Exposure measurement error: A short methodological primer and playlist

Contributors: Emma Nichols, Erik Meijer, Yao-Yi Chiang, Alden L. Gross, Eleanor Hayes-Larson, Sindana D. Ilango, Kayleigh Keller, Regina A. Shih, Adam A. Szpiro, Jennifer Weuve, Sara D. Adar

Studying the exposome, or the totality of exposures that individuals experience across the life course requires measuring features of the exposome. Although researchers seek to optimize measurement, some measurement error is unavoidable. Understanding, quantifying, and accounting for error in exposome measurement is an important methodological consideration in exposome research. While error in an outcome value is handled directly by many modeling approaches (with exceptions), most traditional methods assume no measurement error in the exposure. Error in measuring the exposome can have different impacts on the estimation of health effects depending on the type of measurement error, the relationship between measurement error and other variables in the model (e.g., if measurement error is associated with the outcome, referred to as differential measurement error), and the modeling framework used. This primer and publication playlist seeks to briefly explain key concepts in defining, quantifying, and accounting for measurement error, while embedding and summarizing key citations that cover these concepts in substantial depth.

Simple additive measurement error structures

We first focus on two most common structures of simple additive measurement error:

• Classical measurement error: This form of measurement error occurs when the observed value varies randomly around the true value. This often arises when a measurement tool is noisy. Conceptually, this is akin to using a slightly imprecise scale to measure weight: sometimes weight is overestimated and sometimes underestimated, with an average measurement error of 0. Error is independent of the true value and other variables. In equation form:

  X_observed = X_true + random error

• Berkson measurement error: This form of measurement error occurs when the true value varies randomly around the observed value. This often arises when a single value is assigned to a group of individuals, but the true value randomly fluctuates around the assignment. For example, values from central monitors for environmental conditions (e.g., air pollution concentrations) in a community may be assigned to everyone in that community. Though the assigned exposure is assumed to capture the condition of interest for everyone, the actual exposure varies from that common level randomly between people. Error is independent of the observed value and other variables. In equation form:

  X_true = X_observed + random error

When the mean measurement error is 0 and measurement error is unrelated to the true value (for classical error) or observed value (for Berkson error) in addition to being unrelated to other variables in the model, bias from either classical measurement error or Berkson measurement error is well understood in linear regression models. In these settings with a single continuous exposure variable:

Classical measurement error:

• Regression coefficients from linear models will be biased towards the null (i.e., towards zero).

There is a large body of research within psychology, epidemiology, and other fields that focuses on this form of bias, typically called “attenuation bias.” A number of procedures have been proposed to use estimates of reliability (a ratio that depends on the magnitude of exposure variation and measurement error variation) to correct for attenuation given a set of assumptions.

Additional approaches in the econometrics literature to adjust for classical measurement error include:

• Estimating bounds on coefficients. No additional information or assumptions are necessary (Schennach, 2020; Section 3.9).
• When a variable is available that is correlated with the true measure but not with measurement error, and is not included in the model, this can be used as an instrumental variable (IV) (Schennach, 2020; Section 3.10.2).

Berkson measurement error:

• Regression coefficients from linear models will be unbiased. Variance will be inflated, but no correction is required for valid inference.

More complex measurement error structures

When models are nonlinear, or when measurement error structures are more complex (e.g., associated with the outcome or errors in other variables), other approaches are needed. Given the common occurrence of complex measurement error patterns, existing literature cautions researchers against the use of simple heuristics described above, which apply to non-differential, independent classical measurement error of continuous variables, but do not apply to more complex situations. We review a few of the most common complex forms of measurement error below, and use directed acyclic graphs (DAGs) to illustrate these structures. DAGs are adapted from Hernan & Robins, 2020, which covers a variety of topics in causal inference, including measurement error.

In all following DAGs, we consider exposure (X) and outcome (Y). We use a * to denote a mismeasured variable and U to denote measurement error. In one instance, we also include a second exposure (W). For example, classical measurement error in the exposure would be encoded as:

Complex, alternative structures include:

• Differential measurement error: an error structure where measurement error is associated with another variable in the model, such as the outcome.

