#### Pages within the "support" section:

# 15.2 Quantitative evaluation of uncertainties

· Deterministic methods include “worst case” analysis, interval analysis, and sensitivity or scenario analysis. These explore the impact on the assessment of alternative inputs or assumptions, without attempting to quantify their relative probability. They therefore generate a range of possible values for the outcome (e.g. the net health impact), without quantifying their relative likelihood. This approach is used in Tiers 2 and 3 of the Brafo framework (Hoekstra et al. submitted) and can be implemented with the Qalibra framework and software by performing repeat assessments with different inputs to explore alternative assumptions.

· Probabilistic methods go beyond this by using probability distributions to represent the relative likelihood of alternative inputs or assumptions. These distributions are then “propagated” through the assessment, to generate a probability distribution for the outcome (e.g. net health impact) which represents the combined effect of all the quantified uncertainties. Probabilistic methods are included in Brafo Tier 4, and are available as an option for quantifying uncertainty in the Qalibra software.

In probabilistic modelling generally, distributions are used to represent variability (real differences in a parameter, e.g. in the intake for different individuals) and/or uncertainty (due to lack of precise knowledge about the parameter, e.g. due to measurement errors or sampling variation). Distributions for input parameters are “propagated” through the model or assessment, e.g. using Monte Carlo simulation, to generate a probability distribution for the net health impact that represents the combined effect of all the quantified uncertainties.

Monte Carlo simulation repeats the whole assessment calculations thousands of times (until the output distributions stabilise), each time with different parameter values drawn from their distributions. The results from each iteration are combined to form probability distributions for the outputs. These output distributions may be presented graphically, and/or used to put confidence intervals on numerical outputs. They can also be used to derive estimates for the probability of achieving particular outcomes, e.g. the probability that the change in net health impact is positive, or the probability it exceeds some critical level of interest to the policy-maker.

Probabilistic modelling requires a high level of statistical expertise in addition to the other disciplines that are already required for risk-benefit assessment. Probabilistic methods are increasingly used for exposure assessment (e.g. Cullen and Frey 1999, EFSA 2007) and are beginning to be used for risk characterisation (e.g. van der Voet and Slob 2007) and risk-benefit analysis (e.g. van der Voet et al. 2007, www.beneris.eu and this project). Some general principles for Monte Carlo risk assessment have been published by the US EPA (1997).