Biomarker data with measurement error in medical research: A literature review

Abstract

A biomarker is a measurable indicator of the severity or presence of a disease or medical condition in biomedical or epidemiological research. Biomarkers may help in early diagnosis and prevention of diseases. Several biomarkers have been identified for many diseases such as carbohydrate antigen 19-9 for pancreatic cancer. However, biomarkers may be measured with errors due to many reasons such as specimen collection or day-to-day within-subject variability of the biomarker, among others. Measurement error in the biomarker leads to bias in the regression parameter estimation for the association of the biomarker with disease in epidemiological studies. In addition, measurement error in the biomarkers may affect standard diagnostic measures to evaluate the performance of biomarkers such as the receiver operating characteristic (ROC) curve, area under the ROC curve, sensitivity, and specificity. Measurement error may also have an effect on how to combine multiple cancer biomarkers as a composite predictor for disease diagnosis. In follow-up studies, biomarkers are often collected intermittently at examination times, which may be sparse and typically biomarkers are not observed at the event times. Joint modeling of longitudinal and time-to-event data is a valid approach to account for measurement error in the analysis of repeatedly measured biomarkers and time-to-event outcomes. In this article, we provide a literature review on existing methods to correct for estimation in regression analysis, diagnostic measures, and joint modeling of longitudinal biomarkers and survival outcomes when the biomarkers are measured with errors.

This article is categorized under:

• Statistical and Graphical Methods of Data Analysis > Robust Methods

• Statistical and Graphical Methods of Data Analysis > EM Algorithm

• Statistical Models > Survival Models

1 |. INTRODUCTION

In epidemiological research, biomarkers may be helpful in early diagnosis and prevention of diseases. For example, carbohydrate antigen 19-9 (CA19-9) and carcinoembryonic antigen (CEA) are the most commonly used biomarkers for diagnosis and management of patients with pancreatic cancer (Poruk et al., 2013). In HIV studies, researchers are often interested in the association between CD4 counts, or viral loads, and time to AIDS or death (Tsiatis et al., 1995). For cardiovascular heart disease, blood pressure is an important risk factors. However, in addition to a possible error from the instrument, blood pressure has seasonal, daily, or even hourly variation. In practice, a long-term (or short-term) average of an infinite number of blood pressures of a subject is a more meaningful covariate variable than an observed measure of the average of a few observed measurements (Hughes et al., 1995). In this example, it is not feasible to measure all the variations for a period of time due to many reasons, such as testing conditions, or biological fluctuations.

We consider a simple example to demonstrate an effect of covariate measurement error on regression analysis. Assume $Y_i={\beta }_0+{\beta }_1{X}_i+e_i$, where $X_i$ has mean 0 and variance ${\sigma }_x^2$, and $e_i$ is independent of $X_i$, has mean zero and variance 0.5, $i=1,\dots ,n$. Suppose the true $X_i$ is not observed, but a surrogate variable $W_i=X_i+U_i$ is observed, where $U_i$ is independent of $X_i$ and $e_i$, such that $E\left(U_i\right)=0$, and $\mathrm{v}\mathrm{a}\mathrm{r}\left(U_i\right)={\sigma }_u^2$. Let ${\widehat{\beta }}_N$ be a naive estimator that is obtained from an ordinary least squares regression of $Y_i$ on $W_i$. Denote the slope estimator of the ${\beta }_1$ by ${\widehat{\beta }}_{N,1}$. Under a bivariate normal assumption of $\left(X_i,U_i\right)$, by calculating the conditional expectation of $X_i$ given $W_i$, we can obtain that ${\widehat{\beta }}_{N,1}$ converges to $\lambda {\beta }_1$, where $\lambda ={\sigma }_x^2/\left({\sigma }_x^2+{\sigma }_u^2\right)<1$. That is, measurement error has an attenuation effect. The attenuating factor, $\lambda$, is also called reliability ratio. In Figure 1, we show the bias problem when the measurement error problem is ignored. We generated $\left(X_i,Y_i,W_i\right),i=1,\dots ,n$, via the data description given above with $n=500,{\sigma }_x=1$, and ${\sigma }_u=1$. In this case, the naive estimate of the slope is about $0.5{\beta }_1$. The scatter points of $\left(X_i,Y_i\right)$ and $\left(W_i,Y_i\right)$ are presented in the figure but with different point types and colors. In the figure, we also included the fitted regression lines of $Y_i$ on $X_i$ and $W_i$, respectively. The attenuation effect of measurement error on the naive estimator can be seen since the slope of the line via the naive estimator is smaller than the slope of the line via the regression analysis using $\left(X_i,Y_i\right),i=1,\dots ,n$.

FIGURE 1.

Impact of measurement error on naive estimation. The sample size $n=500$, the variance of $X$ is 1, and the variance of measurement error $U$ is 1.

Behavioral factors, such as diet and physical activity, may play an important role in the prevention of chronic diseases. However, measuring dietary intake or physical activity is challenging. In this problem, biomarker data have been studied to understand these measures. Traditional methods for diet and physical activity are typically based on self-report data. For example, food frequency questionnaires (FFQs), repeated food records, and 24-h recalls are standard methods in the estimation of dietary intake. Typically, an additive measurement error model may be considered to address the issue of measurement error (Freedman et al., 1991). The bias on the effect estimation of dietary intake without adjusting for measurement error is often attenuation. However, attenuation may not always be the case (Carroll et al., 2006, chap. 3). In many clinical trials, both FFQ and food records are available at least for a subset of the study cohort. A general approach is to assume that repeated dietary measurements from food records consist of a participant’s underlying true long-term (or short-term) average of dietary intake with an addition of independent measurement errors. Then, dietary measurements from an FFQ are assumed to be linearly associated with the participant’s long-term dietary intake. But, the main concern is that it is not well justified that measurements from food records are better than those from FFQs. Another issue is that researchers may be concerned that measurement errors may be associated with an individual’s characteristics, such as education or body mass index (Heitman & Lissner, 1995). To reduce systematic errors, an approach is to study biomarker data for nutrition and physical activity. An example is the National Cancer Institute’s Observing Protein and Energy Nutrition (OPEN) Study, which involved doubly-labeled water and urinary nitrogen assessments. Other than biomarker data, questionnaires, and 24-h recalls are collected for 261 men and 223 women in Maryland. For the OPEN study, Kipnis et al. (2001, 2003) provided details on measurement error modeling of the data. Another example for dietary measurement error is the Nutrient Biomarker Study (NBS) within the Dietary Modification Trial component of the Women’s Health Initiative (WHI). Doubly-labeled water as a biomarker was used for the assessment of energy consumption and urinary nitrogen assessment of protein consumption. See, for example, Neuhouser et al., 2008, for related research. Sugar et al. (2007) investigated statistical models and methods for this general problem with the use of biomarkers and self-report data.

In addition to regression analysis, biomarkers have been commonly studied as diagnostic tools to distinguish subjects between the diseased and non-diseased groups. The ability of a diagnostic biomarker to discriminate between cases and controls can be based on some well-known criteria such as the receiver operating characteristic (ROC) curve, area under the ROC curve (AUC), sensitivity, or specificity. However, measurement error in biomarker assays can cause biased estimation of a diagnostic accuracy measure. That is, biomarker measurement error may lead to bias estimation of the efficacy of a diagnostic biomarker; as discussed in Coffin and Sukhatme (1997). A real example of biomarker measurement error is from a research topic in Early Detection Research Network (EDRN). The EDRN laboratories have developed several candidate biomarkers for the detection of early-stage pancreatic ductal adenocarcinoma (PDAC). See, for example, Liu et al. (2020). The goal is to find biomarkers that could improve upon the performance of the current best marker, CA19-9 for early detection of PDAC. An EDRN pancreatic cancer bakeoff study has a primary aim to compare the performance of several candidate biomarkers for discriminating resectable PDAC from benign pancreatic disease. The evaluation may be based on a new biomarker alone, or a combination of the new biomarker with CA19-9. Most research assays use relatively a small volume of blood, but they could be associated with larger measurement errors. In comparison, most clinical grade assays use relatively a larger volume of blood but with the advantage that they typically have smaller measurement errors. In general, the triage decision has to be made on performance of a research assay to determine the potential diagnostic accuracy of a clinical assay. Hence, taking into account assay measurement error will likely play an important role of selecting a good biomarker in cancer diagnosis and prevention.

