estimator of ${\Sigma }_{{\beta }_1}$.

Proof

We need to derive asymptotic for ${\hat{\beta }}_1$ and then apply 3. The asymptotic of ${\hat{\beta }}_1$ can be derived as below:

$$\begin{array}{l}0=\sum _{i=1}^{n_1} U_i({\hat{\beta }}_1)\\ =\sum _{i=1}^{n_1} \left\{U_i({\beta }_1^{\mathrm{*}})+{\frac{\partial U}{\partial {\beta }_1}|}_{{\beta }_1^{\mathrm{*}}}({\hat{\beta }}_1-{\beta }_1^{\mathrm{*}})+o(\Vert {\hat{\beta }}_1-{\beta }_1^{\mathrm{*}}\Vert )\right\}.\end{array}$$

where $U_i({\beta }_1^{\mathrm{*}})={\left(1,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}{X}_i^{\mathrm{*}}-{\left(1,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}\left(1,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }}\right){\beta }_1^{\mathrm{*}}$.

With ULLN,

$${n_1^{-1}\sum _{i=1}^{n_1} \frac{\partial U_i\left({\beta }_1\right)}{\partial {\beta }_1}|}_{{\hat{\beta }}_1}=E\left({\frac{\partial U}{\partial {\beta }_1}|}_{{\hat{\beta }}_1}\right)+o_p(1)$$

and with continuous mapping theorem, we have

$$E\left({\frac{\partial U}{\partial {\beta }_1}|}_{{\hat{\beta }}_1}\right)=E\left({\frac{\partial U}{\partial {\beta }_1}|}_{{\beta }_1^{\mathrm{*}}}\right)+o_p(1)$$

So we have

$${n_1^{-1}\sum _{i=1}^{n_1} \frac{\partial U_i\left({\beta }_1\right)}{\partial {\beta }_1}|}_{{\hat{\beta }}_1}=-I_{{\beta }_1}+o_p\left(1\right).$$

Plug in the Taylor expansion, notice that $EU({\beta }_1^{\mathrm{*}})=0$, we have

$$\begin{array}{l}0=n_1^{-1}\sum _{i=1}^{n_1} \{U_i({\beta }_1^{\mathrm{*}})-EU({\beta }_1^{\mathrm{*}})\}-I_{{\beta }_1}({\hat{\beta }}_1-{\beta }_1^{\mathrm{*}})\\ +o_p(1)\Vert {\hat{\beta }}_1-{\beta }_1^{\mathrm{*}}\Vert .\end{array}$$

So we have

$$\sqrt{n_1}({\hat{\beta }}_1-{\beta }_1^{\mathrm{*}})=I_{{\beta }_1}^{-1}\left\{n_1^{-1/2}\sum _{i=1}^{n_1} U_i({\beta }_1^{\mathrm{*}})\right\}+o_p\left(1\right).$$

By CLT, we have

$$n_1^{-1/2}\sum _{i=1}^{n_2} \{U_i({\beta }_1^{\mathrm{*}})-EU({\beta }_1^{\mathrm{*}})\}{\Rightarrow }_{d}N(0,Var(U_i({\beta }_1^{\mathrm{*}})))$$

So we have

$${\Sigma }_{{\beta }_1}=I_{{\beta }_1}^{-1}\left\{Var\left(U\left({\beta }_1\right)\right)\right\}{I}_{{\beta }_1}^{-\mathrm{\top }}.$$

By assumption, ${\stackrel{\mathrm{ˆ}}{\Sigma }}_{{\beta }_1}={\Sigma }_{{\beta }_1}+o_p(1)$ which is a consistent estimator of ${\Sigma }_{{\beta }_1}$.

Then by applying Lemma 3, we have

$${\hat{\Sigma }}_n={\hat{I}}_{{\gamma }_1}^{-1}\left({\hat{I}}_{{\gamma }_1}+\frac{n_2}{n_1}{\hat{I}}_{{\beta }_1}{\hat{\Sigma }}_{{\beta }_1}{\hat{I}}_{{\beta }_1}^{\mathrm{\top }}\right){\hat{I}}_{{\gamma }_1}^{-\mathrm{\top }}$$

Lemma 5

Assume $n_2/n_1\to C_1<\mathrm{\infty }$ and ${\gamma }_2^{\mathrm{*}}$ is the unique solution to $EU({\gamma }_2,{\beta }_2^{\mathrm{*}})=0$ for an estimating function $U\left({\gamma }_2,{\beta }_2\right)$. If ${\hat{\gamma }}_2$ solve the estimating equation $0=n_2^{-1}{\sum }_{i=1}^{n_2} U_i({\gamma }_2,{\hat{\beta }}_2)$ where $\sqrt{n_1}({\hat{\beta }}_2-{\beta }_2^{\mathrm{*}})\to N(0,{\Sigma }_{{\beta }_2})$, then we have $\sqrt{n_2}\left({\hat{\gamma }}_2-{\gamma }_2^{\mathrm{*}}\right)\to N\left(0,{\Sigma }_{{\gamma }_2}\right)$ where

$${\Sigma }_{{\gamma }_2}=I_{{\gamma }_2}^{-1}\left(I_{{\gamma }_2}+C_1{I}_{{\beta }_2}{\mathrm{\Sigma }}_{{\beta }_2}{I}_{{\beta }_2}^{\mathrm{\top }}\right)I_{{\gamma }_2}^{-\mathrm{\top }},$$

where $I_{{\gamma }_2}=E\left(-{\frac{\partial U\left({\gamma }_2,{\beta }_2\right)}{\partial {\gamma }_2}|}_{{\gamma }_2^{\mathrm{*}},{\beta }_2^{\mathrm{*}}}\right)$ and $I_{{\beta }_2}=E\left(-{\frac{\partial U\left({\gamma }_2,{\beta }_2\right)}{\partial {\beta }_2}|}_{{\gamma }_2^{\mathrm{*}},{\beta }_2^{\mathrm{*}}}\right)$. This variance can be consistently estimated by

$${\stackrel{\mathrm{ˆ}}{\mathrm{\Sigma }}}_{{\gamma }_2}={\hat{I}}_{{\gamma }_2}^{-1}\left({\hat{I}}_{{\gamma }_2}+\frac{n_2}{n_1}{\hat{I}}_{{\beta }_2}{\widehat{{\beta }_2}}_{{\hat{I}}_{{\beta }_2}}^{\mathrm{\top }}\right){\hat{I}}_{{\gamma }_2}^{-\mathrm{\top }},$$

where ${\hat{Y}}_{{\gamma }_2}=n_2^{-1}{\sum }_{i=1}^{n_2} \left(-{\frac{\partial U_i\left({\gamma }_2,{\beta }_2\right)}{\partial {\gamma }_2}|}_{{\hat{\gamma }}_2,{\hat{\beta }}_2}\right),{\hat{I}}_{{\beta }_2}=n_2^{-1}{\sum }_{i=1}^{n_2} \left(-{\frac{\partial U_i\left({\gamma }_2,{\beta }_2\right)}{\partial {\hat{\beta }}_2}|}_{{\hat{\gamma }}_2,{\hat{\beta }}_2}\right)$ and ${\hat{\Sigma }}_{{\beta }_2}$ is a consistent estimator of ${\Sigma }_{{\beta }_2}$.

Proof

Define ${\beta }_2=\frac{{\beta }_1}{BF}$.

