Regression calibration with more surrogates than mismeasured variables

In a recent paper (Weller EA, Milton DK, Eisen EA, Spiegelman D. Regression calibration for logistic regression with multiple surrogates for one exposure. Journal of Statistical Planning and Inference 2007; 137: 449‐461), the authors discussed fitting logistic regression models when a scalar main explanatory variable is measured with error by several surrogates, that is, a situation with more surrogates than variables measured with error. They compared two methods of adjusting for measurement error using a regression calibration approximate model as if it were exact. One is the standard regression calibration approach consisting of substituting an estimated conditional expectation of the true covariate given observed data in the logistic regression. The other is a novel two‐stage approach when the logistic regression is fitted to multiple surrogates, and then a linear combination of estimated slopes is formed as the estimate of interest. Applying estimated asymptotic variances for both methods in a single data set with some sensitivity analysis, the authors asserted superiority of their two‐stage approach. We investigate this claim in some detail. A troubling aspect of the proposed two‐stage method is that, unlike standard regression calibration and a natural form of maximum likelihood, the resulting estimates are not invariant to reparameterization of nuisance parameters in the model. We show, however, that, under the regression calibration approximation, the two‐stage method is asymptotically equivalent to a maximum likelihood formulation, and is therefore in theory superior to standard regression calibration. However, our extensive finite‐sample simulations in the practically important parameter space where the regression calibration model provides a good approximation failed to uncover such superiority of the two‐stage method. We also discuss extensions to different data structures. Copyright © 2012 John Wiley & Sons, Ltd.     

Some aspects of measurement error in explanatory variables for continuous and binary regression models

A simple form of measurement error model for explanatory variables is studied incorporating classical and Berkson cases as particular forms, and allowing for either additive or multiplicative errors. The work is motivated by epidemiological problems, and therefore consideration is given not only to continuous response variables but also to logistic regression models. The possibility that different individuals in a study have errors of different types is also considered. The relatively simple estimation procedures proposed for use with cohort data and case-control data are checked by simulation, under the assumption of various error structures. The results show that even in situations where conventional analysis yields slope estimates that are on average attenuated by a factor of approximately 50 per cent, estimates obtained using the proposed amended likelihood functions are within 5 per cent of their true values. The work was carried out to provide a method for the analysis of lung cancer risk following residential radon exposure, but it should be applicable to a wide variety of situations. © 1998 John Wiley & Sons, Ltd.     

A conditional likelihood approach for regression analysis using biomarkers measured with batch‐specific error

Measurement error is common in epidemiological and biomedical studies. When biomarkers are measured in batches or groups, measurement error is potentially correlated within each batch or group. In regression analysis, most existing methods are not applicable in the presence of batch‐specific measurement error in predictors. We propose a robust conditional likelihood approach to account for batch‐specific error in predictors when batch effect is additive and the predominant source of error, which requires no assumptions on the distribution of measurement error. Although a regression model with batch as a categorical covariable yields the same parameter estimates as the proposed conditional likelihood approach for linear regression, this result does not hold in general for all generalized linear models, in particular, logistic regression. Our simulation studies show that the conditional likelihood approach achieves better finite sample performance than the regression calibration approach or a naive approach without adjustment for measurement error. In the case of logistic regression, our proposed approach is shown to also outperform the regression approach with batch as a categorical covariate. In addition, we also examine a ‘hybrid’ approach combining the conditional likelihood method and the regression calibration method, which is shown in simulations to achieve good performance in the presence of both batch‐specific and measurement‐specific errors. We illustrate our method by using data from a colorectal adenoma study. Copyright © 2012 John Wiley & Sons, Ltd.     

Residential radon and risk of lung cancer: a combined analysis of 7 North American case-control studies

BACKGROUND: Underground miners exposed to high levels of radon have an excess risk of lung cancer. Residential exposure to radon is at much lower levels, and the risk of lung cancer with residential exposure is less clear. We conducted a systematic analysis of pooled data from all North American residential radon studies. METHODS: The pooling project included original data from 7 North American case-control studies, all of which used long-term alpha-track detectors to assess residential radon concentrations. A total of 3662 cases and 4966 controls were retained for the analysis. We used conditional likelihood regression to estimate the excess risk of lung cancer. RESULTS: Odds ratios (ORs) for lung cancer increased with residential radon concentration. The estimated OR after exposure to radon at a concentration of 100 Bq/m3 in the exposure time window 5 to 30 years before the index date was 1.11 (95% confidence interval = 1.00-1.28). This estimate is compatible with the estimate of 1.12 (1.02-1.25) predicted by downward extrapolation of the miner data. There was no evidence of heterogeneity of radon effects across studies. There was no apparent heterogeneity in the association by sex, educational level, type of respondent (proxy or self), or cigarette smoking, although there was some evidence of a decreasing radon-associated lung cancer risk with age. Analyses restricted to subsets of the data with presumed more accurate radon dosimetry resulted in increased estimates of risk. CONCLUSIONS: These results provide direct evidence of an association between residential radon and lung cancer risk, a finding predicted using miner data and consistent with results from animal and in vitro studies.     

