Extensions to multiple covariates measured with error
In this thesis, I have focused on the setting wherein a single exposure is measured with error. However, there will be many applications wherein there are multiple exposures measured with error. Even with classical measurement error in linear regression, this scenario can result in bias in any direction if the error is ignored [1,5]. The Bayesian methods outlined in this thesis should all extend, in theory, to accommodate multiple exposures measured with error.
Extension of RC to multiple covariates measures with error would depend on the correlation between the errors [1]. For specific dietary applications, there exists an extension of the EC model to a bivariate model applicable to the estimation of both dietary intake and overall caloric consumption [65]. It is unlikely that RC would be easily extended to many covariates measured with error if there exists a complex correlation structure.
If an additive error model is specified for each covariate measured with error and validation data are available for each, then MI with fully conditional specification (or chained equations) can be used without alteration to its use in the missing data context by treating each covariates as subject to missingness.
Muff et al demonstrated the application of INLA for measurement error correction for two covariates measured with error given a complex measurement error model (a mix of classical error and Berkson error, i.e. 𝑋 = 𝑊 + 𝑈) [152]. It has therefore been demonstrated that this approach can be used for multiple covariates measured with error.
Bayesian analysis with MCMC could undoubtedly accommodate multiple covariates measured with error when the functional form of each latent covariate is known. When the functional form is unknown, the method of Bayesian variable selection that I extended to BTS with measurement error, SSVS, has been previously extended for use in the multivariate setting [61].
181 The Bayesian RC methods may most easily accommodate multiple covariates measured with error whatever the error structure between them. In fact, this approach is most similar to the two- stage analysis often used in environmental epidemiology wherein multiple data sources may be used in the first stage to estimate exposures and these estimated exposures are in the second stage used in the exposure-outcome model [149].
Extensions to survival analysis
All methods presented in this thesis could be extended for use in survival analysis. The general application of RC in the context of survival models has been previously asserted [72,73]; an approximation is required. The Bayesian implementations of RC require no further extension from what was presented in Section 4.4.4 and 6.4.2 to be applied in the context of survival analysis but they require the same approximation as frequentist RC.
When applied with the Cox proportional hazards model, the fractional polynomial method has been found to have a slightly elevated type I error rate in simulation studies [153]. Given this and the need for an approximation, any future work which combines survival analysis, RC (including Bayesian RC), and fractional polynomials should include an evaluation of the type I error. Muff et al demonstrated the use of INLA for measurement error correction for an untransformed error-prone exposure with a survival outcome model [152]. INLA can accommodate survival outcomes through either the Cox proportional hazards model or the Weibull model [154]. While Bayesian analysis may be performed with the Cox proportional hazards model, this has been found to result in very slow convergence of the model [37]. Fully parametric models, such as the Weibull model, may be used as an alternative and results in faster convergence.
8.5 L
IMITATIONSThe shapes of association simulated in this thesis did not include features found to be challenging in the fractional polynomial method. The shapes were either monotonic in the exposure range or had a single maxima or minima. These relatively simple association shapes are examples of the types of shapes best fit by fractional polynomials [155]. More complex non-linear exposure- outcome relationships may require the use of alternative non-parametric methods. Additionally, the range of exposure values in the simulation studies did not include zeros or negative values. While negative values can be transformed to be in the positive range by the addition of a uniform value, the specific choice of value may result in quite disparate model selection introducing additional analytical challenge [8].
Due to the time involved in the convergence of MCMC methods, particularly those incorporating logistic regression, simulation studies in this thesis were limited to 100 – 400 simulated data sets.
182 This limits the extent to which bias and coverage may be assessed. The MC errors for bias were relatively small compared to the bias being addressed by measurement error correction. The MC error for coverage were significantly wider and only moderate to severe under or overcoverage could be distinguished.
In this thesis, only very limited exploration of the effects of misspecification of the error model were explored in a sensitivity analysis for a subset of the methods. While no sensitivity analysis was performed for BTS with measurement error correction wherein the measurement error model was misspecified, it has previously been shown that misspecification of the subsidiary model in a Bayesian joint modelling approach results in more biased estimates than sequential approaches [149,156].