A conditional likelihood approach for regression analysis using biomarkers measured with batch‐specific error

Measurement error is common in epidemiological and biomedical studies. When biomarkers are measured in batches or groups, measurement error is potentially correlated within each batch or group. In regression analysis, most existing methods are not applicable in the presence of batch‐specific measurement error in predictors. We propose a robust conditional likelihood approach to account for batch‐specific error in predictors when batch effect is additive and the predominant source of error, which requires no assumptions on the distribution of measurement error. Although a regression model with batch as a categorical covariable yields the same parameter estimates as the proposed conditional likelihood approach for linear regression, this result does not hold in general for all generalized linear models, in particular, logistic regression. Our simulation studies show that the conditional likelihood approach achieves better finite sample performance than the regression calibration approach or a naive approach without adjustment for measurement error. In the case of logistic regression, our proposed approach is shown to also outperform the regression approach with batch as a categorical covariate. In addition, we also examine a ‘hybrid’ approach combining the conditional likelihood method and the regression calibration method, which is shown in simulations to achieve good performance in the presence of both batch‐specific and measurement‐specific errors. We illustrate our method by using data from a colorectal adenoma study. Copyright © 2012 John Wiley & Sons, Ltd.     

Maximum likelihood, multiple imputation and regression calibration for measurement error adjustment

In epidemiologic studies of exposure-disease association, often only a surrogate measure of exposure is available for the majority of the sample. A validation sub-study may be conducted to estimate the relation between the surrogate measure and true exposure levels. In this article, we discuss three methods of estimation for such a main study/validation study design: (i) maximum likelihood (ML), (ii) multiple imputation (MI) and (iii) regression calibration (RC). For logistic regression, we show how each method depends on a different numerical approximation to the likelihood, and we adapt standard software to compute both MI and ML estimates. We use simulation to compare the performance of the estimators for both realistic and extreme settings, and for both internal and external validation designs. Our results indicate that with large measurement error or large enough sample sizes, ML performs as well as or better than MI and RC. However, for smaller measurement error and small sample sizes, either ML or RC may have the advantage. Interestingly, in most cases the relative advantage of RC versus ML was determined by the relative variance rather than the bias of the estimators. Software code for all three methods in SAS is provided.     

Estimating and testing interactions in linear regression models when explanatory variables are subject to classical measurement error

Estimating and testing interactions in a linear regression model when normally distributed explanatory variables are subject to classical measurement error is complex, since the interaction term is a product of two variables and involves errors of more complex structure. Our aim is to develop simple methods, based on the method of moments (MM) and regression calibration (RC) that yield consistent estimators of the regression coefficients and their standard errors when the model includes one or more interactions. In contrast to previous work using structural equations models framework, our methods allow errors that are correlated with each other and can deal with measurements of relatively low reliability. Using simulations, we show that, under the normality assumptions, the RC method yields estimators with negligible bias and is superior to MM in both bias and variance. We also show that the RC method also yields the correct type I error rate of the test of the interaction. However, when the true covariates are not normally distributed, we recommend using MM. We provide an example relating homocysteine to serum folate and B12 levels.     

Regression calibration to correct correlated errors in outcome and exposure

Measurement error arises through a variety of mechanisms. A rich literature exists on the bias introduced by covariate measurement error and on methods of analysis to address this bias. By comparison, less attention has been given to errors in outcome assessment and nonclassical covariate measurement error. We consider an extension of the regression calibration method to settings with errors in a continuous outcome, where the errors may be correlated with prognostic covariates or with covariate measurement error. This method adjusts for the measurement error in the data and can be applied with either a validation subset, on which the true data are also observed (eg, a study audit), or a reliability subset, where a second observation of error prone measurements are available. For each case, we provide conditions under which the proposed method is identifiable and leads to consistent estimates of the regression parameter. When the second measurement on the reliability subset has no error or classical unbiased measurement error, the proposed method is consistent even when the primary outcome and exposures of interest are subject to both systematic and random error. We examine the performance of the method with simulations for a variety of measurement error scenarios and sizes of the reliability subset. We illustrate the method's application using data from the Women's Health Initiative Dietary Modification Trial.     