As a demonstration, we consider a data set for pancreatic cancer biomarkers. Several candidate biomarkers for the detection of early-stage PDAC have been developed in the past. With effective biomarkers, a significant advantage is to reduce the need for surgery biopsy or long-term follow-up. Figure 2 shows the association between a clinical assay and research assay measures of CA19-9 from two labs. The two plots on the left portion are from the original data, and the two plots on the right portion are from log-transformed data. The $R^2$ values are provided on the top of the plots; the $R^2$ values from log-transformed data of the two labs are higher than those from the original data. The association between the clinical assays and research assays from each of the two labs is approximately linear when the data are log-transformed. Measurement error in the biomarkers can often attenuate diagnostic efficacy. For the purpose of a biomarker development, it is essential to take into account the loss of diagnostic efficacy of a biomarker due to measurement errors.

FIGURE 2.

Left: Clinic assay versus research assay from 2 labs. Right: logarithmic transform of clinic assay versus logarithmic transform of research assay from 2 labs.

To take into account biomarker measurement error, Coffin and Sukhatme (1997) developed a nonparametric correction method for AUC estimation via kernel smoothers. When biomarkers and measurement errors are from a normal distribution among the control and case groups, respectively, Faraggi (2000) derived an exact relationship between the observed AUC and the true AUC. Based on the relationship provided in Faraggi (2000), ignoring measurement error can typically attenuate AUC. Hence, a naive estimator will likely under-estimate the efficacy of a diagnostic biomarker. In a related biomarker measurement error setting, White and Xie (2013) developed correction estimators for sensitivity and specificity.

In a follow-up study, biomarkers are often measured over time. Joint modeling of longitudinal and survival data provides a tool to investigate the relationship between a repeatedly measured biomarker that may be error-prone, such as systolic blood pressure (SBP), and the time to an event of interest, such as time to non-fatal stroke; see, for example, Crowther et al. (2013). Another follow-up study example is AIDS Clinical Trials Group (ACTG) 175 trial. In the ACTG 175 trial, zidovudine alone, zidovudine plus didanosine, zidovudine plus zalcitabine, or didanosine alone are compared in HIV-infected subjects on the basis of time to progression to AIDS or death. In the study, a total of 2467 individuals were recruited between December 1991 and October 1992, and they were followed until November 1994. There were a total of 308 events. In the study, CD4 counts were also collected on each participant about once every 12 weeks after randomization. It is well known that biomarkers such as CD4 may be associated with intra-individual variations and measurement errors. In this problem, Tsiatis et al. (1995) proposed a joint modeling approach to assess the effect of longitudinal CD4 counts on mortality.

In this article, we review methods for biomarker measurement error in biomedical research. A few general research areas will be discussed. The article is organized as follows. In Section 2, we review methods for biomarker measurement error in regression analysis. In Section 3, we focus on the methods for covariate measurement error in regression analysis when combining biomarker and self-reported data for the underlying unobserved exposure. One application is measurement error in diet and physical activity, and the other application is measurement error in radiation exposure. In Section 4 we review methods for diagnostic measures, such as sensitivity, specificity, and AUC, when biomarkers are measured with errors. We also discuss how to combine multiple cancer biomarkers as a composite predictor for disease diagnosis or prevention when the biomarkers are measured with errors. In Section 5, we review methods for longitudinal biomarkers that may be measured with errors for time-to-event outcomes. In Section 6, we summarize our review and discuss potential future developments in the area of biomarker measurement error.

2 |. BIOMARKER MEASUREMENT ERROR IN REGRESSION ANALYSIS

Biomarker measurement error may happen when the biomarker is an exposure variable, or it may be the outcome of interest in regression analysis. Many methods have been proposed in a number of research fields to deal with measurement errors in the data (Buonaccorsi, 2010; Carroll et al., 2006; Yi, 2017). In this section, we will first review methods for biomarker measurement error when it is an exposure variable in the regression. In this problem, we assume that the biomarker is error-prone and there is no gold standard measurement for the biomarker. Let the sample size be denoted by $n$, and the outcome variable of the regression model be denoted by $Y_i$, where $i=1,\dots ,n$. For example, $Y_i=1$ denotes a pancreatic cancer case, $Y_i=0$ denotes a healthy control. Let the vector of biomarker variables be denoted by ${\mathit{X}}_i$. Let ${\mathit{Z}}_i$ be a vector of covariates that are observed without measurement errors, such as age, gender, and body mass index. For generalized linear models, we assume that there is a function such that

$$E\left(Y\mid {\mathit{X}}_i,{\mathit{Z}}_i\right)=\mathrm{g}\left({\beta }_0+{\beta }_1^{\prime }{\mathit{X}}_i+{\beta }_2^{\prime }{\mathit{Z}}_i\right),$$ (1)

where $\mathit{\beta }={\left({\beta }_0,{\mathit{\beta }}_1^{\prime },{\mathit{\beta }}_2^{\prime }\right)}^{\prime }$ is the vector of parameters of interest, and $g(\cdot )$ is the mean function of $Y$ given $\mathit{X}$ and $\mathit{Z}$. If $Y_i={\beta }_0+{\mathit{\beta }}_1^{\prime }{\mathit{X}}_i+{\mathit{\beta }}_2^{\prime }{\mathit{Z}}_i+e_i$, where $e_i$ is the residual, then the $g(\cdot )$ function is the identify function. For logistic regression, the mean function can be written as

$$\mathrm{p}\mathrm{r}(Y=1\mid \mathit{X},\mathit{Z})=H\left({\beta }_0+{\beta }_1^{\prime }\mathit{X}+{\beta }_2^{\prime }\mathit{Z}\right)\text{, where }H\left(u\right)=\{1+\text{exp}\left(-u\right){\}}^{-1}.$$

The relative risk parameter ${\mathit{\beta }}_1$ has the advantage of being easy to interpret. For example, if ${\beta }_1=\mathrm{l}\mathrm{n}\left(2\right)$, then the odds ratio for the exposure at $X=x_0+1$ versus $X=x_0$ is 2. If ${\mathit{X}}_i$ is observed in linear regression, the least square estimator is the same as the maximum likelihood (ML) estimator under some general conditions. For the generalized linear model in (1), if $\mathit{X}$ is available, a consistent estimator of $\mathit{\beta }$ can be obtained by solving

$$\sum _{i=1}^n{\left(1,{\mathit{X}}_i^{\prime },{\mathit{Z}}_i^{\prime }\right)}^{\prime }\left\{Y_i-g\left({\beta }_0+{\mathit{\beta }}_1^{\prime }{\mathit{X}}_i+{\mathit{\beta }}_2^{\prime }{\mathit{Z}}_i\right)\right\}=\mathbf{0}.$$ (2)

The estimating equation given above in (2) is unbiased and hence a consistent estimator can be obtained. However, because $\mathit{X}$ is not observed, the estimating equation given above cannot be applied. To be able to evaluate the measurement error magnitude, a common approach is to obtain repeated surrogate variables for biomarkers $\mathit{X}$. A vector $\mathit{W}$ is a surrogate for $\mathit{X}$ if $\mathit{W}$ is conditionally independent of the outcome $Y$ given $(\mathit{X},\mathit{Z})$. Assume that observed surrogates

$${\mathit{W}}_{ij}={\mathit{X}}_i+{\mathit{U}}_{ij},j=1,\dots ,k_i,$$

where $k_i$ is the number of replicates for the $i$th subject. A naive estimator of the regression coefficient can be obtained by applying an ordinary regression with covariate data ${\stackrel{-}{\mathit{W}}}_{i·}$, where ${\stackrel{-}{\mathit{W}}}_{i·}={\sum }_{j=1}^{k_i}{\mathit{W}}_{ij}/k_i$. However, it is well-known (e.g., Carroll et al., 2006, chap. 2) that the naive estimator may have a serious bias problem.

2.1 |. Likelihood via estimating equation and regression calibration

The ML estimator of $\mathit{\beta }$ can be obtained by taking the conditional expectation of the correctly measured data likelihood given the observed data. One simple way for the calculations is the expected estimating equation (EEE) approach of Wang et al. (2008). The ML estimator is based on an additional assumption on the distribution of $X_i$ and ${\tilde{U}}_i$, where ${\tilde{\mathit{U}}}_i=\left({\mathit{U}}_{i1},\cdots ,{\mathit{U}}_{i{k}_i}\right)$. Let ${\tilde{\mathit{W}}}_i=\left({\mathit{W}}_{i1},\dots ,{\mathit{W}}_{i{k}_i}\right)$ be the replicates of the error-prone biomarkers. Under the reasonable assumption that the distribution of $(\tilde{\mathit{W}},\mathit{X})$ does not involve $\beta$, then

$$\frac{\partial }{\partial \beta }\text{log}P\left(Y∣\tilde{\mathit{W}},\mathit{Z}\right)=\frac{\left(\partial /\partial \beta \right)\int P\left(Y∣\mathit{X},\mathit{Z}\right)P\left(\mathit{W}∣\mathit{X},\mathit{Z}\right)P\left(\mathit{X}∣\mathit{Z}\right)d\mathit{X}}{P\left(Y,\mathit{W}∣\mathit{Z}\right)}=E\left\{\left.\frac{\partial }{\partial \beta }\text{log}P\left(Y∣\mathit{X},\mathit{Z}\right)\right|Y,\tilde{\mathit{W}},\mathit{Z}\right\}.$$

This indicates that the likelihood-based score of the observed data can be obtained by the conditional expectation of the likelihood-based score of the correctly measured true data given the observed data. To calculate the conditional expectation, one needs to model the distribution of ${\mathit{X}}_i$ and ${\tilde{\mathit{U}}}_i$. If we knew the nuisance parameters in modeling ${\mathit{X}}_i$ and ${\tilde{\mathit{U}}}_i$, then the ML estimate of $\mathit{\beta }$ solves

$$\sum _{i=1}^{n}E\left\{S_{\beta }\left(Y_i,{\mathit{X}}_i,{\mathit{Z}}_i\right)\mid Y_i,{\tilde{\mathit{W}}}_i,{\mathit{Z}}_i\right\}=0,$$

where $S_{\beta }\left(Y_i,{\mathit{X}}_i,{\mathit{Z}}_i\right)$ is the full data score given in (2). Here, the full data score means the estimating score when all variables involved in the regression model (including ${\mathit{X}}_i$) are all available.

