Covariate measurement error adjustment for matched case-control studies

We propose a conditional scores procedure for obtaining bias-corrected estimates of log odds ratios from matched case-control data in which one or more covariates are subject to measurement error. The approach involves conditioning on sufficient statistics for the unobservable true covariates that are treated as fixed unknown parameters. For the case of Gaussian nondifferential measurement error, we derive a set of unbiased score equations that can then be solved to estimate the log odds ratio parameters of interest. The procedure successfully removes the bias in naive estimates, and standard error estimates are obtained by resampling methods. We present an example of the procedure applied to data from a matched case-control study of prostate cancer and serum hormone levels, and we compare its performance to that of regression calibration procedures.     

Correcting for measurement error in fractional polynomial models using Bayesian modelling and regression calibration,with an application to alcohol and mortality

Exposure measurement error can result in a biased estimate of the association between an exposure and outcome. When the exposure–outcome relationship is linear on the appropriate scale (e.g. linear, logistic) and the measurement error is classical, that is the result of random noise, the result is attenuation of the effect. When the relationship is non‐linear, measurement error distorts the true shape of the association. Regression calibration is a commonly used method for correcting for measurement error, in which each individual's unknown true exposure in the outcome regression model is replaced by its expectation conditional on the error‐prone measure and any fully measured covariates. Regression calibration is simple to execute when the exposure is untransformed in the linear predictor of the outcome regression model, but less straightforward when non‐linear transformations of the exposure are used. We describe a method for applying regression calibration in models in which a non‐linear association is modelled by transforming the exposure using a fractional polynomial model. It is shown that taking a Bayesian estimation approach is advantageous. By use of Markov chain Monte Carlo algorithms, one can sample from the distribution of the true exposure for each individual. Transformations of the sampled values can then be performed directly and used to find the expectation of the transformed exposure required for regression calibration. A simulation study shows that the proposed approach performs well. We apply the method to investigate the relationship between usual alcohol intake and subsequent all‐cause mortality using an error model that adjusts for the episodic nature of alcohol consumption.     

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.     

Nonparametric correction for covariate measurement error in a stratified Cox model

Stratified Cox regression models with large number of strata and small stratum size are useful in many settings, including matched case-control family studies. In the presence of measurement error in covariates and a large number of strata, we show that extensions of existing methods fail either to reduce the bias or to correct the bias under nonsymmetric distributions of the true covariate or the error term. We propose a nonparametric correction method for the estimation of regression coefficients, and show that the estimators are asymptotically consistent for the true parameters. Small sample properties are evaluated in a simulation study. The method is illustrated with an analysis of Framingham data.     

Approximate maximum likelihood estimation for logistic regression with covariate measurement error

In nutritional epidemiology, dietary intake assessed with a food frequency questionnaire is prone to measurement error. Ignoring the measurement error in covariates causes estimates to be biased and leads to a loss of power. In this paper, we consider an additive error model according to the characteristics of the European Prospective Investigation into Cancer and Nutrition (EPIC)‐InterAct Study data, and derive an approximate maximum likelihood estimation (AMLE) for covariates with measurement error under logistic regression. This method can be regarded as an adjusted version of regression calibration and can provide an approximate consistent estimator. Asymptotic normality of this estimator is established under regularity conditions, and simulation studies are conducted to empirically examine the finite sample performance of the proposed method. We apply AMLE to deal with measurement errors in some interested nutrients of the EPIC‐InterAct Study under a sensitivity analysis framework.     

Correcting for measurement error in individual-level covariates in nonlinear mixed effects models

The nonlinear mixed effects model is used to represent data in pharmacokinetics, viral dynamics, and other areas where an objective is to elucidate associations among individual-specific model parameters and covariates; however, covariates may be measured with error. For additive measurement error, we show substitution of mismeasured covariates for true covariates may lead to biased estimators for fixed effects and random effects covariance parameters, while regression calibration may eliminate bias in fixed effects but fail to correct that in covariance parameters. We develop methods to take account of measurement error that correct this bias and may be implemented with standard software, and we demonstrate their utility via simulation and application to data from a study of HIV dynamics.     

