Logistic regression with exposure biomarkers and flexible measurement error

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

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 MethodsStatistical and Graphical Methods of Data Analysis > EM AlgorithmStatistical Models > Survival Models.

Semiparametric regression calibration for general hazard models in survival analysis with covariate measurement error; surprising performance under linear hazard

Observational epidemiological studies often confront the problem of estimating exposure-disease relationships when the exposure is not measured exactly. Regression calibration (RC) is a common approach to correct for bias in regression analysis with covariate measurement error. In survival analysis with covariate measurement error, it is well known that the RC estimator may be biased when the hazard is an exponential function of the covariates. In the paper, we investigate the RC estimator with general hazard functions, including exponential and linear functions of the covariates. When the hazard is a linear function of the covariates, we show that a risk set regression calibration (RRC) is consistent and robust to a working model for the calibration function. Under exponential hazard models, there is a trade-off between bias and efficiency when comparing RC and RRC. However, one surprising finding is that the trade-off between bias and efficiency in measurement error research is not seen under linear hazard when the unobserved covariate is from a uniform or normal distribution. Under this situation, the RRC estimator is in general slightly better than the RC estimator in terms of both bias and efficiency. The methods are applied to the Nutritional Biomarkers Study of the Women's Health Initiative.

Parametric Regression Analysis with Covariate Misclassification in Main Study/Validation Study Designs

Measurement error and misclassification have long been a concern in many fields, including medicine, administrative health care data, epidemiology, and survey sampling. It is known that measurement error and misclassification may seriously degrade the quality of estimation and inference, and should be avoided whenever possible. However, in practice, it is inevitable that measurements contain error for a variety of reasons. It is thus necessary to develop statistical strategies to cope with this issue. Although many inference methods have been proposed in the literature to address mis-measurement effects, some important issues remain unexplored. Typically, it is generally unclear how the available methods may perform relative to each other. In this paper, capitalizing on the unique feature of discrete variables, we consider settings with misclassified binary covariates and investigate issues concerning covariate misclassification; our development parallels available strategies for handling measurement error in continuous covariates. Under a unified framework, we examine a number of valid inferential procedures for practical settings where a validation study, either internal or external, is available besides a main study. Furthermore, we compare the relative performance of these methods and make practical recommendations.

Robust best linear estimator for Cox regression with instrumental variables in whole cohort and surrogates with additive measurement error in calibration sample

Biomedical researchers are often interested in estimating the effect of an environmental exposure in relation to a chronic disease endpoint. However, the exposure variable of interest may be measured with errors. In a subset of the whole cohort, a surrogate variable is available for the true unobserved exposure variable. The surrogate variable satisfies an additive measurement error model, but it may not have repeated measurements. The subset in which the surrogate variables are available is called a calibration sample. In addition to the surrogate variables that are available among the subjects in the calibration sample, we consider the situation when there is an instrumental variable available for all study subjects. An instrumental variable is correlated with the unobserved true exposure variable, and hence can be useful in the estimation of the regression coefficients. In this paper, we propose a nonparametric method for Cox regression using the observed data from the whole cohort. The nonparametric estimator is the best linear combination of a nonparametric correction estimator from the calibration sample and the difference of the naive estimators from the calibration sample and the whole cohort. The asymptotic distribution is derived, and the finite sample performance of the proposed estimator is examined via intensive simulation studies. The methods are applied to the Nutritional Biomarkers Study of the Women's Health Initiative.

Moment reconstruction and moment-adjusted imputation when exposure is generated by a complex, nonlinear random effects modeling process

For the classical, homoscedastic measurement error model, moment reconstruction (Freedman et al., 2004, 2008) and moment-adjusted imputation (Thomas et al., 2011) are appealing, computationally simple imputation-like methods for general model fitting. Like classical regression calibration, the idea is to replace the unobserved variable subject to measurement error with a proxy that can be used in a variety of analyses. Moment reconstruction and moment-adjusted imputation differ from regression calibration in that they attempt to match multiple features of the latent variable, and also to match some of the latent variable's relationships with the response and additional covariates. In this note, we consider a problem where true exposure is generated by a complex, nonlinear random effects modeling process, and develop analogues of moment reconstruction and moment-adjusted imputation for this case. This general model includes classical measurement errors, Berkson measurement errors, mixtures of Berkson and classical errors and problems that are not measurement error problems, but also cases where the data-generating process for true exposure is a complex, nonlinear random effects modeling process. The methods are illustrated using the National Institutes of Health-AARP Diet and Health Study where the latent variable is a dietary pattern score called the Healthy Eating Index-2005. We also show how our general model includes methods used in radiation epidemiology as a special case. Simulations are used to illustrate the methods.

