Exposure misclassification: bias in category specific Poisson regression coefficients

In epidemiologic studies of the association between exposure and disease, misclassification of exposure is common and known to induce bias in the effect estimates. The nature of the bias is difficult to foretell. For this purpose, we present a simple method to assess the bias in Poisson regression coefficients for a categorical exposure variable subject to misclassification. We derive expressions for the category specific coefficients from the regression on the error-prone exposure (naive coefficients) in terms of the coefficients from the regression on the true exposure (true coefficients). These expressions are similar for crude and adjusted models, if we assume that the covariates are measured without error and that it is independence between the misclassification probabilities and covariate values. We find that the bias in the naive coefficient for one category of the exposure variable depends on all true category specific coefficients weighted by misclassification probabilities. On the other hand, misclassification of an exposure variable does not induce bias in the estimates of the coefficients of the (perfectly measured) covariates. Similarities with linear regression models are pointed out. For selected scenarios of true exposure-disease associations and selected patterns of misclassification, we illustrate the inconsistency in naive Poisson regression coefficients and show that it can be difficult to intuitively characterize the nature of the bias. Both the magnitude and the direction of the bias may vary between categories of an exposure variable.     

Full-likelihood approaches to misclassification of a binary exposure in matched case-control studies

We consider analysis of matched case-control studies where a binary exposure is potentially misclassified, and there may be a variety of matching ratios. The parameter of interest is the ratio of odds of case exposure to control exposure. By extending the conditional model for perfectly classified data via a random effects or Bayesian formulation, we obtain estimates and confidence intervals for the misclassified case which reduce back to standard analytic forms as the error probabilities reduce to zero. Several examples are given, highlighting different analytic phenomena. In a simulation study, using mixed matching ratios, the coverage of the intervals are found to be good, although point estimates are slightly biased on the log scale. Extensions of the basic model are given allowing for uncertainty in the knowledge of misclassification rates, and the inclusion of prior information about the parameter of interest.     

Measurement error in the explanatory variable of a binary regression: regression calibration and integrated conditional likelihood in studies of residential radon and lung cancer

In epidemiology, one approach to investigating the dependence of disease risk on an explanatory variable in the presence of several confounding variables is by fitting a binary regression using a conditional likelihood, thus eliminating the nuisance parameters. When the explanatory variable is measured with error, the estimated regression coefficient is biased usually towards zero. Motivated by the need to correct for this bias in analyses that combine data from a number of case-control studies of lung cancer risk associated with exposure to residential radon, two approaches are investigated. Both employ the conditional distribution of the true explanatory variable given the measured one. The method of regression calibration uses the expected value of the true given measured variable as the covariate. The second approach integrates the conditional likelihood numerically by sampling from the distribution of the true given measured explanatory variable. The two approaches give very similar point estimates and confidence intervals not only for the motivating example but also for an artificial data set with known properties. These results and some further simulations that demonstrate correct coverage for the confidence intervals suggest that for studies of residential radon and lung cancer the regression calibration approach will perform very well, so that nothing more sophisticated is needed to correct for measurement error.     

Bayesian mixed hidden Markov models: a multi‐level approach to modeling categorical outcomes with differential misclassification

Questionnaire‐based health status outcomes are often prone to misclassification. When studying the effect of risk factors on such outcomes, ignoring any potential misclassification may lead to biased effect estimates. Analytical challenges posed by these misclassified outcomes are further complicated when simultaneously exploring factors for both the misclassification and health processes in a multi‐level setting. To address these challenges, we propose a fully Bayesian mixed hidden Markov model (BMHMM) for handling differential misclassification in categorical outcomes in a multi‐level setting. The BMHMM generalizes the traditional hidden Markov model (HMM) by introducing random effects into three sets of HMM parameters for joint estimation of the prevalence, transition, and misclassification probabilities. This formulation not only allows joint estimation of all three sets of parameters but also accounts for cluster‐level heterogeneity based on a multi‐level model structure. Using this novel approach, both the true health status prevalence and the transition probabilities between the health states during follow‐up are modeled as functions of covariates. The observed, possibly misclassified, health states are related to the true, but unobserved, health states and covariates. Results from simulation studies are presented to validate the estimation procedure, to show the computational efficiency due to the Bayesian approach and also to illustrate the gains from the proposed method compared to existing methods that ignore outcome misclassification and cluster‐level heterogeneity. We apply the proposed method to examine the risk factors for both asthma transition and misclassification in the Southern California Children's Health Study. Copyright © 2013 John Wiley & Sons, Ltd.     

