Berkson Error Research Articles

Occupational exposure to nickel or hexavalent chromium and the risk of lung cancer (SYNERGY)

Abstract Background Hexavalent chromium (Cr(VI)) and nickel (Ni) are known occupational carcinogens, with frequent exposures, for example, in welding-related activities. However, limited evidence exists on the exposure-response relationship of Cr(VI) or Ni with lung-cancer risk. We estimated lung-cancer risks based on quantitative indices of occupational exposure to each metal, and their interaction with smoking habits. Methods Fourteen case-control studies from Europe and Canada (16,901 cases, 20,965 control subjects) were pooled. A measurement-based job-exposure matrix was used to estimate year- and region-specific exposure levels for Cr(VI) and Ni, which were linked to the study subjects’ occupational histories. Odds ratios (OR) and 95% confidence intervals (CI) were calculated, adjusting for study, age group, smoking, and exposure to other occupational lung carcinogens. Results The OR for the highest quartile (>99.5 μg/m3-years) of cumulative exposure to Cr(VI) in men was 1.32 (95% CI 1.19-1.47) and for nickel (highest quartile >78.1 μg/m3-years) OR = 1.29 (95% CI 1.15-1.45). Corresponding effect estimates in women were slightly lower. The joint effects of Cr(VI) and nickel with smoking were generally greater than additive. Conclusions We observed positive exposure-response relationships in the low-dose range of Cr(VI) and Ni exposures, based on a large study population including detailed information on smoking habits and a measurement-based job-exposure matrix. Although, we cannot rule out a combined classical measurement and Berkson error structure that may have caused bias in our risk estimates, our results provide empirical evidence to further specify preventive measures at the workplace in order to protect workers from hazardous substances. Key messages • Relatively low cumulative levels of occupational exposure to Cr(VI) and Ni were associated with increased ORs for lung cancer, particularly in men. • Important empirical insights for the prevention of cancer at the workplace can be drawn from the results.

Measurement error–tolerant Poisson regression for Valley Fever incidence prediction

The incidence of Valley Fever can be predicted using a Poisson regression model, which links the expected incidence count to influential covariates with model coefficients. In real-world practice, the estimated model coefficients are sensitive to the inevitable measurement errors in the covariates, which are usually manifested as classical errors and/or Berkson errors. While conventional methods have separately addressed each type of error, the joint consideration of both, referred to as mixture measurement error in this article, is currently absent from the existing literature, leaving model coefficient estimators with significant biases. This article presents a measurement-error tolerant Poisson regression to mitigate the impacts of the mixture measurement error in covariates of health-care data analysis. An error-structureadapted quasi-likelihood estimation method is proposed to enhance the accuracy of model estimation. The effectiveness of the proposed method is demonstrated through a numerical case study built upon a Valley Fever investigation, validating its improved estimation accuracy and robustness.

Dealing with area‐to‐point spatial misalignment in species distribution models

Species distribution models (SDMs) are extensively used to estimate species–environment relationships (SERs) and predict species distribution across space and time. For this purpose, it is key to choose relevant spatial grains for predictor and response variables at the onset of the modelling process. However, environmental variables are often derived from large‐scale climate models at a grain that can be coarser than the one of the response variable. Such area‐to‐point spatial misalignment can bias estimates of SER and jeopardise the robustness of predictions. We used a virtual species approach, running simulations across different levels of area‐to‐point spatial misalignment to seek statistical solutions to this problem. We specifically compared accuracy of SER estimates and predictive performances, assessed across different degrees of spatial heterogeneity in environmental conditions, of three SDMs: a GLM, a spatial GLM and a Berkson error model (BEM) that accounts for fine‐grain environmental heterogeneity within coarse‐grain cells. Only the BEM accurately estimates SER from relatively coarse‐grain environmental data (up to 50 times coarser than the response grain), while the two GLMs provide flattened SER. However, all three models perform poorly when predicting from coarse‐grain data, particularly in environments that are more heterogeneous than the training conditions. Conversely, decreasing environmental heterogeneity relative to the training dataset reduces the predictive biases. Because predictions are made from covariate‐grain data, the BEM displays lower predictive performance than the two GLMs. Thus, standard model selection methods would fail to select the model that best estimates SERs (here, the BEM), which could lead to false interpretations about the environmental drivers of species distributions. Overall, we conclude that the BEM, because it can robustly estimate SER at the response grain, holds great promise to overcome area‐to‐point misalignment.