Adjust quality scores from alignment and improve sequencing accuracy

In shotgun sequencing, statistical reconstruction of a consensus from alignment requires a model of measurement error. Churchill and Waterman proposed one such model and an expectation–maximization (EM) algorithm to estimate sequencing error rates for each assembly matrix. Ewing and Green defined Phred quality scores for base-calling from sequencing traces by training a model on a large amount of data. However, sample preparations and sequencing machines may work under different conditions in practice and therefore quality scores need to be adjusted. Moreover, the information given by quality scores is incomplete in the sense that they do not describe error patterns. We observe that each nucleotide base has its specific error pattern that varies across the range of quality values. We develop models of measurement error for shotgun sequencing by combining the two perspectives above. We propose a logistic model taking quality scores as covariates. The model is trained by a procedure combining an EM algorithm and model selection techniques. The training results in calibration of quality values and leads to a more accurate construction of consensus. Besides Phred scores obtained from ABI sequencers, we apply the same technique to calibrate quality values that come along with Beckman sequencers.     

Bias correction of bounded location error in binary data

Binary regression models for spatial data are commonly used in disciplines such as epidemiology and ecology. Many spatially referenced binary data sets suffer from location error, which occurs when the recorded location of an observation differs from its true location. When location error occurs, values of the covariates associated with the true spatial locations of the observations cannot be obtained. We show how a change of support (COS) can be applied to regression models for binary data to provide coefficient estimates when the true values of the covariates are unavailable, but the unknown location of the observations are contained within nonoverlapping arbitrarily shaped polygons. The COS accommodates spatial and nonspatial covariates and preserves the convenient interpretation of methods such as logistic and probit regression. Using a simulation experiment, we compare binary regression models with a COS to naive approaches that ignore location error. We illustrate the flexibility of the COS by modeling individual-level disease risk in a population using a binary data set where the locations of the observations are unknown but contained within administrative units. Our simulation experiment and data illustration corroborate that conventional regression models for binary data that ignore location error are unreliable, but that the COS can be used to eliminate bias while preserving model choice.     

A longitudinal measurement error model with a semicontinuous covariate

Covariate measurement error in regression is typically assumed to act in an additive or multiplicative manner on the true covariate value. However, such an assumption does not hold for the measurement error of sleep-disordered breathing (SDB) in the Wisconsin Sleep Cohort Study (WSCS). The true covariate is the severity of SDB, and the observed surrogate is the number of breathing pauses per unit time of sleep, which has a nonnegative semicontinuous distribution with a point mass at zero. We propose a latent variable measurement error model for the error structure in this situation and implement it in a linear mixed model. The estimation procedure is similar to regression calibration but involves a distributional assumption for the latent variable. Modeling and model-fitting strategies are explored and illustrated through an example from the WSCS.     

Evaluation of a two‐part regression calibration to adjust for dietary exposure measurement error in the Cox proportional hazards model: A simulation study

Dietary questionnaires are prone to measurement error, which bias the perceived association between dietary intake and risk of disease. Short‐term measurements are required to adjust for the bias in the association. For foods that are not consumed daily, the short‐term measurements are often characterized by excess zeroes. Via a simulation study, the performance of a two‐part calibration model that was developed for a single‐replicate study design was assessed by mimicking leafy vegetable intake reports from the multicenter European Prospective Investigation into Cancer and Nutrition (EPIC) study. In part I of the fitted two‐part calibration model, a logistic distribution was assumed; in part II, a gamma distribution was assumed. The model was assessed with respect to the magnitude of the correlation between the consumption probability and the consumed amount (hereafter, cross‐part correlation), the number and form of covariates in the calibration model, the percentage of zero response values, and the magnitude of the measurement error in the dietary intake. From the simulation study results, transforming the dietary variable in the regression calibration to an appropriate scale was found to be the most important factor for the model performance. Reducing the number of covariates in the model could be beneficial, but was not critical in large‐sample studies. The performance was remarkably robust when fitting a one‐part rather than a two‐part model. The model performance was minimally affected by the cross‐part correlation.     

