Use of the Bayesian family of methods to correct for effects of exposure measurement error in polynomial regression models

Measurement error in a continuous exposure, if ignored, may cause bias in the estimation of the relationship between exposure and outcome. This presents a significant challenge for understanding exposure-outcome associations in many areas of research, including economic, social, medical and epidemiological research. The presence of classical, i.e. random, measurement error in a continuous exposure has been shown to lead to underestimation of a simple linear relationship. When the functional form of the exposure within a regression model is not linear, i.e. when transformations of the exposure are included, measurement error obscures the true shape of the relationship by making the association appear more linear. Bias in this case will be unknown in direction and vary by exposure level. The most commonly used method for measurement error correction is regression calibration, but this requires an approximation for logistic and survival regression models and does not extend easily to more complex error models. This work investigates three methods for measurement error correction from the Bayesian family of methods: Bayesian analysis using Markov chain Monte Carlo (MCMC), integrated nested Laplace approximations (INLA), and multiple imputation (MI). These have been proposed for measurement error correction but have not been extensively compared, extended for use in several important scenarios, or applied to flexible parametric models. The focus on Bayesian methods was motivated by their flexibility to accommodate complex measurement error models and non-linear exposure-outcome associations. Polynomial regression models are widely used and are often the most interpretable models. In order for measurement error correction methods to be widely implemented, they should be able to accommodate known polynomial transformations as well as model selection procedures when the functional form of the error-prone exposure is unknown. Therefore, in this thesis, correction methods are integrated with the fractional polynomial method, a flexible polynomial model-building procedure for positive continuous variables. In this thesis, I perform a large simulation study comparing proposed methods for measurement error correction from the Bayesian family (i.e. MCMC, INLA, and MI) to the most common method of measurement error correction. Extensions of INLA and MI are presented in order to accommodate both a validation study setting wherein the error-free exposure is measured in a subgroup as well as a replicate study setting wherein there are multiple measures of the error-prone exposure. In order to accommodate unknown polynomial transformations of the error-prone variable, two approaches not used before in this context are proposed and explored in simulation studies alongside more standard methods. The first approach uses Bayesian posterior means in lieu of maximum likelihood estimates within regression calibration. The second approach adapts methods of Bayesian variable selection to the selection of the best polynomial transformation of the error-prone exposure while accommodating measurement error. Successful methods are applied to a motivating example, fitting the non-linear association between alcohol intake and all-cause mortality. By combining measurement error correction adaptable to complex error models with polynomial regression inclusive of model-selection, this work fills a niche which will facilitate wider use of measurement error correction techniques.