First note that,

$$\begin{array}{r}Var(\tilde{X}V,W)={\Omega }_1=\frac{1}{n_1}\sum _i {\left\{{\tilde{X}}_i-{\left(1,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}{\beta }_1\right\}}^2,\\ Var(\tilde{X}V)={\Omega }_2=\frac{1}{n_1}\sum _i {\left\{{\tilde{X}}_i-\left(1,V_i^{\mathrm{\top }}\right){\beta }_t\right\}}^2,\end{array}$$

where we let ${\Omega }_{1i}={\left\{{\tilde{X}}_i-\left(1,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }}\right){\beta }_1\right\}}^2$ and ${\Omega }_{2i}={\left({\tilde{X}}_i-\left(1,V_i^{\mathrm{\top }}\right){\beta }_t\right)}^2$.

Second, the estimating equations considered are

$$\begin{array}{r}{U}_{121i}={\left(1,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}{\Omega }_1^{-1}({\tilde{X}}_i-\left(1,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }}\right){\beta }_1),\\ U_{122i}={\left(1,V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}{\Omega }_2^{-1}({\tilde{X}}_i-(1,V_i^{\mathrm{\top }}){\beta }_t).\\ U_{123i}={({\tilde{X}}_i-{\left(1,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}\beta )}^2-{\Omega }_1,\\ U_{124i}={({\tilde{X}}_i-\left(1,V_i^{\mathrm{\top }}\right){\beta }_t)}^2-{\Omega }_2,\end{array}$$

Third, we can derive the asymptotic normal distribution for $\beta ,$ ${\beta }_t,$ ${\Omega }_1$ and ${\Omega }_2$ as

$$\sqrt{n_1}\left[\left(\begin{array}{c}{\hat{\beta }}_1\\ {\hat{\beta }}_t\\ {\widehat{\Omega }}_1\\ {\stackrel{\mathrm{ˆ}}{\Omega }}_2\end{array}\right)-\left(\begin{array}{c}\beta \\ {\beta }_t\\ {\Omega }_1\\ {\Omega }_2\end{array}\right)\right]\to N\left(0,I^{-1}J{I}^{-\mathrm{\top }}\right),$$

where $J$ is the variance covariance matrix of the above four estimating equations and $I$ is a matrix composed by the expectation of derivatives of each estimating equation with respect to $\beta ,{\beta }_t,{\Omega }_1$ and ${\Omega }_2$, respectively. Specifically,

$$I=\left(\begin{array}{cccc}\frac{1}{n_1}\sum _i {\left({\mathbb{X}}_i\right)}^{\mathrm{\top }}\left({\mathbb{X}}_i\right)& 0& 0& 0\\ 0& \frac{1}{n_1}\sum _i {\left({\mathbb{X}}_{ti}\right)}^{\mathrm{\top }}\left({\mathbb{X}}_{ti}\right)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$

where ${\mathbb{X}}_i={\left(1,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}$ and ${\mathbb{X}}_{ti}={\left(1,V_i^{\mathrm{\top }}\right)}^{\mathrm{\top }}$

Fourth, the asymptotic normal distribution for ${\hat{\beta }}_1$ and $\widehat{BF}$ jointly can be derived using delta method.

$$\sqrt{n_1}\left[(\begin{array}{c}\hat{\beta }\\ \widehat{BF}\end{array})-(\begin{array}{c}\beta \\ BF\end{array})\right]\to N(0,C{I}^{-1}J{I}^{-\mathrm{\top }}{C}^{\mathrm{\top }}),$$

where $\mathrm{C}$ is a matrix derived by taking derivative of $\beta$ and $\mathrm{B}\mathrm{F}$ each with respect to $\beta ,$ ${\beta }_t,$ $V_1$ and $V_2$ respectively. For example,

$$\frac{\partial BF}{\partial \beta }=0,\frac{\partial BF}{\partial {\beta }_t}=0,\frac{\partial BF}{\partial {\Omega }_1}=\frac{-1}{{\Omega }_2-{\tilde{\sigma }}_x^2},\frac{\partial BF}{\partial V_2}=\frac{{\Omega }_1-{\tilde{\sigma }}_x^2}{{\left({\Omega }_2-{\tilde{\sigma }}_x^2\right)}^2}$$

Fifth, ${\hat{\beta }}_2=\frac{{\hat{\beta }}_1}{BF}$ can be derived using delta method.

$$\sqrt{n_1}\left({\hat{\beta }}_2-{\beta }_2\right)\to N(0,C^{\prime }(C{I}^{-1}J{I}^{-\mathrm{\top }}{C}^{\mathrm{\top }})C^{\mathrm{\top }\mathrm{\top }}),$$

where C’ is a matrix derived by taking derivative of ${\beta }_2$ each with respect to $\beta$ and $BF$, respectively. That is,

$$\frac{\partial {\beta }_2}{\partial \beta }=\frac{1}{BF},\frac{\partial {\beta }_2}{\partial BF}=-\frac{\beta }{B{F}^2}$$

Then we have estimating equation for ${\beta }_2$ as below:

$$\begin{array}{l}0=\sum _{i=1}^{n_1} U_i({\hat{\beta }}_2)\\ =\sum _{i=1}^{n_1} \left\{U_i({\beta }_2^{\mathrm{*}})+{\left.\frac{\partial U}{\partial {\beta }_2}\right|}_{{\beta }_2^{\mathrm{*}}}({\hat{\beta }}_2-{\beta }_2^{\mathrm{*}})+o(\Vert {\hat{\beta }}_2-{\beta }_2^{\mathrm{*}}\Vert )\right\}.\end{array}$$

With ULLN,

$${n_1^{-1}\sum _{i=1}^{n_1} \frac{\partial U_i\left({\beta }_2\right)}{\partial {\beta }_2}|}_{{\hat{\beta }}_2}=E\left({\frac{\partial U}{\partial {\beta }_2}|}_{{\hat{\beta }}_2}\right)+o_p(1)$$

and with continuous mapping theorem, we have

$$E\left({\frac{\partial U}{\partial {\beta }_1}|}_{{\hat{\beta }}_2}\right)=E\left({\frac{\partial U}{\partial {\beta }_1}|}_{{\beta }_2^{\mathrm{*}}}\right)+o_p(1)$$

So we have

$${n_1^{-1}\sum _{i=1}^{n_1} \frac{\partial U_i\left({\beta }_2\right)}{\partial {\beta }_1}|}_{{\hat{\beta }}_2}=-I_{{\beta }_1}+o_p\left(1\right).$$

Plug in the Taylor expansion, notice that $EU({\beta }_2^{\mathrm{*}})=0$, we have

$$\begin{array}{l}0=n_1^{-1}\sum _{i=1}^{n_1} \{U_i({\beta }_2^{\mathrm{*}})-EU({\beta }_2^{\mathrm{*}})\}-I_{{\beta }_1}({\hat{\beta }}_2-{\beta }_2^{\mathrm{*}})\\ +o_p(1)\Vert {\hat{\beta }}_2-{\beta }_2^{\mathrm{*}}\Vert .\end{array}$$

So we have

$$\sqrt{n_1}({\hat{\beta }}_2-{\beta }_2^{\mathrm{*}})=I_{{\beta }_2}^{-1}\left\{n_1^{-1/2}\sum _{i=1}^{n_1} U_i({\beta }_2^{\mathrm{*}})\right\}+o_p\left(1\right).$$

By CLT, we have

$$n_1^{-1/2}\sum _{i=1}^{n_2} \{U_i({\beta }_2^{\mathrm{*}})-EU({\beta }_2^{\mathrm{*}})\}{\Rightarrow }_{d}N(0,Var(U_i({\beta }_2^{\mathrm{*}})))$$

So we have

$${\mathrm{\Sigma }}_{{\beta }_2}=I_{{\beta }_2}^{-1}\left\{Var\left(U\left({\beta }_2\right)\right)\right\}{I}_{{\beta }_2}^{-\mathrm{\top }}.$$

By assumption, ${\stackrel{\mathrm{ˆ}}{\Sigma }}_{{\beta }_2}={\Sigma }_{{\beta }_2}+o_p(1)$ which is a consistent estimator of ${\Sigma }_{{\beta }_2}$.

