Modelling unit to unit variation¶
In the basic model for an acute exposure assessment, it is assumed that the concentration of the substance displays the variation of residues between units in the marketplace. In general, both monitoring data and controlled field trial data are obtained using composite samples and, as a result, some of the unit to unit variation is averaged out. The model for unit variability aims to adjust the composite sample mean such that sampled concentrations represent the originally unit to unit variation of the units in the compositie sample.
MCRA offers three distributions to sample from:
the beta distribution,
and the bernoulli distribution.
The beta distribution simulates values for a unit in the composite sample and requires knowledge of the number of units in a composite sample and of the variability between units. The lognormal distribution simulates values for a new unit in the batch and requires only knowledge of the variability between units. The bernoulli distribution is considered as a limiting case of the beta distribution when knowledge of the variability between units is lacking and only the number of units in the composite sample is known. For the beta and lognormal distribution, estimates of unit variability are realistic (no censoring at the value of the monitoring residue) or conservative (unit values are left-censored at the value of the monitoring residue). For the lognormal distribution, sampled concentrations have no upper limit whereas for the beta distribution, sampled concentration values for a unit are never higher than the monitoring residue * the number of units in the composite sample.
Variability between units is specified using a variability factor v (defined as 97.5th percentile divided by mean) or a coefficient of variation cv (standard deviation divided by mean). Following FAO/WHO recommendations, for small crops (unit weight < 25 g), a default variability factor \(v\) = 1 is used, for large crops (unit weight ≥ 25 g), a variability factor \(v\) = 5 is used. For foods which are processed in large batches, e.g. juicing, marmalade/jam, sauce/puree, bulking/blending a variability factor \(v\) = 1 is proposed.
Estimation of intake values using the concept of unit variability¶
A composite sample for food \(k\) is composed of \(nu_{k}\) units with nominal unit weight \(wu_{k}\). The weight of a composite sample is \(wm_{k} = nu_{k} \cdot wu_{k}\) with mean residue value \(cm_{k}\).
For each iteration \(i\) in the MC-simulation, obtain for each food k a simulated intake \(x_{ik}\), and a simulated composite sample concentration \(cm_{ik}\).
Calculate the number of unit intakes \(nux_{ik}\) in \(x_{ik}\) (round upwards) and set weights \(w_{ikl}\) equal to unit weight \(wu_{k}\), except for the last partial intake, which has weight \(w_{ikl} = x_{ik} - (nux_{ik} - 1) wu_{k}\) .
For the beta or bernoulli distribution: draw \(nux_{ik}\) simulated values \(bc_{ikl}\) from a beta or bernoulli distribution. Calculate concentration values as \(c_{ikl} = bc_{ikl} \cdot cm_{ik, max} = bc_{ikl} \cdot cm_{ik} \cdot nu_{k} = \mathit{svf}_{ikl} \cdot cm_{ik}\), where \(nu_{k}\) is the number of units in a composite sample of food \(k\), and \(svf_{ikl}\) is the stochastic variability factor for this simulated unit, i.e. the ratio between simulated concentration \(c_{ikl}\) and the simulated composite sample concentration \(cm_{ik}\). Sum to obtain the simulated concentration in the consumed portion:
For the lognormal distribution: draw \(nux_{ik}\) simulated logconcentration values \(\mathit{lc}_{ikl}\) from a normal distribution with (optional) a biased mean \(\mu = ln(cm_{ik})\) or (default) unbiased mean \(\mu = ln(cm_{ik}) - 1/2 \sigma^2\) and standard deviation \(\sigma\). Calculate concentration values as
where \(\mathit{svf}_{ikl}\) is the stochastic variability factor for this simulated unit, i.e. the ratio between simulated concentration \(c_{ikl}\) and the simulated composite sample concentration \(cm_{ik}\). Back transform and sum to obtain the simulated concentration in the consumed portion:
For cumulative exposure assessments, a sensitivity analysis may be performed by specifying a full correlation between concentrations from different substances on the same unit. As a result, high (or low) concentrations from different substances occur together on the same unit. In MCRA, for each unit the random sequence is repeatedly used to generate concentration values for all substances.