On Method of Moments Estimation in Linear Mixed Effects Models with Measurement Error on Covariates and Response with Application to a Longitudinal Study of Gene-Environment Interaction

We study a linear mixed effects model for longitudinal data, where the response variable and covariates with fixed effects are subject to measurement error. We propose a method of moment estimation that does not require any assumption on the functional forms of the distributions of random effects and other random errors in the model. For a classical measurement error model we apply the instrumental variable approach to ensure identifiability of the parameters. Our methodology, without instrumental variables, can be applied to Berkson measurement errors. Using simulation studies, we investigate the finite sample performances of the estimators and show the impact of measurement error on the covariates and the response on the estimation procedure. The results show that our method performs quite satisfactory, especially for the fixed effects with measurement error (even under misspecification of measurement error model). This method is applied to a real data example of a large birth and child cohort study.     

A smoothed corrected score approach for proportional hazards model with misclassified discretized covariates induced by error-contaminated continuous time-dependent exposure

We consider the proportional hazards model in which the covariates include the discretized categories of a continuous time-dependent exposure variable measured with error. Naively ignoring the measurement error in the analysis may cause biased estimation and erroneous inference. Although various approaches have been proposed to deal with measurement error when the hazard depends linearly on the time-dependent variable, it has not yet been investigated how to correct when the hazard depends on the discretized categories of the time-dependent variable. To fill this gap in the literature, we propose a smoothed corrected score approach based on approximation of the discretized categories after smoothing the indicator function. The consistency and asymptotic normality of the proposed estimator are established. The observation times of the time-dependent variable are allowed to be informative. For comparison, we also extend to this setting two approximate approaches, the regression calibration and the risk-set regression calibration. The methods are assessed by simulation studies and by application to data from an HIV clinical trial.     

Simulation‐selection‐extrapolation: Estimation in high‐dimensional errors‐in‐variables models

Errors‐in‐variables models in high‐dimensional settings pose two challenges in application. First, the number of observed covariates is larger than the sample size, while only a small number of covariates are true predictors under an assumption of model sparsity. Second, the presence of measurement error can result in severely biased parameter estimates, and also affects the ability of penalized methods such as the lasso to recover the true sparsity pattern. A new estimation procedure called SIMulation‐SELection‐EXtrapolation (SIMSELEX) is proposed. This procedure makes double use of lasso methodology. First, the lasso is used to estimate sparse solutions in the simulation step, after which a group lasso is implemented to do variable selection. The SIMSELEX estimator is shown to perform well in variable selection, and has significantly lower estimation error than naive estimators that ignore measurement error. SIMSELEX can be applied in a variety of errors‐in‐variables settings, including linear models, generalized linear models, and Cox survival models. It is furthermore shown in the Supporting Information how SIMSELEX can be applied to spline‐based regression models. A simulation study is conducted to compare the SIMSELEX estimators to existing methods in the linear and logistic model settings, and to evaluate performance compared to naive methods in the Cox and spline models. Finally, the method is used to analyze a microarray dataset that contains gene expression measurements of favorable histology Wilms tumors.     

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.     

Effects of covariate measurement error in the initial level and rate of change of an exposure variable

When repeated measures of an exposure variable are obtained on individuals, it can be of epidemiologic interest to relate the slope of this variable over time to a subsequent response. Subject-specific estimates of this slope are measured with error, as are corresponding estimates of the level of exposure, i.e., the intercept of a linear regression over time. Because the intercept is often correlated with the slope and may also be associated with the outcome, each error-prone covariate (intercept and slope) is a potential confounder, thereby tending to accentuate potential biases due to measurement error. Under a familiar mixed linear model for the exposure measurements, we present closed-form estimators for the true parameters of interest in the case of a continuous outcome with complete and equally timed follow-up for all subjects. Generalizations to handle incomplete follow-up, other types of outcome variables, and additional fixed covariates are illustrated via maximum likelihood. We provide examples using data from the Multicenter AIDS Cohort Study. In these examples, substantial adjustments are made to uncorrected parameter estimates corresponding to the health-related effects of exposure variable slopes over time. We illustrate the potential impact of such adjustments on the interpretation of an epidemiologic analysis.     