Sensitivity analysis for misclassification in logistic regression via likelihood methods and predictive value weighting

The potential for bias due to misclassification error in regression analysis is well understood by statisticians and epidemiologists. Assuming little or no available data for estimating misclassification probabilities, investigators sometimes seek to gauge the sensitivity of an estimated effect to variations in the assumed values of those probabilities. We present an intuitive and flexible approach to such a sensitivity analysis, assuming an underlying logistic regression model. For outcome misclassification, we argue that a likelihood‐based analysis is the cleanest and the most preferable approach. In the case of covariate misclassification, we combine observed data on the outcome, error‐prone binary covariate of interest, and other covariates measured without error, together with investigator‐supplied values for sensitivity and specificity parameters, to produce corresponding positive and negative predictive values. These values serve as estimated weights to be used in fitting the model of interest to an appropriately defined expanded data set using standard statistical software. Jackknifing provides a convenient tool for incorporating uncertainty in the estimated weights into valid standard errors to accompany log odds ratio estimates obtained from the sensitivity analysis. Examples illustrate the flexibility of this unified strategy, and simulations suggest that it performs well relative to a maximum likelihood approach carried out via numerical optimization. Copyright © 2010 John Wiley & Sons, Ltd.     

Regression calibration with more surrogates than mismeasured variables

In a recent paper (Weller EA, Milton DK, Eisen EA, Spiegelman D. Regression calibration for logistic regression with multiple surrogates for one exposure. Journal of Statistical Planning and Inference 2007; 137: 449‐461), the authors discussed fitting logistic regression models when a scalar main explanatory variable is measured with error by several surrogates, that is, a situation with more surrogates than variables measured with error. They compared two methods of adjusting for measurement error using a regression calibration approximate model as if it were exact. One is the standard regression calibration approach consisting of substituting an estimated conditional expectation of the true covariate given observed data in the logistic regression. The other is a novel two‐stage approach when the logistic regression is fitted to multiple surrogates, and then a linear combination of estimated slopes is formed as the estimate of interest. Applying estimated asymptotic variances for both methods in a single data set with some sensitivity analysis, the authors asserted superiority of their two‐stage approach. We investigate this claim in some detail. A troubling aspect of the proposed two‐stage method is that, unlike standard regression calibration and a natural form of maximum likelihood, the resulting estimates are not invariant to reparameterization of nuisance parameters in the model. We show, however, that, under the regression calibration approximation, the two‐stage method is asymptotically equivalent to a maximum likelihood formulation, and is therefore in theory superior to standard regression calibration. However, our extensive finite‐sample simulations in the practically important parameter space where the regression calibration model provides a good approximation failed to uncover such superiority of the two‐stage method. We also discuss extensions to different data structures. Copyright © 2012 John Wiley & Sons, Ltd.     

Sensitivity of regression calibration to non‐perfect validation data with application to the Norwegian Women and Cancer Study

Measurement error occurs when we observe error‐prone surrogates, rather than true values. It is common in observational studies and especially so in epidemiology, in nutritional epidemiology in particular. Correcting for measurement error has become common, and regression calibration is the most popular way to account for measurement error in continuous covariates. We consider its use in the context where there are validation data, which are used to calibrate the true values given the observed covariates. We allow for the case that the true value itself may not be observed in the validation data, but instead, a so‐called reference measure is observed. The regression calibration method relies on certain assumptions.This paper examines possible biases in regression calibration estimators when some of these assumptions are violated. More specifically, we allow for the fact that (i) the reference measure may not necessarily be an ‘alloyed gold standard’ (i.e., unbiased) for the true value; (ii) there may be correlated random subject effects contributing to the surrogate and reference measures in the validation data; and (iii) the calibration model itself may not be the same in the validation study as in the main study; that is, it is not transportable. We expand on previous work to provide a general result, which characterizes potential bias in the regression calibration estimators as a result of any combination of the violations aforementioned. We then illustrate some of the general results with data from the Norwegian Women and Cancer Study. Copyright © 2015 John Wiley & Sons, Ltd.     

