On latent-variable model misspecification in structural measurement error models for binary response

Summary .  We consider structural measurement error models for a binary response. We show that likelihood-based estimators obtained from fitting structural measurement error models with pooled binary responses can be far more robust to covariate measurement error in the presence of latent-variable model misspecification than the corresponding estimators from individual responses. Furthermore, despite the loss in information, pooling can provide improved parameter estimators in terms of mean-squared error. Based on these and other findings, we create a new diagnostic method to detect latent-variable model misspecification in structural measurement error models with individual binary response. We use simulation and data from the Framingham Heart Study to illustrate our methods.     

SIMEX variance component tests in generalized linear mixed measurement error models

In the analysis of clustered data with covariates measured with error, a problem of common interest is to test for correlation within clusters and heterogeneity across clusters. We examined this problem in the framework of generalized linear mixed measurement error models. We propose using the simulation extrapolation (SIMEX) method to construct a score test for the null hypothesis that all variance components are zero. A key feature of this SIMEX score test is that no assumptions need to be made regarding the distributions of the random effects and the unobserved covariates. We illustrate this test by analyzing Framingham heart disease data and evaluate its performance by simulation. We also propose individual SIMEX score tests for testing the variance components separately. Both tests can be easily implemented using existing statistical software.     

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.     

Control for confounding in the presence of measurement error in hierarchical models

Hierarchical modeling is becoming increasingly popular in epidemiology, particularly in air pollution studies. When potential confounding exists, a multilevel model yields better power to assess the independent effects of each predictor by gathering evidence across many sub-studies. If the predictors are measured with unknown error, bias can be expected in the individual substudies, and in the combined estimates of the second-stage model. We consider two alternative methods for estimating the independent effects of two predictors in a hierarchical model. We show both analytically and via simulation that one of these gives essentially unbiased estimates even in the presence of measurement error, at the price of a moderate reduction in power. The second avoids the potential for upward bias, at the price of a smaller reduction in power. Since measurement error is endemic in epidemiology, these approaches hold considerable potential. We illustrate the two methods by applying them to two air pollution studies. In the first, we re-analyze published data to show that the estimated effect of fine particles on daily deaths, independent of coarse particles, was downwardly biased by measurement error in the original analysis. The estimated effect of coarse particles becomes more protective using the new estimates. In the second example, we use published data on the association between airborne particles and daily deaths in 10 US cities to estimate the effect of gaseous air pollutants on daily deaths. The resulting effect size estimates were very small and the confidence intervals included zero.     

Predicting and correcting bias caused by measurement error in line transect sampling using multiplicative error models

Line transect sampling is one of the most widely used methods for animal abundance assessment. Standard estimation methods assume certain detection on the transect, no animal movement, and no measurement errors. Failure of the assumptions can cause substantial bias. In this work, the effect of error measurement on line transect estimators is investigated. Based on considerations of the process generating the errors, a multiplicative error model is presented and a simple way of correcting estimates based on knowledge of the error distribution is proposed. Using beta models for the error distribution, the effect of errors and of the proposed correction is assessed by simulation. Adequate confidence intervals for the corrected estimates are obtained using a bootstrap variance estimate for the correction and the delta method. As noted by Chen (1998, Biometrics 54, 899-908), even unbiased estimators of the distances might lead to biased density estimators, depending on the actual error distribution. In contrast with the findings of Chen, who used an additive model, unbiased estimation of distances, given a multiplicative model, lead to overestimation of density. Some error distributions result in observed distance distributions that make efficient estimation impossible, by removing the shoulder present in the original detection function. This indicates the need to improve field methods to reduce measurement error. An application of the new methods to a real data set is presented.     

Humidity as a source of measurement error in osteometrics

A simple experiment was conducted to test the hypothesis that variation in humidity causes expansion of bone and, thereby, affects measurements of dried, preserved skulls. The experiment consisted of subjecting ten macaque skulls to increased humidity for 24 hours. Measurements of nine skull dimensions taken immediately before and after humidification revealed a statistically significant treatment effect of increased skull size with increased humidity. The length of the molar tooth row increased by about 0.1 mm (0.50%) while the greatest length of the skull increased by about 0.9 mm (0.57%). The specimens returned to their original dimensions within 1-2 days after being removed from the humidity chamber. These results confirm the impression gained by the practical experience of measuring museum specimens in different locations and environments. It appears that bony changes associated with humidity differences represent a real, though minor, source of measurement error in osteometrics.     

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.     

On corrected score approach for proportional hazards model with covariate measurement error

In the presence of covariate measurement error with the proportional hazards model, several functional modeling methods have been proposed. These include the conditional score estimator (Tsiatis and Davidian, 2001, Biometrika 88, 447-458), the parametric correction estimator (Nakamura, 1992, Biometrics 48, 829-838), and the nonparametric correction estimator (Huang and Wang, 2000, Journal of the American Statistical Association 95, 1209-1219) in the order of weaker assumptions on the error. Although they are all consistent, each suffers from potential difficulties with small samples and substantial measurement error. In this article, upon noting that the conditional score and parametric correction estimators are asymptotically equivalent in the case of normal error, we investigate their relative finite sample performance and discover that the former is superior. This finding motivates a general refinement approach to parametric and nonparametric correction methods. The refined correction estimators are asymptotically equivalent to their standard counterparts, but have improved numerical properties and perform better when the standard estimates do not exist or are outliers. Simulation results and application to an HIV clinical trial are presented.     

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.     

The Youden Index and the optimal cut-point corrected for measurement error

Random measurement error can attenuate a biomarker's ability to discriminate between diseased and non-diseased populations. A global measure of biomarker effectiveness is the Youden index, the maximum difference between sensitivity, the probability of correctly classifying diseased individuals, and 1-specificity, the probability of incorrectly classifying health individuals. We present an approach for estimating the Youden index and associated optimal cut-point for a normally distributed biomarker that corrects for normally distributed random measurement error. We also provide confidence intervals for these corrected estimates using the delta method and coverage probability through simulation over a variety of situations. Applying these techniques to the biomarker thiobarbituric acid reaction substance (TBARS), a measure of sub-products of lipid peroxidation that has been proposed as a discriminating measurement for cardiovascular disease, yields a 50% increase in diagnostic effectiveness at the optimal cut-point. This result may lead to biomarkers that were once naively considered ineffective becoming useful diagnostic devices.     

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.     

Multistage evaluation of measurement error in a reliability study

We introduce sequential testing procedures for the planning and analysis of reliability studies to assess an exposure's measurement error. The designs allow repeated evaluation of reliability of the measurements and stop testing if early evidence shows the measurement error is within the level of tolerance. Methods are developed and critical values tabulated for a number of two-stage designs. The methods are exemplified using an example evaluating the reliability of biomarkers associated with oxidative stress.