Then by applying Lemma 3, we have

$${\hat{\Sigma }}_{{\gamma }_2}={\hat{I}}_{{\gamma }_2}^{-1}\left({\hat{I}}_{{\gamma }_2}+\frac{n_2}{n_1}{\hat{I}}_{{\beta }_2}{\stackrel{\mathrm{ˆ}}{\mathrm{\Sigma }}}_{{\beta }_2}{\hat{I}}_{{\beta }_2}^{\mathrm{\top }}\right){\hat{I}}_{{\gamma }_2}^{-\mathrm{\top }}$$

Lemma 6

Assume $n_2/n_1\to C_1<\mathrm{\infty }$ and ${\gamma }_3^{\mathrm{*}}$ is the unique solution to $EU({\gamma }_3,{\beta }_3^{\mathrm{*}})=0$ for an estimating function $U\left({\gamma }_3,{\beta }_3\right)$. If ${\gamma }_3$ solve the estimating equation $0=n_2^{-1}{\sum }_{i=1}^{n_2} U_i({\gamma }_3,{\hat{\beta }}_3)$ where $\sqrt{n_1}({\hat{\beta }}_3-{\beta }_3^{\mathrm{*}})\to N(0,{\Sigma }_{{\beta }_3})$, then we have $\sqrt{n_2}\left({\gamma }_3-{\gamma }_3^{\mathrm{*}}\right)\to N\left(0,{\Sigma }_{{\gamma }_3}\right)$ where

$${\Sigma }_{{\gamma }_3}=I_{{\gamma }_3}^{-1}\left(I_{{\gamma }_3}+C_1{I}_{{\beta }_3}{\Sigma }_{{\beta }_3}{I}_{{\beta }_3}^{\mathrm{\top }}\right)I_{{\gamma }_3}^{-\mathrm{\top }},$$

where $I_{{\gamma }_3}=E\left(-{\frac{\partial U\left({\gamma }_3,{\beta }_3\right)}{\partial {\gamma }_3}|}_{{\gamma }_3^{\mathrm{*}},{\beta }_3^{\mathrm{*}}}\right)$ and $I_{{\beta }_3}=E\left(-{\frac{\partial U\left({\gamma }_3,{\beta }_3\right)}{\partial {\beta }_3}|}_{{\gamma }_3^{\mathrm{*}},{\beta }_3^{\mathrm{*}}}\right)$. This variance can be consistently estimated by

$${\hat{\Sigma }}_{{\gamma }_3}={\hat{I}}_{{\gamma }_3}^{-1}\left({\hat{I}}_{{\gamma }_3}+\frac{n_2}{n_1}{\hat{I}}_{{\beta }_3}{\hat{\Sigma }}_{{\beta }_3}{\hat{I}}_{{\beta }_3}^{\mathrm{\top }}\right){\hat{I}}_{{\gamma }_3}^{-\mathrm{\top }},$$

where ${\hat{I}}_{{\gamma }_3}=n_2^{-1}{\sum }_{i=1}^{n_2} \left(-{\frac{\partial U_i\left({\gamma }_3,{\beta }_3\right)}{\partial {\gamma }_3}|}_{{\gamma }_3,{\hat{\beta }}_3}\right),{\hat{I}}_{{\beta }_3}=n_2^{-1}{\sum }_{i=1}^{n_2} \left(-{\frac{\partial U_i\left({\gamma }_3,{\beta }_3\right)}{\partial {\beta }_3}|}_{{\hat{\gamma }}_3,{\hat{\beta }}_3}\right)$ and ${\hat{\Sigma }}_{{\beta }_3}$ is a consistent estimator of ${\Sigma }_{{\beta }_3}$.

Proof

We can derive the asymptotic for ${\hat{\beta }}_3$ as below:

$$\begin{array}{l}0={\sum }_{i=1}^{n_1} U_i({\hat{\beta }}_3)\\ ={\sum }_{i=1}^{n_1} \left\{U_i({\beta }_3^{\mathrm{*}})+{\frac{\partial U}{\partial {\beta }_3}|}_{{\beta }_3^{\mathrm{*}}}({\hat{\beta }}_3-{\beta }_3^{\mathrm{*}})+o(\Vert {\hat{\beta }}_3-{\beta }_3^{\mathrm{*}}\Vert )\right\}.\end{array}$$

where $U_i({\beta }_3^{\mathrm{*}})={\left(1,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }},Q_i\right)}^{\mathrm{\top }}{\tilde{X}}_i-{\left(1,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }},Q_i\right)}^{\mathrm{\top }}\left(1,W_i^{\mathrm{\top }},V_i^{\mathrm{\top }},Q_i\right){\beta }_3^{\mathrm{*}}$.

With ULLN,

$${n_1^{-1}\sum _{i=1}^{n_1} \frac{\partial U_i\left({\beta }_3\right)}{\partial {\beta }_3}|}_{{\hat{\beta }}_3}=E\left({\frac{\partial U}{\partial {\beta }_3}|}_{{\hat{\beta }}_3}\right)+o_p(1)$$

and with continuous mapping theorem, we have

$$E\left({\frac{\partial U}{\partial {\beta }_3}|}_{{\hat{\beta }}_3}\right)=E\left({\frac{\partial U}{\partial {\beta }_3}|}_{{\beta }_3^{\mathrm{*}}}\right)+o_p(1)$$

So we have

$${n_1^{-1}\sum _{i=1}^{n_1} \frac{\partial U_i\left({\beta }_3\right)}{\partial {\beta }_3}|}_{{\hat{\beta }}_3}=-I_{{\beta }_3}+o_p\left(1\right).$$

Plug in the Taylor expansion, notice that $EU({\beta }_3^{\mathrm{*}})=0$, we have

$$\begin{array}{l}0=n_1^{-1}\sum _{i=1}^{n_1} \{U_i({\beta }_3^{\mathrm{*}})-EU({\beta }_3^{\mathrm{*}})\}-I_{{\beta }_3}({\hat{\beta }}_3-{\beta }_3^{\mathrm{*}})\\ +o_p(1)\Vert {\hat{\beta }}_3-{\beta }_3^{\mathrm{*}}\Vert .\end{array}$$

So we have

$$\sqrt{n_1}({\hat{\beta }}_3-{\beta }_3^{\mathrm{*}})=I_{{\beta }_3}^{-1}\left\{n_1^{-1/2}\sum _{i=1}^{n_1} U_i({\beta }_3^{\mathrm{*}})\right\}+o_p(1).$$

By CLT, we have

$${{\Sigma }_{{\beta }_3}=I_{{\beta }_3}^{-1}\left\{Var\left(U\left({\beta }_3\right)\right)\right\}}_{{\beta }_3}^{-\mathrm{\top }}.$$

By assumption, ${\hat{\Sigma }}_{{\beta }_3}={\Sigma }_{{\beta }_3}+o_p(1)$ which is a consistent estimator of ${\Sigma }_{{\beta }_3}$.

