Chronic exposure models¶

Using the person-day exposures MCRA uses one of the following models to calculate the distribution of usual exposure at the person level:

The observed individual means observed individual means (OIM) model;
The logisticnormal-normal (LNN) model, in a full version that includes the estimation of correlation between exposure frequency and amount, and in a simpler version without this estimation;
The betabinomial-normal (BBN) model;
The discrete/semi-parametric model known as the Iowa State University Foods (ISUF) model. For this model, an equal number of days per individual is assumed.

In modelling usual exposure, two situations can be distinguished. Foods are consumed on a daily basis or foods are episodically consumed. For the logisticnormal-normal model and the betabinomial-normal model, the latter requires fitting of a two-part model,

a model for the frequency of consumption, and
a model for the exposure amount on consumption days.

In the final step, both models are integrated in order to obtain the usual exposure distribution. For daily consumed foods, fitting of the frequency of consumption is skipped and modelling resorts to fitting the model to daily exposure amounts only. Note that the distinction between BNN and LNN disappears and modelling will give equivalent results.

Observed individual means (OIM)¶

The usual exposure distribution for a population is estimated with the empirical distribution of individual means. Each mean is the average of all single-day exposures for an individual. The mean value for an individual still contains a considerable amount of within-individual variation. As a consequence, the distribution of within-individual means has larger variance than the true usual exposure distribution and estimates using the OIM-method are biased, leading to a too high estimate of the fraction of the population with a usual exposure above some standard. Despite its known tendency to over-estimate high-tail exposures, the OIM method is the method to be used in EFSA (2012) [EFSA, 2012] basic assessments.

Model based and model assisted¶

Following Kipnis et al. [Kipnis et al., 2009], some of the models available in MCRA are extended to predict individual usual exposures. This model assisted approach has been added to BBN and LNN when used without correlation) and may be a useful extension in evaluating the relationship between health outcomes and individual usual exposures of foods. In contrast, the estimation of the usual exposure distribution in the general population is called the model based approach. Summarizing, we get Table 85:

Table 85 Model based and assisted approach available for chronic exposure models¶
Model based approach	Model assisted approach
	observed individual means (OIM)
betabinomial-normal (BBN)	betabinomial-normal (BBN)
logisticnormal-normal (LNN) without correlation	logisticnormal-normal (LNN) without correlation
logisticnormal-normal (LNN) with correlation
Iowa State University Foods (ISUF)

The model assisted approach builds on the proposal of Kipnis et al. [Kipnis et al., 2009], but is modified to ensure that the population mean and variance are better represented. The method is based on shrinkage of the observed individual means (modified BLUP estimates) and shrinkage of the observed exposure frequencies. The model-assisted usual exposure distribution applies to the population for which the consumption data are representative, and automatically integrates over any covariates present in the model. Model-assisted exposures are not yet available for LNN, and when a covariable is modelled by a spline function of degree higher than 1. In case of a model with covariates the usual exposure is presented in graphs and tables as a function of the covariates (conditional usual exposure distributions).

Betabinomial-Normal model (BBN)¶

The Betabinomial-Normal (BBN) model for chronic risk assessment is described in [de Boer et al., 2009], including its near-identity to the STEM-II model presented in [Slob, 2006]. The BBN model combines a betabinomial model for the exposure frequencies with a normal model for transformed positive exposures.

Logisticnormal-Normal model (LNN with and without correlation)¶

In the logisticnormal-normal (LNN) model, exposure frequencies are modelled by a logistic normal distribution. In notation, for probability \(p\):

logit(\(p\)) = log(\(p/1-p\)) = \(\mu-{i} + \underline {c}_{i}\)

where \(\mu_{i}\) represents the person specific fixed effect model and \(\underline {c}_{i}\) represent person specific random effects with estimated variance component \(\sigma_{between}^2\). Similarly as in the BBN model, the positive exposure amounts are modelled, after transformation (logarithmic or Box-Cox), with a normal distribution. This model is referred to as the LogisticNormal-Normal (LNN) model. The full LNN model model includes the estimation of a correlation between exposure frequency and exposure amount. This is similar to the NCI model described in Tooze et al. [Tooze et al., 2006]. A simple and computationally less demanding version of the LNN method does not estimate the correlation between frequency and amount. The models are fitted by maximum likelihood, employing Gauss-Hermite integration.

For chronic models amounts are usually transformed before the statistical model is fit. The power transformation, given by \(y^p\), has been replaced by the equivalent Box-Cox transformation. The Box-Cox transformation is a linear function of the power transformation, given by \((y^p-1)/p\), and has a better numerical stability. Gauss-Hermite integration is used for back-transformation (see also Box Cox power transformation).

Discrete/semi-parametric model (ISUF)¶

Nusser et al. [Nusser et al., 1996] described how to assess chronic risks for data sets with positive exposures (a small fraction of zero exposures was allowed, but then replaced by a small positive value). The modelling allowed for heterogeneity of variance, e.g. the concept that some people are more variable than others with respect to their consumption habits. However, a disadvantage of the method was the restricted use to contaminated foods which were consumed on an almost daily basis, e.g. dioxin in fish, meat or diary products. The estimation of usual exposure from data sets with a substantial amount of zero exposures became feasible by modelling separately zero exposure on part or all of the days via the estimation of exposure probabilities as detailed in Nusser et al. [Nusser et al., 1997] and Dodd [Dodd, 1996]. In MCRA, a discrete/semi-parametric model is implemented allowing for zero exposure and heterogeneity of variance following the basic ideas of Nusser et al. and Dodd ([Nusser et al., 1996], [Nusser et al., 1997], [Dodd, 1996]). This implementation of the ISUF model for chronic risk assessment is fully described in de Boer et al. [de Boer et al., 2009].