Considerations for analysis of time‐to‐event outcomes measured with error: Bias and correction with SIMEX

For time‐to‐event outcomes, a rich literature exists on the bias introduced by covariate measurement error in regression models, such as the Cox model, and methods of analysis to address this bias. By comparison, less attention has been given to understanding the impact or addressing errors in the failure time outcome. For many diseases, the timing of an event of interest (such as progression‐free survival or time to AIDS progression) can be difficult to assess or reliant on self‐report and therefore prone to measurement error. For linear models, it is well known that random errors in the outcome variable do not bias regression estimates. With nonlinear models, however, even random error or misclassification can introduce bias into estimated parameters. We compare the performance of 2 common regression models, the Cox and Weibull models, in the setting of measurement error in the failure time outcome. We introduce an extension of the SIMEX method to correct for bias in hazard ratio estimates from the Cox model and discuss other analysis options to address measurement error in the response. A formula to estimate the bias induced into the hazard ratio by classical measurement error in the event time for a log‐linear survival model is presented. Detailed numerical studies are presented to examine the performance of the proposed SIMEX method under varying levels and parametric forms of the error in the outcome. We further illustrate the method with observational data on HIV outcomes from the Vanderbilt Comprehensive Care Clinic.     

The number needed to treat (NNT) can be adjusted for bias when the outcome is measured with error

BACKGROUND AND OBJECTIVE: We consider the number needed to treat (NNT) when the event of interest is defined by dichotomizing a continuous response at a threshold level. If the response is measured with error, the resulting NNT is biased. We consider methods to reduce this bias. METHODS: Bias adjustment was studied using simulations in which we varied the distributions of the underlying response and measurement error, including both normal and nonnormal distributions. We studied a maximum likelihood estimate (MLE) based on normality assumptions, and also considered a simulation-extrapolation estimate (SIMEX) without such assumptions. The treatment effect across all potential thresholds was summarized using an NNT threshold curve. RESULTS: Crude NNT estimation was substantially biased due to measurement error. The MLE performed well under normality, and it continued to perform well with nonnormal measurement error, but when the underlying response was nonnormal the MLE was unacceptably biased and was outperformed by the SIMEX estimate. The simulation results were also reflected in empirical data from a randomized study of cholesterol-lowering therapy. CONCLUSION: Ignoring measurement error can lead to substantial bias in NNT, which can have an important practical effect on the interpretation of analyses. Analysis methods that adjust for measurement error bias can be used to assess the sensitivity of NNT estimates to this effect.     

A comparison of regression calibration, moment reconstruction and imputation for adjusting for covariate measurement error in regression

Regression calibration (RC) is a popular method for estimating regression coefficients when one or more continuous explanatory variables, X, are measured with an error. In this method, the mismeasured covariate, W, is substituted by the expectation E(X|W), based on the assumption that the error in the measurement of X is non-differential. Using simulations, we compare three versions of RC with two other 'substitution' methods, moment reconstruction (MR) and imputation (IM), neither of which rely on the non-differential error assumption. We investigate studies that have an internal calibration sub-study. For RC, we consider (i) the usual version of RC, (ii) RC applied only to the 'marker' information in the calibration study, and (iii) an 'efficient' version (ERC) in which the estimators (i) and (ii) are combined. Our results show that ERC is preferable when there is non-differential measurement error. Under this condition, there are cases where ERC is less efficient than MR or IM, but they rarely occur in epidemiology. We show that the efficiency gain of usual RC and ERC over the other methods can sometimes be dramatic. The usual version of RC carries similar efficiency gains to ERC over MR and IM, but becomes unstable as measurement error becomes large, leading to bias and poor precision. When differential measurement error does pertain, then MR and IM have considerably less bias than RC, but can have much larger variance. We demonstrate our findings with an analysis of dietary fat intake and mortality in a large cohort study.     

