Introduction

Heterogeneity of estimates of associations produced by epidemiologic analyses present a challenge for synthesis of evidence. One plausible cause of such heterogeneity is rooted in differences in the accuracy of exposure assessment since misclassification of a binary exposure may influence the location and uncertainty of the effect estimate. If non-differential misclassification of exposure is ignored, the association between exposure and disease is expected (for an infinite sample size) to be attenuated towards the null. Such non-differential misclassification can reduce power and lead to missed associations as well as false positives [1•]. In the case of differential exposure misclassification, effect estimates can be expected to be biased towards or away from the null, making interpretation particularly perilous. Differential misclassification with respect to outcome is of particular concern in studies where exposure assessment is collected following disease ascertainment. A typical approach to minimizing it involves separating the methods of data collection for the health outcome and determinants of exposure. However, differential exposure misclassification can also result from dichotomizing a continuous exposure that is measured with error. This type of misclassification has been called “differential due to dichotomization” (DDD) misclassification [2, 3•]. A detailed discussion of how differential exposure misclassification arises from non-differential measurement error is outlined in Appendix A, following the derivation given in Gustafson [2]. Differential misclassification due to dichotomization can only occur when the exposure and outcome are associated, through either causality or confounding. Since we conduct studies with the idea that a relationship may exist between exposure and outcome, we should more carefully consider the possibility of differential misclassification in our analyses. Differential misclassification from this source cannot be remedied by common study design strategies, such as blinding the exposure assessment from the health status.

Many methods have been proposed to account for misclassification in the context of retrospective case-control studies where misclassification parameters (sensitivity, Sn, and specificity, Sp) are not known exactly. Such approaches are motivated by the observation that misclassification-corrected odds ratios obtained using a particular point estimate of Sn and Sp can be highly sensitive to small differences between the actual and guessed values [4, 5]. More reliable estimation is obtained when we take into account the uncertainty regarding the actual values of the misclassification parameters. This can be accomplished by implementing a Bayesian approach where one posits prior distributions for the Sn and Sp [4]. Intuitively, the procedure samples from these distributions and corrects for misclassification over many iterations. The procedure can distinguish between “good” and “poor” guesses of Sn and Sp by reconciling them with observed data and other parameters and models involved via evaluation of likelihood. Such methods obtain samples from the posterior distribution of the true (misclassification-corrected) effect estimates relating exposure to outcome. A detailed description of the approach, Bayesian Markov chain Monte Carlo (MCMC) algorithms, can be found in Gustafson [2]. Applications of these methods can be used to reconcile heterogeneity in effect estimates between different effect estimates obtained within a study using different exposure metrics [6•], and this holds promise for resolving between-study heterogeneity as well. Luta et al. [7•] presented and illustrated these methods in an accessible form that aids their dissemination to epidemiologists in general.

We apply Bayesian methods for correction of exposure misclassification in two population-based case-control studies of the association between occupational asthmagen (agents that can trigger asthma) exposure and autism spectrum disorder (ASD): the Study to Explore Early Development (SEED) conducted in the USA [8] and a study nested within the Danish registers [9]. In both studies, we estimated maternal occupational asthmagen exposure job codes with an asthma-specific job exposure matrix (JEM) [10]. We also illustrate the Bayesian evolution of evidence from a first hypothesis about exposure-ASD association and quality of exposure assessment postulated before the conduct of our first analysis in the smaller US sample [8], followed by integration of results from our first analysis, about both the exposure-ASD relationship and quality of exposure assessment, into the second, much larger, Danish study [9]. We realize that knowledge derived from the USA regarding Sn and Sp of the JEM may not be entirely applicable to the study in Denmark due to differences in how occupational histories were collected, and address this by adjusting priors on the misclassification parameters to reflect this added uncertainty. We show the resulting effect size estimates and model of exposure misclassification that represent what we have learnt about the influence of maternal asthmagen exposure on risk of ASD and quality of exposure assessment tools.