By applying Lemma 3, we have

$${\hat{\Sigma }}_{{\gamma }_3}={\hat{I}}_{{\gamma }_3}^{-1}\left({\hat{I}}_{{\gamma }_3}+\frac{n_2}{n_1}{\hat{I}}_{{\beta }_3}{\hat{\Sigma }}_{{\beta }_3}{\hat{I}}_{{\beta }_3}^{\mathrm{\top }}\right){\hat{I}}_{{\gamma }_3}^{-\mathrm{\top }}$$

Lemma 7

Assume ${\gamma }_4^{\mathrm{*}}$ is the unique solution to $EU\left({\gamma }_4\right)=0$ for an estimating function $U\left({\gamma }_4\right)$. Solving the estimating equation $0=n_1^{-1}{\sum }_{i=1}^{n_1} U_i\left({\hat{\gamma }}_4\right)$, we have $\sqrt{n_1}\left({\hat{\gamma }}_4-{\gamma }_4^{\mathrm{*}}\right)\to N\left(0,{\Sigma }_{{\gamma }_4}\right)$ where

$${\mathrm{\Sigma }}_{{\gamma }_4}=I_{{\gamma }_4}^{-1}\left(Var\left(U\left({\gamma }_4\right)\right)\right)I_{{\gamma }_4}^{-\mathrm{\top }},$$

where $I_{{\gamma }_4}=E\left(-{\frac{\partial U\left({\gamma }_4\right)}{\partial {\gamma }_4}|}_{{\gamma }_4}\right)$. This variance can be consistently estimated by

$${\stackrel{\mathrm{ˆ}}{\mathrm{\Sigma }}}_{{\gamma }_4}={\hat{I}}_{{\gamma }_4}^{-1}\left(Var\left(U\left({\gamma }_4\right)\right)\right){\stackrel{\mathrm{ˆ}}{1}}_{{\gamma }_4}^{-\mathrm{\top }},$$

where ${\hat{I}}_{{\gamma }_4}=n_1^{-1}{\sum }_{i=1}^{n_1} \left(-\frac{\partial U_i\left({\gamma }_4\right)}{\partial {\gamma }_4}\mid {\hat{\gamma }}_4\right)$ and ${\hat{\Sigma }}_{{\gamma }_4}$ is a consistent estimator of ${\Sigma }_{{\gamma }_4}$.

Proof

Derive asymptotic ${\hat{\gamma }}_4$ for method 4 directly

$$\begin{array}{l}0=\sum _{i=1}^{n_1} U_i\left({\hat{\gamma }}_4\right)\\ =\sum _{i=1}^{n_1} \left\{U_i\left({\gamma }_4^{\mathrm{*}}\right)+{\frac{\partial U}{\partial {\gamma }_4}|}_{{\gamma }_4^{\mathrm{*}}}\left({\hat{\gamma }}_4-{\gamma }_4^{\mathrm{*}}\right)+o\left(∥{\hat{\gamma }}_4-{\gamma }_4^{\mathrm{*}}∥\right)\right\}.\end{array}$$

where $U_i\left({\gamma }_4^{\mathrm{*}}\right)={\left(1,V_i^{\mathrm{\top }},Q_i\right)}^{\mathrm{\top }}{\tilde{X}}_i-{\left(1,V_i^{\mathrm{\top }},Q_i\right)}^T\left(1,V_i^{\mathrm{\top }},Q_i\right){\gamma }_4^{\mathrm{*}}$. With ULLN,

$${n_1^{-1}\sum _{i=1}^{n_1} \frac{\partial U_i\left({\gamma }_4\right)}{\partial {\gamma }_4}|}_{{\gamma }_4}=E\left({\frac{\partial U}{\partial {\gamma }_4}|}_{{\gamma }_4}\right)+o_p(1)$$

and with continuous mapping theorem, we have

$$E\left({\frac{\partial U}{\partial {\gamma }_4}|}_{{\gamma }_4}\right)=E\left({\frac{\partial U}{\partial {\gamma }_4}|}_{{\gamma }_4}\right)+o_p(1)$$

So we have

$${n_1^{-1}\sum _{i=1}^{n_1} \frac{\partial U_i\left({\gamma }_4\right)}{\partial {\gamma }_4}|}_{{\hat{\gamma }}_4}=-I_{{\gamma }_4}+o_p\left(1\right).$$

Plug in the Taylor expansion, notice that $EU\left({\gamma }_4^{\mathrm{*}}\right)=0$, we have

$$\begin{array}{l}0=n_1^{-1}\sum _{i=1}^{n_1} \left\{U_i\left({\gamma }_4^{\mathrm{*}}\right)-EU\left({\gamma }_4^{\mathrm{*}}\right)\right\}-I_{{\gamma }_4}\left({\hat{\gamma }}_4-{\gamma }_4^{\mathrm{*}}\right)\\ +o_p(1)∥{\hat{\gamma }}_4-{\gamma }_4^{\mathrm{*}}∥.\end{array}$$

So we have

$$\sqrt{n_1}\left({\hat{\gamma }}_4-{\gamma }_4^{\mathrm{*}}\right)=I_{{\gamma }_4}^{-1}\left\{n_1^{-1/2}\sum _{i=1}^{n_1} U_i\left({\gamma }_4^{\mathrm{*}}\right)\right\}+o_p(1)$$

By CLT, we have

$$n_1^{-1/2}\sum _{i=1}^{n_2} \left\{U_i\left({\gamma }_4^{\mathrm{*}}\right)-EU\left({\gamma }_4^{\mathrm{*}}\right)\right\}{\Rightarrow }_{d}N\left(0,Var\left(U_i\left({\gamma }_4^{\mathrm{*}}\right)\right)\right)$$

So we have

$${\Sigma }_{{\eta }_4}=I_{{\gamma }_4}^{-1}\left\{Var\left(U\left({\gamma }_4\right)\right)\right\}{I}_{{\gamma }_4}^{-\mathrm{\top }}$$

By assumption, ${\Sigma }_{{\gamma }_4}={\Sigma }_{{\gamma }_4}+o_p(1)$ which is a consistent estimator of ${\Sigma }_{{\gamma }_4}$.

Combine different steps together, we obtain the final asymptotics result.

Theorem 1

Provide the asymptotic for method 1–4 under Cox model

With $\frac{{\mu }_3}{{\mu }_2}\to C_2$ and $\frac{m_2}{n_1}\to C_1$, we have $\sqrt{n_3}({\theta }_1-{\theta }_1^{\mathrm{*}})\to N\left(0,{\Sigma }_{{\theta }_1}\right)$ where ${\Sigma }_{{\theta }_1}=I_{{\theta }_1}^{-1}\left(I_{{\theta }_1}+C_2{I}_n{\Sigma }_n{l}_n^{\mathrm{\top }}\right)I_{{\theta }_1}^{-\mathrm{\top }}$ can be consistently estimated by ${\hat{\Sigma }}_{{\theta }_1}={\hat{I}}_{{\theta }_1}^{-1}({\hat{I}}_{{\theta }_1}+\frac{n_3}{n_2}{\hat{I}}_n{\hat{\Sigma }}_n{\dot{I}}_n^{\mathrm{\top }}){\hat{I}}_{{\theta }_1}^{-\mathrm{\top }}$ for method 1.