A generalisation of the method of regression calibration and comparison with Bayesian and frequentist model averaging methods

For many cancer sites low-dose risks are not known and must be extrapolated from those observed in groups exposed at much higher levels of dose. Measurement error can substantially alter the dose–response shape and hence the extrapolated risk. Even in studies with direct measurement of low-dose exposures measurement error could be substantial in relation to the size of the dose estimates and thereby distort population risk estimates. Recently, there has been considerable attention paid to methods of dealing with shared errors, which are common in many datasets, and particularly important in occupational and environmental settings. In this paper we test Bayesian model averaging (BMA) and frequentist model averaging (FMA) methods, the first of these similar to the so-called Bayesian two-dimensional Monte Carlo (2DMC) method, and both fairly recently proposed, against a very newly proposed modification of the regression calibration method, the extended regression calibration (ERC) method, which is particularly suited to studies in which there is a substantial amount of shared error, and in which there may also be curvature in the true dose response. The quasi-2DMC with BMA method performs well when a linear model is assumed, but very poorly when a linear-quadratic model is assumed, with coverage probabilities both for the linear and quadratic dose coefficients that are under 5% when the magnitude of shared Berkson error is large (50%). For the linear model the bias is generally under 10%. However, using a linear-quadratic model it produces substantially biased (by a factor of 10) estimates of both the linear and quadratic coefficients, with the linear coefficient overestimated and the quadratic coefficient underestimated. FMA performs as well as quasi-2DMC with BMA when a linear model is assumed, and generally much better with a linear-quadratic model, although the coverage probability for the quadratic coefficient is uniformly too high. However both linear and quadratic coefficients have pronounced upward bias, particularly when Berkson error is large. By comparison ERC yields coverage probabilities that are too low when shared and unshared Berkson errors are both large (50%), although otherwise it performs well, and coverage is generally better than the quasi-2DMC with BMA or FMA methods, particularly for the linear-quadratic model. The bias of the predicted relative risk at a variety of doses is generally smallest for ERC, and largest for the quasi-2DMC with BMA and FMA methods (apart from unadjusted regression), with standard regression calibration and Monte Carlo maximum likelihood exhibiting bias in predicted relative risk generally somewhat intermediate between ERC and the other two methods. In general ERC performs best in the scenarios presented, and should be the method of choice in situations where there may be substantial shared error, or suspected curvature in the dose response.

Adjusting for Berkson error in exposure in ordinary and conditional logistic regression and in Poisson regression

BackgroundINTEROCC is a seven-country cohort study of occupational exposures and brain cancer risk, including occupational exposure to electromagnetic fields (EMF). In the absence of data on individual exposures, a Job Exposure Matrix (JEM) may be used to construct likely exposure scenarios in occupational settings. This tool was constructed using statistical summaries of exposure to EMF for various occupational categories for a comparable group of workers.MethodsIn this study, we use the Canadian data from INTEROCC to determine the best EMF exposure surrogate/estimate from three appropriately chosen surrogates from the JEM, along with a fourth surrogate based on Berkson error adjustments obtained via numerical approximation of the likelihood function. In this article, we examine the case in which exposures are gamma-distributed for each occupation in the JEM, as an alternative to the log-normal exposure distribution considered in a previous study conducted by our research team. We also study using those surrogates and the Berkson error adjustment in Poisson regression and conditional logistic regression.ResultsSimulations show that the introduced methods of Berkson error adjustment for non-stratified analyses provide accurate estimates of the risk of developing tumors in case of gamma exposure model. Alternatively, and under some technical assumptions, the arithmetic mean is the best surrogate when a gamma-distribution is used as an exposure model. Simulations also show that none of the present methods could provide an accurate estimate of the risk in case of stratified analyses.ConclusionWhile our previous study found the geometric mean to be the best exposure surrogate, the present study suggests that the best surrogate is dependent on the exposure model; the arithmetic means in case of gamma-exposure model and the geometric means in case of log-normal exposure model. However, we could present a better method of Berkson error adjustment for each of the two exposure models. Our results provide useful guidance on the application of JEMs for occupational exposure assessments, with adjustment for Berkson error.

