In epidemiology many exposures of interest are measured with error. Random, or 'classical', error in exposure measurements attenuates linear exposure-disease associations. However, its precise effects on different nonlinear associations are not well known. We use simulation studies to assess how classical measurement error affects observed association shapes and power to detect nonlinearity. We focus on a proportional hazards model for the exposure-disease association and consider six true association shapes of relevance in epidemiology: linear, threshold, U-shaped, Jshaped, increasing quadratic, asymptotic. The association shapes are modeled using three popular methods: grouped exposure analyses, fractional polynomials, P-splines. Under each true association shape and each method we illustrate the effects of classical exposure measurement error, considering varying degrees of random error. We also assess what we refer to as MacMahon's method for correcting for classical exposure measurement error under grouped exposure analyses, which uses replicate measurements to estimate usual exposure within observed exposure groups. The validity of this method for nonlinear associations has not previously been investigated. Under nonlinear exposure-disease associations, classical measurement error results in increasingly linear shapes and not always an attenuated association at a given exposure level. Fractional polynomials and P-splines give similar results and offer advantages over grouped exposure analyses by providing realistic models. P-splines offer greatest power to detect nonlinearity, however random exposure measurement error results in a potential considerable loss of power to detect nonlinearity under all methods. MacMahon's method performs well for quadratic associations, but does not in general recover nonlinear shapes.

To investigate the association between a continuous exposure and an outcome it is common to categorize the exposure and estimate the relative associations between categories. Error in measurement of the continuous exposure results in misclassification when the exposure is categorized. In this paper we investigate methods for correcting for this misclassification. We consider applications of methods for continuous exposures and for fundamentally categorical exposures. A particular challenge is that even nondifferential error in the underlying continuous exposure can result in differential misclassification in the categorized exposure, i.e. misclassification dependent on the outcome. For continuous exposures, there exist a range of methods for correcting for the effects of exposure measurement error on the exposure-outcome association, including regression calibration (RC), multiple imputation (MI), moment reconstruction (MR) and simulation extrapolation (SIMEX). There are also correction methods for use with genuinely categorical exposures, using estimated misclassification probabilities. Alongside simple methods using estimated misclassification probabilities, we also consider two RC based methods, MI and MR of the continuous exposure followed by categorization, and a new SIMEX method. Simulation studies are used to compare the methods when the true exposure is available in a validation study and the more common situation in which replicate or additional error-prone exposure measurements are available in a subsample. We restrict attention to the case where the underlying association between the continuous exposure and the outcome is linear on the appropriate scale. RC and SIMEX methods fail to correct adequately for bias. However, MI and MR perform well. Methods using estimated misclassification probabilities also perform well, provided differential misclassification is assumed, however these methods are restricted to estimation of odds ratios and have other practical drawbacks. MI and MR have the benefit of being flexible for use with different analysis models, with quantile-based cutpoints, and more easily accommodate covariate adjustment. In summary, we found that MI and MR can be applied to correct exposureoutcome associations for the effects of misclassification error when the association is linear. Extending MI and MR for use with categorized continuous exposures under nonlinear exposureoutcome associations is now an important area for further research.

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