Methods

Case-Control Studies

The first is the Study to Explore Early Development (SEED), a US multi-site, case-control study designed to investigate risk factors, co-morbidities, and phenotypes of ASD [11]. We focus on comparisons between the ASD cases (N = 463) and population (POP) controls (N = 710). Participants were required to be born and reside in one of six study catchment areas in California, Colorado, Georgia, Maryland, North Carolina, and Pennsylvania between September 1, 2003 and August 31, 2006. Children in the ASD group were ascertained through service and educational providers for children with developmental disabilities, whereas POP children were identified through random sampling of vital records [11]. ASD classification was based on results from the Autism Diagnostic Observation Study (ADOS) and the Autism Diagnostic Interview Revised (ADI-R) when children were 30 to 68 months of age. Maternal job histories were collected as part of a computer-assisted telephone interview shortly after enrollment in the study. Mothers reported jobs held for 1 month or more for at least 10 h per week from 3 months prior to the end of the pregnancy until the child was born or the mother stopped breastfeeding. As detailed in Singer et al. [8], we coded jobs according to the International Labor Organization’s International Standard Classification of Occupations 1988 (ISCO-88) [12]. Analyses were restricted to mothers reporting at least one job that overlapped with the pregnancy.

The second case-control study is nested in the Danish registers [9]. The sample consists of 29,359 controls and 6706 ASD cases singleton births born in Denmark between January 1, 1993, and December 31, 2007, with an employed mother in the year representing the majority of the pregnancy. Details of the selection of the study sample are described elsewhere [9]. ASD cases were defined as having a reported ICD-10 diagnosis of autism spectrum disorder (ICD-10 codes: F84.0, F84.1, F84.5, F84.8, and F84.9) in the Danish Central Psychiatric Register (DPCR) [13] from January 1, 1995 into April 2013. We linked the ASD cases and controls to maternal occupation (Danish International Standard Classification of Occupation, DISCO-88) and industry codes (NACE codes) from the Employment Classification Module (AKM) [14] to determine maternal occupation and industry in the year that overlapped most with the pregnancy.

The specific details of the exposure assessment using asthma-specific JEM [10] for each study are described elsewhere [8, 9], but they broadly followed the procedures prescribed by Kennedy et al. (2000). The JEM produces a binary indicator regarding presence of exposure to a compound or mixture known or strongly suspected of causing occupational asthma. Sensitivity and specificity of the JEM were previously evaluated in studies of occupational asthma under the assumption of non-differential exposure misclassification [15, 16], which revealed evidence in support of higher Sp compared to Sn, as intended by the creators of the JEM.

Correction for Misclassification of Exposure

Overview

The goal of this analysis is to correct for exposure misclassification generated by using an asthma JEM to classify exposure based on job codes into a binary exposure indicator. We begin with correction for exposure misclassification in an individual-level analysis that includes adjustment for covariates in the SEED study. For purely illustrative (not inferential) purposes, we contrast with a model where we assume near-perfect exposure classification, as is typically done in occupational epidemiology studies [8], to those in which we assume exposure misclassification based on prior knowledge about the performance of the JEM in different settings. We allow the differential misclassification by case status but let data inform its extent by examining posterior distributions of Sn and Sp for cases and controls. Thus, we present two different models (Table 1): [1] assuming almost perfect classification of exposure, allowing for differential misclassification (model S_1), and [2] setting priors on misclassification parameters based on previous studies, allowing for differential exposure misclassification (model S_2). It is important to note that while we allow for differential misclassification of exposure, we do not force it to be differential, because the priors on Sn and Sp are the same for cases and controls.

Table 1 Posterior distributions, medians, and 95% credible intervals after allowing for differential exposure misclassification and learning from study to study about association between exposure to asthmagens and autism spectrum disorders
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In the second portion of the analysis, we apply misclassification correction using data from a two-by-two contingency table of maternal occupational asthmagen exposure by ASD case-control status from the Danish study, with Sn and Sp priors informed by the SEED study. (We do not conduct individual-level analysis in the Danish study because of logistical challenges in access to individual-level data.) We set priors on Sn, Sp, and the odds ratio based on posterior distributions from the SEED model S_2 (model D_1). We also acknowledge that performance of the JEM in SEED may have been different than that in the Danish analysis and therefore also conduct an additional analysis with priors on Sn and Sp derived on the basis of SEED for model D_1 to have the same location but greater variance, leading to model D_2.