Measurement error models with interactions

An important use of measurement error models is to correct regression models for bias due to covariate measurement error. Most measurement error models assume that the observed error-prone covariate (WW ) is a linear function of the unobserved true covariate (X) plus other covariates (Z) in the regression model. In this paper, we consider models for W that include interactions between X and Z. We derive the conditional distribution of X given W and Z and use it to extend the method of regression calibration to this class of measurement error models. We apply the model to dietary data and test whether self-reported dietary intake includes an interaction between true intake and body mass index. We also perform simulations to compare the model to simpler approximate calibration models.

Functional and Structural Methods with Mixed Measurement Error and Misclassification in Covariates

Covariate measurement imprecision or errors arise frequently in many areas. It is well known that ignoring such errors can substantially degrade the quality of inference or even yield erroneous results. Although in practice both covariates subject to measurement error and covariates subject to misclassification can occur, research attention in the literature has mainly focused on addressing either one of these problems separately. To fill this gap, we develop estimation and inference methods that accommodate both characteristics simultaneously. Specifically, we consider measurement error and misclassification in generalized linear models under the scenario that an external validation study is available, and systematically develop a number of effective functional and structural methods. Our methods can be applied to different situations to meet various objectives.

Risk of using logistic regression to illustrate exposure-response relationship of infectious diseases

Background

In most biological experiments, especially infectious disease, the exposure-response relationship is interrelated by a multitude of factors rather than many independent factors. Little is known about the suitability of ordinary, categorical exposures, and logarithmic transformation which have been presented in logistic regression models to assess the likelihood of an infectious disease as a function of a risk or exposure. This study aims to examine and compare the current approaches.

Methods

A simulated human immunodeficiency virus (HIV) population, dynamic infection data for 100,000 individuals with 1% initial prevalence and 2% infectivity, was created. Using the Monte Carlo method (computational algorithm) to repeat random sampling to obtain numerical results, linearity between log odds and exposure, and suitability in practice were examined in the three model approaches.

Results

Despite diverse population prevalence, the linearity was not satisfied between log odds and raw exposures. Logarithmic transformation of exposures improved the linearity to a certain extent, and categorical exposures satisfied the linear assumption (which was important for modelling). When the population prevalence was low (assumed < 10%), performances of the three models were significantly different. Comparing to ordinary logistic regression, the logarithmic transformation approach demonstrated better accuracy of estimation except that at the two inflection points: likelihood of infection increased from slowly to sharply, then slowly again. The approach using categorical exposures had better estimations around the real values, but the measurement was coarse due to categorization.

Conclusions

It is not suitable to directly use ordinary logistic regression to explore the exposure-response relationship of HIV as an infectious disease. This study provides some recommendations for practical implementations including: 1) utilize categorical exposure if a large sample size and low population prevalence are provided; 2) utilize a logarithmic transformed exposure if the sample size is insufficient or the population prevalence is too high (such as 30%).

Electronic supplementary material

The online version of this article (doi:10.1186/1471-2334-14-540) contains supplementary material, which is available to authorized users.

Estimated Distributions of Usual Physical Activity during Recess

PURPOSE

This study aimed to estimate distributions of usual physical activity during recess in schools in low-income areas using measurement error models and to compare model-adjusted distributions to unadjusted distributions based on a single day of measurement.

METHODS

A randomized study of the Playworks program was conducted in 29 schools from six U.S. cities. A sample of 365 fourth- and fifth-grade students in 26 of the study schools wore accelerometers during their recess periods on two school days. Estimates for the percentage of time spent in moderate to vigorous physical activity (MVPA) during recess were constructed from the accelerometer data for each school day. Using measurement error models, distributions for the usual amount of time spent in MVPA during recess were estimated for intervention and control groups of males and females. Unadjusted distributions for these same groups were also constructed using data from a single school day.

RESULTS

There is considerable intraindividual variability in the students' physical activity, which accounts for 67%-83% of the overall variability, depending on the study group. Unadjusted single-day distributions are much wider and have more weight in the tails than model-adjusted distributions owing to this large intraindividual variability in the data.

CONCLUSIONS

Using measurement error models to analyze physical activity data collected from recess periods will allow for more accurate and reliable inferences on students' physical activity.