We now review a simple method that might be relatively practical to address the issue of biomarker measurement error. We consider a special case if $Y_i={\beta }_0+{\mathit{\beta }}_1^{\prime }{\mathit{X}}_i+{\mathit{\beta }}_2^{\prime }{\mathit{Z}}_i+e_i$. Then given the observed data ${\tilde{\mathit{W}}}_i,{\mathit{Z}}_i,E\left(Y_i\mid {\tilde{\mathit{W}}}_i,{\mathit{Z}}_i\right)={\beta }_0+{\beta }_1^{\prime }E\left({\mathit{X}}_i\mid {\tilde{\mathit{W}}}_i,{\mathit{Z}}_i\right)+{\mathit{\beta }}_2^{\prime }{\mathit{Z}}_i$. This implies that under linear regression, if ${\mathit{X}}_i$ is measured with errors, then replacing ${\mathit{X}}_i$ by $E\left({\mathit{X}}_i\mid {\tilde{\mathit{W}}}_i,{\mathit{Z}}_i\right)$ will lead to consistent estimation of $\mathit{\beta }$. The RC method is a useful tool in measurement error problems (Carroll et al., 2006, chap. 4) even though it is not consistent for nonlinear regression. In Cox regression, Prentice (1982) proposed the idea of the RC estimator to estimate the effect of radiation exposure to cancer due to atomic bombs. It has the advantages of (i) being conceptually simple; (ii) a simple version of it can be easily implemented; (iii) being efficient and it tends to have small bias as long as the relative risk is not too large (Wang et al., 1997).

The RC estimator may replace $\mathit{X}$ by $E(\mathit{X}\mid \tilde{\mathit{W}},\mathit{Z})$ by using a parametric approach, which would involve many nuisance parameters, such as the regression coefficients and the first two moments in the joint modeling of $\tilde{\mathit{W}},\mathit{X},\mathit{Z}$ via a multivariate normal distribution. The nuisance parameters can be estimated by the method of moments; see Carroll et al. (2006, chap. 6) and Wang et al. (2017). However, instead of the above method, the RC can be implemented by a more robust semiparametric approach (Wang & Song, 2021). Suppose we have replicates ${\mathit{W}}_{i1}$ and ${\mathit{W}}_{i2}$. We note that we can use $E\left({\mathit{W}}_{i1}\mid {\mathit{W}}_{i2},{\mathit{Z}}_i\right)$ for $E\left({\mathit{X}}_i\mid {\mathit{W}}_{i2},{\mathit{Z}}_i\right)$, by noticing that $\mathit{W}$ given $\mathit{X}$ follows a classical additive error model. As an illustration, we could consider a regression model for ${\mathit{W}}_{i1}$ given $\left({\mathit{W}}_{i2},{\mathit{Z}}_i\right)$. For example, if $W$ is scalar and it may be modeled via $E\left(W_{i1}\mid W_{i2},{\mathit{Z}}_i\right)={\alpha }_0+{\alpha }_1{W}_{i2}+{\alpha }_2^{\prime }{\mathit{Z}}_i$ assuming the association is appropriate. Then the semiparametric RC estimator can be obtained by replacing $X_i$ with ${\widehat{\alpha }}_0+{\widehat{\alpha }}_1{W}_{i2}+{\widehat{\mathit{\alpha }}}_2{\mathit{Z}}_i$, where ${\widehat{\alpha }}_0,{\widehat{\alpha }}_1$, and ${\widehat{\alpha }}_2$ are consistent estimates for ${{\alpha }_0,\alpha }_1$, and ${\alpha }_2$. Wang and Song (2021) showed that for censored outcome regression, if the hazard function is a linear function of $\mathit{X}$ and $\mathit{Z}$, then the semiparametric RC estimator is consistent even if the association of ${\mathit{W}}_{i1}$ given ${\mathit{W}}_{i2}$ and ${\mathit{Z}}_i$ is misspecified. For example, under linear hazard function, if the true association of ${\mathit{W}}_{i1}$ given ${\mathit{W}}_{i2}$ and ${\mathit{Z}}_i$ is quadratic but is modeled by a linear regression, then the semiparametric RC estimator is still consistent.

2.2 |. Conditional scores and corrected scores

For linear regression, the RC estimator and the ML estimator may be robust to the distribution assumptions of $\mathit{X}$ and $\mathit{U}$. However, for nonlinear regression, typically they are not robust to the distributional assumption of $\mathit{X}$ or $\mathit{U}$. Stefanski and Carroll (1985, 1987) proposed functional methods, which does not need the model assumption of $\mathit{X}$, by using sufficiency and conditional scores. We consider the situation when $Y_i$ given $\left({\mathit{X}}_i,{\mathit{Z}}_i\right)$ is linear with mean ${\beta }_0+{\mathit{\beta }}_1^{\prime }{\mathit{X}}_i+{\mathit{\beta }}_2^{\prime }{\mathit{Z}}_i$ and the situation when $Y_i$ is binary with mean $H\left({\beta }_0+{\mathit{\beta }}_1^{\prime }{\mathit{X}}_i+{\mathit{\beta }}_2^{\prime }{\mathit{Z}}_i\right)$. Suppose the measurement error has a normal distribution and we treat ${\mathit{X}}_i$ as parameters, then the statistic ${\mathrm{\Delta }}_i={\stackrel{-}{\mathit{W}}}_i+Y_i\left({\mathrm{\Sigma }}_u/k_i\right){\mathit{\beta }}_1$ is complete and sufficient for ${\mathit{X}}_i$, where ${\mathrm{\Sigma }}_u$ is the variance–covariance matrix of ${\mathit{U}}_{ij}$. As a result, the distribution of $Y_i$ given ${\mathrm{\Delta }}_i$ depends only on $Y_i,\stackrel{-}{W_i}$, and $\beta$, but not on ${\mathit{X}}_i$. If $Y_i$ given $\left({\mathit{X}}_i,{\mathit{Z}}_i\right)$ is linear with mean ${\beta }_0+{\mathit{\beta }}_1^{\prime }{\mathit{X}}_i+{\mathit{\beta }}_2^{\prime }{\mathit{Z}}_i$ and variance ${\sigma }_e^2$, then $E\left(Y_i\mid {\mathrm{\Delta }}_i,{\mathit{Z}}_i\right)=\left(1+{\mathit{\beta }}_1^{\prime }{\mathrm{\Delta }}_i+{\mathit{\beta }}_2^{\prime }{\mathit{Z}}_i\right)/\left\{1+\left({\mathit{\beta }}_1^{\prime }{\mathrm{\Sigma }}_u{\mathit{\beta }}_1/k_i\right)/{\sigma }_e^2\right\}$. If $Y$ is logistic, then $E\left(Y\mid {\mathrm{\Delta }}_i,{\mathit{Z}}_i\right)=H\left({\beta }_0+{\mathit{\beta }}_1^{\prime }{\mathrm{\Delta }}_i^{*}+{\mathit{\beta }}_2^{\prime }{\mathit{Z}}_i\right)$, where ${\mathrm{\Delta }}_i^{*}={\mathrm{\Delta }}_i-\left({\mathrm{\Sigma }}_u{\mathit{\beta }}_1/2k_i\right)$. The estimating equation for the sufficiency score estimator is

$$\sum _{i=1}^n{\left(1,{\mathrm{\Delta }}_i^{*},{\mathit{Z}}_i^{\prime }\right)}^{\prime }\left\{Y_i-H\left({\beta }_0+{\mathit{\beta }}_1^{\prime }{\mathrm{\Delta }}_i^{*}+{\mathit{\beta }}_2^{\prime }{\mathit{Z}}_i\right)\right\}=0.$$ (3)

We may treat the sufficiency score estimator as another type of RC estimator by replacing ${\mathit{X}}_i$ by ${\mathrm{\Delta }}_i^{*}$, but note that ${\mathrm{\Delta }}_i^{*}$ is a function of $Y_i$ and ${\mathit{\beta }}_1$. Stefanski and Carroll (1985) showed that the sufficiency score is optimal for a particular structural model. They also extended the sufficiency score estimation to a class of conditional score estimators. For Cox regression, conditional score estimation was developed by Tsiatis and Davidian (2001). The details are omitted here.

