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.     

A Bayesian approach for instrumental variable analysis with censored time‐to‐event outcome

Instrumental variable (IV) analysis has been widely used in economics, epidemiology, and other fields to estimate the causal effects of covariates on outcomes, in the presence of unobserved confounders and/or measurement errors in covariates. However, IV methods for time‐to‐event outcome with censored data remain underdeveloped. This paper proposes a Bayesian approach for IV analysis with censored time‐to‐event outcome by using a two‐stage linear model. A Markov chain Monte Carlo sampling method is developed for parameter estimation for both normal and non‐normal linear models with elliptically contoured error distributions. The performance of our method is examined by simulation studies. Our method largely reduces bias and greatly improves coverage probability of the estimated causal effect, compared with the method that ignores the unobserved confounders and measurement errors. We illustrate our method on the Women's Health Initiative Observational Study and the Atherosclerosis Risk in Communities Study. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Modeling zero‐inflated count data using a covariate‐dependent random effect model

In various medical related researches, excessive zeros, which make the standard Poisson regression model inadequate, often exist in count data. We proposed a covariate‐dependent random effect model to accommodate the excess zeros and the heterogeneity in the population simultaneously. This work is motivated by a data set from a survey on the dental health status of Hong Kong preschool children where the response variable is the number of decayed, missing, or filled teeth. The random effect has a sound biological interpretation as the overall oral health status or other personal qualities of an individual child that is unobserved and unable to be quantified easily. The overall measure of oral health status, responsible for accommodating the excessive zeros and also the heterogeneity among the children, is covariate dependent. This covariate‐dependent random effect model allows one to distinguish whether a potential covariate has an effect on the conceived overall oral health condition of the children, that is, the random effect, or has a direct effect on the magnitude of the counts, or both. We proposed a multiple imputation approach for estimation of the parameters. We discussed the choice of the imputation size. We evaluated the performance of the proposed estimation method through simulation studies, and we applied the model and method to the dental data. Copyright © 2012 John Wiley & Sons, Ltd.     

A Bayesian mixture of semiparametric mixed‐effects joint models for skewed‐longitudinal and time‐to‐event data

In longitudinal studies, it is of interest to investigate how repeatedly measured markers in time are associated with a time to an event of interest, and in the mean time, the repeated measurements are often observed with the features of a heterogeneous population, non‐normality, and covariate measured with error because of longitudinal nature. Statistical analysis may complicate dramatically when one analyzes longitudinal–survival data with these features together. Recently, a mixture of skewed distributions has received increasing attention in the treatment of heterogeneous data involving asymmetric behaviors across subclasses, but there are relatively few studies accommodating heterogeneity, non‐normality, and measurement error in covariate simultaneously arose in longitudinal–survival data setting. Under the umbrella of Bayesian inference, this article explores a finite mixture of semiparametric mixed‐effects joint models with skewed distributions for longitudinal measures with an attempt to mediate homogeneous characteristics, adjust departures from normality, and tailor accuracy from measurement error in covariate as well as overcome shortages of confidence in specifying a time‐to‐event model. The Bayesian mixture of joint modeling offers an appropriate avenue to estimate not only all parameters of mixture joint models but also probabilities of class membership. Simulation studies are conducted to assess the performance of the proposed method, and a real example is analyzed to demonstrate the methodology. The results are reported by comparing potential models with various scenarios. Copyright © 2015 John Wiley & Sons, Ltd.     

Regression analysis with covariates that have heteroscedastic measurement error

We consider the estimation of the regression of an outcome Y on a covariate X, where X is unobserved, but a variable W that measures X with error is observed. A calibration sample that measures pairs of values of X and W is also available; we consider calibration samples where Y is measured (internal calibration) and not measured (external calibration). One common approach for measurement error correction is Regression Calibration (RC), which substitutes the unknown values of X by predictions from the regression of X on W estimated from the calibration sample. An alternative approach is to multiply impute the missing values of X given Y and W based on an imputation model, and then use multiple imputation (MI) combining rules for inferences. Most of current work assumes that the measurement error of W has a constant variance, whereas in many situations, the variance varies as a function of X. We consider extensions of the RC and MI methods that allow for heteroscedastic measurement error, and compare them by simulation. The MI method is shown to provide better inferences in this setting. We also illustrate the proposed methods using a data set from the BioCycle study.     