With $\frac{{\mu }_3}{{\mu }_2}\to C_2$ and $\frac{m_2}{n_1}\to C_1$, we have $\sqrt{n_3}({\hat{\theta }}_2-{\theta }_2^{\mathrm{*}})\to N\left(0,{\Sigma }_{{\theta }_2}\right)$ where ${\Sigma }_{{\theta }_2}=I_{{\theta }_2}^{-1}(I_{{\theta }_2}+C_2{I}_{{\gamma }_2}{\Sigma }_{{\gamma }_2}{l}_{{\gamma }_2}^{\mathrm{\top }})I_{{\theta }_2}^{-\mathrm{\top }}$ can be consistently estimated by ${\hat{\Sigma }}_{{\theta }_2}={\hat{I}}_{{\theta }_2}^{-1}({\hat{I}}_{{\theta }_2}+\frac{n_3}{n_2}{\hat{I}}_{n_2}{\hat{\Sigma }}_{n_2}{I}_{{\gamma }_2}^T){\hat{I}}_{{\theta }_2}^{-T}$ for method 2.

With $\frac{H_1}{m_2}\to C_2$ and $\frac{m_2}{n_1}\to C_1$, we have $\sqrt{m_3}({\hat{\theta }}_3-{\theta }_3^{\mathrm{*}})\to N\left(0,{\Sigma }_{{\theta }_3}\right)$ where ${\Sigma }_{{\theta }_3}=I_{{\theta }_3}^{-1}(I_{{\theta }_3}+C_2{I}_{{\gamma }_3}{\Sigma }_{{\gamma }_3}{l}_{{\gamma }_3}^{\mathrm{\top }})I_{{\theta }_3}^{-\mathrm{\top }}$ can be consistently estimated by ${\hat{\Sigma }}_{{\theta }_3}={\hat{I}}_{{\theta }_3}^{-1}({\hat{I}}_{{\theta }_3}+\frac{n_3}{n_2}{\hat{I}}_n{\hat{\Sigma }}_{n_3}{\hat{I}}_{{\gamma }_3}^{\mathrm{\top }}){\hat{I}}_{{\theta }_3}^{-\mathrm{\top }}$ for method 3.

With $\frac{m_3}{{\mu }_2}\to C_2$ and $\frac{m_2}{n_1}\to C_1$, we have $\sqrt{n_3}({\hat{\theta }}_4-{\theta }_4^{\mathrm{*}})\to N\left(0,{\Sigma }_{{\theta }_4}\right)$ where ${\Sigma }_{{\theta }_4}=I_{{\theta }_4}^{-1}(I_{{\theta }_4}+C_2{I}_{{\gamma }_4}{\Sigma }_{{\gamma }_4}{I}_{{\gamma }_4}^{\mathrm{\top }})I_{{\theta }_4}^{-\mathrm{\top }}$ can be consistently estimated by ${\hat{\Sigma }}_{{\theta }_4}={\hat{I}}_{{\theta }_4}^{-1}({\hat{I}}_{{\theta }_4}+\frac{n_3}{n_2}{\hat{I}}_2{\hat{\Sigma }}_2{\hat{I}}_{z_4}^{\mathrm{\top }}){\hat{I}}_{{\theta }_4}^{-\mathrm{\top }}$ for methad 4.

Proof By applying Lemma 2 and Lemma 4, the asymptotic ${\Sigma }_{{\theta }_1}$ can be derived as ${\hat{\Sigma }}_{{\theta }_1}={\hat{I}}_{{\theta }_1}^{-1}({\hat{I}}_{{\theta }_1}+\frac{n_3}{n_2}{\hat{I}}_n{\hat{\Sigma }}_{{\gamma }_1}{\hat{I}}_n^{\top }){\hat{I}}_{{\theta }_1}^{-\top }$. By applying Lemma 2 and Lemma 5, the asymptotic ${\Sigma }_{{\theta }_2}$ can be derived as ${\hat{\Sigma }}_{{\theta }_2}={\hat{I}}_{{\theta }_2}^{-1}({\hat{I}}_{{\theta }_2}+\frac{n3}{n_2}{\hat{I}}_{{\gamma }_2}{\hat{\Sigma }}_{{\gamma }_2}{\hat{I}}_{{\gamma }_2}^{\top }){\hat{I}}_{{\theta }_2}^{-\top }$. By applying Lemma 2 and Lemma 6, the asymptotic ${\Sigma }_{{\theta }_3}$ can be derived as ${\hat{\Sigma }}_{{\theta }_3}={\hat{I}}_{{\theta }_3}^{-1}({\hat{I}}_{{\theta }_3}+\frac{n_3}{n_2}{\hat{I}}_{{\gamma }_3}{\hat{\Sigma }}_{{\gamma }_3}{\hat{I}}_{{\gamma }_3}^{\top }){\hat{I}}_{{\theta }_3}^{-\top }$. By applying Lemma 2 and Lemma 7, the asymptotic ${\Sigma }_{{\theta }_4}$ can be derived as ${\hat{\Sigma }}_{{\mathrm{\theta }}_4}={\stackrel{\mathrm{ˆ}}{\mathrm{I}}}_{{\mathrm{\theta }}_4}^{-1}({\hat{I}}_{{\theta }_4}+\frac{n_3}{n_2}{\hat{I}}_{{\gamma }_4}{\hat{\Sigma }}_{{\gamma }_4}{I}_{{\gamma }_4}^{\top }){\hat{I}}_{{\theta }_4}^{-\top }$.

A.3: Efficiency Comparison

In this part, we heuristically compare the efficiency of method 3 and method 4 by comparing the average variation in the estimation of $Z$. For simplicity, we can assume without loss of generality that all variables are centered and only compare the efficiency in estimating $\hat{Z}$ because the variance of $\hat{\theta }$ is a monotone function of the variance of $\hat{Z}$. We compare the expected variance under fixed design. For method 4,

$$E[Var({\hat{Z}}_4\mid Q,V)]=E[n_1^{-1}\left(1,Q,V^{\mathrm{\top }}\right){\{{\left(1,Q,V^{\mathrm{\top }}\right)}^{\mathrm{\top }}\left(1,Q,V^{\mathrm{\top }}\right)\}}^{-1}{\left(1,Q,V^{\mathrm{\top }}\right)}^{\mathrm{\top }}Var(\bar{X}\mid Q,V)],$$

and for method 3,

$$\begin{array}{l}E[Var({\hat{Z}}_3\mid Q,V)]\\ =E[n_2^{-1}\left(1,Q,V^{\mathrm{\top }}\right){\left\{{\left(1,Q,V^{\mathrm{\top }}\right)}^{\mathrm{\top }}\left(1,Q,V^{\mathrm{\top }}\right)\right\}}^{-1}{\left(1,Q,V^{\mathrm{\top }}\right)}^{\mathrm{\top }}Var(\bar{X}\mid Q,V)]\\ +\left(n_1^{-1}-n_2^{-1}\right)\left(1,Q,V^{\mathrm{\top }}\right){\left\{{\left(1,Q,V^{\mathrm{\top }}\right)}^{\mathrm{\top }}\left(1,Q,V^{\mathrm{\top }}\right)\right\}}^{-1}{\left(1,Q,V^{\mathrm{\top }}\right)}^{\mathrm{\top }}Var(\bar{X}\mid Q,V,W)\\ =n_1^{-1}\left(1,Q,V^{\mathrm{\top }}\right){\left\{{\left(1,Q,V^{\mathrm{\top }}\right)}^{\mathrm{\top }}\left(1,Q,V^{\mathrm{\top }}\right)\right\}}^{-1}{\left(1,Q,V^{\mathrm{\top }}\right)}^{\mathrm{\top }}Var(\bar{X}\mid Q,V)\\ \times \left\{\frac{n_1}{n_2}+(1-\frac{n_1}{n_2})(1-R_{(X,W)\mid Q,V}^2)\right\}.\end{array}$$