To review the idea of corrected score estimation, we first consider the special case when $Y_i$ given $\left({\mathit{X}}_i,{\mathit{Z}}_i\right)$ is linear with mean ${\beta }_0+{\mathit{\beta }}_1^{\prime }{\mathit{X}}_i+{\mathit{\beta }}_2^{\prime }{\mathit{Z}}_i$ and if there is a surrogate variable ${\mathit{W}}_i={\mathit{X}}_i+{\mathit{U}}_i$ for the unobserved ${\mathit{X}}_i$. The estimating equation (2) is unbiased, but replacing ${\mathit{X}}_i$ with ${\mathit{W}}_i$ in the estimating equation will have a bias. By some calculations, an unbiased estimating equation can be obtained:

$$\sum _{i=1}^n{\left(1,{\mathit{Z}}_i^{\prime },{\mathit{W}}_i^{\prime }\right)}^{\prime }\left\{Y_i-{\beta }_0-{\mathit{\beta }}_1^{\prime }{\mathit{W}}_i+{\mathit{\beta }}_2^{\prime }{\mathit{Z}}_i\right\}+{\left(\mathrm{0,0},{\mathit{\beta }}_1^{\prime }{\mathrm{\Sigma }}_u\right)}^{\prime }=0.$$

This is the corrected score for linear regression given above. We now consider this situation when replicates ${\mathit{W}}_{ij},j=1,\dots ,k_i$ are available. In the corrected score estimating equation above, if we use ${\stackrel{-}{\mathit{W}}}_{i·}^{\prime }={\sum }_{j=1}^{k_i}{\mathit{W}}_{ij}/k_i$ to replace ${\mathit{W}}_i$, then the adjustment term, ${\mathit{\beta }}_1^{\prime }{\mathrm{\Sigma }}_u$ should be replaced with ${\mathit{\beta }}_1^{\prime }{\mathrm{\Sigma }}_u/k_i$. Corrected score estimation for measurement error in generalized linear models was investigated by Nakamura (1990) and Stefanski (1989). Let $S\left(Y_i,{\mathit{X}}_i,\mathit{\beta }\right)$ be the estimating score when $X_i\mathrm{s}$ are available for all subjects $i=1,\dots ,n$. A function $\mathrm{\Psi }\left(Y_i,{\tilde{\mathit{W}}}_i,{\mathit{Z}}_i,\mathit{\beta }\right)$ is often called a corrected score if it satisfies

$$E\left\{\mathrm{\Psi }\left(Y_i,{\tilde{\mathit{W}}}_i,{\mathit{Z}}_i,\mathit{\beta }\right)\mid Y_i,{\mathit{X}}_i,{\mathit{Z}}_i\right\}=S\left(Y_i,{\mathit{X}}_i,{\mathit{Z}}_i,\mathit{\beta }\right).$$

Huang and Wang (2000, 2001) proposed estimating functions for Cox and logistic regression so that consistency can be assured. The consistency result does not require the distributional assumption of either $\mathit{X}$ or measurement error $\mathit{U}$. Generally, write the estimating equation as $S\left(\mathit{\beta }\right)=0$, where

$$S\left(\mathit{\beta }\right)=\widehat{ℰ}\left\{S\left(Y_i,{\mathit{X}}_i,{\mathit{Z}}_i,\mathit{\beta }\right)\right\}.$$

and $\widehat{ℰ}$ denotes the empirical mean over the sample. A general choice for $S\left(Y_i,X_i,{\mathit{Z}}_i,\mathit{\beta }\right)$ is the score obtained from the log-likelihood function. However, for logistic regression if the original score is used, then there does not exist a $\mathrm{\Psi }$ that satisfies (2.2) (Stefanski, 1989). Huang and Wang (2001) discussed a new idea of correction methods in logistic regression with covariate measurement error. In order to find a correction estimator to accommodate measurement error, they considered a class of weighted score function for the $S\left(Y_i,{\mathit{X}}_i,{\mathit{Z}}_i,\mathit{\beta }\right)$ described above. The details are given in Huang and Wang (2001). In Cox regression with covariate measurement error, Song and Wang (2014) proposed a generalized method of moments nonparametric correction estimator under the situation when a surrogate variable and an instrumental variable are available.

2.3 |. Simulation extrapolation

Simulation extrapolation (SIMEX, Cook & Stefanski, 1995) is another popular approach for covariate measurement error, which can be useful for regression analysis when biomarkers are measured with errors. Similar to RC, the implementation of SIMEX is not too complicated. But, the main ideas behind the two popular methods are very different. The main idea of RC is to replace $\mathit{X}$ by $E(\mathit{X}\mid \mathit{W},\mathit{Z})$. But the main idea of SIMEX is to augment data based a range of parameters, and then the estimator is obtained from extrapolation. In general, SIMEX is considered as functional modeling because nothing is assumed about the distribution of the unobserved covariates. Nevertheless, one caution is that bias may occur from the extrapolation process. See Carroll et al. (2006, chap. 4) for a general review of this method.

There are two steps involved in the SIMEX: simulation and extrapolation. In the simulation step, a measurement error with a larger standard deviation than that in the original surrogate is repeatedly added to the observed $\stackrel{-}{\mathit{W}}$ values. The data set resampled has a larger variance, say $(1+\xi ){\mathrm{\Sigma }}_u/k_i$, where $\xi$ is a positive number. Note that we may treat that the naive estimator using ${\stackrel{-}{\mathit{W}}}_i$ as covariates as the case when $\xi =0$. Consider these extra $M$ simulation points, say $\xi =\mathrm{0.2,0.4},\dots ,2$ in which $M=10$. At each $\xi$ point, taking a fixed positive number, say $B$, for repeated experiments. For $b=1,\dots ,B$, one generates

$${\mathit{W}}_{\xi ,b,i}={\stackrel{-}{\mathit{W}}}_i+\sqrt{\xi }{\mathit{U}}_{b,i},$$

where $U_{b,i}$ is typically generated from a normal error with mean 0 and variance ${\mathrm{\Sigma }}_u/k_i$. Let the resulting naive estimate be denoted by ${\tilde{\mathit{\beta }}}_b\left(\xi \right)$. Then at each $\xi$ point, we calculate a pseudo estimates ${\widehat{\mathit{\beta }}}_{\xi }=B^{-1}{\sum }_{b=1}^B{\tilde{\mathit{\beta }}}_b\left(\xi \right)$. As a result, this step has obtained $(M+1)$ estimates, including the naive estimate using ${\bar{W}}_i$, in which $\xi =0$.

Then we would have obtained a sequence of estimates ${\widehat{\mathit{\beta }}}_{\xi }$. It is seen that our goal is to approximate the case when $\xi =-1$; this is like the situation when the true ${\mathit{X}}_i$ were used as covariates. In the extrapolation step, a practical extrapolant function is the quadratic function $𝒢(\xi ,\gamma )={\gamma }_0+{\gamma }_1\xi +{\gamma }_2{\xi }^2$. We can apply a quadratic regression model of ${\widehat{\mathit{\beta }}}_{\xi }$ on $\xi \text{'}\text{s}$, which will lead to ${\widehat{\gamma }}_0,{\widehat{\gamma }}_1,{\widehat{\gamma }}_2$. As a result, the SIMEX estimate is ${\widehat{\gamma }}_0-{\widehat{\gamma }}_1+{\widehat{\gamma }}_2$. A standard bootstrap procedure may be applied to estimate the covariance of the SIMEX estimator.

SIMEX has been later studied by Carroll et al. (1996). Wang and Huang (2003) applied SIMEX to an error in timing problem. Apanasovich et al. (2009) developed the basic theory for SIMEX in semiparametric problems using kernel-based estimation methods. Tapsoba et al. (2019) applied SIMEX to Cox regression analysis when some covariates are possibly contaminated with a mixture of Berkson and classical errors.

3 |. METHODS FOR BIOMARKER AND SELF-REPORTED DATA

Self-reporting is a common approach for gathering data in medical and epidemiologic research, among other research areas. One advantage of self-report is because of easy analysis and efficient data processing. However, self-report data may also have some disadvantages. For example, self-report answers may be exaggerated, or respondents may not want to reveal private details. Many types of reporting biases may affect the results, such as social desirability bias. In the following subsections, we will review methods to make use biomarker data and self-report data together to adjust for potential systematic measurement errors due to self-report data.