Correcting for multivariate measurement error by regression calibration in meta‐analyses of epidemiological studies

Within‐person variability in measured values of multiple risk factors can bias their associations with disease. The multivariate regression calibration (RC) approach can correct for such measurement error and has been applied to studies in which true values or independent repeat measurements of the risk factors are observed on a subsample. We extend the multivariate RC techniques to a meta‐analysis framework where multiple studies provide independent repeat measurements and information on disease outcome. We consider the cases where some or all studies have repeat measurements, and compare study‐specific, averaged and empirical Bayes estimates of RC parameters. Additionally, we allow for binary covariates (e.g. smoking status) and for uncertainty and time trends in the measurement error corrections. Our methods are illustrated using a subset of individual participant data from prospective long‐term studies in the Fibrinogen Studies Collaboration to assess the relationship between usual levels of plasma fibrinogen and the risk of coronary heart disease, allowing for measurement error in plasma fibrinogen and several confounders. Copyright © 2009 John Wiley & Sons, Ltd.     

The impact of covariate measurement error on risk prediction

In the development of risk prediction models, predictors are often measured with error. In this paper, we investigate the impact of covariate measurement error on risk prediction. We compare the prediction performance using a costly variable measured without error, along with error‐free covariates, to that of a model based on an inexpensive surrogate along with the error‐free covariates. We consider continuous error‐prone covariates with homoscedastic and heteroscedastic errors, and also a discrete misclassified covariate. Prediction performance is evaluated by the area under the receiver operating characteristic curve (AUC), the Brier score (BS), and the ratio of the observed to the expected number of events (calibration). In an extensive numerical study, we show that (i) the prediction model with the error‐prone covariate is very well calibrated, even when it is mis‐specified; (ii) using the error‐prone covariate instead of the true covariate can reduce the AUC and increase the BS dramatically; (iii) adding an auxiliary variable, which is correlated with the error‐prone covariate but conditionally independent of the outcome given all covariates in the true model, can improve the AUC and BS substantially. We conclude that reducing measurement error in covariates will improve the ensuing risk prediction, unless the association between the error‐free and error‐prone covariates is very high. Finally, we demonstrate how a validation study can be used to assess the effect of mismeasured covariates on risk prediction. These concepts are illustrated in a breast cancer risk prediction model developed in the Nurses' Health Study. Copyright © 2015 John Wiley & Sons, Ltd.     

Measurement error in the explanatory variable of a binary regression: regression calibration and integrated conditional likelihood in studies of residential radon and lung cancer

In epidemiology, one approach to investigating the dependence of disease risk on an explanatory variable in the presence of several confounding variables is by fitting a binary regression using a conditional likelihood, thus eliminating the nuisance parameters. When the explanatory variable is measured with error, the estimated regression coefficient is biased usually towards zero. Motivated by the need to correct for this bias in analyses that combine data from a number of case-control studies of lung cancer risk associated with exposure to residential radon, two approaches are investigated. Both employ the conditional distribution of the true explanatory variable given the measured one. The method of regression calibration uses the expected value of the true given measured variable as the covariate. The second approach integrates the conditional likelihood numerically by sampling from the distribution of the true given measured explanatory variable. The two approaches give very similar point estimates and confidence intervals not only for the motivating example but also for an artificial data set with known properties. These results and some further simulations that demonstrate correct coverage for the confidence intervals suggest that for studies of residential radon and lung cancer the regression calibration approach will perform very well, so that nothing more sophisticated is needed to correct for measurement error.     

A bayesian approach to logistic regression models having measurement error following a mixture distribution

To estimate the parameters in a logistic regression model when the predictors are subject to random or systematic measurement error, we take a Bayesian approach and average the true logistic probability over the conditional posterior distribution of the true value of the predictor given its observed value. We allow this posterior distribution to consist of a mixture when the measurement error distribution changes form with observed exposure. We apply the method to study the risk of alcohol consumption on breast cancer using the Nurses Health Study data. We estimate measurement error from a small subsample where we compare true with reported consumption. Some of the self-reported non-drinkers truly do not drink. The resulting risk estimates differ sharply from those computed by standard logistic regression that ignores measurement error.     

