Correcting for measurement error in fractional polynomial models using Bayesian modelling and regression calibration, with an application to alcohol and mortality

Abstract

Exposure measurement error can result in a biased estimate of the association between an exposure and outcome. When the exposure–outcome relationship is linear on the appropriate scale (e.g. linear, logistic) and the measurement error is classical, that is the result of random noise, the result is attenuation of the effect. When the relationship is non-linear, measurement error distorts the true shape of the association. Regression calibration is a commonly used method for correcting for measurement error, in which each individual’s unknown true exposure in the outcome regression model is replaced by its expectation conditional on the error-prone measure and any fully measured covariates. Regression calibration is simple to execute when the exposure is untransformed in the linear predictor of the outcome regression model, but less straightforward when non-linear transformations of the exposure are used. We describe a method for applying regression calibration in models in which a non-linear association is modelled by transforming the exposure using a fractional polynomial model. It is shown that taking a Bayesian estimation approach is advantageous. By use of Markov chain Monte Carlo algorithms, one can sample from the distribution of the true exposure for each individual. Transformations of the sampled values can then be performed directly and used to find the expectation of the transformed exposure required for regression calibration. A simulation study shows that the proposed approach performs well. We apply the method to investigate the relationship between usual alcohol intake and subsequent all-cause mortality using an error model that adjusts for the episodic nature of alcohol consumption.

Generalized measurement error: Intrinsic and incidental measurement error

“…The total standard error of the statistic is the positive square root of the total variance. When the measurement protocol ρ generates only trivial RVVMs (calibrated or not), the total variance collapses to only the first term of (13), which is also the naive sampling variance considered in (12). We can refer to this first component of ( 13) as the classical or sampling variance of the statistic.…”

“…It is important to realize that classical notions of measurement error only capture the type of uncertainty described in the first component of (13). This is a bit counterintuitive given the classical terminology, but can be better understood as follows.…”

“…All expectations here are with respect to the usual sampling distribution of the statistic, so this is really just a decomposition of the first component of (13). Indeed, the second component of the total variance is always zero since the classical framework does not encompass uncertainty due to intrinsic measurement error.…”

“…These can then be used to validate a proposed measurement error model relating X to X* improving inferences dependent on X when using only the observed values of X* on the entire sample [10,11]. Replication studies require two or more error-prone proxies for X which can then be combined to improve inferences dependent on X via some kind of statistical model [12][13][14]. Notably, this paradigm is the one most commonly employed in factor analysis for the study of latent variables, ubiquitous in the scale development and survey literature of the behavioural sciences [15][16][17].…”

STRATOS guidance document on measurement error and misclassification of variables in observational epidemiology: Part 2—More complex methods of adjustment and advanced topics

“…Note, Keogh and White described an MI approach for use in the setting of a replicates study, assuming availability of repeated measures of the error‐prone covariate in some individuals, and assuming classical error. More recently, another approach for the setting of a validation or replicates study has been described based on a modification of the substantive model compatible imputation approach for missing data described by Bartlett et al, and accompanying software is available in R …”