A generalisation of the method of regression calibration

There is direct evidence of risks at moderate and high levels of radiation dose for highly radiogenic cancers such as leukaemia and thyroid cancer. For many cancer sites, however, it is necessary to assess risks via extrapolation from groups exposed at moderate and high levels of dose, about which there are substantial uncertainties. Crucial to the resolution of this area of uncertainty is the modelling of the dose–response relationship and the importance of both systematic and random dosimetric errors for analyses in the various exposed groups. It is well recognised that measurement error can alter substantially the shape of this relationship and hence the derived population risk estimates. Particular attention has been devoted to the issue of shared errors, common in many datasets, and particularly important in occupational settings. We propose a modification of the regression calibration method which is particularly suited to studies in which there is a substantial amount of shared error, and in which there may also be curvature in the true dose response. This method can be used in settings where there is a mixture of Berkson and classical error. In fits to synthetic datasets in which there is substantial upward curvature in the true dose response, and varying (and sometimes substantial) amounts of classical and Berkson error, we show that the coverage probabilities of all methods for the linear coefficient alpha are near the desired level, irrespective of the magnitudes of assumed Berkson and classical error, whether shared or unshared. However, the coverage probabilities for the quadratic coefficient beta are generally too low for the unadjusted and regression calibration methods, particularly for larger magnitudes of the Berkson error, whether this is shared or unshared. In contrast Monte Carlo maximum likelihood yields coverage probabilities for beta that are uniformly too high. The extended regression calibration method yields coverage probabilities that are too low when shared and unshared Berkson errors are both large, although otherwise it performs well, and coverage is generally better than these other three methods. A notable feature is that for all methods apart from extended regression calibration the estimates of the quadratic coefficient beta are substantially upwardly biased.

Estimation in Linear-Rate Simple Survival Models with Measurement Errors and Censoring

A simple exponential regression model is considered where the rate parameter of the response variable linearly depends on the explanatory variable. We consider complications of the model: censoring of the response variable (either upper censoring or interval observations), the additive classical error or multiplicative Berkson error in the explanatory variable, or a combination of censoring with Berkson errors. We construct or use already-known estimators in the models, and verify their performance in simulations.

Issues in Implementing Regression Calibration Analyses.

Regression calibration is a popular approach for correcting biases in estimated regression parameters when exposure variables are measured with error. This approach involves building a calibration equation to estimate the value of the unknown true exposure given the error-prone measurement and other covariates. The estimated, or calibrated, exposure is then substituted for the unknown true exposure in the health outcome regression model. When used properly, regression calibration can greatly reduce the bias induced by exposure measurement error. Here, we first provide an overview of the statistical framework for regression calibration, specifically discussing how a special type of error, called Berkson error, arises in the estimated exposure. We then present practical issues to consider when applying regression calibration, including: 1) how to develop the calibration equation and which covariates to include; 2) valid ways to calculate standard errors of estimated regression coefficients; and 3) problems arising if one of the covariates in the calibration model is a mediator of the relationship between the exposure and outcome. Throughout, we provide illustrative examples using data from the Hispanic Community Health Study/Study of Latinos (United States, 2008-2011) and simulations. We conclude with recommendations for how to perform regression calibration.

Generalized measurement error: Intrinsic and incidental measurement error.

In this paper, we generalize the notion of measurement error on deterministic sample datasets to accommodate sample data that are random-variable-valued. This leads to the formulation of two distinct kinds of measurement error: intrinsic measurement error, and incidental measurement error. Incidental measurement error will be recognized as the traditional kind that arises from a set of deterministic sample measurements, and upon which the traditional measurement error modelling literature is based, while intrinsic measurement error reflects some subjective quality of either the measurement tool or the measurand itself. We define calibrating conditions that generalize common and classical types of measurement error models to this broader measurement domain, and explain how the notion of generalized Berkson error in particular mathematicizes what it means to be an expert assessor or rater for a measurement process. We then explore how classical point estimation, inference, and likelihood theory can be generalized to accommodate sample data composed of generic random-variable-valued measurements.

Using SIMEX and regression calibration to correct for mixtures of classical and Berkson measurement error

BACKGROUND AND AIM Measurement error in modelled air pollution data is complex consisting of classical and Berkson error components. As part of the MELONS project, we investigated the adequacy of regression calibration and SIMEX in correcting health effect estimates when measurement error contains both components and may also be spatially correlated. METHODS In a simulation study of 5,000 subjects, we considered scenarios based on the amount of total measurement error (either 20% or 40% of the variance in the true annual average pollutant concentrations); the proportion of error that was classical (0%,20%,40%,60%, 80% or 100%); and whether total measurement error was homoscedastic or heteroscedastic. Within each simulation, we obtained a dataset of true, and error prone annual average pollutant concentrations and outcome data linked to the true by a hazard ratio of 1.041 per 10µg/m³. The error prone pollutant concentrations were then used as the exposure variable in a Cox regression, and for error correction, a 5% hypothetical validation sub-sample of the data were randomly selected and used to estimate the classical and Berkson error variances. Based on 1,600 simulations per scenario we investigated mean percent bias in the estimated log hazard ratio and the ability of regression calibration and SIMEX to correct for this bias. RESULTS Mean percent bias in the uncorrected log hazard ratio was typically negative (i.e., towards the null) ranging from -27.4% to +2.0% across scenarios. Mean percent bias in the corrected estimates ranged from -4.0% to +5.7% for regression calibration and -6.0% to +4.2% for SIMEX. CONCLUSIONS Across a range of scenarios varying the mix of classical and Berkson error and allowing total measurement error to vary spatially, SIMEX and regression calibration produced corrected log hazard ratios with relatively small mean percent bias. KEYWORDS Measurement Error, SIMEX, Regression Calibration

