Modelling unit-to-unit variation
The basic model for an acute exposure assessment assumes 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. 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 composite 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. It 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. It 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 either 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 times the number of units in the composite sample.
Variability between units is specified using a variability factor
Estimation of intake values using the concept of unit variability
A composite sample for food
For each iteration
in the MC-simulation, obtain for each food k a simulated intake , and a simulated composite sample concentration .Calculate the number of unit intakes
in (round upwards) and set weights equal to unit weight , except for the last partial intake, which has weight .For the beta or bernoulli distribution: draw
simulated values from a beta or bernoulli distribution. Calculate concentration values as , where is the number of units in a composite sample of food , and is the stochastic variability factor for this simulated unit, i.e. the ratio between simulated concentration and the simulated composite sample concentration . Sum to obtain the simulated concentration in the consumed portion:
For the lognormal distribution: draw
simulated logconcentration values from a normal distribution with (optional) a biased mean or (default) unbiased mean and standard deviation . Calculate concentration values as
where
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.
Beta distribution
Under the beta model simulated unit values are drawn from a bounded distribution on the interval (0,
and the variance
or, as applied in MCRA, by the mean
For the simulated unit values in each iteration of the program we require an expected value
and
From the second formula it can be seen that
instead of a coefficient of variation
for
Sampled values from the beta distribution are rescaled by multiplication with
Lognormal distribution
The lognormal distribution is characterised by
The variability factor is defined as the 97.5th percentile of the concentration in the individual measurements divided by the corresponding mean concentration seen in the composite sample. A variability factor
with
Solving for
with roots for
The smallest positive root is taken as an estimate for
Bernoulli distribution
The bernoulli model is a limiting case of the beta model, which can be used if no information on unit variability is available, but only the number of units in a composite sample is known (see van der Voet et al. 2001).
As a worst case approach we may take the coefficient of variation
or
for the value 0 and probability
or
for the value
In MCRA values 0 are actually replaced by