3.1 |. Biomarker data for radiation exposure error estimation

We first review an application of biomarker measurement error in the study of long-term health effects of exposure to radiation among Japanese atomic bomb survivors. The research is conducted by the Radiation Effects Research Foundation (RERF). The RERF studies are one of the most important sources of information for the effects of radiation exposure on diseases such as cancer or cardiovascular diseases. However, the dose estimates may be quite uncertain. The linear excess relative risk (ERR) model is popular for exposure effect estimation in environmental or occupational epidemiology (Thomas, 1981). An important reason for this model is that in many applications (such as radiation effects), the exposure effect is more linear than exponential. See Pierce et al. (1992) and Pierce et al. (2008) for error correction techniques in the RERF data. However, the RERF dosimetry system provided only a single point estimate of dose for each individual, but not a set of replicate estimates. To address the problem of identifying intro-individual variance, an additional variable that is associated with the unobserved true dose may provide information about the error in the dose estimate. The additional variable is considered as an instrumental variable (IV). In general, a variable is an IV if it is correlated with the unobserved exposure, independent of the measurement error of the surrogate variable for the true exposure, and independent of the outcome variable given the true covariates. Hence, an approach to estimating the measurement error property of the dosimetry system is via the use of biomarker data serving as instrumental variables. In this research, Stram et al. (1993) conducted analyses of the effect of radiation exposure and other covariates They found significant radiation dose–response relationships for percent stable chromosome aberration (%SCA) among the A bomb survivors. Their research provided compelling evidence that %SCA is a useful, long-term biomarker of radiation exposure.

Suppose the sample size of the cohort of interest is $n$. In the RERF radiation exposure problem, a typical model for the radiation exposure is via the classical additive measurement error model, $W_i=X_i+U_i$, in which $W_i(i=1,\dots ,n)$ is the radiation estimate from the dosimetry system, and $X_i$ is the true but unobserved radiation exposure, $E\left(U_i\mid X_i,{\mathit{Z}}_i\right)=0$, and $\mathrm{v}\mathrm{a}\mathrm{r}\left(U_i\mid X_i,{\mathit{Z}}_i\right)$ may not be a constant. We consider time-to-event modeling in this problem. Let $T_i^0$ be the survival time of the $i\mathrm{t}\mathrm{h}$ individual, and $C_i$ be the censoring time. The response consists of observed variables $T_i\equiv \mathrm{m}\mathrm{i}\mathrm{n}\left(T_i^0,C_i\right)$ and ${\delta }_i\equiv I\left(T_i^0\le C_i\right)$, where $I(\cdot )$ is the indicator function. We are interested in the relationship between survival time $T_i^0$ and $X_i$ (biomarker) and ${\mathit{Z}}_i$ (other covariates), but $T_i^0$ may be subject to censoring. We assume that $T_i^0$ is independent of $C_i$ given $X_i$ and ${\mathit{Z}}_i$, and the hazard function $\lambda (\cdot )$ of $T_i^0$ given $\left(X_i,{\mathit{Z}}_i\right)$ follows the ERR hazard model

$$\lambda \left(t\mid X_i,{\mathit{Z}}_i\right)={\lambda }_0\left(t\right)r\left(\mathit{\beta },X_i,{\mathit{Z}}_i\right),\text{ where }r\left(\mathit{\beta },X_i,{\mathit{Z}}_i\right)=e^{{\mathit{\beta }}_2^{\prime }{\mathit{Z}}_i}\left(1+{\beta }_1{X}_i{e}^{{\mathit{\beta }}_3^{\prime }{\mathit{Z}}_i}\right),$$ (4)

and ${\left({\beta }_1,{\mathit{\beta }}_2^{\prime },{\mathit{\beta }}_3^{\prime }\right)}^{\prime }\equiv \mathit{\beta }$ is the vector of regression parameters, and ${\lambda }_0(\cdot )$ is the unknown baseline hazard function. We assume that $1+{\beta }_1{X}_i{e}^{{\mathit{\beta }}_3^{\prime }{\mathit{Z}}_i}>0$, which in general holds since radiation dose has been shown to be positively associated with cancer risk $\left({\beta }_1\ge 0\right)$ based on most findings from radiation research. In the ERR model (4), ${\mathit{\beta }}_2$ is for the background disease rate as a function of covariate $\mathit{Z},{\mathit{\beta }}_3$ is treated as the effect modification, and ${\beta }_1$ is the ERR per unit dose for the effect modification $(Z=0)$ at baseline. We note that the vector of covariates $\mathit{Z}$ may be time-dependent. In the RERF example, Wang et al. (2017) considered %SCA as an IV. Denote the IV by $Q_i$, which satisfies

$$Q_i=h\left(X_i,Z_i\right)+V_i,\text{ where }E\left(V_i\mid X_i,Z_i\right)=0,$$ (5)

for any function $h(X,\mathit{Z}),V_i$ is independent of the outcome $\left(T_i,{\delta }_i\right)$, and $\mathrm{v}\mathrm{a}\mathrm{r}\left(V_i\mid X_i,{\mathit{Z}}_i\right)$ may not be a constant. A special example for the model above is when $h\left(X_i,{\mathit{Z}}_i\right)={\alpha }_0+{\alpha }_1{X}_i+{\alpha }_2{X}_i^2+{\mathit{\alpha }}_3^{\prime }{\mathit{Z}}_i+{\mathit{\alpha }}_4^{\prime }{X}_i{\mathit{Z}}_i+{\mathit{\alpha }}_5{X}_i^2{\mathit{Z}}_i$, which is a linear-quadratic relation between %SCA and gamma marrow dose. Given that $X_i$ is not observable, Equation (5) itself cannot be identified (even if there are repeated $Q_i$ measurements). Nevertheless, the parameters in (5) can be identified and consistently estimated when the observed data (including the observed time outcomes and surrogates $W_i$) are used in the analysis.

The RC estimator in this problem may replace $X$ by $E(X\mid W,Q,\mathit{Z})$ by using the a conditional expectation formula by assuming the observed data are from a multivariate normal distribution. The nuisance parameters can be estimated by the method of moments. However, it may involve some computational challenges. In addition, there could be model violation issues. Instead, the RC can be implemented by a more robust approach to calculate $E(W\mid Q,\mathit{Z})$ for $E(X\mid Q,\mathit{Z})$, by noticing that $W$ given $X$ follows a classical additive error model. We could consider a regression model for $W$ given $(Q,\mathit{Z})$. For example, $W$ may be modeled as linear, such as $E(W\mid Q,\mathit{Z})={\alpha }_0+{\alpha }_{1}Q+{\mathit{\alpha }}_2^{\prime }\mathit{Z}$ if the association is appropriate. Biomarker data with measurement error in medical research: A literature review The RC estimator for this example is to $X_i$ with ${\widehat{\alpha }}_0+{\widehat{\alpha }}_{1}Q+{\widehat{\mathit{\alpha }}}_2{\mathit{Z}}_i$, where ${\widehat{\alpha }}_0,{\widehat{\alpha }}_1$, and ${\widehat{\mathit{\alpha }}}_2$ are consistent estimates for ${\alpha }_0,{\alpha }_1$, and ${\mathit{\alpha }}_2$. This RC estimator is semiparametric in the sense that we only assume the regression model of $W$ given $(Q,\mathit{Z})$, but not a covariate distribution. Details of robustness of the semiparametric RC estimator can be seen in Wang and Song (2021). In addition to the RC estimator, a generalized method of moments can be seen in Song and Wang (2014), and a joint nonparametric correction score estimator can be seen in Wang et al. (2017).