Recovering true risks when multilevel exposure and covariables are both misclassified.

The authors extend previous results on nondifferential exposure misclassification to the situation in which multilevel exposure and covariables are both misclassified. They show that if misclassification is nondifferential and the predictive value matrices are independent of other predictor variables it is possible to recover the true relative risks as a function of the biased estimates and the misclassification matrices alone. If the covariable is a confounder, the true relative risks may be recovered from the apparent relative risks derived from misclassified data and the misclassification matrix for the exposure variable with respect to its surrogate. If the covariable is an effect modifier, the true relative risk matrix may be recovered from the apparent relative risk matrix and misclassification matrices for both the exposure variable with respect to its surrogate and the covariable with respect to its surrogate. By varying the misclassification matrices, the sensitivity of published relative risk estimates to different patterns of misclassification can be analyzed. If it is not possible to design a study protocol that is free of misclassification, choosing surrogate variables whose predictive value is constant with respect to other predictors appears to be a desirable design objective.     

Adjustment for non-differential misclassification error in the generalized linear model

It is well known that estimates of association between an outcome variable and a set of categorical covariates, some of which are measured with misclassification, tend to be biased upon application of the usual methods of estimation that ignore the classification error. We propose a method to adjust for misclassification in covariates when one applies the generalized linear model. In the case where one can observe some true covariates only through surrogates, we combine a latent class analysis with the approach to incorporate multiple surrogates into the model. We include discussion on the efficacy of repeated measurements which one can view as a special case of multiple surrogates with identical distribution. We provide two examples to demonstrate the applicability of the method and the efficacy of multiple replicates for a covariate subject to misclassification in a regression framework.     

Transitions in an imperfectly observed binary variable: depressive symptomatology in adolescents

We describe a methodology for analysing transitions over time in a binary outcome variable that is subject to misclassification (that is, measurement error). Logistic regression models for transition events in the true underlying state are combined with estimates of probabilities of misclassification of the underlying state. The model is based on the Markovian assumption that the probabilities of transition in the underlying state at a given time depend only on the underlying state at the previous time. Hence we estimate odds-ratio effects for transitions that are adjusted for the effect of misclassification. Comparing these adjusted estimates with estimates that are obtained without taking misclassification into account indicates that the latter can be biased either toward or away from the null. For the estimates to exist, certain restrictions on the observed data and misclassification probabilities need to be met. If these restrictions are not satisfied then the conclusion from the analysis is that all observed transition events can be explained solely by the error in outcome assessment, in which case it is likely that an aspect of the model is incorrect. The motivation for this work comes from an analysis of transitions in depression status for a cohort of Australian teenagers participating in a longitudinal study of adolescent health.     

Accounting for error due to misclassification of exposures in case-control studies of gene-environment interaction

We consider analysis of data from an unmatched case-control study design with a binary genetic factor and a binary environmental exposure when both genetic and environmental exposures could be potentially misclassified. We devise an estimation strategy that corrects for misclassification errors and also exploits the gene-environment independence assumption. The proposed corrected point estimates and confidence intervals for misclassified data reduce back to standard analytical forms as the misclassification error rates go to zero. We illustrate the methods by simulating unmatched case-control data sets under varying levels of disease-exposure association and with different degrees of misclassification. A real data set on a case-control study of colorectal cancer where a validation subsample is available for assessing genotyping error is used to illustrate our methods.     

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.     