Since we use individual-level data for the SEED and only a contingency table as data for the Denmark Bayesian analyses, the model specifications are different. However, all Bayesian analyses that account for exposure misclassification or measurement error share common features. We must specify three models: (a) an exposure model, (b) a measurement (misclassification) model, and (c) an outcome model, following Gustafson [2]. The details of model specification particular to our studies as well as details of sampling for the posterior distributions are given in Appendix B. Code is presented in Appendix C. We focus next on specifics of our analyses that deal with elucidation and specification of the prior distributions.

SEED: Specification of Priors

We express the typical implicit assumption of almost perfect classification of exposure by setting prior distributions of beta (1000,1) on Sn and Sp. These distributions have a mean of 0.999 and standard deviation of 0.001 and are constrained between 0 and 1 by the properties of the beta distribution. These results should be roughly equivalent to typical uncorrected analyses.

We derived realistic priors on Sn and Sp of the JEM from previous literature. We detail here the origin of these priors, which were initially elucidated through expert opinion and then were updated with data from two analyses. Liu et al. [15] asked experts (two occupational physicians and one occupational hygienist) to report the best guess, upper bound, and lower bound of true value for the Sn and Sp of the asthma-specific JEM. Liu et al. [15] obtained posterior distributions for the Sn and Sp by updating these priors using data from workers’ compensation claims and physician billing records from Alberta, Canada, to examine the association between asthmagen exposure and new adult-onset asthma. Beach et al. [16] further updated the Sn and Sp posteriors from Liu et al. [15] in a comparable analysis using similar data from British Columbia, Canada. We used the posterior distributions of sensitivity and specificity of the JEM from Beach et al. [16] as a basis for our prior distributions in this analysis because they reflect the synthesis of evidence regarding exposure misclassification by the JEM prior to the SEED analysis.

Beach et al. [16] reported Sn and Sp posterior distributions for 16 categories of asthmagen exposures. Since there was not much variability across the asthmagen categories for these misclassification parameters, we averaged the medians, 2.5th percentiles, and 97.5th percentiles to obtain a best guess of the mode, 2.5th percentile, and 97.5th percentile for Sn and Sp distributions for any asthmagen. The best guess of the 2.5th percentile, mode, and 97.5th percentile was 0.133, 0.381, and 0.728 for the Sn and 0.990, 0.992, and 0.994 for the Sp, respectively. We used the betaExpert function in R to determine parameters for beta distributions corresponding to the above percentiles and then used these beta distributions as priors for Sn and Sp. Based on this, we set the prior distribution for Sn as beta (3.6, 5.2) and the prior distribution for Sp as beta (1000, 9.1); the priors on Sn and Sp were centered on means of 0.41 and 0.991, respectively.

We assumed a normally distributed uninformative prior with a mean of 0 and variance of 0.5 for all log odds ratios in the outcome and measurement models, except for the intercept of the outcome model and the coefficient for sex. For these two coefficients, we set an uninformative normally distributed prior with a mean of 0 and a variance of 1. The N (0, 0.5) distribution implies that the true odds ratio is roughly between 1/4 and 4 while a N (0, 1) distribution implies that the true odds ratio is roughly between 1/7 and 7. We judged these to be sufficiently vague and realistic while not informing the direction the associations. This was consistent with this being the first ever analysis of the association in question.

Denmark: Specification of Priors

In the second series of Bayesian analyses, we update beliefs from the SEED study with a contingency table of data from a Denmark register study. Crude and adjusted odds ratios for the association between maternal asthmagen exposure and ASD in the Denmark register study are similar [9], so we are not concerned that confounders are not considered in these Bayesian analyses. Because we learnt about the magnitude of the odds ratio (θ = OR) relating the exposure and outcome from the SEED analysis, we set prior distributions on the prevalence of exposure among controls (r0) and θ to induce a prior distribution on the true exposure prevalence among cases:

