Single value risks from individual risks

In this option, a percentage point can be specified for the chosen risk metric (margin of exposure (\(\operatorname{MOE}\)) or hazard index (\(\operatorname{HI}\))). The corresponding percentile is calculated from the distribution of individual risks. The default percentiles are a margin of exposure at 0.1% or a hazard index at 99.9%. Specify whether the risk metric is calculated using the inverse distribution or not. This option is provided because percentile calculation in small data sets is asymmetric in both tails. When this option is set, the percentile is calculated as the inverse of the complementary percentage of the inverse distribution. E.g., the \(p_{0.1}\) of the \(\operatorname{MOE}\) distribution is calculated as 1/(\(p_{99.9}\) of 1/\(\operatorname{MOE}\) distribution); the \(p_{99.9}\) of the \(\operatorname{HI}\) distribution is calculated as 1/(\(p_{0.1}\) of 1/\(\operatorname{HI}\) distribution).

Adjustment factors and uncertainty specification

Many sources of uncertainty that may affect input data, model assumptions and assessment methodology do not enter the assessment. In EFSA (2020a) and EFSA (2020b), thirty-four sources of uncertainty were identified and the impact of each source on the \(\operatorname{MOE}\) was quantified. Some uncertainties tend to overestimate the \(\operatorname{MOE}\), others tend to underestimate it. Following the guidance of the EFSA Scientific Committee, specific \(\operatorname{MOE}\) and/or \(\operatorname{HI}\) percentiles are adjusted using adjustment factors for exposure and hazard, e.g. from expert elicitation. They may be available as fixed values or as parametric uncertainty distributions. In the nominal run, the percentile is adjusted with the median of the uncertainty distribution. In each uncertainty run, adjustment factors are sampled from the uncertainty distribution. In the MCRA interface, for both exposure and hazard distribution separately, a fixed value or a parametric uncertainty distribution is specified. The available parametric uncertainty distributions are the same as available in the SHELF package that was used by EFSA. The SHeffield ELicitation Framework (SHELF) is a package of documents, templates and software to carry out elicitation of probability distributions for uncertain quantities from a group of experts (http://www.tonyohagan.co.uk/shelf/).

To summarize, in the nominal run, the median of the uncertainty distribution is taken as adjustment factor. In each uncertainty run, the adjustment factor value is sampled from the uncertainty distribution. The adjustment factors for exposure and hazard are multiplied yielding the overall adjustment factor.

Options for specifying uncertainty distributions are:

• Gamma(a, b) with offset c, with shape and rate parameters a and b > 0.

• Beta(a, b) scaled to the interval [\(c\), d], with shape parameters a and b > 0.

• Lognormal(\(\mu\), \(s\)) with offset \(c\). Parameters \(\mu\) and \(s\) specify the mean and standard deviation of the underlying normal.

• Log Student t(\(\mu\), \(s\), \(\nu\)) with offset d. Parameters \(\mu\) and \(s\) specify the mean and standard deviation of the underlying normal, \(\nu\) the degrees of freedom, \(\nu\) > 0.

In the figures below, the consensus distributions of the experts for the combined impact of the quantified uncertainties affecting exposure are shown. When the larger part of the distribution is above 1, it is more likely that resolving the uncertainties will increase the median estimate. The vertical red line marks f = 1, where resolving the uncertainties would not change the calculated estimate.

Figure 109 Scaled gamma (a=3.26, b=3.56, offset=0.9), table 6, EFSA (2020a).

Figure 110 Scaled beta (a=2.37, b=4.26, lowerbound=0.5, upperbound=6), table 7, EFSA (2020a).

Figure 111 Scaled lognormal (\(\mu=0.705\), \(s=0.566\), offset=1), table 8, EFSA (2020b).

Figure 112 Scaled logstudents t (\(\mu=-0.593\), \(s=0.367\), \(\nu=3\), offset=0.5), table 9, EFSA (2020b).

Background-only adjustment factor

When exposures are calculated by combining focal food/substance concentrations with background concentrations, it may be appropriate to have a separate adjustment for the foreground and background. A pragmatic solution agreed with EFSA is to estimate the contribution of the foreground in the tail above the selected percentile. Suppose this contribution is \(c\). Note that \(c\) will vary in uncertainty runs. Then, the adjustment factor should be multiplied by \((1-c)\), i.e. no adjustment for the focal part.

The calculation proceeds as follows:

\[\begin{split}\begin{array}{ll} p_{\operatorname{MOE},\mathtt{adjusted}} & = p_{\operatorname{MOE}} \cdot (c + (1-c) \cdot \mathtt{AdjustmentFactor}_{\mathtt{exposure}} \cdot \mathtt{AdjustmentFactor}_{\mathtt{hazard}} )\\ p_{\operatorname{HI},\mathtt{adjusted}} & = \displaystyle {p_{\operatorname{HI}} \over c + (1-c) \cdot \mathtt{AdjustmentFactor}_{\mathtt{exposure}} \cdot \mathtt{AdjustmentFactor}_{\mathtt{hazard}} } \end{array}\end{split}\]

Note that when the focal substance measurements are converted to active substances using substance conversions or deterministic substance conversions, then \(c\) is the sum of the contributions of the focal food in and all active substances to which the substance translates.

In Figure 113, an example is shown where the margin of exposure is adjusted for the exposure and hazard distribution based on expert elicitation. The median adjustment factors for exposure and hazard are respectively, 1.77 and 3.01. The overall adjustment factor is 5.33.

Figure 113 Margin of exposure (model) and adjusted margin of exposure (model + expert) with uncertainty bounds.