3.2 |. Biomarker data for diet and physical activity error estimation

Some recent dietary or physical activity studies have been proposed to understand the measurement biases from the use of self-report data. In the Introduction section, we briefly described the National Cancer Institute’s OPEN Study which included doubly-labeled water and urinary nitrogen assessments, along with questionnaires and 24-h recalls for 261 men and 223 women in Maryland. Doubly-labeled water was used for assessment of energy consumption and urinary nitrogen assessment of protein consumption; see, for example, Neuhouser et al. (2008) and Prentice et al. (2009). Hence, in the diet or physical activity application, biomarker data (such as doubly-labeled water data) are treated as unbiased surrogates for the true exposure, while self-reported data are treated as IVs. We assume that $T_i^0$ is independent of $C_i$ given $X_i$ and ${\mathit{Z}}_i$, and the hazards function $\lambda \left(t\right)$ of $T_i^0$ given $\left(X_i,{\mathit{Z}}_i\right)$ follows the Cox (1972) proportional hazards model

$$\lambda \left(t,X_i,{\mathit{Z}}_i\right)={\lambda }_0\left(t\right)\text{exp}\left({\beta }_1{X}_i+{\mathit{\beta }}_2^{\prime }{\mathit{Z}}_i\right),$$ (6)

where ${\left({\beta }_1,{\mathit{\beta }}_2^{\prime }\right)}^{\prime }\equiv \mathit{\beta }$ is the vector of parameters of interest. In practice, $X_i$, such as long-term (or sometimes short-term) fat intake, is not observable and hence we will not be able to apply the partial likelihood estimating method directly. We assume that a biomarker cohort is available that includes biomarker data (such as doubly-labeled water data for energy expenditure) for $X$ that follow the classical measurement error model $W_i=X_i+U_i$, where $E\left(U_i\mid X_i\right)=0$. Here we assume there is only one $W_i$, but the methods can be easily applied to the situation when replicates are available. In this section, we use notation $Q_i$ for a routinely available self-report measure of $X_i$. We assume that the self-report data follow the following flexible measurement error model $Q_i=h\left(X_i,{\mathit{Z}}_i\right)+V_i$, where $h(X,\mathit{Z})$ is a known function but with unknown parameters. Also, $V_i$ is a random error such that $E\left(V_i\mid X_i,{\mathit{Z}}_i\right)=0$. For example, $Q_i={\gamma }_0+{\gamma }_1{X}_i+{\mathit{\gamma }}_2^{\prime }{\mathit{Z}}_i+{\mathit{\gamma }}_3^{\prime }{X}_i{\mathit{Z}}_i+V_i$, where ${\gamma }_0,{\gamma }_1,{\mathit{\gamma }}_2^{\prime }$, and ${\mathit{\gamma }}_3$ are unknown coefficients. We also assume that $V_i$ has mean 0 and is independent of ${\mathit{Z}}_i$.

In the WHI diet biomarker study, the NBS is a subset with diet biomarker and self-report data. In the application, we can call the NBS data as a calibration sample. The non-calibration sample in this example contains self-report diet data, but not the biomarker data. The methods that we described in the RERF example above can be applied to the WHI data with some modification in the modeling. Other than the methods described, Wang and Song (2016) proposed a best linear estimator to combine estimators from the calibration data and the non-calibration data. We omit the details of the estimating procedure.

4 |. DIAGNOSIS MEASURE WITH BIOMARKER MEASUREMENT ERRORS

An important role of biomarkers is that biomarkers can often help researchers or doctors diagnose diseases and other health conditions, identify health risks in a person, among other purposes. However, measurement error in biomarkers may affect estimation of the diagnostic accuracy measure. We now consider statistical models to include not only the situation when replicates of a biomarker are available, but also the situation with two different test measures of the same biomarker, such as research assays and clinical assays for CA19-9. One goal of this research is to understand the degree of improvement of a clinical assay over a research assay. In addition, the models can also be applied to the situation when two different test measures of the same biomarker that may have linear association. We assume that the total sample size is $n$. Let the total sample size be $n$. For individual $i$, let the disease status be denoted by $Y_i,i=1,\dots ,n$, in which $Y_i=0$ for a non-disease control, and $Y_i=1$ for a case. Let $W_i$ be a biomarker assay measure from individual $i$, and $X_i$ be the true underlying biomarker. We assume there is another assay measure, $M_i$, for the same biomarker. We assume the following models:

$$\begin{gathered} W_i=X_i+U_i,E\left(U_i\mid X_i\right)=0, \\ {M}_i={\alpha }_0+{\alpha }_1{X}_i+V_i,E\left(V_i\mid X_i\right)=0, \end{gathered}$$ (7)

where $U_i$ is the measurement error from biomarker assay $W_i,V_i$ is the measurement error from biomarker assay $M_i$, and $U_i$ and $V_i$ are independent. We use notation ${\mu }_x$ and ${\sigma }_x$ to denote the mean and standard deviation of any random variable $X$, respectively. In (7), $M_i$ is a biased surrogate; see Wang and Feng (2023) for details regarding estimation of the parameters associated with $\left(W_i,M_i\right)$. An application of model (7) is for the situation when replicates are available, and under this situation $\left({\alpha }_0,{\alpha }_1\right)=\left(\mathrm{0,1}\right)$ and ${\sigma }_u={\sigma }_v$, in which ${\sigma }_u$ is the standard deviation of $U$, and ${\sigma }_v$ is the standard deviation of $V$. The second application of model (7) is when $W_i$ is an unbiased measure of one biomarker (true $X$ plus an error), but $M_i$ is a biased measure of the same biomarker with the assumption that $M_i$ is a linear function of $X_i$. This application is common since many research assays use a different technology (such as mass spectrometry) from that used for clinical assays (such as ELISA).

We use notation ${\mu }_{x,0}$ for the mean of $X$ among controls, and notation ${\mu }_{x,1}$ for the mean of $X$ among cases, respectively. In addition, we assume larger values of a biomarker are associated with disease. That is, we assume ${\mu }_{x,1}\ge {\mu }_{x,0}$. For a potential cutoff point $c$ of the continuous biomarker, an individual is classified as a diseased case if $X_i\ge c$, or classified as a non-diseased control if $X_i<c$. By definition, sensitivity of biomarker $X$ is the true positive rate; specificity is the true negative rate. In this section, we assume a fixed cutoff point $c$ that has been determined in advance. In addition, we assume there are $n_0$ controls and $n_1$ cases; that is, ${\sum }_{i=1}^n{Y}_i=n_1$. For the $i\mathrm{t}\mathrm{h}$ control, let $X_{\left(0\right),i},i=1,\dots ,n_0$, be the $i\mathrm{t}\mathrm{h}X$ biomarker. For the $i\mathrm{t}\mathrm{h}$ case, let $X_{\left(1\right),i},i=1,\dots ,n_1$ be the $i$ th $X$ biomarker. We use notation $U_{\left(0\right),i}$ and $U_{\left(1\right),i}$ to denote the measurement errors in both groups, respectively. As in Bamber (1975), the AUC of biomarker $X$ is the same as $pr\left(X_{\left(1\right)}>X_{\left(0\right)}\right)$. Therefore, the AUC is a measure to show how well the biomarker distinguishes between the cases and the controls. We use notation ${𝒜}_x$ to denote the AUC when $X$ is the true biomarker, ${\sigma }_{x_0}^2$ and ${\sigma }_{x_1}^2$ be the variances of $X$ among controls and cases, respectively, and ${\sigma }_{u,0}^2$ and ${\sigma }_{u,1}^2$ be the variances of $U$ among controls and cases, respectively. We assume that $X$ and $U$ are independent, which is reasonable in general applications. If ${\sigma }_{x,0}^2={\sigma }_{x,1}^2={\sigma }_x^2,{\sigma }_{u,0}^2={\sigma }_{u,1}^2={\sigma }_u^2$, then ${\lambda }^2\equiv {\sigma }_u^2/{\sigma }_x^2$ is the intra- versus inter-individual variance ratio. This ratio typically provides a standardized measure of the size of measurement error. If $X$ is normally distributed among the controls and among the cases, respectively, Faraggi (2000) showed that the AUC with the true $X$ can be represented as a function of the AUC with the error-prone $W$ with the following relationship:

$${𝒜}_x=\mathrm{\Phi }\left\{{\mathrm{\Phi }}^{-1}\left({𝒜}_w\right)\sqrt{1+{\lambda }^2}\right\},$$ (8)

in which $\mathrm{\Phi }(\cdot )$ is the cumulative distribution function of the standard normal distribution, and ${𝒜}_w$ is the AUC based on the observed biomarker $W$. The correction method via (8) given above provides a simple adjustment for the AUC estimation under the situation when the measurement error variance is known. As a special example, if the AUC estimate of an error-prone biomarker is 0.75, and if ${\sigma }_u$ is the same as ${\sigma }_x$, then the AUC from the true assay without measurement error will be about 0.83. This type of adjustment is often called a correction for attenuation (CFA) estimator. If the biomarker variances among the controls and cases are different, ${\sigma }_{x,0}$ may be different from ${\sigma }_{x,1}$, or ${\sigma }_{u,0}$ may be different from ${\sigma }_{u,1}$, then Reiser (2000) showed that the correction relationship is basically the same form as (8), but the ${\lambda }^2$ will need to be replaced with ${\lambda }^2=\left({\sigma }_{u,0}^2+{\sigma }_{u,1}^2\right)/\left({\sigma }_{x,0}^2+{\sigma }_{x,1}^2\right)$.