Multiple augmentation for interval-censored data with measurement error

There has been substantial effort devoted to the analysis of censored failure time with covariates that are subject to measurement error. Previous studies have focused on right-censored survival data, but interval-censored survival data with covariate measurement error are yet to be investigated. Our study is partly motivated by analysis of the HIV clinical trial AIDS Clinical Trial Group (ACTG) 175 data, where the occurrence time of AIDS is interval censored and the covariate CD4 count is subject to measurement error. We assume that the data are realized from a proportional hazards model. A multiple augmentation approach is proposed to convert interval-censored data to right-censored data, and the conditional score approach is then employed to account for measurement error. The proposed approach is easy to implement and can be readily extended to other semiparametric models. Extensive simulations show that the proposed approach has satisfactory finite-sample performance. The ACTG 175 data are then analyzed.     

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.     

Covariate measurement error and the estimation of random effect parameters in a mixed model for longitudinal data

We explore the effects of measurement error in a time-varying covariate for a mixed model applied to a longitudinal study of plasma levels and dietary intake of beta-carotene. We derive a simple expression for the bias of large sample estimates of the variance of random effects in a longitudinal model for plasma levels when dietary intake is treated as a time-varying covariate subject to measurement error. In general, estimates for these variances made without consideration of measurement error are biased positively, unlike estimates for the slope coefficients which tend to be ‘attenuated’ If we can assume that the residuals from a longitudinal fit for the time-varying covariate behave like measurement errors, we can estimate the original parameters without the need for additional validation or reliability studies. We propose a method to test this assumption and show that the assumption is reasonable for the example data. We then use a likelihood-based method of estimation that involves a simple extension of existing methods for fitting mixed models. Simulations illustrate the properties of the proposed estimators. © 1998 John Wiley & Sons, Ltd.     

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.     

Statistical methods for multivariate interval-censored recurrent events

Multi-type recurrent event data arise when two or more different kinds of events may occur repeatedly over a period of observation. The scientific objectives in such settings are often to describe features of the marginal processes and to study the association between the different types of events. Interval-censored multi-type recurrent event data arise when the precise event times are unobserved, but intervals are available during which the events are known to have occurred. This type of data is common in studies of patients with advanced cancer, for example, where the events may represent the development of different types of metastatic lesions which are only detectable by conducting bone scans of the entire skeleton. In this setting it is of interest to characterize the incidence of the various types of bone lesions, to estimate the impact of treatment and other covariate effects on the development of new lesions, and to understand the relationship between the processes generating the bone lesions. We develop joint models for multi-type interval-censored recurrent events which accommodate dependencies between different types of events and enable one to examine the covariate effects via regression. However, since the marginal likelihood resulting from the multivariate random effect model is intractable, we describe a Gibbs sampling algorithm to facilitate model fitting and inference. We use generalized estimating equations for estimation and inference based on marginal models. The finite sample properties of the marginal approach are studied via simulation. The estimates of both the regression coefficients and the variance-covariance parameters are shown to have negligible bias and 95 per cent confidence intervals based on the asymptotic variance formula are shown to have excellent empirical coverage probabilities in all of the settings considered. The application of these methods to data from a trial of women with advanced breast cancer provides insight into the clinical course of bone metastases in this population.     