Mediation analysis when a continuous mediator is measured with error and the outcome follows a generalized linear model

Mediation analysis is a popular approach to examine the extent to which the effect of an exposure on an outcome is through an intermediate variable (mediator) and the extent to which the effect is direct. When the mediator is mis‐measured, the validity of mediation analysis can be severely undermined. In this paper, we first study the bias of classical, non‐differential measurement error on a continuous mediator in the estimation of direct and indirect causal effects in generalized linear models when the outcome is either continuous or discrete and exposure–mediator interaction may be present. Our theoretical results as well as a numerical study demonstrate that in the presence of non‐linearities, the bias of naive estimators for direct and indirect effects that ignore measurement error can take unintuitive directions. We then develop methods to correct for measurement error. Three correction approaches using method of moments, regression calibration, and SIMEX are compared. We apply the proposed method to the Massachusetts General Hospital lung cancer study to evaluate the effect of genetic variants mediated through smoking on lung cancer risk. Copyright © 2014 John Wiley & Sons, Ltd.     

A permutation test for inference in logistic regression with small- and moderate-sized data sets

Inference based on large sample results can be highly inaccurate if applied to logistic regression with small data sets. Furthermore, maximum likelihood estimates for the regression parameters will on occasion not exist, and large sample results will be invalid. Exact conditional logistic regression is an alternative that can be used whether or not maximum likelihood estimates exist, but can be overly conservative. This approach also requires grouping the values of continuous variables corresponding to nuisance parameters, and inference can depend on how this is done. A simple permutation test of the hypothesis that a regression parameter is zero can overcome these limitations. The variable of interest is replaced by the residuals from a linear regression of it on all other independent variables. Logistic regressions are then done for permutations of these residuals, and a p-value is computed by comparing the resulting likelihood ratio statistics to the original observed value. Simulations of binary outcome data with two independent variables that have binary or lognormal distributions yield the following results: (a) in small data sets consisting of 20 observations, type I error is well-controlled by the permutation test, but poorly controlled by the asymptotic likelihood ratio test; (b) in large data sets consisting of 1000 observations, performance of the permutation test appears equivalent to that of the asymptotic test; and (c) in small data sets, the p-value for the permutation test is usually similar to the mid-p-value for exact conditional logistic regression.     

Correcting for multivariate measurement error by regression calibration in meta‐analyses of epidemiological studies

Within‐person variability in measured values of multiple risk factors can bias their associations with disease. The multivariate regression calibration (RC) approach can correct for such measurement error and has been applied to studies in which true values or independent repeat measurements of the risk factors are observed on a subsample. We extend the multivariate RC techniques to a meta‐analysis framework where multiple studies provide independent repeat measurements and information on disease outcome. We consider the cases where some or all studies have repeat measurements, and compare study‐specific, averaged and empirical Bayes estimates of RC parameters. Additionally, we allow for binary covariates (e.g. smoking status) and for uncertainty and time trends in the measurement error corrections. Our methods are illustrated using a subset of individual participant data from prospective long‐term studies in the Fibrinogen Studies Collaboration to assess the relationship between usual levels of plasma fibrinogen and the risk of coronary heart disease, allowing for measurement error in plasma fibrinogen and several confounders. Copyright © 2009 John Wiley & Sons, Ltd.     

Evaluating model‐based imputation methods for missing covariates in regression models with interactions

Imputation strategies are widely used in settings that involve inference with incomplete data. However, implementation of a particular approach always rests on assumptions, and subtle distinctions between methods can have an impact on subsequent analyses. In this research article, we are concerned with regression models in which the true underlying relationship includes interaction terms. We focus in particular on a linear model with one fully observed continuous predictor, a second partially observed continuous predictor, and their interaction. We derive the conditional distribution of the missing covariate and interaction term given the observed covariate and the outcome variable, and examine the performance of a multiple imputation procedure based on this distribution. We also investigate several alternative procedures that can be implemented by adapting multivariate normal multiple imputation software in ways that might be expected to perform well despite incompatibilities between model assumptions and true underlying relationships among the variables. The methods are compared in terms of bias, coverage, and CI width. As expected, the procedure based on the correct conditional distribution performs well across all scenarios. Just as importantly for general practitioners, several of the approaches based on multivariate normality perform comparably with the correct conditional distribution in a number of circumstances, although interestingly, procedures that seek to preserve the multiplicative relationship between the interaction term and the main‐effects are found to be substantially less reliable. For illustration, the various procedures are applied to an analysis of post‐traumatic stress disorder symptoms in a study of childhood trauma. Copyright © 2015 John Wiley & Sons, Ltd.     

