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.     

A flexible approach to measurement error correction in case-control studies

SUMMARY: We investigate the use of prospective likelihood methods to analyze retrospective case-control data where some of the covariates are measured with error. We show that prospective methods can be applied and the case-control sampling scheme can be ignored if one adequately models the distribution of the error-prone covariates in the case-control sampling scheme. Indeed, subject to this, the prospective likelihood methods result in consistent estimates and information standard errors are asymptotically correct. However, the distribution of such covariates is not the same in the population and under case-control sampling, dictating the need to model the distribution flexibly. In this article, we illustrate the general principle by modeling the distribution of the continuous error-prone covariates using the skewnormal distribution. The performance of the method is evaluated through simulation studies, which show satisfactory results in terms of bias and coverage. Finally, the method is applied to the analysis of two data sets which refer, respectively, to a cholesterol study and a study on breast cancer.     

Causal mediation analysis with a latent mediator

Health researchers are often interested in assessing the direct effect of a treatment or exposure on an outcome variable, as well as its indirect (or mediation) effect through an intermediate variable (or mediator). For an outcome following a nonlinear model, the mediation formula may be used to estimate causally interpretable mediation effects. This method, like others, assumes that the mediator is observed. However, as is common in structural equations modeling, we may wish to consider a latent (unobserved) mediator. We follow a potential outcomes framework and assume a generalized structural equations model (GSEM). We provide maximum‐likelihood estimation of GSEM parameters using an approximate Monte Carlo EM algorithm, coupled with a mediation formula approach to estimate natural direct and indirect effects. The method relies on an untestable sequential ignorability assumption; we assess robustness to this assumption by adapting a recently proposed method for sensitivity analysis. Simulation studies show good properties of the proposed estimators in plausible scenarios. Our method is applied to a study of the effect of mother education on occurrence of adolescent dental caries, in which we examine possible mediation through latent oral health behavior.     

Measurement error in covariates in the marginal hazards model for multivariate failure time data

We consider measurement error in covariates in the marginal hazards model for multivariate failure time data. We explore the bias implications of normal additive measurement error without assuming a distribution for the underlying true covariate. To correct measurement-error-induced bias in the regression coefficient of the marginal model, we propose to apply the SIMEX procedure and demonstrate its large and small sample properties for both known and estimated measurement error variance. We illustrate this method using the Lipid Research Clinics Coronary Primary Prevention Trial data with total cholesterol as the covariate measured with error and time until angina and time until nonfatal myocardial infarction as the correlated outcomes of interest.     

Shared uncertainty in measurement error problems, with application to Nevada Test Site fallout data

In radiation epidemiology, it is often necessary to use mathematical models in the absence of direct measurements of individual doses. When complex models are used as surrogates for direct measurements to estimate individual doses that occurred almost 50 years ago, dose estimates will be associated with considerable error, this error being a mixture of (a) classical measurement error due to individual data such as diet histories and (b) Berkson measurement error associated with various aspects of the dosimetry system. In the Nevada Test Site(NTS) Thyroid Disease Study, the Berkson measurement errors are correlated within strata. This article concerns the development of statistical methods for inference about risk of radiation dose on thyroid disease, methods that account for the complex error structure inherence in the problem. Bayesian methods using Markov chain Monte Carlo and Monte-Carlo expectation-maximization methods are described, with both sharing a key Metropolis-Hastings step. Regression calibration is also considered, but we show that regression calibration does not use the correlation structure of the Berkson errors. Our methods are applied to the NTS Study, where we find a strong dose-response relationship between dose and thyroiditis. We conclude that full consideration of mixtures of Berkson and classical uncertainties in reconstructed individual doses are important for quantifying the dose response and its credibility/confidence interval. Using regression calibration and expectation values for individual doses can lead to a substantial underestimation of the excess relative risk per gray and its 95% confidence intervals.     

A zero‐augmented generalized gamma regression calibration to adjust for covariate measurement error: A case of an episodically consumed dietary intake