Significance testing for correlated binary outcome data

Multiple logistic regression is a commonly used multivariate technique for analyzing data with a binary outcome. One assumption needed for this method of analysis is the independence of outcome for all sample points in a data set. In ophthalmologic data and other types of correlated binary data, this assumption is often grossly violated and the validity of the technique becomes an issue. A technique has been developed (Rosner, 1984) that utilizes a polychotomous logistic regression model to allow one to look at multiple exposure variables in the context of a correlated binary data structure. This model is an extension of the beta-binomial model, which has been widely used to model correlated binary data when no covariates are present. In this paper, a relationship is developed between the two techniques, whereby it is shown that use of ordinary logistic regression in the presence of correlated binary data can result in true significance levels that are considerably larger than nominal levels in frequently encountered situations. This relationship is explored in detail in the case of a single dichotomous exposure variable. In this case, the appropriate test statistic can be expressed as an adjusted chi-square statistic based on the 2 X 2 contingency table relating exposure to outcome. The test statistic is easily computed as a function of the ordinary chi-square statistic and the correlation between eyes (or more generally between cluster members) for outcome and exposure, respectively. This generalizes some previous results obtained by Koval and Donner (1987, in Festschrift for V. M. Joshi, I. B. MacNeill (ed.), Vol. V, 199-224.(ABSTRACT TRUNCATED AT 250 WORDS)     

Use of Two-Part Regression Calibration Model to Correct for Measurement Error in Episodically Consumed Foods in a Single-Replicate Study Design: EPIC Case Study

In epidemiologic studies, measurement error in dietary variables often attenuates association between dietary intake and disease occurrence. To adjust for the attenuation caused by error in dietary intake, regression calibration is commonly used. To apply regression calibration, unbiased reference measurements are required. Short-term reference measurements for foods that are not consumed daily contain excess zeroes that pose challenges in the calibration model. We adapted two-part regression calibration model, initially developed for multiple replicates of reference measurements per individual to a single-replicate setting. We showed how to handle excess zero reference measurements by two-step modeling approach, how to explore heteroscedasticity in the consumed amount with variance-mean graph, how to explore nonlinearity with the generalized additive modeling (GAM) and the empirical logit approaches, and how to select covariates in the calibration model. The performance of two-part calibration model was compared with the one-part counterpart. We used vegetable intake and mortality data from European Prospective Investigation on Cancer and Nutrition (EPIC) study. In the EPIC, reference measurements were taken with 24-hour recalls. For each of the three vegetable subgroups assessed separately, correcting for error with an appropriately specified two-part calibration model resulted in about three fold increase in the strength of association with all-cause mortality, as measured by the log hazard ratio. Further found is that the standard way of including covariates in the calibration model can lead to over fitting the two-part calibration model. Moreover, the extent of adjusting for error is influenced by the number and forms of covariates in the calibration model. For episodically consumed foods, we advise researchers to pay special attention to response distribution, nonlinearity, and covariate inclusion in specifying the calibration model.     

A simulation study of measurement error correction methods in logistic regression

Measurement error models in logistic regression have received considerable theoretical interest over the past 10-15 years. In this paper, we present the results of a simulation study that compares four estimation methods: the so-called regression calibration method, probit maximum likelihood as an approximation to the logistic maximum likelihood, the exact maximum likelihood method based on a logistic model, and the naive estimator, which is the result of simply ignoring the fact that some of the explanatory variables are measured with error. We have compared the behavior of these methods in a simple, additive measurement error model. We show that, in this situation, the regression calibration method is a very good alternative to more mathematically sophisticated methods.     

Efficient measurement error correction with spatially misaligned data

Association studies in environmental statistics often involve exposure and outcome data that are misaligned in space. A common strategy is to employ a spatial model such as universal kriging to predict exposures at locations with outcome data and then estimate a regression parameter of interest using the predicted exposures. This results in measurement error because the predicted exposures do not correspond exactly to the true values. We characterize the measurement error by decomposing it into Berkson-like and classical-like components. One correction approach is the parametric bootstrap, which is effective but computationally intensive since it requires solving a nonlinear optimization problem for the exposure model parameters in each bootstrap sample. We propose a less computationally intensive alternative termed the "parameter bootstrap" that only requires solving one nonlinear optimization problem, and we also compare bootstrap methods to other recently proposed methods. We illustrate our methodology in simulations and with publicly available data from the Environmental Protection Agency.