Extending methods for investigating the relationship between treatment effect and baseline risk from pairwise meta‐analysis to network meta‐analysis

Baseline risk is a proxy for unmeasured but important patient‐level characteristics, which may be modifiers of treatment effect, and is a potential source of heterogeneity in meta‐analysis. Models adjusting for baseline risk have been developed for pairwise meta‐analysis using the observed event rate in the placebo arm and taking into account the measurement error in the covariate to ensure that an unbiased estimate of the relationship is obtained. Our objective is to extend these methods to network meta‐analysis where it is of interest to adjust for baseline imbalances in the non‐intervention group event rate to reduce both heterogeneity and possibly inconsistency. This objective is complicated in network meta‐analysis by this covariate being sometimes missing, because of the fact that not all studies in a network may have a non‐active intervention arm. A random‐effects meta‐regression model allowing for inclusion of multi‐arm trials and trials without a ‘non‐intervention’ arm is developed. Analyses are conducted within a Bayesian framework using the WinBUGS software. The method is illustrated using two examples: (i) interventions to promote functional smoke alarm ownership by households with children and (ii) analgesics to reduce post‐operative morphine consumption following a major surgery. The results showed no evidence of baseline effect in the smoke alarm example, but the analgesics example shows that the adjustment can greatly reduce heterogeneity and improve overall model fit. Copyright © 2012 John Wiley & Sons, Ltd.     

A mechanistic nonlinear model for censored and mismeasured covariates in longitudinal models,with application in AIDS studies

When modeling longitudinal data, the true values of time‐varying covariates may be unknown because of detection‐limit censoring or measurement error. A common approach in the literature is to empirically model the covariate process based on observed data and then predict the censored values or mismeasured values based on this empirical model. Such an empirical model can be misleading, especially for censored values since the (unobserved) censored values may behave very differently than observed values due to the underlying data‐generation mechanisms or disease status. In this paper, we propose a mechanistic nonlinear covariate model based on the underlying data‐generation mechanisms to address censored values and mismeasured values. Such a mechanistic model is based on solid scientific or biological arguments, so the predicted censored or mismeasured values are more reasonable. We use a Monte Carlo EM algorithm for likelihood inference and apply the methods to an AIDS dataset, where viral load is censored by a lower detection limit. Simulation results confirm that the proposed models and methods offer substantial advantages over existing empirical covariate models for censored and mismeasured covariates.     

MEBoost: Variable selection in the presence of measurement error

We present a novel method for variable selection in regression models when covariates are measured with error. The iterative algorithm we propose, M easurement E rror Boost ing (MEBoost), follows a path defined by estimating equations that correct for covariate measurement error. We illustrate the use of MEBoost in practice by analyzing data from the Box Lunch Study, a clinical trial in nutrition where several variables are based on self-report and, hence, measured with error, where we are interested in performing model selection from a large data set to select variables that are related to the number of times a subject binge ate in the last 28 days. Furthermore, we evaluated our method and compared its performance to the recently proposed Convex Conditioned Lasso and to the “naive” Lasso, which does not correct for measurement error through a simulation study. Increasing the degree of measurement error increased prediction error and decreased the probability of accurate covariate selection, but this loss of accuracy occurred to a lesser degree when using MEBoost. Through simulations, we also make a case for the consistency of the model selected.     

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.     

Model misspecification and robustness in causal inference: comparing matching with doubly robust estimation

In this paper, we compare the robustness properties of a matching estimator with a doubly robust estimator. We describe the robustness properties of matching and subclassification estimators by showing how misspecification of the propensity score model can result in the consistent estimation of an average causal effect. The propensity scores are covariate scores, which are a class of functions that removes bias due to all observed covariates. When matching on a parametric model (e.g., a propensity or a prognostic score), the matching estimator is robust to model misspecifications if the misspecified model belongs to the class of covariate scores. The implication is that there are multiple possibilities for the matching estimator in contrast to the doubly robust estimator in which the researcher has two chances to make reliable inference. In simulations, we compare the finite sample properties of the matching estimator with a simple inverse probability weighting estimator and a doubly robust estimator. For the misspecifications in our study, the mean square error of the matching estimator is smaller than the mean square error of both the simple inverse probability weighting estimator and the doubly robust estimators.     