$$ {r}_1={\theta r}_0/\left(1-{r}_0+{\theta r}_0\right). $$

In the first model (model D_1), we assume that exposure assessment was the same in the two studies, so we set priors on sensitivity as beta (11.0, 38.5) for controls and beta (8.3, 17.2) for cases; the prior on specificity for both cases and controls was beta (718, 8.2). These distributions were derived from the posterior distribution from the SEED analysis. In the second model (model D_2), we inflate the variances of Sn and Sp to reflect the possibility that that differences in the collection of exposure history in the two studies impacted exposure assessment. We achieved this by setting the upper percentiles of priors on Sn to be 20% greater than that in model D_1, which roughly doubled the variance. Thus, for model D_2, priors for Sn were beta (5.9, 19.5) for controls and beta (4.2, 8.1) for cases. We set a beta (46.6, 3.4) prior for Sp, which assumes a prior mean of 0.93 and a standard deviation of 0.04. We set a uniform prior only on the true exposure prevalence among the controls, r0, and set an informative Gaussian prior with a mean of − 0.34 with a variance of 0.33 on the log odds ratio based on the posterior from our analysis of SEED.

Results

The adjusted OR for maternal occupational asthmagen exposure comparing ASD cases to population controls in the SEED study was 1.39 (95% confidence interval (CI) 0.96–2.02), following a typical JEM exposure classification approach [8]. When we assumed near-perfect classification of exposure, which is essentially equivalent to not adjusting for exposure misclassification, we observed a posterior OR of 1.37 (95% CrI 0.96–1.96) (model S_1) (Table 1). In the analysis allowing for differential exposure misclassification by the JEM (model S_2), the median of the posterior of the adjusted OR was 0.71 (95% CrI 0.23–2.42). The results allowing for differential exposure misclassification by the JEM suggest that the analyses not adjusted for exposure misclassification are positively biased, although the confidence intervals are overlapping.

The posterior distribution of sensitivity is concentrated among smaller values compared to its prior, but priors and posteriors were similar for the specificity (model S_2, Table 1). We also observe that the median of the posterior distribution for the Sn in cases was higher than the median of the Sn posterior distribution for the controls. Though the posterior Sn distributions produced for Sn for cases and controls from Model S_2 overlap, this suggests that measurable differential misclassification could be at play.

In the Danish case-control analysis, we found an inverse association between maternal asthmagen exposure and ASD using typical analytic approaches (crude OR 0.92, 95% CI 0.86–0.99) [9]. The posterior distributions for Sn and Sp resulting from models D_1 are similar to the posteriors from SEED model S_2 (Table 1). In model D_1 where we set an informative prior on the odds ratio based on the posterior from SEED model S_2, we generate a posterior odds ratio with a median at 0.64 (95% CrI 0.23–1.94). The posterior odds ratio is pulled towards the SEED result by inclusion of this prior. The posterior distribution for the Denmark model D_1 is similar to SEED model S_2 and only a little more concentrated around the median. Thus, despite the large sample size from this second study, we learn little new about the effect estimate of interest. When we admit additional uncertainty about Sn and Sp in analysis of Danish study in model D_2, the posterior distribution for the odds ratio (median 0.68, 95% CrI 0.23–1.97) is similar to the posterior for model D_1 (Table 1). We illustrate how our knowledge of the asthmagen-ASD association and Sn changed from before analysis of SEED through to misclassification correction of the Danish study in Fig. 1.

Fig. 1
figure 1

Illustration of prior and posterior distributions for analysis of SEED (model S_2) and study nested in Denmark (model D_1); posterior distributions from model S_2 were used to set priors for model D_1

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Discussion

In this paper, we illustrate a Bayesian method for correcting for exposure misclassification in the context of two studies examining the association between maternal occupational asthmagen exposure and ASD in the children. Inferentially, our models suggest that there is no measurable association between maternal asthmagen occupational exposure around the time of pregnancy and ASD. This conclusion was consistent with and without exposure misclassification adjustment, although the effect size estimates and confidence limits did fluctuate. We illustrate that it is hard to predict how misclassification of exposure affects every specific analysis. We also argue that it is more difficult than commonly realized to be sure that such misclassification is non-differential with respect to health outcome, even when exposure assessment is blind to outcome. We describe use of Bayesian tools that make correction for misclassification accessible to epidemiologists who collaborate with statisticians. We highlight the importance of setting defensible priors that capture knowledge that existed before the data in any given study is collected. This is particularly important when the data does not allow us to learn about all parameters of interest, as is typically the case with differential exposure misclassification. In doing so, we must take care to avoid showing over-confidence, as would arise from not considering a range of plausible priors. We consider these methodological matters in details below.