Hazard ratio estimation for biomarker-calibrated dietary exposures

Uncertainty concerning the measurement error properties of self-reported diet has important implications for the reliability of nutritional epidemiology reports. Biomarkers based on the urinary recovery of expended nutrients can provide an objective measure of short-term nutrient consumption for certain nutrients and, when applied to a subset of a study cohort, can be used to calibrate corresponding self-report nutrient consumption assessments. A nonstandard measurement error model that makes provision for systematic error and subject-specific error, along with the usual independent random error, is needed for the self-report data. Three estimation procedures for hazard ratio (Cox model) parameters are extended for application to this more complex measurement error structure. These procedures are risk set regression calibration, conditional score, and nonparametric corrected score. An estimator for the cumulative baseline hazard function is also provided. The performance of each method is assessed in a simulation study. The methods are then applied to an example from the Women's Health Initiative Dietary Modification Trial.

Robust best linear estimation for regression analysis using surrogate and instrumental variables

We investigate methods for regression analysis when covariates are measured with errors. In a subset of the whole cohort, a surrogate variable is available for the true unobserved exposure variable. The surrogate variable satisfies the classical measurement error model, but it may not have repeated measurements. In addition to the surrogate variables that are available among the subjects in the calibration sample, we assume that there is an instrumental variable (IV) that is available for all study subjects. An IV is correlated with the unobserved true exposure variable and hence can be useful in the estimation of the regression coefficients. We propose a robust best linear estimator that uses all the available data, which is the most efficient among a class of consistent estimators. The proposed estimator is shown to be consistent and asymptotically normal under very weak distributional assumptions. For Poisson or linear regression, the proposed estimator is consistent even if the measurement error from the surrogate or IV is heteroscedastic. Finite-sample performance of the proposed estimator is examined and compared with other estimators via intensive simulation studies. The proposed method and other methods are applied to a bladder cancer case-control study.

Evaluation and comparison of food records, recalls, and frequencies for energy and protein assessment by using recovery biomarkers

The food frequency questionnaire approach to dietary assessment is ubiquitous in nutritional epidemiology research. Food records and recalls provide approaches that may also be adaptable for use in large epidemiologic cohorts, if warranted by better measurement properties. The authors collected (2007-2009) a 4-day food record, three 24-hour dietary recalls, and a food frequency questionnaire from 450 postmenopausal women in the Women's Health Initiative prospective cohort study (enrollment, 1994-1998), along with biomarkers of energy and protein consumption. Through comparison with biomarkers, the food record is shown to provide a stronger estimate of energy and protein than does the food frequency questionnaire, with 24-hour recalls mostly intermediate. Differences were smaller and nonsignificant for protein density. Food frequencies, records, and recalls were, respectively, able to "explain" 3.8%, 7.8%, and 2.8% of biomarker variation for energy; 8.4%, 22.6%, and 16.2% of biomarker variation for protein; and 6.5%, 11.0%, and 7.0% of biomarker variation for protein density. However, calibration equations that include body mass index, age, and ethnicity substantially improve these numbers to 41.7%, 44.7%, and 42.1% for energy; 20.3%, 32.7%, and 28.4% for protein; and 8.7%, 14.4%, and 10.4% for protein density. Calibration equations using any of the assessment procedures may yield suitable consumption estimates for epidemiologic study purposes.

Biomarker-calibrated energy and protein consumption and cardiovascular disease risk among postmenopausal women

BACKGROUND

Nutritional epidemiology cohort studies primarily use food frequency questionnaires (FFQs). In part because FFQs are more reliable for nutrient densities than for absolute nutrient consumption, reports from association studies typically present only nutrient density measures in relation to disease risk.

METHODS

We used objective biomarkers to correct FFQ assessments for measurement error, and examined absolute energy and protein consumption in relation to cardiovascular disease incidence. FFQs and subsequent physician-adjudicated cardiovascular disease incidence were assessed for 80,370 postmenopausal women in the age range 50-79 years at enrollment in the comparison group of the Dietary Modification Trial or the prospective Observational Study in the Women's Health Initiative. Urinary recovery biomarkers of energy and protein were obtained from a subsample of 544 women, with concurrent FFQ information.

RESULTS

After biomarker correction, energy consumption was positively associated with coronary heart disease incidence (hazard ratio = 1.18; 95% confidence interval = 1.04-1.33, for 20% consumption increment) and protein density was inversely associated (0.85 [0.75-0.97]). The positive energy association appeared to be mediated by body fat accumulation. Ischemic stroke incidence was inversely associated with energy and protein consumption, but not with protein density.

CONCLUSIONS

A positive association between energy and coronary heart disease risk can be attributed to body mass accumulation. Ischemic stroke risk is inversely associated with energy and protein consumption, possibly due to correlations between consumption and physical activity.