The trend‐renewal process: a useful model for medical recurrence data

Time‐to‐event data analysis has a long tradition in applied statistics. Many models have been developed for data where each subject or observation unit experiences at most one event during its life. In contrast, in some applications, the subjects may experience more than one event. Recurrent events appear in science, medicine, economy, and technology. Often the events are followed by a repair action in reliability or a treatment in life science. A model to deal with recurrent event times for incomplete repair of technical systems is the trend‐renewal process. It is composed of a trend and a renewal component. In the present paper, we use a Weibull process for both of these components. The model is extended to include a Cox type covariate term to account for observed heterogeneity. A further extension includes random effects to account for unobserved heterogeneity. We fit the suggested version of the trend‐renewal process to a data set of hospital readmission times of colon cancer patients to illustrate the method for application to clinical data. Copyright © 2012 John Wiley & Sons, Ltd.     

Inference on cancer screening exam accuracy using population‐level administrative data

This paper develops a model for cancer screening and cancer incidence data, accommodating the partially unobserved disease status, clustered data structures, general covariate effects, and dependence between exams. The true unobserved cancer and detection status of screening participants are treated as latent variables, and a Markov Chain Monte Carlo algorithm is used to estimate the Bayesian posterior distributions of the diagnostic error rates and disease prevalence. We show how the Bayesian approach can be used to draw inferences about screening exam properties and disease prevalence while allowing for the possibility of conditional dependence between two exams. The techniques are applied to the estimation of the diagnostic accuracy of mammography and clinical breast examination using data from the Ontario Breast Screening Program in Canada. Copyright © 2015 John Wiley & Sons, Ltd.     

Bayesian adjustment for covariate measurement errors: A flexible parametric approach

In most epidemiological investigations, the study units are people, the outcome variable (or the response) is a health‐related event, and the explanatory variables are usually environmental and/or socio‐demographic factors. The fundamental task in such investigations is to quantify the association between the explanatory variables (covariates/exposures) and the outcome variable through a suitable regression model. The accuracy of such quantification depends on how precisely the relevant covariates are measured. In many instances, we cannot measure some of the covariates accurately. Rather, we can measure noisy (mismeasured) versions of them. In statistical terminology, mismeasurement in continuous covariates is known as measurement errors or errors‐in‐variables. Regression analyses based on mismeasured covariates lead to biased inference about the true underlying response–covariate associations. In this paper, we suggest a flexible parametric approach for avoiding this bias when estimating the response–covariate relationship through a logistic regression model. More specifically, we consider the flexible generalized skew‐normal and the flexible generalized skew‐t distributions for modeling the unobserved true exposure. For inference and computational purposes, we use Bayesian Markov chain Monte Carlo techniques. We investigate the performance of the proposed flexible parametric approach in comparison with a common flexible parametric approach through extensive simulation studies. We also compare the proposed method with the competing flexible parametric method on a real‐life data set. Though emphasis is put on the logistic regression model, the proposed method is unified and is applicable to the other generalized linear models, and to other types of non‐linear regression models as well. Copyright © 2009 John Wiley & Sons, Ltd.     

Incorporating baseline measurements into the analysis of crossover trials with time‐to‐event endpoints

Two‐period two‐treatment (2×2) crossover designs are commonly used in clinical trials. For continuous endpoints, it has been shown that baseline (pretreatment) measurements collected before the start of each treatment period can be useful in improving the power of the analysis. Methods to achieve a corresponding gain for censored time‐to‐event endpoints have not been adequately studied. We propose a method in which censored values are treated as missing data and multiply imputed using prespecified parametric event time models. The event times in each imputed data set are then log‐transformed and analyzed using a linear model suitable for a 2×2 crossover design with continuous endpoints, with the difference in period‐specific baselines included as a covariate. Results obtained from the imputed data sets are synthesized for point and confidence interval estimation of the treatment ratio of geometric mean event times using model averaging in conjunction with Rubin's combination rule. We use simulations to illustrate the favorable operating characteristics of our method relative to two other methods for crossover trials with censored time‐to‐event data, ie, a hierarchical rank test that ignores the baselines and a stratified Cox model that uses each study subject as a stratum and includes period‐specific baselines as a covariate. Application to a real data example is provided.