Our results nonetheless illustrate key points regarding exposure misclassification in epidemiologic studies. First, we demonstrate here that the odds ratio estimate can move in unexpected ways when we allow for misclassification by the JEM to be differential. In the epidemiologic literature, authors often assert the belief that exposure misclassification is non-differential and thus results are biased to the null, and argue that reported associations are likely not “spurious” under this rationale. However, in practice, it may be difficult to know the true extent of differential misclassification. In the SEED study, as in any case-control study, we may suspect that differential recall for many self-reported variables because mothers who have a child with an ASD may recall the pregnancy differently than mothers of typically developing controls. However, mathematically, as discussed in Appendix A, the focus of this paper is the possibility for differential misclassification to occur through the process of dichotomizing a continuous exposure measured with error, regardless of the timing of exposure query [2, 3•]. The extent to which exposure misclassification is non-differential by dichotomizing a continuous exposure measure depends on the strength of exposure-outcome association [3•]. However, it can also be a product of uncontrolled confounding. Thus, we assert that it is important to consider the potential impact of differential exposure misclassification. If the true association does not exist and there is no uncontrolled confounding, then we do not expect exposure misclassification to deviate from non-differential. The fact that we observe little evidence for measurable differential misclassification in the Denmark study is concordant with the observation of no measurable association between exposure and outcome.

Second, we illustrate that even in situations of relatively large sample sizes, posterior effect estimates are affected by misclassification bias. In our example, we observed a precise protective effect estimate of occupational asthmagen exposure on ASD risk in the Denmark study, using typical analytic approaches. When we corrected for exposure misclassification based on prior evidence, we observed a more protective tendency in point estimate, but the credible intervals widened.

This illustrates the challenges faced in occupational epidemiology. The JEM we chose is among the best of its kind, yet the low sensitivity limits our ability to confidently identify new associations. The inability to improve our estimates and precision in the very large Danish study illustrates that without improving exposure measurements and assessment methods, we have perhaps reached the limit of what we can discover with tools like JEMs when the associations are weak. Though we specifically focus on the JEM example here, this concern regarding exposure misclassification exists in any epidemiology study where we classify a continuous exposure measured with error.

One important caveat regarding these models is that we must be careful in setting prior distributions, especially given problems with non-identifiability. Recall that pi = ri*Sn + (1-ri)*(1-Sp) where pi is observed exposure prevalence and ri is the true exposure prevalence; i = 0 denotes controls and i = 1 denotes cases. If we observe an exposure prevalence, pi, of 0.21, and specificity, Sp, is approximately 1, then there are two possible solutions for (ri, Sn): (0.3, 0.7) and (0.7, 0.3). The priors will determine the solution that is selected. Thus, if we place a prior on sensitivity that is concentrated on 0.3, the solution for the true exposure prevalence will converge upon 0.7. Since we have prior knowledge of the performance of the JEM and not true exposure prevalence in the selected samples, we place informative priors on the Sn and Sp instead of the true exposure prevalence. In our analysis, the particular identifiability issue only emerges when the specificity is close to one, but illustrates that use of these models should be guided by knowledge. Prior knowledge may also be complicated by the fact model parameters may not be directly transportable across different study populations, suggesting the importance of perhaps considering a few plausible priors.

Conclusion

In epidemiologic studies, we often present results as if there is no measurement error despite the fact that it exists. We illustrate the sensitivity of effect size estimates to exposure misclassification in a limited series of studies and show how Bayesian procedures can be readily applied to address exposure misclassification. These methods accommodate our uncertainty regarding the amount of misclassification. We illustrated these methods through use of WinBUGS and R, although there are alternative packages that can be used for Bayesian inference, including OpenBUGS (an open-source version of BUGS software) [17], Just Another Gibbs Sampler (JAGS) [18], and STAN [19]. We argue that analyses that account for this misclassification should become more commonplace within the epidemiologic literature in general. Though ultimately, these misclassification error methods cannot replace investing in the development of better exposure assessment tools and validating the quality of currently existing exposure assessment tools.