We now use notation $S{e}_x$ and $S{e}_w$ to denote the sensitivity of $X$ and $W$. We also use notation $S{p}_x$ and $S{p}_w$ to denote the specificity of $X$ and $W$, respectively. Suppose $X$ is from a normal distribution, then the sensitivity for $X$ can be shown to be $1-\mathrm{\Phi }\left(\left(c-{\mu }_{x,1}\right)/{\sigma }_{x,1}\right)$, and the specificity can be shown to be $\mathrm{\Phi }\left(\left(c-{\mu }_{x,0}\right)/{\sigma }_{x,0}\right)$. White and Xie (2013) showed the following approximation:

$$S{e}_x\approx 1-\mathrm{\Phi }\left\{{\mathrm{\Phi }}^{-1}\left(1-S{e}_w\right)\sqrt{1+{\lambda }_1^2}\right\},S{p}_x\approx \mathrm{\Phi }\left\{{\mathrm{\Phi }}^{-1}\left(S{p}_w\right)\sqrt{1+{\lambda }_0^2}\right\},$$ (9)

in which ${\lambda }_1^2={\sigma }_{u,1}^2/{\sigma }_{x,1}^2$, and ${\lambda }_0^2={\sigma }_{u,0}^2/{\sigma }_{x,0}^2$. The approximation in (9) will become equality if the sample size increases to infinity. Hence, under the normality assumption given above, sensitivity and specificity of a biomarker will in general be attenuated if the biomarker measurement is measured with errors. Numerically, (9) provides CFA estimation for sensitivity and specification that may work well when the biomarker data are symmetric.

The CFA estimator described above is under the assumption that both the true biomarker data and measurement errors are normally distributed. From our data example in the introduction section, biomarkers are often skewed. Hence, Wang and Feng (2023) proposed to correct for estimation of sensitivity, specificity and AUC due to measurement error by modeling the distribution for the unobserved biomarkers based on a class of skew-normal (SN) distributions (Azzalini, 1985). If $X$ is from the SN distribution with parameters $(\xi ,\omega ,\kappa )$, then density of $X$ can be written as

$$f\left(x,\xi ,\omega ,\kappa \right)=\frac{2}{\omega }\varphi \left(\frac{x-\xi }{\omega }\right)\mathrm{\Phi }\left(\kappa \frac{x-\xi }{\omega }\right),$$

where $\xi$ and $\omega$ are the location and scale parameters, respectively, and $\kappa$ is the skewness parameter. When $\kappa =0$, the specific SN distribution is a normal distribution. A logarithmic transformation for skewed data may reduce the skewness, but the transformed data may still be skewed. One limitation of the SN correction estimator is that it is not consistent for the class of all distributions. Nevertheless, with the consideration that biomarkers are often skewed in the distribution, the SN correction estimator may be useful in many general applications.

Measurement error may appear in multiple biomarkers. A panel of biomarkers is necessary to improve sensitivity/specificity over the use of a single biomarker. A case study is a recent EDRN pancreatic cancer bakeoff study to compare different biomarker panels for PDAC early detection. Most labs have CA19-9, the most common biomarker for PDAC, in their panels together with other novel biomarkers where only research assays are available (e.g., TIMP1, LRG1) (Liu et al., 2020). When there is no measurement error (i.e., ${\mathit{X}}_i$ is available), Su and Liu (1993), Huang et al. (2011) investigated the optimal linear combination that maximizes the AUC. Taking into account measurement errors from multiple biomarkers will provide a more precise estimate of the diagnostic accuracy measure. Further research is important to address the issue of effectively combining multiple cancer biomarkers which are measured with errors.

In a related cancer biomarker research, Wang and Feng (2010) proposed methods for boosting with missing high dimensional predictors. It is well known that boosting is one of the most important developments in classification methodology. It is a general method to combine the performance of many weak classifiers to produce a powerful classification procedure. Boosting was developed in the machine learning theory literature (Schapire, 1990) and it has been analyzed from statistical perspectives; see Schapire et al. (1998) and Friedman et al. (2000). The work of Wang and Feng (2010) was to find an efficient approach that combines multiple, and often high-dimensional, biomarker predictors for classification. Their approach can be applied when the predictors may be subject to a complex missing data mechanism. If biomarkers are measured with errors, but there is a validation set in which measurement error in the biomarkers is very close zero, then the imputation approach of Wang and Feng (2010) can be applied. However, if a validation subset is not available, then further research will be needed to address the issue of measurement error in high dimensional biomarkers.

5 |. JOINT MODELING OF LONGITUDINAL BIOMARKERS AND TIME-TO-EVENT OUTCOMES

In this section, we review is the situation when biomarkers are longitudinal, and the time to event outcomes may be censored. Tsiatis et al. (1995) investigated a joint modeling problem motivated from the analysis of ACTG 175 trial data. In the ACTG 175 study, treatment assignment, demographic information, and physiological characteristics, are recorded at baseline. Immunologic and virologic status including CD4 count and viral RNA copy number are taken at subsequent clinic visits. For each participant, time to progression to AIDS or death is also recorded, except that some individuals may withdraw early from the study or does not experience the event by the time of study closure. We assume that there are $n$ individuals in the study. For subject $i,i=1,\dots ,n$, let $X_i\left(t\right)$ be an unobserved true biomarker at time $t$ that may be measured with an error. Let $W_i\left(t\right)$ be the observed surrogate for $X_i\left(t\right)$ such that $W_i\left(t\right)=X_i\left(t\right)+U_i\left(t\right)$, in which $U_i\left(t\right)$ is the measurement error. In this section, $X_i\left(t\right)$ is assumed to be a scalar for the purpose of simplicity of presentation. In most cases, the methods to be described can be extended to the situation when ${\mathit{X}}_i\left(t\right)$ is a vector. We assume $t\in [0,\tau ]$, where 0 denotes baseline, while $\tau$ the end of follow-up time. Tsiatis et al. (1995) and Wang et al. (2000) considered a random effects model for the observed longitudinal data $W_i\left(t\right)$ such that $W_i\left(t\right)={\mathit{D}}_i(t{)}^{\prime }{\mathit{\alpha }}_i+U_i(t)$, where ${\mathit{D}}_i\left(t\right)$ is a vector of observed covariates at time $t$, and ${\mathit{\alpha }}_i$ is a vector of random coefficients. For example, ${\mathit{D}}_i\left(t\right)=(1,t{)}^{\prime }$, and ${\mathit{\alpha }}_i=\left({\alpha }_{i0},{\alpha }_{i1}\right)$, and under this situation $X_i\left(t\right)={\alpha }_{0i}+{\alpha }_{i1}t$ for unknown parameters ${\alpha }_{0i}$ and ${\alpha }_{1i}$. We assume that the longitudinal measurements $W_{ij}$ are measured at $k_i$ different times $t_{ij}$, such that $W_{ij}\equiv W_i\left(t_{ij}\right),j=1,\dots ,k_i$. We use notation ${\tilde{W}}_i$ to denote the column vector of the observed longitudinal $W_{ij}$ for subject $i$. We may write $W_{ij}={\mathit{D}}_{ij}^{\prime }{\mathit{\alpha }}_i+U_{ij}$, where ${\mathit{D}}_{ij}={\mathit{D}}_i\left(t_{ij}\right)$ and $U_{ij}=U_i\left(t_{ij}\right)$. Let ${\mathit{Z}}_i$ be the vector of other observed covariates (without measurement error).

For participant $i$, we let $T_i^0$ be the survival time, and $C_i$ be the censoring time. The outcome variable consists of observed variables $T_i\equiv \text{min}\left(T_i^0,C_i\right)$ and ${\delta }_i\equiv I\left(T_i^0\le C_i\right)$, in which $I(\cdot )$ is the indicator function. We assume that the hazards function $\lambda \left(t\right)$ of $T_i^0$ given $\left(X_i,{\mathit{Z}}_i\right)$ follows the Cox (1972) proportional hazards model

$$\lambda \left(t,X_i,{\mathit{Z}}_i\right)={\lambda }_0\left(t\right)\text{exp}\left({\beta }_1{X}_i\left(0\right)+{\beta }_2{X}_i\left(t\right)+{\mathit{\beta }}_3{\mathit{Z}}_i\right),$$ (10)

where ${\left({\beta }_1,{\beta }_2,{\mathit{\beta }}_3^{\prime }\right)}^{\prime }\equiv \mathit{\beta }$ is the vector of parameters of interest, and ${\lambda }_0(\cdot )$ is an unspecified baseline hazard function. In our problem, instead of observing $X_i\left(0\right)$ and $X_i\left(t\right)$, we observe longitudinal $W_i$ values. Here $X_i\left(0\right)$ is the baseline true biomarker value, and $X_i\left(t\right)$ is the true biomarker value at time $t$. Under the Cox model (10), the hazards ratio in the analysis is $exp\left({\beta }_1{X}_i\left(0\right)+{\beta }_2{X}_i\left(t\right)+{\mathit{\beta }}_3^{\prime }\mathit{Z}\right)$. Our goal is to develop valid estimation methods for the regression coefficients ${\beta }_1,{\beta }_2$, and ${\mathit{\beta }}_3$. If $X_i\left(t\right)={\alpha }_{0i}+{\alpha }_{1i}t$, then the hazard function (10) is the same as ${\lambda }_0\left(dt\right)\mathrm{e}\mathrm{x}\mathrm{p}\left({\beta }_1{\alpha }_{0i}+{\beta }_2{\alpha }_{1i}\left(t\right)+{\mathit{\beta }}_3^{\prime }{\mathit{Z}}_i\right)$ in which we are interested in the effects of the intercept and slope of the linear trajectory of the biomarkers.

