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CPS1916 Erica P. et al.
Such formulation was included in the model by adding a latent variable for
the exposure, namely a normally distributed variable with mean equal to 0 and
variance equal to the error variance, and modelling the error variance with a
Berkson framework. The same priors were used as in the classical case.
Another source of imprecision and possible bias in the assessment of the
association between omic signals and TRAP exposure is potentially given by
the formulation of independent models for each omic feature. Dependency
across metabolic features is very likely to occur in practice, first of all because
5749 different features are sampled and analysed from the same 60
individuals, and second because they all reflect metabolic pathways and
phenomena that are highly correlated in each individual. The resulting model
was formulated as follows:

where the response variable followed a multivariate normal distribution
~ (0, ∑ ) , with ∑ denoting the covariance matrix of the omics signals

and all variances were given a gamma prior with shape and scale equal to 0.01,
in consistency with the univariate model.
Finally, the Bayesian hierarchical structures were further generalized to
account for correlation among exposure to different pollutants. The resulting
model was formulated as follows:

3. Result
As expected, the inclusion of a classical measurement error term resulted
in different estimates of the association between omic signals and TRAP
measurements, compared to the naive model which does not include such
term. Note that the presence of classical measurement error in pollutant
measures can cause bias in different directions, and that the effect, as well as
the direction of the error correction, is not evident a priori.
On the other hand, including a Berkson measurement error did not change
the estimates quantitatively, as expected from theory. See Figure (1) for the
distribution of regression coefficients using the three models, using JAGS.

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