A Bayesian measurement error approach to QT interval correction and prolongation

Assessment of QT interval prolongation is an integral part of clinical studies in drug development because a prolonged QT interval can cause sudden cardiac death. Traditionally a linear or non-linear regression method is applied to estimate subject- or group-specific heart rate corrected QT intervals (QTc) on which comparisons are based among treatment groups. These regression models rely on a fundamental assumption that the predictor variable (RR interval) is measured without error. However, the fact is that both QT and RR intervals measured in electrocardiogram (ECG) are subject to not only measurement error, but also fluctuation that is caused by physiological and biological factors. Hence the assumption in the regression models is most likely violated. In this paper we propose a Bayesian hierarchical measurement error model to evaluate QTc interval and prolongation. The proposed approach is illustrated using a real data set. Simulation studies show that our proposed Bayesian measurement error approach outperforms the current most commonly used frequentist methods.

Statistical Aspects of the Use of Biomarkers in Nutritional Epidemiology Research

Few strong and consistent associations have arisen from observational studies of dietary consumption in relation to chronic disease risk. Measurement error in self-reported dietary assessment may be obscuring many such associations. Attempts to correct for measurement error have mostly used a second self-report assessment in a subset of a study cohort to calibrate the self-report assessment used throughout the cohort, under the dubious assumption of uncorrelated measurement errors between the two assessments. The use, instead, of objective biomarkers of nutrient consumption to produce calibrated consumption estimates provides a promising approach to enhance study reliability. As summarized here, we have recently applied this nutrient biomarker approach to examine energy, protein, and percent of energy from protein, in relation to disease incidence in Women's Health Initiative cohorts, and find strong associations that are not evident without biomarker calibration. A major bottleneck for the broader use of a biomarker-calibration approach is the rather few nutrients for which a suitable biomarker has been developed. Some methodologic approaches to the development of additional pertinent biomarkers, including the possible use of a respiratory quotient from indirect calorimetry for macronutrient biomarker development, and the potential of human feeding studies for the evaluation of a range of urine- and blood-based potential biomarkers, will briefly be described.

Biomarker-calibrated energy and protein consumption and increased cancer risk among postmenopausal women

The authors previously reported equations, derived from the Nutrient Biomarker Study within the Women's Health Initiative, that produce calibrated estimates of energy, protein, and percentage of energy from protein consumption from corresponding food frequency questionnaire estimates and data on other factors, such as body mass index, age, and ethnicity. Here, these equations were applied to yield calibrated consumption estimates for 21,711 women enrolled in the Women's Health Initiative dietary modification trial comparison group and 59,105 women enrolled in the observational study. These estimates were related prospectively to total and site-specific invasive cancer incidence (1993-2005). In combined cohort analyses that do not control for body mass, uncalibrated energy was not associated with total cancer incidence or site-specific cancer incidence for most sites, whereas biomarker-calibrated energy was positively associated with total cancer (hazard ratio = 1.18, 95% confidence interval: 1.10, 1.27, for 20% consumption increase), as well as with breast, colon, endometrial, and kidney cancer (respective hazard ratios of 1.24, 1.35, 1.83, and 1.47). Calibrated protein was weakly associated, and calibrated percentage of energy from protein was inversely associated, with total cancer. Calibrated energy and body mass index associations were highly interdependent. Implications for the interpretation of nutritional epidemiology studies are described.

Observational studies, clinical trials, and the women's health initiative

The complementary roles fulfilled by observational studies and randomized controlled trials in the population science research agenda is illustrated using results from the Women's Health Initiative (WHI). Comparative and joint analyses of clinical trial and observational study data can enhance observational study design and analysis choices, and can augment randomized trial implications. These concepts are described in the context of findings from the WHI randomized trials of postmenopausal hormone therapy and of a low-fat dietary pattern, especially in relation to coronary heart disease, stroke, and breast cancer. The role of biomarkers of exposure and outcome, including high-dimensional genomic and proteomic biomarkers, in the elucidation of disease associations, will also be discussed in these same contexts.

Epidemiologic Methods Developments: A Look Forward to the Year 2032

This article responds to a request by the Editors for a perspective on potential epidemiologic methods developments between now and the year 2032 when the American College of Epidemiology will have its 50th Annual Meeting. The response begins by describing the need for enhanced methods in epidemiologic research and goes on to suggest some approaches to satisfying such needs. The suggested approaches include the more extensive use of biomarkers for exposure assessment, the greater standardization of data analysis and reporting methods, and enhancement of the interplay between observational studies and randomized controlled trials. It is argued that a phased approach to epidemiologic hypothesis evaluation may often be needed, with hypotheses that are promising in observational studies subjected to controlled trials having well-selected intermediate outcomes. It is also argued that a multidisciplinary, coordinated community of scientists interested in disease risk estimation and disease prevention will be needed for epidemiologic research to fulfill its potential over the next 25 years.