Comparison of alternative regression models for paired binary data

We used simulated data, derived from real ophthalmologic examples, to evaluate the performance of alternative logistic regression approaches for paired binary data. Approaches considered were: standard logistic regression (ignoring the correlation between fellow eyes, treating individuals classified on the basis of their more impaired eye as the unit of analysis, or considering only right eyes); marginal logistic regression models fitted by the maximum likelihood approach of Lipsitz, Laird and Harrington or the estimating equation approach of Liang and Zeger; and conditional logistic regression models fitted by the maximum likelihood approach of Rosner or the estimating equation approach of Connolly and Liang. Taylor series approximations were used to compare conditional and marginal parameter estimates. Consideration of type I and II error rates found application of standard logistic regression to be inferior to methods that treated the eye as the unit of analysis and accounted for the correlation between fellow eyes. Among these latter approaches, none was uniformly superior to the others across the range of conditions considered.     

Correction of logistic regression relative risk estimates and confidence intervals for systematic within-person measurement error

Errors in the measurement of exposure that are independent of disease status tend to bias relative risk estimates and other measures of effect in epidemiologic studies toward the null value. Two methods are provided to correct relative risk estimates obtained from logistic regression models for measurement errors in continuous exposures within cohort studies that may be due to either random (unbiased) within-person variation or to systematic errors for individual subjects. These methods require a separate validation study to estimate the regression coefficient lambda relating the surrogate measure to true exposure. In the linear approximation method, the true logistic regression coefficient beta* is estimated by beta/lambda, where beta is the observed logistic regression coefficient based on the surrogate measure. In the likelihood approximation method, a second-order Taylor series expansion is used to approximate the logistic function, enabling closed-form likelihood estimation of beta*. Confidence intervals for the corrected relative risks are provided that include a component representing error in the estimation of lambda. Based on simulation studies, both methods perform well for true odds ratios up to 3.0; for higher odds ratios the likelihood approximation method was superior with respect to both bias and coverage probability. An example is provided based on data from a prospective study of dietary fat intake and risk of breast cancer and a validation study of the questionnaire used to assess dietary fat intake.     

Linear mixed models for replication data to efficiently allow for covariate measurement error

It is well known that measurement error in the covariates of regression models generally causes bias in parameter estimates. Correction for such biases requires information concerning the measurement error, which is often in the form of internal validation or replication data. Regression calibration (RC) is a popular approach to correct for covariate measurement error, which involves predicting the true covariate using error‐prone measurements. Likelihood methods have previously been proposed as an alternative approach to estimate the parameters in models affected by measurement error, but have been relatively infrequently employed in medical statistics and epidemiology, partly because of computational complexity and concerns regarding robustness to distributional assumptions. We show how a standard random‐intercepts model can be used to obtain maximum likelihood (ML) estimates when the outcome model is linear or logistic regression under certain normality assumptions, when internal error‐prone replicate measurements are available. Through simulations we show that for linear regression, ML gives more efficient estimates than RC, although the gain is typically small. Furthermore, we show that RC and ML estimates remain consistent even when the normality assumptions are violated. For logistic regression, our implementation of ML is consistent if the true covariate is conditionally normal given the outcome, in contrast to RC. In simulations, this ML estimator showed less bias in situations where RC gives non‐negligible biases. Our proposal makes the ML approach to dealing with covariate measurement error more accessible to researchers, which we hope will improve its viability as a useful alternative to methods such as RC. Copyright © 2009 John Wiley & Sons, Ltd.     

A bayesian approach to logistic regression models having measurement error following a mixture distribution

To estimate the parameters in a logistic regression model when the predictors are subject to random or systematic measurement error, we take a Bayesian approach and average the true logistic probability over the conditional posterior distribution of the true value of the predictor given its observed value. We allow this posterior distribution to consist of a mixture when the measurement error distribution changes form with observed exposure. We apply the method to study the risk of alcohol consumption on breast cancer using the Nurses Health Study data. We estimate measurement error from a small subsample where we compare true with reported consumption. Some of the self-reported non-drinkers truly do not drink. The resulting risk estimates differ sharply from those computed by standard logistic regression that ignores measurement error.     

