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.     

Efficacy of repeated measures in regression models with measurement error.

Ignoring measurement error may cause bias in the estimation of regression parameters. When the true covariates are unobservable, multiple imprecise measurements can be used in the analysis to correct for the associated bias. We suggest a simple estimating procedure that gives consistent estimates of regression parameters by using the repeated measurements with error. The relative Pitman efficiency of our estimator based on models with and without measurement error has been found to be a simple function of the number of replicates and the ratio of intra- to inter-variance of the true covariate. The procedure thus provides a guide for deciding the number of repeated measurements in the design stage. An example from a survey study is presented.     

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.     

Analysis of longitudinal data with unmeasured confounders.

Confounding in longitudinal or clustered data creates special problems and opportunities because the relationship between the confounder and covariate of interest may differ across and within individuals or clusters. A well-known example of such confounding in longitudinal data is the presence of cohort and period effects in models of aging in epidemiologic research. We first formulate a data-generating model with confounding and derive the distribution of the response variable unconditional on the confounder. We then examine the properties of the regression coefficient for some analytic approaches when the confounder is omitted from the fitted model. The expected value of the regression coefficient differs in across- and within-individual regression. In the multivariate case, within- and between-individual information is combined and weighted according to the assumed covariance structure. We assume compound symmetry in the fitted covariance matrix and derive the variance, bias, and mean squared error of the slope estimate as a function of the fitted within-individual correlation. We find that even in this simplest multivariate case, the trade-off between bias and variance depends on a large number of parameters. It is generally preferable to fit correlations somewhat above the true correlation to minimize the effect of between-individual confounders or cohort effects. Period effects can lead to situations where it is advantageous to fit correlations that are below the true correlation. The results highlight the trade-offs inherent in the choice of method for analysis of longitudinal data, and show that an appropriate choice can be made only after determining whether within- or between-individual confounding is the major concern.     

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.     

When should one adjust for measurement error in baseline variables in observational studies?

Previously, we showed that in randomised experiments, correction for measurement error in a baseline variable induces bias in the estimated treatment effect, and conversely that ignoring measurement error avoids bias. In observational studies, non-zero baseline covariate differences between treatment groups may be anticipated. Using a graphical approach, we argue intuitively that if baseline differences are large, failing to correct for measurement error leads to a biased estimate of the treatment effect. In contrast, correction eliminates bias if the true and observed baseline differences are equal. If this equality is not satisfied, the corrected estimator is also biased, but typically less so than the uncorrected estimator. Contrasting these findings, we conclude that there must be a threshold for the true baseline difference, above which correction is worthwhile. We derive expressions for the bias of the corrected and uncorrected estimators, as functions of the correlation of the baseline variable with the study outcome, its reliability, the true baseline difference, and the sample sizes. Comparison of these expressions defines a theoretical decision threshold about whether to correct for measurement error. The results show that correction is usually preferred in large studies, and also in small studies with moderate baseline differences. If the group sample sizes are very disparate, correction is less advantageous. If the equivalent balanced sample size is less than about 25 per group, one should correct for measurement error if the true baseline difference is expected to exceed 0.2-0.3 standard deviation units. These results are illustrated with data from a cohort study of atherosclerosis.     

Biases incurred from nonrandom repeat testing of haemoglobin levels in blood donors: Selective testing and its implications

To help prevent anaemia, it is a requisite for blood donors to undergo a haemoglobin test to ensure levels are not too low before donation. It is therefore important to have an accurate testing device and strategy to ensure donors are not being inappropriately bled. A recent study in blood donors used a selective testing strategy where if a donor's haemoglobin level is below the level required for donation, then another reading is taken and if this occurs again, a third and final reading is used. This strategy can reduce the average number of readings required per donor compared to taking three measurements for all donors. However, the final decision‐making measurement will on average be higher than a single measurement. In this paper, a selective testing strategy is compared against other strategies. Individual‐level biases are derived for the selective strategy and are shown to depend on how close a donor's true haemoglobin level is to the donation threshold and the magnitude of error in the testing device. A simulation study was conducted using the distribution of haemoglobin levels from a large donor population to investigate the effects different strategies have on population performance. We consider scenarios based on varying the measurement device bias and error, including differential biases that depend on the underlying haemoglobin level. Discriminatory performance is shown to be affected when using the selective testing strategies, especially when measurement error is large and when differential bias is present in the device. We recommend that the average of a number of readings should be used in preference to selective testing strategies if multiple measurements are available.     

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.     