• Dependent measurement error: an error structure where measurement errors in multiple variables are associated with each other. Measurement error in the exposure and outcome can be dependent (Panel A, below) but in exposome research with multiple exposures of interest, there may also be dependence in the measurement error across the multiple exposure variables (Panel B, below).

• Measurement error in categorical variables: if the error-prone variable is categorical, the error is often referred to as misclassification, which can lead to bias even in the case of classical measurement error.

These forms of error can also occur in combination. For instance, it is possible to have differential, dependent measurement error.

Quantifying and adjusting for more complex measurement error structures

Oftentimes, validation data (i.e., external data or a subset of internal data containing both mismeasured and true measurements) or repeated measurements can be useful in estimating bias or adjusting findings in the presence of complex measurement error structures. We highlight key topic areas and papers related to various types of measurement error structures, and approaches to adjust for measurement error:

Regression calibration: Regression calibration uses validation data to predict the true value (i.e., the value without measurement error) based on the potentially mis-measured value and a set of covariates. Alternatively, regression calibration can leverage information about error variance from other sources, such as from repeated noisy measurements. Information from the calibration model is then used to adjust the biased regression coefficient. Standard errors can be estimated using a bootstrap or via an asymptotic variance estimator.

Multiple imputation for measurement error correction (MIME): MIME treats measurement error as a missing data problem. Internal validation data is needed, and these data are used to derive an imputation model leveraging covariate and outcome data as well as the mismeasured variable.

Simulation extrapolation (SIMEX): SIMEX relies on a good estimate of error variance, which, as in the case of regression calibration, can come from validation data or repeated noisy measurements. The SIMEX procedure involves repeatedly simulating additional error in the mismeasured variable and re-estimating the association of interest. These simulations enable the estimation of a smooth curve describing the relationship between the magnitude of error and estimated coefficient, which can then be extrapolated to estimate what the coefficient would be without measurement error (i.e., under the condition where the measurement error variance is 0). This procedure is a generalization of bootstrap-based measurement error bias correction.

Simulation analysis: Simulations are a flexible tool that can be used to ask questions about the impact of measurement error on findings in scenarios where the truth is known. Simulation analysis to probe the effects of measurement error is often discussed in the context of literature on quantitative bias analysis, though this umbrella term describes a broader suite of methods and approaches, including approaches to quantify the impacts of other forms of bias.

Joint modeling: Joint models for the exposure-outcome relationship and the measurement error process can also be used to directly account for exposure measurement error with or without validation data. These models typically simultaneously estimate a model for the relationship between the outcome and the observed (with error) exposure alongside a model for the relationship between the true exposure and the observed (with error) exposure, with the models linked by a latent variable representing true exposure. Estimation is possible by bringing in additional information about the magnitude of the measurement error (typically the error variance). Hierarchical Bayesian estimation approaches are well suited for estimating these joint models and are often used in this setting.

Measurement error in geospatial exposures based on prediction models: Measurement error in spatial exposures resulting from prediction models (e.g., air pollution, heat) are complex, as the prediction process typically results in highly correlated measurement error with features of classical and Berkson errors. Existing approaches to adjust for this form of measurement error typically require re-estimation of the exposure model, which may not be feasible for researchers due to data available restrictions.

Latent variable modeling: Latent variable models can be used when the construct of interest is not directly observed but can be inferred from a larger number of variables hypothesized to be caused by the latent variable. In a concrete example, cognitive functioning can be conceptualized as a latent variable, and modeled using observed responses to a battery of cognitive tests. In addition to estimating the measurement structure of cognition or other latent variables, latent variable models can be embedded in a broader structural equation modeling framework. By simultaneous estimating a measurement model for the potentially mismeasured variable and the health effects model in the same modeling framework, this approach can explicitly consider and account for measurement error.

Conclusion

Measurement error in capturing features of the exposome is an underappreciated aspect of exposome research that can have important impacts on research findings. We note that measurement error in other components of health outcome models, including in covariates and in outcomes is beyond the scope of this primer, but can have important (and different) impacts. We also did not consider the impacts of measurement error on more complex model structures, including machine learning models or mixtures models that consider many potentially mismeasured exposures simultaneously, which are important areas for future research investment. Nonetheless, this primer can serve increase awareness and understanding of both basic concepts in measurement error and available approaches that can help researchers quantify the impacts of measurement error in research and correct findings for potential bias due to measurement error.