Estimation of a Heterogeneous Demand Function with Berkson Errors

Abstract Berkson errors are commonplace in empirical microeconomics. In consumer demand, this form of measurement error occurs when the price an individual pays is measured by the (weighted) average price paid by individuals in a group (e.g., a county) rather than the true transaction price. We show the importance of Berkson errors for demand estimation with nonseparable unobserved heterogeneity. We develop a consistent estimator using external information on the true price distribution. Examining gasoline demand in the United States, we document substantial within-market price variability. Accounting for Berkson errors is quantitatively important. Imposing the Slutsky shape constraint reduces sensitivity to Berkson errors.

The Impact of Exposure Measurement Error on the Estimated Concentration-Response Relationship between Long-Term Exposure to and Mortality.

Background:Exposure measurement error is a central concern in air pollution epidemiology. Given that studies have been using ambient air pollution predictions as proxy exposure measures, the potential impact of exposure error on health effect estimates needs to be comprehensively assessed.Objectives:We aimed to generate wide-ranging scenarios to assess direction and magnitude of bias caused by exposure errors under plausible concentration–response relationships between annual exposure to fine particulate matter [PM in aerodynamic diameter ()] and all-cause mortality.Methods:In this simulation study, we use daily predictions at spatial resolution to estimate annual exposures and their uncertainties for ZIP Codes of residence across the contiguous United States between 2000 and 2016. We consider scenarios in which we vary the error type (classical or Berkson) and the true concentration–response relationship between exposure and mortality (linear, quadratic, or soft-threshold—i.e., a smooth approximation to the hard-threshold model). In each scenario, we generate numbers of deaths using error-free exposures and confounders of concurrent air pollutants and neighborhood-level covariates and perform epidemiological analyses using error-prone exposures under correct specification or misspecification of the concentration–response relationship between exposure and mortality, adjusting for the confounders.Results:We simulate 1,000 replicates of each of 162 scenarios investigated. In general, both classical and Berkson errors can bias the concentration–response curve toward the null. The biases remain small even when using three times the predicted uncertainty to generate errors and are relatively larger at higher exposure levels.Discussion:Our findings suggest that the causal determination for long-term exposure and mortality is unlikely to be undermined when using high-resolution ambient predictions given that the estimated effect is generally smaller than the truth. The small magnitude of bias suggests that epidemiological findings are relatively robust against the exposure error. In practice, the use of ambient predictions with a finer spatial resolution will result in smaller bias. https://doi.org/10.1289/EHP10389

Nonparametric estimation of cumulative distribution function from noisy data in the presence of Berkson and classical errors

Nonparametric estimation of cumulative distribution function from noisy data in the presence of Berkson and classical errors

Estimation in a linear errors-in-variables model under a mixture of classical and Berkson errors

A linear structural regression model is studied, where the covariate is observed with a mixture of the classical and Berkson measurement errors. Both variances of the classical and Berkson errors are assumed known. Without normality assumptions, consistent estimators of model parameters are constructed and conditions for their asymptotic normality are given. The estimators are divided into two asymptotically independent groups.

A monthly spatio-temporal modelling of patterns for children on maintenance grant in Namibia

BACKGROUND: Child support grant (CSG) is one of the social protection strategies which is today widely seen as an intervention contributing to poverty reduction. However, despite substantial expenses, it has been documented that Namibia’s social protection benefits do not reach intended beneficiaries in an efficient manner. AIMS: This study aims to determine factors associated with spatial and temporal variation in maintenance child grant and as well as identify regions with elevated incidence rate ratios of maintenance grant in Namibia. METHODS: We fitted a Bayesian spatio-temporal regression model on maintenance grant data available over 9 years in Namibia. RESULTS: The number of children on grant has almost tripled between March 2007 and October 2015 (it changed from 50596 in 2007 to 132840). Unemployment and orphanhood were significantly associated with the incidence rate ratio of maintenance grant (CI:(1.634, 2.627) and CI: (1.000, 1.004), respectively. The adjustment of measurement error in orphanhood through the Berkson error model has ensured the stability of its effect. CONCLUSIONS: This study has shown the strength of using measurement error models for analysing child grant data. Furthermore, the study has demonstrated that the northern regions of Namibia have the highest child incidence rate ratio of maintenance grant whereas the regions in central and south are at low incidence rate at present. The maps produced in this study can be particularly helpful in allocating efficiently limited resources in poor settings.