Bayesian inference for unidirectional misclassification of a binary response trait

When assessing association between a binary trait and some covariates, the binary response may be subject to unidirectional misclassification. Unidirectional misclassification can occur when revealing a particular level of the trait is associated with a type of cost, such as a social desirability or financial cost. The feasibility of addressing misclassification is commonly obscured by model identification issues. The current paper attempts to study the efficacy of inference when the binary response variable is subject to unidirectional misclassification. From a theoretical perspective, we demonstrate that the key model parameters possess identifiability, except for the case with a single binary covariate. From a practical standpoint, the logistic model with quantitative covariates can be weakly identified, in the sense that the Fisher information matrix may be near singular. This can make learning some parameters difficult under certain parameter settings, even with quite large samples. In other cases, the stronger identification enables the model to provide more effective adjustment for unidirectional misclassification. An extension to the Poisson approximation of the binomial model reveals the identifiability of the Poisson and zero‐inflated Poisson models. For fully identified models, the proposed method adjusts for misclassification based on learning from data. For binary models where there is difficulty in identification, the method is useful for sensitivity analyses on the potential impact from unidirectional misclassification.     

Bayesian adjustment for exposure misclassification in case–control studies

Poor measurement of explanatory variables occurs frequently in observational studies. Error‐prone observations may lead to biased estimation and loss of power in detecting the impact of explanatory variables on the response. We consider misclassified binary exposure in the context of case–control studies, assuming the availability of validation data to inform the magnitude of the misclassification. A Bayesian adjustment to correct the misclassification is investigated. Simulation studies show that the Bayesian method can have advantages over non‐Bayesian counterparts, particularly in the face of a rare exposure, small validation sample sizes, and uncertainty about whether exposure misclassification is differential or non‐differential. The method is illustrated via application to several real studies. Copyright © 2010 John Wiley & Sons, Ltd.     

Bayesian measurement‐error‐driven hidden Markov regression model for calibrating the effect of covariates on multistate outcomes: Application to androgenetic alopecia

Multistate Markov regression models used for quantifying the effect size of state‐specific covariates pertaining to the dynamics of multistate outcomes have gained popularity. However, the measurements of multistate outcome are prone to the errors of classification, particularly when a population‐based survey/research is involved with proxy measurements of outcome due to cost consideration. Such a misclassification may affect the effect size of relevant covariates such as odds ratio used in the field of epidemiology. We proposed a Bayesian measurement‐error‐driven hidden Markov regression model for calibrating these biased estimates with and without a 2‐stage validation design. A simulation algorithm was developed to assess various scenarios of underestimation and overestimation given nondifferential misclassification (independent of covariates) and differential misclassification (dependent on covariates). We applied our proposed method to the community‐based survey of androgenetic alopecia and found that the effect size of the majority of covariate was inflated after calibration regardless of which type of misclassification. Our proposed Bayesian measurement‐error‐driven hidden Markov regression model is practicable and effective in calibrating the effects of covariates on multistate outcome, but the prior distribution on measurement errors accrued from 2‐stage validation design is strongly recommended.     

Statistical analysis with missing exposure data measured by proxy respondents: a misclassification problem within a missing‐data problem

In studies of older adults, researchers often recruit proxy respondents, such as relatives or caregivers, when study participants cannot provide self‐reports (e.g., because of illness). Proxies are usually only sought to report on behalf of participants with missing self‐reports; thus, either a participant self‐report or proxy report, but not both, is available for each participant. Furthermore, the missing‐data mechanism for participant self‐reports is not identifiable and may be nonignorable. When exposures are binary and participant self‐reports are conceptualized as the gold standard, substituting error‐prone proxy reports for missing participant self‐reports may produce biased estimates of outcome means. Researchers can handle this data structure by treating the problem as one of misclassification within the stratum of participants with missing self‐reports. Most methods for addressing exposure misclassification require validation data, replicate data, or an assumption of nondifferential misclassification; other methods may result in an exposure misclassification model that is incompatible with the analysis model. We propose a model that makes none of the aforementioned requirements and still preserves model compatibility. Two user‐specified tuning parameters encode the exposure misclassification model. Two proposed approaches estimate outcome means standardized for (potentially) high‐dimensional covariates using multiple imputation followed by propensity score methods. The first method is parametric and uses maximum likelihood to estimate the exposure misclassification model (i.e., the imputation model) and the propensity score model (i.e., the analysis model); the second method is nonparametric and uses boosted classification and regression trees to estimate both models. We apply both methods to a study of elderly hip fracture patients. Copyright © 2014 John Wiley & Sons, Ltd.     