Measurement error in exposure variables is a serious impediment in epidemiological studies that relate exposures to health outcomes. In nutritional studies, interest could be in the association between long‐term dietary intake and disease occurrence. Long‐term intake is usually assessed with food frequency questionnaire (FFQ), which is prone to recall bias. Measurement error in FFQ‐reported intakes leads to bias in parameter estimate that quantifies the association. To adjust for bias in the association, a calibration study is required to obtain unbiased intake measurements using a short‐term instrument such as 24‐hour recall (24HR). The 24HR intakes are used as response in regression calibration to adjust for bias in the association. For foods not consumed daily, 24HR‐reported intakes are usually characterized by excess zeroes, right skewness, and heteroscedasticity posing serious challenge in regression calibration modeling. We proposed a zero‐augmented calibration model to adjust for measurement error in reported intake, while handling excess zeroes, skewness, and heteroscedasticity simultaneously without transforming 24HR intake values. We compared the proposed calibration method with the standard method and with methods that ignore measurement error by estimating long‐term intake with 24HR and FFQ‐reported intakes. The comparison was done in real and simulated datasets. With the 24HR, the mean increase in mercury level per ounce fish intake was about 0.4; with the FFQ intake, the increase was about 1.2. With both calibration methods, the mean increase was about 2.0. Similar trend was observed in the simulation study. In conclusion, the proposed calibration method performs at least as good as the standard method.     

Filtered kriging for spatial data with heterogeneous measurement error variances

When predicting values for the measurement-error-free component of an observed spatial process, it is generally assumed that the process has a common measurement error variance. However, it is often the case that each measurement in a spatial data set has a known, site-specific measurement error variance, rendering the observed process nonstationary. We present a simple approach for estimating the semivariogram of the unobservable measurement-error-free process using a bias adjustment of the classical semivariogram formula. We then develop a new kriging predictor that filters the measurement errors. For scenarios where each site's measurement error variance is a function of the process of interest, we recommend an approach that also uses a variance-stabilizing transformation. The properties of the heterogeneous variance measurement-error-filtered kriging (HFK) predictor and variance-stabilized HFK predictor, and the improvement of these approaches over standard measurement-error-filtered kriging are demonstrated using simulation. The approach is illustrated with climate model output from the Hudson Strait area in northern Canada. In the illustration, locations with high or low measurement error variances are appropriately down- or upweighted in the prediction of the underlying process, yielding a realistically smooth picture of the phenomenon of interest.     

A new method for dealing with measurement error in explanatory variables of regression models

We introduce a new method, moment reconstruction, of correcting for measurement error in covariates in regression models. The central idea is similar to regression calibration in that the values of the covariates that are measured with error are replaced by "adjusted" values. In regression calibration the adjusted value is the expectation of the true value conditional on the measured value. In moment reconstruction the adjusted value is the variance-preserving empirical Bayes estimate of the true value conditional on the outcome variable. The adjusted values thereby have the same first two moments and the same covariance with the outcome variable as the unobserved "true" covariate values. We show that moment reconstruction is equivalent to regression calibration in the case of linear regression, but leads to different results for logistic regression. For case-control studies with logistic regression and covariates that are normally distributed within cases and controls, we show that the resulting estimates of the regression coefficients are consistent. In simulations we demonstrate that for logistic regression, moment reconstruction carries less bias than regression calibration, and for case-control studies is superior in mean-square error to the standard regression calibration approach. Finally, we give an example of the use of moment reconstruction in linear discriminant analysis and a nonstandard problem where we wish to adjust a classification tree for measurement error in the explanatory variables.     

Logistic regression with exposure biomarkers and flexible measurement error

Regression calibration, refined regression calibration, and conditional scores estimation procedures are extended to a measurement model that is motivated by nutritional and physical activity epidemiology. Biomarker data, available on a small subset of a study cohort for reasons of cost, are assumed to adhere to a classical measurement error model, while corresponding self-report nutrient consumption or activity-related energy expenditure data are available for the entire cohort. The self-report assessment measurement model includes a person-specific random effect, the mean and variance of which may depend on individual characteristics such as body mass index or ethnicity. Logistic regression is used to relate the disease odds ratio to the actual, but unmeasured, dietary or physical activity exposure. Simulation studies are presented to evaluate and contrast the three estimation procedures, and to provide insight into preferred biomarker subsample size under selected cohort study configurations.     

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.     

Adjusting body mass for measurement error with invalid validation data

We propose a new method for using validation data to correct self-reported weight and height in surveys that do not measure respondents. The standard correction in prior research regresses actual measures on reported values using an external validation dataset, and then uses the estimated coefficients to predict actual measures in the primary dataset. This approach requires the strong assumption that the expectations of measured weight and height conditional on the reported values are the same in both datasets. In contrast, we use percentile ranks rather than levels of reported weight and height. Our approach requires the weaker assumption that the conditional expectations of actual measures are increasing in reported values in both samples. This makes our correction more robust to differences in measurement error across surveys as long as both surveys represent the same population. We examine three nationally representative datasets and find that misreporting appears to be sensitive to differences in survey context. When we compare predicted BMI distributions using the two validation approaches, we find that the standard correction is affected by differences in misreporting while our correction is not. Finally, we present several examples that demonstrate the potential importance of our correction for future econometric analyses and estimates of obesity rates.