Appendix B: Details of Simulation Settings and Additional Results

B.1: Details of Simulation Settings

For each setting, we evaluate the strength of biomarker, self-reported data as well as observed prediction strength from each stage. Specifically, we consider the following quantities, coefficient and partial coefficient of multiple determination on long-term dietary intake by biomarker, mathematically, $R_{ZWV}^2=1-\frac{Var(Z|W,V)}{Var(Z)},$ $R_{ZW\mid V}^2=1-\frac{Var(Z\mid W,V)}{Var(Z\mid V)}$; coefficient and partial coefficient of multiple determination on long-term dietary intake by self-reported data, mathematically, $R_{ZQV}^2=1-\frac{Var(Z\mid Q,V)}{Var(Z)},$ $R_{ZQ\mid V}^2=1-\frac{Var(Z\mid Q,V)}{Var(Z\mid V)}$; coefficient and partial coefficient of multiple determination on long-term dietary intake by self-reported data and biomarker, mathematically, $R_{ZQWV}^2=1-\frac{Var(Z\mid W,Q,V)}{Var(Z)}$, $R_{ZQW\mid V}^2=1-\frac{Var(Z|W,Q,V)}{Var(Z\mid V)}$; coefficient and partial coefficient of multiple determination on consumed dietary intake by biomaker, mathematically, $R_{\tilde{X}WV}^2=1-\frac{Var(\tilde{X}\mid W,V)}{Var(\tilde{X})},$ $R_{\tilde{X}W\mid V}^2=1-\frac{Var(\tilde{X}\mid W,V)}{Var(\tilde{X}\mid V)}$; coefficient and partial coefficient of multiple determination on consumed dietary intake by biomaker and self-reported data, mathematically, $R_{\tilde{X}WV}^2=1-\frac{Var(\tilde{X}|W,Q,V)}{Var(\tilde{X})}$, $R_{\tilde{X}WQV}^2=1-\frac{Var(\tilde{X}\mid ,W,Q,V)}{Var(\tilde{X}\mid V)}$, coefficient and partial coefficient of multiple determination on estimated dietary intake with Method 2 by self-reported data, mathematically, $R_{{\hat{X}}_2}^{2}QV=1-\frac{Var({\hat{X}}_2\mid Q,V]}{Var({\hat{X}}_2)}$ and $R_{{\hat{X}}_{2}Q|V}^2=1-\frac{Var({\hat{X}}_2\mid Q|V)}{Var({\hat{X}}_2\mid V)}$.

In settings 1, 2 and 3, we fixed the effect on $Q$ by setting $a_0=4,a_1=1.5$ and ${\sigma }_q=3$. In setting 4, 5 and 6, we set $a_0=0.4,$ $a_1=2$ and ${\sigma }_q=4$. In addition, we decrease the coefficient of $X$ on $W$ from 1.3 to 0.8 in the first three settings while we decrease the coefficient of $X$ on $W$ from 1.1 to 0.5 in the last three settings. Table 5 displayed all different types of $R^2$ mentioned above for all six settings. To be more specific, by fixing the strength of self-reported data and correlation between true dietary intake and personal characteristic, we gradually decreased the strength of biomarker in the first three settings. In the last three settings, the correlation between true dietary intake and personal characteristic is set to be 0. The strength of self-reported data is again fixed but at a different level compared to the first three settings. We again decreased the strength of biomarker gradually in the last three settings. Below is the list of the six settings with varying parameters.

Table 5:

List of $R^2$ and censoring rates (CR) under mixed censoring (Mix), uniform censoring (Unif) and exponential censoring (Exp) among different measurements

	Setting 1	Setting 2	Setting 3	Setting 4	Setting 5	Setting 6
$R_{ZWV}^2$	0.68	0.64	0.55	0.53	0.38	0.19
$R_{ZW|V}^2$	0.49	0.41	0.28	0.53	0.38	0.19
$R_{ZQV}^2$	0.46	0.46	0.46	0.20	0.20	0.20
$R_{ZQ|V}^2$	0.13	0.13	0.13	0.20	0.20	0.20
$R_{ZQWV}^2$	0.71	0.67	0.60	0.58	0.46	0.33
$R_{ZQW|V}^2$	0.53	0.46	0.35	0.58	0.46	0.33
$R_{\tilde{X}WV}^2$	0.56	0.52	0.44	0.44	0.32	0.16
$R_{\tilde{X}W|V}^2$	0.38	0.32	0.21	0.44	0.32	0.16
$R_{\tilde{X}WQV}^2$	0.58	0.54	0.48	0.48	0.38	0.26
$R_{\tilde{X}WQ|V}^2$	0.40	0.35	0.26	0.48	0.38	0.26
$R_{{\widehat{X}}_{2}QV}^2$	0.59	0.92	0.98	0.11	0.08	0.04
$R_{{\widehat{X}}_{2}Q|V}^2$	0.07	0.06	0.04	0.11	0.08	0.04
$\text{CR}\left({\lambda }_0\right(t)=0.002,\text{Mix})$	94%	94%	94%	95%	95%	95%
$\text{CR}\left({\lambda }_0\right(t)=0.002,\text{Unif})$	97%	97%	97%	97%	97%	97%
$\text{CR}\left({\lambda }_0\right(t)=0.002,\text{Exp})$	94%	94%	94%	95%	95%	95%
$\text{CR}\left({\lambda }_0\right(t)=0.004,\text{Mix})$	89%	89%	89%	90%	90%	90%
$\text{CR}\left({\lambda }_0\right(t)=0.004,\text{Unif})$	94%	94%	94%	95%	95%	95%
$\text{CR}\left({\lambda }_0\right(t)=0.004,\text{Exp})$	89%	89%	89%	90%	90%	90%
$\text{CR}\left({\lambda }_0\right(t)=0.0004t,\text{Mix})$	94%	94%	94%	95%	95%	95%
$\text{CR}\left({\lambda }_0\right(t)=0.0004t,\text{Unif})$	91%	91%	91%	92%	92%	92%
$\text{CR}\left({\lambda }_0\right(t)=0.0004t,\text{EXP})$	76%	76%	76%	77%	77%	77%

$b_1=1.3,\rho =0.6,a_0=4,a_1=1.5,{\sigma }_q=3$ (Setting 1);

$b_1=1.1,\rho =0.6,a_0=4,a_1=1.5,{\sigma }_q=3$ (Setting 2);

$b_1=0.8,\rho =0.6,a_0=4,a_1=1.5,{\sigma }_q=3$ (Setting 3);

$b_1=1.1,\rho =0,a_0=0.4,a_1=2,{\sigma }_q=4$ (Setting 4);

$b_1=1.1,\rho =0,a_0=0.4,a_1=2,{\sigma }_q=4$ (Setting 5);

$b_1=0.5,\rho =0,a_0=0.4,a_1=2,{\sigma }_q=4$ (Setting 6); The values of $R^2\mathrm{s}$ are listed in table 5.