Bayesian approach to average power calculations for binary regression models with misclassified outcomes

We develop a simulation‐based procedure for determining the required sample size in binomial regression risk assessment studies when response data are subject to misclassification. A Bayesian average power criterion is used to determine a sample size that provides high probability, averaged over the distribution of potential future data sets, of correctly establishing the direction of association between predictor variables and the probability of event occurrence. The method is broadly applicable to any parametric binomial regression model including, but not limited to, the popular logistic, probit, and complementary log–log models. We detail a common medical scenario wherein ascertainment of true disease status is impractical or otherwise impeded, and in its place the outcome of a single binary diagnostic test is used as a surrogate. These methods are then extended to the two diagnostic test setting. We illustrate the method with categorical covariates using one example that involves screening for human papillomavirus. This example coupled with results from simulated data highlights the utility of our Bayesian sample size procedure with error prone measurements. Copyright © 2008 John Wiley & Sons, Ltd.     

Correcting for misclassification for a monotone disease process with an application in dental research

Motivated by a longitudinal oral health study, we evaluate the performance of binary Markov models in which the response variable is subject to an unconstrained misclassification process and follows a monotone or progressive behavior. Theoretical and empirical arguments show that the simple version of the model can be used to estimate the prevalence, incidences, and misclassification parameters without the need of external information and that the incidence estimators associated with the model outperformed approaches previously proposed in the literature. We propose an extension of the simple version of the binary Markov model to describe the relationship between the covariates and the prevalence and incidence allowing for different classifiers. We implemented a Bayesian version of the extended model and show that, under the settings of our motivating example, the parameters can be estimated without any external information. Finally, the analyses of the motivating problem are presented.     

Bayesian analysis of a matched case–control study with expert prior information on both the misclassification of exposure and the exposure–disease association

We propose a Bayesian adjustment for the misclassification of a binary exposure variable in a matched case–control study. The method admits a priori knowledge about both the misclassification parameters and the exposure–disease association. The standard Dirichlet prior distribution for a multinomial model is extended to allow separation of prior assertions about the exposure–disease association from assertions about other parameters. The method is applied to a study of occupational risk factors for new‐onset adult asthma. Copyright © 2009 John Wiley & Sons, Ltd.     

Bayesian analyses of longitudinal binary data using Markov regression models of unknown order

We present non-homogeneous Markov regression models of unknown order as a means to assess the duration of autoregressive dependence in longitudinal binary data. We describe a subject's transition probability evolving over time using logistic regression models for his or her past outcomes and covariates. When the initial values of the binary process are unknown, they are treated as latent variables. The unknown initial values, model parameters, and the order of transitions are then estimated using a Bayesian variable selection approach, via Gibbs sampling. As a comparison with our approach, we also implement the deviance information criterion (DIC) for the determination of the order of transitions. An example addresses the progression of substance use in a community sample of n = 242 American Indian children who were interviewed annually four times. An extension of the Markov model to account for subject-to-subject heterogeneity is also discussed.     

Does nondifferential misclassification of exposure always bias a true effect toward the null value?

The authors present some examples to demonstrate that in certain nondifferential misclassification conditions with polychotomous exposure variables, estimates of odds ratios for categories at intermediate level of risk can be biased away from the null or can change direction. In addition, the authors present two examples to demonstrate that the slope of the dose-response trend for the true distributions can change direction, creating a false inverse trend, even if the misclassification is nondifferential.     

Curious phenomena in Bayesian adjustment for exposure misclassification

Many epidemiologic investigations involve some discussion of exposure misclassification, but rarely is there an attempt to adjust for misclassification formally in the statistical analysis. Rather, investigators tend to rely on intuition to comment qualitatively on how misclassification might impact their findings. We point out several ways in which intuition might fail, in the context of unmatched case-control analysis with non-differential exposure misclassification. Particularly, we focus on how intuition can conflict with the results of a Bayesian analysis that accounts for the various uncertainties at hand. First, the Bayesian adjustment for misclassification can weaken the evidence about the direction of an exposure-disease association. Second, admitting uncertainty about the misclassification parameters can lead to narrower interval estimates concerning the association. We focus on the simple setting of unmatched case-control analysis with binary exposure and without adjustment for confounders, though much of our discussion should be relevant more generally.