Simultaneous inference of the partially linear model with a multivariate unknown function

In this paper, we conduct simultaneous inference of the non-parametric part of a partially linear model when the non-parametric component is a multivariate unknown function. Based on semi-parametric estimates of the model, we construct a simultaneous confidence region of the multivariate function for simultaneous inference. The developed methodology is applied to perform simultaneous inference for the U.S. gasoline demand where the income and price variables are contaminated by Berkson errors. The empirical results strongly suggest that the linearity of the U.S. gasoline demand is rejected. The results are also used to propose an alternative form for the demand.

STRATOS guidance document on measurement error and misclassification of variables in observational epidemiology: Part 2-More complex methods of adjustment and advanced topics.

We continue our review of issues related to measurement error and misclassification in epidemiology. We further describe methods of adjusting for biased estimation caused by measurement error in continuous covariates, covering likelihood methods, Bayesian methods, moment reconstruction, moment-adjusted imputation, and multiple imputation. We then describe which methods can also be used with misclassification of categorical covariates. Methods of adjusting estimation of distributions of continuous variables for measurement error are then reviewed. Illustrative examples are provided throughout these sections. We provide lists of available software for implementing these methods and also provide the code for implementing our examples in the Supporting Information. Next, we present several advanced topics, including data subject to both classical and Berkson error, modeling continuous exposures with measurement error, and categorical exposures with misclassification in the same model, variable selection when some of the variables are measured with error, adjusting analyses or design for error in an outcome variable, and categorizing continuous variables measured with error. Finally, we provide some advice for the often met situations where variables are known to be measured with substantial error, but there is only an external reference standard or partial (or no) information about the type or magnitude of the error.

Bias due to Berkson error: issues when using predicted values in place of observed covariates.

Studies often want to test for the association between an unmeasured covariate and an outcome. In the absence of a measurement, the study may substitute values generated from a prediction model. Justification for such methods can be found by noting that, with standard assumptions, this is equivalent to fitting a regression model for an outcome variable when at least one covariate is measured with Berkson error. Under this setting, it is known that consistent or nearly consistent inference can be obtained under many linear and nonlinear outcome models. In this article, we focus on the linear regression outcome model and show that this consistency property does not hold when there is unmeasured confounding in the outcome model, in which case the marginal inference based on a covariate measured with Berkson error differs from the same inference based on observed covariates. Since unmeasured confounding is ubiquitous in applications, this severely limits the practical use of such measurements, and, in particular, the substitution of predicted values for observed covariates. These issues are illustrated using data from the National Health and Nutrition Examination Survey to study the joint association of total percent body fat and body mass index with HbA1c. It is shown that using predicted total percent body fat in place of observed percent body fat yields inferences which often differ significantly, in some cases suggesting opposite relationships among covariates.

Simultaneous Inference of the Partially Linear Model with a Multivariate Unknown Function

In this paper, we conduct simultaneous inference of the non-parametric part of a partially linear model when the non-parametric component is a multivariate unknown function. Based on semi-parametric estimates of the model, we construct a simultaneous confidence region of the multivariate function for simultaneous inference. The developed methodology is applied to perform simultaneous inference for the U.S. gasoline demand where the income and price variables are contaminated by Berkson errors. The empirical results strongly suggest that the linearity of the U.S. gasoline demand is rejected. The results are also used to propose an alternative form for the demand.

Density Deconvolution with Small Berkson Errors

The present paper studies density deconvolution in the presence of small Berkson errors, in particular, when the variances of the errors tend to zero as the sample size grows. It is known that when the Berkson errors are present, in some cases, the unknown density estimator can be obtained by simple averaging without using kernels. However, this may not be the case when Berkson errors are asymptotically small. By treating the former case as a kernel estimator with the zero bandwidth, we obtain the optimal expressions for the bandwidth. We show that the density of Berkson errors acts as a regularizer, so that the kernel estimator is unnecessary when the variance of Berkson errors lies above some threshold that depends on the shapes of the densities in the model and the number of observations.