B.2: Additional Simulation Results

To better compare the robustness of these methods, we further conduct simulation under the setting where the relationships between the self-reported dietary intake and the short-term dietary intake are different before and after the controlled feeding study. Table 6 summarized simulation results when the correlation structures between the FFQ and the true dietary intakes of the controlled feeding study and the full cohort are different under settings 1 and 4. With the correlation unequal between controlled feeding study and full cohort, method 2 with BF added has shown more robustness in controlling bias compared with method 3 and method 4. With the increase of difference from 10% to 50% in association between $Q$ and $Z$, the performance of both method 3 and method 4 become worse (larger bias) while the performance in controlling bias of the estimator from method 2 is consistently good in most cases. In addition, in most settings, we observe that method 2 has smaller SD compared with method 3–4. The performance of method 2 is adequate even when partial $R^2\mathrm{s}$ (i.e., Setting 3: $R_{\tilde{X}W\mid V}^2=0.21$; Setting 6: $R_{\tilde{X}W\mid V}^2=0.16$) from the biomarker construction step and the calibration equation building step are both low, which suggests that in the real application of method 2, one may not need to be too stringent on the threshold of partial $R^2$.

Table 6:

Simulation results comparing different methods’ performance when the correlation structures between the short term dietary intake and true dietary intakes of the controlled feeding study and the full cohort are different when ${\lambda }_0\left(t\right)=0.002$.

Censor	Setting	Diff	Method	$(n_1,n_2,n_3)=(150,300,5150)$				$(n_1,n_2,n_3)=(300,600,10300)$				$(n_1,n_2,n_3)=(600,1200,20600)$
				Bias	SE	SD	CR	Bias	SE	SD	CR	Bias	SE	SD	CR
Mix	1	0.1	1	0.48	0.56	0.56	0.97	0.40	0.36	0.37	0.87	0.39	0.34	0.33	0.82
		0.1	2	0.05	0.29	0.29	0.97	0.01	0.19	0.19	0.96	0.01	0.18	0.17	0.96
		0.1	3	0.04	0.99	1.2	0.97	0.03	0.20	0.20	0.97	0.03	0.18	0.18	0.97
		0.1	4	0.10	0.35	0.36	0.98	0.06	0.22	0.24	0.97	0.05	0.20	0.20	0.97
		0.3	1	0.48	0.56	0.56	0.97	0.40	0.36	0.37	0.87	0.39	0.34	0.33	0.82
		0.3	2	0.05	0.29	0.29	0.97	0.01	0.19	0.19	0.96	0.01	0.18	0.17	0.96
		0.3	3	0.01	9.06	3.7	0.99	0.08	0.23	0.24	0.98	0.07	0.21	0.20	0.97
		0.3	4	0.25	0.66	0.68	0.99	0.18	0.33	0.37	0.99	0.16	0.26	0.26	0.97
		0.5	1	0.48	0.56	0.56	0.97	0.40	0.36	0.37	0.87	0.39	0.34	0.33	0.82
		0.5	2	0.05	0.29	0.29	0.97	0.01	0.19	0.19	0.96	0.01	0.18	0.17	0.96
		0.5	3	0.28	3.03	1.9	0.99	0.15	0.27	0.29	0.98	0.13	0.24	0.23	0.96
		0.5	4	0.60	12.14	4.63	1.00	0.37	17.9	7.29	1.00	0.40	0.43	0.47	0.97
	4	0.1	1	0.38	0.35	0.34	0.91	0.35	0.24	0.24	0.75	0.35	0.22	0.22	0.67
		0.1	2	0.03	0.19	0.19	0.96	0.01	0.13	0.13	0.96	0.01	0.12	0.12	0.96
		0.1	3	0.04	0.20	0.21	0.98	0.02	0.13	0.14	0.96	0.02	0.12	0.12	0.96
		0.1	4	0.06	0.22	0.22	0.98	0.04	0.14	0.15	0.97	0.04	0.13	0.13	0.95
		0.3	1	0.38	0.35	0.34	0.91	0.35	0.24	0.24	0.75	0.35	0.22	0.22	0.67
		0.3	2	0.03	0.19	0.19	0.96	0.01	0.13	0.13	0.96	0.01	0.12	0.12	0.96
		0.3	3	0.08	0.23	0.27	0.99	0.06	0.15	0.15	0.97	0.06	0.14	0.14	0.95
		0.3	4	0.18	0.37	0.52	1.00	0.14	0.19	0.20	0.97	0.14	0.17	0.17	0.93
		0.5	1	0.38	0.35	0.34	0.91	0.35	0.24	0.24	0.75	0.35	0.22	0.22	0.67
		0.5	2	0.03	0.19	0.19	0.96	0.01	0.13	0.13	0.96	0.01	0.12	0.12	0.96
		0.5	3	0.16	0.35	0.57	0.99	0.11	0.17	0.18	0.97	0.11	0.16	0.16	0.93
		0.5	4	0.47	1.43	1.36	1.00	−0.26	7.08	23.52	0.98	0.33	0.26	0.26	0.87
Unif	1	0.1	1	0.46	0.66	0.67	0.97	0.41	0.43	0.45	0.89	0.41	0.41	0.40	0.85
		0.1	2	0.04	0.34	0.34	0.97	0.02	0.22	0.23	0.96	0.02	0.21	0.21	0.97
		0.1	3	0.02	1.29	1.63	0.98	0.04	0.23	0.24	0.96	0.04	0.22	0.22	0.96
		0.1	4	0.09	0.41	0.43	0.98	0.07	0.26	0.28	0.97	0.06	0.23	0.23	0.96
		0.3	1	0.46	0.66	0.67	0.97	0.41	0.43	0.45	0.89	0.41	0.41	0.40	0.85
		0.3	2	0.04	0.34	0.34	0.97	0.02	0.22	0.23	0.96	0.02	0.21	0.21	0.97
		0.3	3	0.01	8.55	3.47	0.98	0.08	0.27	0.28	0.97	0.08	0.25	0.24	0.96
		0.3	4	0.24	0.74	0.79	0.99	0.19	0.37	0.42	0.98	0.17	0.30	0.30	0.96
		0.5	1	0.46	0.66	0.67	0.97	0.41	0.43	0.45	0.89	0.41	0.41	0.40	0.85
		0.5	2	0.04	0.34	0.34	0.97	0.02	0.22	0.23	0.96	0.02	0.21	0.21	0.97
		0.5	3	0.28	3.3	2.05	0.99	0.15	0.32	0.34	0.97	0.14	0.28	0.28	0.96
		0.5	4	0.64	10.89	4.01	0.99	0.31	23.52	9.65	0.99	0.42	0.50	0.52	0.97
	4	0.1	1	0.37	0.42	0.40	0.94	0.41	0.43	0.45	0.89	0.35	0.27	0.27	0.77
		0.1	2	0.02	0.23	0.22	0.97	0.01	0.15	0.15	0.97	0.01	0.15	0.15	0.96
		0.1	3	0.03	0.23	0.24	0.97	0.02	0.16	0.16	0.96	0.02	0.15	0.15	0.95
		0.1	4	0.05	0.25	0.25	0.97	0.04	0.17	0.17	0.97	0.04	0.16	0.16	0.95
		0.3	1	0.37	0.42	0.40	0.94	0.41	0.43	0.45	0.89	0.35	0.27	0.27	0.77
		0.3	2	0.02	0.23	0.22	0.97	0.01	0.15	0.15	0.97	0.01	0.15	0.15	0.96
		0.3	3	0.08	0.27	0.31	0.98	0.06	0.17	0.17	0.96	0.06	0.16	0.16	0.95
		0.3	4	0.18	0.42	0.60	0.99	0.14	0.22	0.23	0.96	0.13	0.20	0.20	0.94
		0.5	1	0.37	0.42	0.40	0.94	0.41	0.43	0.45	0.89	0.35	0.27	0.27	0.77
		0.5	2	0.02	0.23	0.22	0.97	0.01	0.15	0.15	0.97	0.01	0.15	0.15	0.96
		0.5	3	0.15	0.40	0.66	0.98	0.11	0.20	0.20	0.97	0.11	0.19	0.19	0.94
		0.5	4	0.44	1.16	0.99	0.99	−0.23	4.78	21.23	0.98	0.32	0.29	0.30	0.91
Exp	1	0.1	1	0.44	0.51	0.52	0.95	0.41	0.33	0.35	0.83	0.39	0.30	0.30	0.79
		0.1	2	0.03	0.26	0.27	0.97	0.01	0.17	0.18	0.96	0.01	0.16	0.16	0.96
		0.1	3	0.02	0.95	1.18	0.97	0.03	0.18	0.19	0.97	0.02	0.17	0.16	0.96
		0.1	4	0.08	0.32	0.33	0.97	0.06	0.21	0.24	0.96	0.05	0.18	0.18	0.97
		0.3	1	0.44	0.51	0.52	0.97	0.41	0.33	0.35	0.83	0.39	0.30	0.30	0.79
		0.3	2	0.03	0.26	0.27	0.97	0.01	0.17	0.18	0.96	0.01	0.16	0.16	0.96
		0.3	3	−0.03	10.60	4.36	0.98	0.08	0.21	0.23	0.97	0.07	0.19	0.18	0.97
		0.3	4	0.03	0.61	0.63	0.98	0.18	0.31	0.37	0.99	0.16	0.24	0.25	0.97
		0.5	1	0.44	0.51	0.52	0.97	0.41	0.33	0.35	0.83	0.39	0.30	0.30	0.79
		0.5	2	0.03	0.26	0.27	0.97	0.01	0.17	0.18	0.96	0.01	0.16	0.16	0.96
		0.5	3	0.23	2.42	1.47	0.98	0.15	0.26	0.29	0.98	0.13	0.22	0.21	0.96
		0.5	4	0.63	9.78	3.56	0.99	0.42	13.46	5.43	1.00	0.40	0.42	0.49	0.98
	4	0.1	1	0.37	0.32	0.31	0.91	0.34	0.22	0.23	0.69	0.34	0.20	0.20	0.63
		0.1	2	0.02	0.18	0.17	0.97	0.00	0.12	0.12	0.96	0.00	0.11	0.11	0.96
		0.1	3	0.03	0.18	0.19	0.97	0.02	0.12	0.12	0.96	0.02	0.11	0.11	0.95
		0.1	4	0.05	0.20	0.20	0.97	0.04	0.13	0.14	0.97	0.03	0.12	0.12	0.95
		0.3	1	0.37	0.32	0.31	0.91	0.34	0.22	0.23	0.69	0.34	0.20	0.20	0.63
		0.3	2	0.02	0.18	0.17	0.97	0.00	0.12	0.12	0.96	0.00	0.11	0.11	0.96
		0.3	3	0.08	0.21	0.25	0.97	0.06	0.14	0.14	0.97	0.05	0.12	0.12	0.94
		0.3	4	0.17	0.35	0.50	0.99	0.14	0.18	0.18	0.98	0.12	0.15	0.15	0.93
		0.5	1	0.37	0.32	0.31	0.91	0.34	0.22	0.23	0.69	0.34	0.20	0.20	0.63
		0.5	2	0.02	0.18	0.17	0.97	0.00	0.12	0.12	0.96	0.00	0.11	0.11	0.96
		0.5	3	0.15	0.33	0.55	0.98	0.11	0.16	0.17	0.97	0.10	0.14	0.14	0.93
		0.5	4	0.46	1.53	1.53	1.00	−0.26	7.08	23.52	0.98	0.31	0.24	0.24	0.88