Assessment of bias with emphasis on method comparison

* Definition of bias - distinct from accuracy, bias is an average deviation from a true value. * Method comparison - a set of specimens is assayed by both an existing method and the new candidate method, and the results compared. The following list describes the testing procedures and data handling required in a method comparison study for the assessment of bias: test material; number and disposition of specimens; summary of findings; the problem with correlation, and the difference plot; statistics of difference; log transformation of the difference plot; statistics of difference with logs; linear regression; Deming and Passing-Bablok models; the value of r in linear regression; choice of statistics; examples of suitable computer programs. * Acceptable bias criteria are discussed. * Linearity and recovery - failing either of these criteria should serve as a warning that method comparison data may conceal an unrecognised bias. * Finally, consideration of all steps in the assessment of bias is required to determine acceptability or not of the method comparison.     

Segmented regression in the presence of covariate measurement error in main study/validation study designs

This article considers the problem of segmented regression in the presence of covariate measurement error in main study/validation study designs. First, we derive a closed and interpretable form for the full likelihood. After that, we use the likelihood results to compute the bias of the estimated changepoint in the case when the measurement error is ignored. We find the direction of the bias in the estimated changepoint to be determined by the design distribution of the observed covariates, and the bias can be in either direction. We apply the methodology to data from a nutritional study that investigates the relation between dietary folate and blood serum homocysteine levels and find that the analysis that ignores covariate measurement error would have indicated a much higher minimum daily dietary folate intake requirement than is obtained in the analysis that takes covariate measurement error into account.     

THE EFFECT OF UNMEASURED CONFOUNDERS ON THE ABILITY TO ESTIMATE A TRUE PERFORMANCE OR SELECTION GRADIENT (AND OTHER PARTIAL REGRESSION COEFFICIENTS)

Multiple regression of observational data is frequently used to infer causal effects. Partial regression coefficients are biased estimates of causal effects if unmeasured confounders are not in the regression model. The sensitivity of partial regression coefficients to omitted confounders is investigated with a Monte‐Carlo simulation. A subset of causal traits is “measured” and their effects are estimated using ordinary least squares regression and compared to their expected values. Three major results are: (1) the error due to confounding is much larger than that due to sampling, especially with large samples, (2) confounding error shrinks trivially with sample size, and (3) small true effects are frequently estimated as large effects. Consequently, confidence intervals from regression are poor guides to the true intervals, especially with large sample sizes. The addition of a confounder to the model improves estimates only 55% of the time. Results are improved with complete knowledge of the rank order of causal effects but even with this omniscience, measured intervals are poor proxies for true intervals if there are many unmeasured confounders. The results suggest that only under very limited conditions can we have much confidence in the magnitude of partial regression coefficients as estimates of causal effects.     

Tree-Structured Exponential Regression Modeling

A method for fitting piecewise exponential regression models to censored survival data is described. Stratification is performed recursively, using a combination of statistical tests and residual analysis. The splitting criterion employed in cross-validation is the average squared error of the residuals. The bootstrap is employed to keep the probability of a type I error (the error of discovering two or more strata when there is only one) of the method close to a predetermined value. The proposed method can thus also serve as a formal goodness-of-fit test for the exponential regression model. Real and simulated data are used for illustration.     

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.     

Statistical estimation of gene expression using multiple laser scans of microarrays

We propose a statistical model for estimating gene expression using data from multiple laser scans at different settings of hybridized microarrays. A functional regression model is used, based on a non-linear relationship with both additive and multiplicative error terms. The function is derived as the expected value of a pixel, given that values are censored at 65 535, the maximum detectable intensity for double precision scanning software. Maximum likelihood estimation based on a Cauchy distribution is used to fit the model, which is able to estimate gene expressions taking account of outliers and the systematic bias caused by signal censoring of highly expressed genes. We have applied the method to experimental data. Simulation studies suggest that the model can estimate the true gene expression with negligible bias. AVAILABILITY: FORTRAN 90 code for implementing the method can be obtained from the authors.     

A robust method using propensity score stratification for correcting verification bias for binary tests

Sensitivity and specificity are common measures of the accuracy of a diagnostic test. The usual estimators of these quantities are unbiased if data on the diagnostic test result and the true disease status are obtained from all subjects in an appropriately selected sample. In some studies, verification of the true disease status is performed only for a subset of subjects, possibly depending on the result of the diagnostic test and other characteristics of the subjects. Estimators of sensitivity and specificity based on this subset of subjects are typically biased; this is known as verification bias. Methods have been proposed to correct verification bias under the assumption that the missing data on disease status are missing at random (MAR), that is, the probability of missingness depends on the true (missing) disease status only through the test result and observed covariate information. When some of the covariates are continuous, or the number of covariates is relatively large, the existing methods require parametric models for the probability of disease or the probability of verification (given the test result and covariates), and hence are subject to model misspecification. We propose a new method for correcting verification bias based on the propensity score, defined as the predicted probability of verification given the test result and observed covariates. This is estimated separately for those with positive and negative test results. The new method classifies the verified sample into several subsamples that have homogeneous propensity scores and allows correction for verification bias. Simulation studies demonstrate that the new estimators are more robust to model misspecification than existing methods, but still perform well when the models for the probability of disease and probability of verification are correctly specified.     