Estimating and testing interactions in linear regression models when explanatory variables are subject to classical measurement error

Estimating and testing interactions in a linear regression model when normally distributed explanatory variables are subject to classical measurement error is complex, since the interaction term is a product of two variables and involves errors of more complex structure. Our aim is to develop simple methods, based on the method of moments (MM) and regression calibration (RC) that yield consistent estimators of the regression coefficients and their standard errors when the model includes one or more interactions. In contrast to previous work using structural equations models framework, our methods allow errors that are correlated with each other and can deal with measurements of relatively low reliability. Using simulations, we show that, under the normality assumptions, the RC method yields estimators with negligible bias and is superior to MM in both bias and variance. We also show that the RC method also yields the correct type I error rate of the test of the interaction. However, when the true covariates are not normally distributed, we recommend using MM. We provide an example relating homocysteine to serum folate and B12 levels.     

A simple significance test for quantile regression

Where OLS regression seeks to model the mean of a random variable as a function of observed variables, quantile regression seeks to model the quantiles of a random variable as functions of observed variables. Tests for the dependence of the quantiles of a random variable upon observed variables have only been developed through the use of computer resampling or based on asymptotic approximations resting on distributional assumptions. We propose an exceedingly simple but heretofore undocumented likelihood ratio test within a logistic regression framework to test the dependence of a quantile of a random variable upon observed variables. Simulated data sets are used to illustrate the rationale, ease, and utility of the hypothesis test. Simulations have been performed over a variety of situations to estimate the type I error rates and statistical power of the procedure. Results from this procedure are compared to (1) previously proposed asymptotic tests for quantile regression and (2) bootstrap techniques commonly used for quantile regression inference. Results show that this less computationally intense method has appropriate type I error control, which is not true for all competing procedures, comparable power to both previously proposed asymptotic tests and bootstrap techniques, and greater computational ease. We illustrate the approach using data from 779 adolescent boys age 12-18 from the Third National Health and Nutrition Examination Survey (NHANES III) to test hypotheses regarding age, ethnicity, and their interaction upon quantiles of waist circumference.     

Residential radon exposure and lung cancer: variation in risk estimates using alternative exposure scenarios

The most direct way to derive risk estimates for residential radon progeny exposure is through epidemiologic studies that examine the association between residential radon exposure and lung cancer. However, the National Research Council concluded that the inconsistency among prior residential radon case-control studies was largely a consequence of errors in radon dosimetry. This paper examines the impact of applying various epidemiologic dosimetry models for radon exposure assessment using a common data set from the Iowa Radon Lung Cancer Study (IRLCS). The IRLCS uniquely combined enhanced dosimetric techniques, individual mobility assessment, and expert histologic review to examine the relationship between cumulative radon exposure, smoking, and lung cancer. The a priori defined IRLCS radon-exposure model produced higher odds ratios than those methodologies that did not link the subject's retrospective mobility with multiple, spatially diverse radon concentrations. In addition, the smallest measurement errors were noted for the IRLCS exposure model. Risk estimates based solely on basement radon measurements generally exhibited the lowest risk estimates and the greatest measurement error. The findings indicate that the power of an epidemiologic study to detect an excess risk from residential radon exposure is enhanced by linking spatially disparate radon concentrations with the subject's retrospective mobility.     

Weighted quantile regression for analyzing health care cost data with missing covariates

Analysis of health care cost data is often complicated by a high level of skewness, heteroscedastic variances and the presence of missing data. Most of the existing literature on cost data analysis have been focused on modeling the conditional mean. In this paper, we study a weighted quantile regression approach for estimating the conditional quantiles health care cost data with missing covariates. The weighted quantile regression estimator is consistent, unlike the naive estimator, and asymptotically normal. Furthermore, we propose a modified BIC for variable selection in quantile regression when the covariates are missing at random. The quantile regression framework allows us to obtain a more complete picture of the effects of the covariates on the health care cost and is naturally adapted to the skewness and heterogeneity of the cost data. The method is semiparametric in the sense that it does not require to specify the likelihood function for the random error or the covariates. We investigate the weighted quantile regression procedure and the modified BIC via extensive simulations. We illustrate the application by analyzing a real data set from a health care cost study. Copyright © 2013 John Wiley & Sons, Ltd.     