Let $N_i\left(t\right)=I\left[T_i\le t,{\delta }_i=1\right]$ be the counting process, $\tau$ be the time limit. By a Taylor series expansion, under the assumptions that: (i) $ℒ(N,Y\mid X,Z,W)=ℒ(N,Y\mid X,Z)$; (ii) ${\beta }_1$ and ${\beta }_2$ are small; and (iii) the disease is rare, the induced relative risk can be approximated by the following:

$$e^{{\mathit{\beta }}_3^{\prime }\mathit{Z}}E\left[\mathrm{e}\mathrm{x}\mathrm{p}\left\{{\beta }_{1}X\left(0\right)+{\beta }_{2}X\left(t\right)\right\}\mid Y\left(t\right)=1,\mathit{Z},\mathit{W}\right]\approx e^{{\mathit{\beta }}_3^{\prime }\mathit{Z}}\text{exp}\left\{{\beta }_{1}E\left(X\left(0\right)∣\mathit{Z},W\right)+{\beta }_{2}E\left(X\left(t\right)∣\mathit{Z},W\right)\right\},$$

in which $Y\left(t\right)=I[T\ge t]$ is the at risk process. Under the conditions and the approximation given above, the regression coefficients can be estimated based on Cox regression partial likelihood estimator but with the replacement of the unobserved $X\left(0\right)$ and $X\left(t\right)$ by $E\left(X\right(0)\mid \mathit{Z},W)$ and $E\left(X\right(t)\mid \mathit{Z},W)$, respectively. Let ${\mathit{D}}_i={\left({\mathit{D}}_{i1},\dots ,{\mathit{D}}_{i{k}_i}\right)}^{\prime }$. Based on this approximation, Wang et al. (2000) proposed an RC estimator by noting the following:

$$\left(\begin{array}{c}{\mathit{\alpha }}_i\\ {\tilde{W}}_i\end{array}\right)~𝒩\left\{\left(\begin{array}{c}{\mathit{\mu }}_{\alpha }\\ {\mathit{D}}_i{\mathit{\mu }}_{\alpha }\end{array}\right),\left(\begin{array}{cc}{\mathrm{\Sigma }}_{\alpha }& {\mathrm{\Sigma }}_{{\alpha }_i}{\mathit{D}}_i^{\prime }\\ {\mathit{D}}_i{\mathrm{\Sigma }}_{\alpha }& {\mathit{D}}_i{\mathrm{\Sigma }}_{\alpha }{\mathit{D}}_i^{\prime }+{\mathrm{\Sigma }}_{ui}\end{array}\right)\right\},$$

where ${\mathrm{\Sigma }}_{\alpha }$ is the variance–covariance matrix of $\mathit{\alpha }$, and ${\mathrm{\Sigma }}_{ui}$ is the variance–covariance matrix of $U_i$. Hence, when $\mathit{Z}$ is not involved in the conditional expectation, the calibration function can be estimated by $E\left({\mathit{\alpha }}_i\mid {\tilde{W}}_i\right)={\mu }_{\alpha }+{\mathrm{\Sigma }}_x{\mathit{D}}_i^{\prime }{\left({\mathit{D}}_i{\mathrm{\Sigma }}_x{\mathit{D}}_i^{\prime }+{\mathrm{\Sigma }}_{ui}\right)}^{-1}\left({\tilde{W}}_i-{\mathit{D}}_i{\mathit{\mu }}_{\alpha }\right)$, where ${\mathrm{\Sigma }}_{ui}$ is the variance–covariance matrix of $U_i$. The RC estimator can then be calculated based on the formula given above.

For the generalized linear model, the conditional score estimation was proposed by Stefanski and Carroll (1987). Motivated by this work, Tsiatis and Davidian (2001) proposed a conditional score estimation in Cox regression when the underlying time dependent covariates follow a linear model. The conditional score estimator does not need the distributional assumption on $X\left(t\right)$. However, the normal assumption on $U\left(t\right)$ is needed. Details of the conditional intensity process given the sufficient statistic can be seen in Tsiatis and Davidian (2001). The conditional score estimation for more general modeling of covariates was further investigated in Song et al. (2002). In addition to the conditional score estimator, another type of function methods for this joint modeling problem is the corrected score estimator. Generally speaking, if $\widehat{\mathrm{\Psi }}(\tilde{\delta },\tilde{T},\tilde{\mathit{X}})$ is a full data score such that $ℰ\left\{\widehat{\mathrm{\Psi }}\right(\tilde{\delta },\tilde{T},\tilde{\mathit{X}}\left)\right\}=0$, then ${\widehat{\mathbf{\Psi }}}_c(\tilde{\delta },\tilde{T},\tilde{\mathit{W}})$ is a corrected score if $ℰ\left\{{\widehat{\mathrm{\Psi }}}_c(\tilde{\delta },\tilde{T},\tilde{\mathit{W}})\mid \tilde{T},\tilde{\delta },\tilde{\mathit{X}}\right\}=\widehat{\mathrm{\Psi }}(\tilde{T},\tilde{\delta },\tilde{\mathit{X}})$. Similar to the conditional score estimator, a corrected score estimator does not need the distributional assumption on $\mathit{X}$, but the normal assumption on $\mathit{U}$ is needed. Details of the corrected score estimator for this joint modeling problem can be seen from Wang (2006).

Further developments in the joint modeling problem have been active. For example, the proportional hazards assumption may be too restrictive in some applications. An example is in AIDS studies. The motivation is that the effect of the antiretroviral treatments may decay after some time. To characterize such covariate effects, a time-varying coefficient proportional hazards model may be appropriate. Song and Wang (2008) proposed a local corrected score estimator and a local conditional score estimator. Song and Wang (2017) proposed spline-based corrected score and conditional score estimators. They found that numerically the conditional score approach has better finite sample performance than the corrected score approach. Huang et al. (2016) applied a generalization of Rao–Blackwell theorem and developed a Monte Carlo error augmentation technique to obtain a more efficient estimator than the conditional and corrected estimators. Song (2017) independently developed a more efficient approach than the conditional score and corrected score methods to fully use the longitudinal information. Huang et al.’s approach adopted the Monte Carlo error augmentation technique, while Song’s approach works even when the longitudinal observation times are informative.

6 |. CONCLUSION

In this article, we review three important research areas on biomarker measurement error. The first area is on the estimation of the regression association between disease outcomes and biomarker data which may be error-prone. We have reviewed some important and well-known methods, but due to space limitation we may have missed some nice approaches in this research of covariate measurement error in regression analysis. Additional methods may be found in the books by Carroll et al. (2006), Buonaccorsi (2010), and Yi (2017).

The second area is on measurement error in the biomarkers in the estimation of diagnostic accuracy such as AUC, sensitivity, and specificity. From this research, we note that it is very challenging to develop nonparametric methods for AUC, sensitivity, or specificity when biomarkers are measured with errors. Another challenging problems in this research is to develop methods to effectively combine multiple disease biomarkers which are measured with errors. For example, for pancreatic cancer prevention, combining multiple biomarkers remains to be a very high impact research. The research could generate new prognostic and predictive tools for clinicians. In practice, single biomarker is often inadequate and a panel of biomarkers is necessary to improve AUC, sensitivity, and specificity. Each biomarker in a panel has its own measurement error and different potential for assay analytical improvement. Hence, further research in this area will likely make significant contributions.

The third research area in this review work is on regression association of longitudinal biomarker data and time-to-event outcomes. The individual data collected throughout patient follow-up constitute important information for assessing the risk of a clinical event. When the longitudinal trajectory is linear, initial level and change of biomarkers may provide risk of a disease. Other than linear trajectory, a piecewise linear trajectory with a subject-specific changepoint (Tapsoba et al., 2011) may be useful in data applications. Clinically, when a cancer-induced changepoint occurs, this change is compared with the subject’s reference trajectory and this information can be helpful for early cancer detection. Therefore, this joint modeling research may provide important contributions in disease prevention and prognosis.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study is not available for public access at this moment, but can be requested from EDRN.