Chronic intake models¶

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

The observed individual means observed individual means (OIM) model;
The logisticnormal-normal model, in a full version that includes the estimation of correlation between intake frequency and amount (LNN), and in a simpler version without this estimation (LNN0);
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 intake, 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 intake amount on consumption days.

In the final step, both models are integrated in order to obtain the usual intake distribution. For daily consumed foods, fitting of the frequency of consumption is skipped and modelling resorts to fitting the model to daily intake amounts only. Note that the distinction between BNN, LNN and LNN0 disappears and modelling will give equivalent results.

Observed individual means (OIM)¶

The usual intake distribution for a population is estimated with the empirical distribution of individual means. Each mean is the average of all single-day intakes 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 intake distribution and estimates using the OIM-method are biased, leading to a too high estimate of the fraction of the population with a usual intake 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) [3] basic assessments.

Model based and model assisted¶

Following Kipnis et al. [31], some of the models available in MCRA are extended to predict individual usual intakes. This model assisted approach has been added to BBN and LNN0 and may be a useful extension in evaluating the relationship between health outcomes and individual usual intakes of foods. In contrast, the estimation of the usual intake distribution in the general population is called the model based approach. Summarizing, we get Table 75:

Table 75 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 without correlation (LNN0)	logisticnormal-normal without correlation (LNN0)
logisticnormal-normal with correlation (LNN)
Iowa State University Foods (ISUF)

The model assisted approach builds on the proposal of Kipnis et al. [31], 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 intake frequencies. The model-assisted usual intake distribution applies to the population for which the consumption data are representative, and automatically integrates over any covariates present in the model. Model-assisted intakes 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 intake is presented in graphs and tables as a function of the covariates (conditional usual intake distributions).

Betabinomial-Normal model (BBN)¶

The Betabinomial-Normal (BBN) model for chronic risk assessment is described in [16], including its near-dentity to the STEM-II model presented in [40].

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

An alternative to the betabinomial modelling of intake frequencies in BBN model is modelling these frequencies 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 \(\sigma_{between}^2\). This model is referred to as the LogisticNormal-Normal (LNN0) model. The full LNN model model includes the estimation of a correlation between intake frequency and intake amount. This is similar to the NCI model described in Tooze et al. [43]. A simple and computationally less demanding version of the LNN method which does not estimate the correlation between frequency and amount is termed LNN0, where the ‘0’ indicates the absence of correlation. 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. [35] described how to assess chronic risks for data sets with positive intakes (a small fraction of zero intakes was allowed, but then replaced by a small positive value). The modeling 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 intake from data sets with a substantial amount of zero intakes became feasible by modeling separately zero intake on part or all of the days via the estimation of intake probabilities as detailed in Nusser et al. [36] and Dodd [17]. In MCRA, a discrete/semi-parametric model is implemented allowing for zero intake and heterogeneity of variance following the basic ideas of Nusser et al. and Dodd ([35], [36], [17]). This implementation of the ISUF model for chronic risk assessment is fully described in de Boer et al. [16].