Estimating human long bone cross-sectional geometric properties: a comparison of noninvasive methods

Cross-sectional properties (areas, second moments of area) have been used extensively for reconstructing the mechanical loading history of long bone shafts. In the absence of a fortuitous break or available computed tomography (CT) facilities, the endosteal and/or periosteal boundaries of a bone may be approximated using alternative noninvasive methods. The present study tests whether cross-sectional geometric properties of human lower limb bones can be adequately estimated using two such techniques: the ellipse model method (EMM), which uses biplanar radiography alone, and the latex cast method (LCM), which involves molding of the subperiosteal contour in combination with biplanar radiography to estimate the contour of the medullary canal. Results of both methods are compared with "true" cross-sectional properties calculated by direct sectioning. The study sample includes matched femora and tibiae of 50 Pecos Pueblo Amerindians. Bone areas and second moments of area were calculated for the midshaft femur and tibia and proximal femoral diaphysis in each individual. Percent differences between methods were derived to evaluate directional (systematic) and absolute (random) error. Multiple regression was also used to investigate the sources of error associated with each method. The results indicate that while the LCM shows generally good correspondence to the true cross-sectional properties, the EMM generally overestimates true parameters. Regression equations are provided to correct this overestimation, and, when applied to another sample, are shown to significantly improve estimates for the femoral midshaft, although corrections are less successful for the other section locations. Our results suggest that the LCM is an adequate substitute for estimating cross-sectional properties when direct sectioning and CT are not feasible. The EMM is a reasonable alternative, although the bias inherent in the method should be corrected if possible, especially when the results of the study are to be compared with data collected using different methods.     

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.     

Effects of exposure misclassification on regression analyses of epidemiologic follow-up study data

In epidemiologic studies, subjects are often misclassified as to their level of exposure. Ignoring this misclassification error in the analysis introduces bias in the estimates of certain parameters and invalidates many hypothesis tests. For situations in which there is misclassification of exposure in a follow-up study with categorical data, we have developed a model that permits consideration of any number of exposure categories and any number of multiple-category covariates. When used with logistic and Poisson regression procedures, this model helps assess the potential for bias when misclassification is ignored. When reliable ancillary information is available, the model can be used to correct for misclassification bias in the estimates produced by these regression procedures.     

Unbiased estimation of relative reproductive success of different groups: evaluation and correction of bias caused by parentage assignment errors

Parentage assignment is widely applied to studies on mating systems, population dynamics and natural selection. However, little is known about the consequence of assignment errors, especially when some parents are not sampled. We investigated the effects of two types of error in parentage assignment, failing to assign a true parent (type A) and assigning an untrue parent (type B), on an estimate of the relative reproductive success (RRS) of two groups of parents. Employing a mathematical approach, we found that (i) when all parents are sampled, minimizing either type A or type B error insures the minimum bias on RRS, and (ii) when a large number of parents is not sampled, type B error substantially biases the estimated RRS towards one. Interestingly, however, (iii) when all parents were sampled and both error rates were moderately high, type A error biased the estimated RRS even more than type B error. We propose new methods to obtain an unbiased estimate of RRS and the number of offspring whose parents are not sampled (zW(z)), by correcting the error effects. Applying them to genotypic data from steelhead trout (Oncorhynchus mykiss), we illustrated how to estimate and control the assignment errors. In the data, we observed up to a 30% assignment error and a strong trade-off between the two types of error, depending on the stringency of the assignment decision criterion. We show that our methods can efficiently estimate an unbiased RRS and zW(z) regardless of assignment method, and how to maximize the statistical power to detect a difference in reproductive success between groups.     

On Estimating Stratified PH Model with Single Covariate from Sparse Data with Application to Brain Metastases Study

The estimation of the unknown parameters in the stratified Cox's proportional hazard model is a typical example of the trade‐off between bias and precision. The stratified partial likelihood estimator is unbiased when the number of strata is large but suffer from being unstable when many strata are non‐informative about the unknown parameters. The estimator obtained by ignoring the heterogeneity among strata, on the other hand, increases the precision of estimates although pays the price for being biased. An estimating procedure, based on the asymptotic properties of the above two estimators, serving to compromise between bias and precision is proposed. Two examples in a radiosurgery for brain metastases study provide some interesting demonstration of such applications.