Accelerated failure time models with covariates subject to measurement error

It has been well known that ignoring measurement error may result in substantially biased estimates in many contexts including linear and nonlinear regressions. For survival data with measurement error in covariates there has been extensive discussion in the literature with the focus being on the Cox proportional hazards models. However, the impact of measurement error on accelerated failure time (AFT) models has received little attention, though AFT models are very useful in survival data analysis. In this paper, we discuss AFT models with error-prone covariates and study the bias induced by the naive approach of ignoring measurement error in covariates. To adjust for such a bias, we describe a simulation and extrapolation method. This method is appealing because it is simple to implement and it does not require modelling the true but error-prone covariate process that is often not observable. Asymptotic normality for the resulting estimators is established. Simulation studies are carried out to evaluate the performance of the proposed method as well as the impact of ignoring measurement error in covariates. The proposed method is applied to analyse a data set arising from the Busselton Health study (Australian J. Public Health 1994; 18:129-135).     

Bayesian quantile regression‐based nonlinear mixed‐effects joint models for time‐to‐event and longitudinal data with multiple features

This article explores Bayesian joint models for a quantile of longitudinal response, mismeasured covariate and event time outcome with an attempt to (i) characterize the entire conditional distribution of the response variable based on quantile regression that may be more robust to outliers and misspecification of error distribution; (ii) tailor accuracy from measurement error, evaluate non‐ignorable missing observations, and adjust departures from normality in covariate; and (iii) overcome shortages of confidence in specifying a time‐to‐event model. When statistical inference is carried out for a longitudinal data set with non‐central location, non‐linearity, non‐normality, measurement error, and missing values as well as event time with being interval censored, it is important to account for the simultaneous treatment of these data features in order to obtain more reliable and robust inferential results. Toward this end, we develop Bayesian joint modeling approach to simultaneously estimating all parameters in the three models: quantile regression‐based nonlinear mixed‐effects model for response using asymmetric Laplace distribution, linear mixed‐effects model with skew‐t distribution for mismeasured covariate in the presence of informative missingness and accelerated failure time model with unspecified nonparametric distribution for event time. We apply the proposed modeling approach to analyzing an AIDS clinical data set and conduct simulation studies to assess the performance of the proposed joint models and method. Copyright © 2016 John Wiley & Sons, Ltd.     

Power and sample size calculations for generalized regression models with covariate measurement error

Covariate measurement error is often a feature of scientific data used for regression modelling. The consequences of such errors include a loss of power of tests of significance for the regression parameters corresponding to the true covariates. Power and sample size calculations that ignore covariate measurement error tend to overestimate power and underestimate the actual sample size required to achieve a desired power. In this paper we derive a novel measurement error corrected power function for generalized linear models using a generalized score test based on quasi-likelihood methods. Our power function is flexible in that it is adaptable to designs with a discrete or continuous scalar covariate (exposure) that can be measured with or without error, allows for additional confounding variables and applies to a broad class of generalized regression and measurement error models. A program is described that provides sample size or power for a continuous exposure with a normal measurement error model and a single normal confounder variable in logistic regression. We demonstrate the improved properties of our power calculations with simulations and numerical studies. An example is given from an ongoing study of cancer and exposure to arsenic as measured by toenail concentrations and tap water samples.     