B.3: Estimation of ${\tilde{\sigma }}_x$

Here we discuss an alternative method to estimate ${\tilde{\sigma }}_x$ when more than one self-reported measurements exist (denoted as $Q_1$ and $Q_2$ for example) and only one of them is available in the first stage sample. For this scenario, using method 3 will limit us to the information of $Q_1$ only and thus might be less efficient comparing to method 2 with the usage of $Q_2$ when $Q_2$ has stronger association with $Z$ than $Q_1$. Specifically, we can first obtain the estimator ${\widehat{\theta }}_1$ and ${\widehat{\theta }}_3$ using $Q_1$ only and then solve the equation ${\widehat{\theta }}_1={\widehat{\theta }}_3/BF$ to obtain ${\widehat{\tilde{\sigma }}}_x$ from

$${\widehat{\tilde{\sigma }}}_x=\sqrt{\frac{{\hat{\theta }}_1}{{\hat{\theta }}_3}\{Var(\tilde{X}\mid W,V)-Var(\tilde{X}\mid V)(1-\frac{{\hat{\theta }}_3}{{\hat{\theta }}_1})\}}$$

Then we can rerun method 2 with $Q_2$ and ${\widehat{\tilde{\sigma }}}_x$ to get a better estimate ${\tilde{\theta }}_2$. Bootstrap can be used to handle the additional variation due to the estimation of ${\widehat{\tilde{\sigma }}}_x$.

To illustrate this, we perform additional simulation under Setting 1 and 4. We modified ${\sigma }_q\mathrm{s}$ for $Q_1$ and $Q_2$ while keeping other parameters unchanged. Table 8 displayed the bias, SD, $R_{ZQV}^2$ and $R_{ZQ\mid V}^2$ under different ${\sigma }_q$ and corresponding $R^2$ and partial $R_{ZQV}^2$ for method 2 and 3. Based on Table 8, we can see Method 2 using $Q_2$ and ${\widehat{\tilde{\sigma }}}_x$ is more efficient (smaller SD) than Method 3 on the estimated associated parameter when the strength of $Q_2$ is large enough.

Table 8:

Simulation results comparing efficiency between method 2 and method 3 with ${\widehat{\tilde{\sigma }}}_x$ used for method 2 when ${\lambda }_0\left(t\right)=0.002$.

					$(n_1,n_2,n_3)=(150,300,5150)$		$(n_1,n_2,n_3)=(300,600,10300)$
Setting	$\left({\sigma }_{q1},{\sigma }_{q2}\right)$	Method	$R_{ZQV}^2$	$R_{ZQ|V}^2$	Bias	SD	Bias	SD
1	(2,3)	2	0.54	0.26	0.00	0.141	0.01	0.121
		3	0.46	0.13	0.06	0.229	0.01	0.132
4	(2,5,4)	2	0.38	0.38	−0.01	0.092	0.00	0.079
		3	0.19	0.19	0.02	0.115	0.00	0.089

Availability of data and material

The data that support the findings of this study are available from the Women’s Health Initiative (WHI) but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available. Data are however available from the authors upon reasonable request in a collaborative mode and with permission of Women’s Health Initiative (WHI).