A simulation-based comparison of techniques to correct for measurement error in matched case-control studies

The presence of measurement errors affecting the covariates in regression models is a relevant topic in many scientific areas, as, for example, in epidemiology. An example is given by an epidemiological population-based matched case-control study on the aetiology of childhood malignancies, which is currently under completion in Italy. This study was aimed at evaluating the effects of childhood exposure to extremely low electromagnetic fields on the risk of disease occurrence by taking into account the possibility of erroneous measures of the exposure. Within this framework, we focus on the application of likelihood methods to correct for measurement error. This approach, which has received less attention in literature with respect to alternatives, is compared with commonly used methods such as regression calibration and SIMEX. The comparison is performed by simulation, under a broad range of measurement error structures.     

Sensitivity of regression calibration to non‐perfect validation data with application to the Norwegian Women and Cancer Study

Measurement error occurs when we observe error‐prone surrogates, rather than true values. It is common in observational studies and especially so in epidemiology, in nutritional epidemiology in particular. Correcting for measurement error has become common, and regression calibration is the most popular way to account for measurement error in continuous covariates. We consider its use in the context where there are validation data, which are used to calibrate the true values given the observed covariates. We allow for the case that the true value itself may not be observed in the validation data, but instead, a so‐called reference measure is observed. The regression calibration method relies on certain assumptions.This paper examines possible biases in regression calibration estimators when some of these assumptions are violated. More specifically, we allow for the fact that (i) the reference measure may not necessarily be an ‘alloyed gold standard’ (i.e., unbiased) for the true value; (ii) there may be correlated random subject effects contributing to the surrogate and reference measures in the validation data; and (iii) the calibration model itself may not be the same in the validation study as in the main study; that is, it is not transportable. We expand on previous work to provide a general result, which characterizes potential bias in the regression calibration estimators as a result of any combination of the violations aforementioned. We then illustrate some of the general results with data from the Norwegian Women and Cancer Study. Copyright © 2015 John Wiley & Sons, Ltd.     

Residential radon and risk of lung cancer in Eastern Germany

BACKGROUND: There is suggestive evidence that residential radon increases lung cancer risk. To elucidate this association further, we conducted a case-control study in Thuringia and Saxony in Eastern Germany during 1990-1997. METHODS: Histologically confirmed lung cancer patients from hospitals and a random sample of population controls matched on age, sex and geographical area were personally interviewed with respect to residential history, smoking, and other risk factors. One-year radon measurements were performed in houses occupied during the 5-35 years prior to the interview. The final analysis included a total of 1,192 cases and 1,640 controls. Odds ratios (OR) and 95% confidence intervals (CI) were estimated by logistic regression. RESULTS: Measurements covered on average 72% of the exposure time window, with mean radon concentrations of 76 Bq/m3 among the cases and 74 Bq/m3 among the controls. The smoking- and asbestos-adjusted ORs for categories of radon (50-80, 80-140 and >140 Bq/m*3, compared with 0-50 Bq/m3) were 0.95 (CI = 0.77 to 1.18), 1.13 (CI = 0.86 to1.50) and 1.30 (CI = 0.88 to 1.93). The excess relative risk per 100 Bq/m? was 0.08 (CI = -0.03 to 0.20) for all subjects and 0.09 (CI = -0.06 to 0.27) for subjects with complete measurements for all 30 years. CONCLUSIONS: Our data indicate a small increase in lung cancer risk as a result of residential radon that is consistent with the findings of previous indoor radon and miner studies.     

Residential radon and lung cancer among never-smokers in Sweden.

In this study, we attempted to reduce existing uncertainty about the relative risk of lung cancer from residential radon exposure among never-smokers. Comprehensive measurements of domestic radon were performed for 258 never-smoking lung cancer cases and 487 never-smoking controls from five Swedish case-control studies. With additional never-smokers from a previous case-control study of lung cancer and residential radon exposure in Sweden, a total of 436 never-smoking lung cancer cases diagnosed in Sweden between 1980 and 1995 and 1,649 never-smoking controls were included. The relative risks (with 95% confidence intervals in parentheses) of lung cancer in relation to categories of time-weighted average domestic radon concentration during three decades, delimited by cutpoints at 50, 80, and 140 Bq m(-3), were 1.08 (0.8--1.5), 1.18 (0.9--1.6), and 1.44 (1.0--2.1), respectively, with average radon concentrations below 50 Bq m(-3) used as reference category and with adjustment for other risk factors. The data suggested that among never-smokers residential radon exposure may be more harmful for those exposed to environmental tobacco smoke. Overall, an excess relative risk of 10% per 100 Bq m(-3) average radon concentration was estimated, which is similar to the summary effect estimate for all subjects in the main residential radon studies to date.