SEGMENTED REGRESSION WITH ERRORS IN PREDICTORS: SEMI-PARAMETRIC AND PARAMETRIC METHODS

We consider the estimation of parameters in a particular segmented generalized linear model with additive measurement error in predictors, with a focus on linear and logistic regression. In epidemiologic studies segmented regression models often occur as threshold models, where it is assumed that the exposure has no influence on the response up to a possibly unknown threshold. Furthermore, in occupational and environmental studies the exposure typically cannot be measured exactly. Ignoring this measurement error leads to asymptotically biased estimators of the threshold. It is shown that this asymptotic bias is different from that observed for estimating standard generalized linear model parameters in the presence of measurement error, being both larger and in different directions than expected. In most cases considered the threshold is asymptotically underestimated. Two standard general methods for correcting for this bias are considered; regression calibration and simulation extrapolation (simex). In ordinary logistic and linear regression these procedures behave similarly, but in the threshold segmented regression model they operate quite differently. The regression calibration estimator usually has more bias but less variance than the simex estimator. Regression calibration and simex are typically thought of as functional methods, also known as semi-parametric methods, because they make no assumptions about the distribution of the unobservable covariate X. The contrasting structural, parametric maximum likelihood estimate assumes a parametric distributional form for X. In ordinary linear regression there is typically little difference between structural and functional methods. One of the major, surprising findings of our study is that in threshold regression, the functional and structural methods differ substantially in their performance. In one of our simulations, approximately consistent functional estimates can be as much as 25 times more variable than the maximum likelihood estimate for a properly specified parametric model. Structural (parametric) modelling ought not be a neglected tool in measurement error models. An example involving dust concentration and bronchitis in a mechanical engineering plant in Munich is used to illustrate the results. © 1997 by John Wiley & Sons, Ltd.     

Two-stage instrumental variable methods for estimating the causal odds ratio: analysis of bias

We present closed-form expressions of asymptotic bias for the causal odds ratio from two estimation approaches of instrumental variable logistic regression: (i) the two-stage predictor substitution (2SPS) method and (ii) the two-stage residual inclusion (2SRI) approach. Under the 2SPS approach, the first stage model yields the predicted value of treatment as a function of an instrument and covariates, and in the second stage model for the outcome, this predicted value replaces the observed value of treatment as a covariate. Under the 2SRI approach, the first stage is the same, but the residual term of the first stage regression is included in the second stage regression, retaining the observed treatment as a covariate. Our bias assessment is for a different context from that of Terza (J. Health Econ. 2008; 27(3):531-543), who focused on the causal odds ratio conditional on the unmeasured confounder, whereas we focus on the causal odds ratio among compliers under the principal stratification framework. Our closed-form bias results show that the 2SPS logistic regression generates asymptotically biased estimates of this causal odds ratio when there is no unmeasured confounding and that this bias increases with increasing unmeasured confounding. The 2SRI logistic regression is asymptotically unbiased when there is no unmeasured confounding, but when there is unmeasured confounding, there is bias and it increases with increasing unmeasured confounding. The closed-form bias results provide guidance for using these IV logistic regression methods. Our simulation results are consistent with our closed-form analytic results under different combinations of parameter settings.     

Fridge: Focused fine‐tuning of ridge regression for personalized predictions

Statistical prediction methods typically require some form of fine‐tuning of tuning parameter(s), with K‐fold cross‐validation as the canonical procedure. For ridge regression, there exist numerous procedures, but common for all, including cross‐validation, is that one single parameter is chosen for all future predictions. We propose instead to calculate a unique tuning parameter for each individual for which we wish to predict an outcome. This generates an individualized prediction by focusing on the vector of covariates of a specific individual. The focused ridge—fridge—procedure is introduced with a 2‐part contribution: First we define an oracle tuning parameter minimizing the mean squared prediction error of a specific covariate vector, and then we propose to estimate this tuning parameter by using plug‐in estimates of the regression coefficients and error variance parameter. The procedure is extended to logistic ridge regression by using parametric bootstrap. For high‐dimensional data, we propose to use ridge regression with cross‐validation as the plug‐in estimate, and simulations show that fridge gives smaller average prediction error than ridge with cross‐validation for both simulated and real data. We illustrate the new concept for both linear and logistic regression models in 2 applications of personalized medicine: predicting individual risk and treatment response based on gene expression data. The method is implemented in the R package fridge.