This paper shows how estimates of means and standard deviations of the approximately log-normal distributions of individual uncertainty factors can be used to estimate percentiles of the distribution of the product of uncertainty factors. 

Acceptable levels of human exposure to noncarcinogenic toxicants in environmental and occupational settings generally are derived by reducing experimental no-observed-adverse-effect levels (NOAELs) or benchmark doses (BDs) by a product of uncertainty factors. The common default value for each uncertainty factor is 10. The author illustrates that an appropriately selected upper percentile, for example, 95th or 99th, of the distribution of the product can be used as a combined uncertainty factor to replace the conventional product of default factors.

This article provides a rationale for the application of uncertainty factors for chronic exposure and illustrates the analysis of the proportion of the population covered by the 10-fold factor for human variability, analysis of special groups (e.g., infants and children, ethnic differences, genetic polymorphisms), analysis of the consequences of intakes in excess of the ADI, TDI or RfD, critical effect and NOAEL and modification of uncertainty factors for determination of acute reference dose values.

This article explains that the establishment of an acute reference dose based on animal studies has to allow for both species differences and inter-individual variability; comparison with the factors used for chronic effects suggests that modification of the usual defaults may be appropriate under certain specific circumstances, but that the usual default of 100 remains appropriate for most cases.

This article uses the case studies of variability in the risks of dying on the ground from a crashing airplane and from the deployment of motor vehicle airbags to demonstrate how better characterization of variability and uncertainty in the risk assessment lead to better risk communication. Analogies to food safety and environmental risks are also discussed. This presentation demonstrates that probabilistic risk assessment has an impact on both risk management and risk communication, and highlights remaining research issues associated with using improved sensitivity and uncertainty analyses in risk assessment.

This paper reviews examples of the application of different methods to the assessment of dietary exposure to food contaminants, including dioxins in seafood, where it was found that the greatest uncertainties relate to toxicity rather than exposure. This article provides an overview of conventional approaches for assessing dietary exposure to contaminants and additives in food. It also provides examples of uncertainty analysis in exposure assessment and also covers uncertainty and the limits of detection. Further work required to implement probabilistic approaches for dietary exposure assessment is also discussed.

This article demonstrates the application of sensitivity analysis to risk assessment models with two-dimensional probabilistic frameworks that distinguish between variability and uncertainty. A microbial food safety process risk (MFSPR) model is used. The process of identifying key controllable inputs and key sources of uncertainty using sensitivity analysis is challenged by typical characteristics of MFSPR models such as nonlinearity, thresholds, interactions, and categorical inputs. Bootstrap simulation is used to quantify uncertainty in ranks of inputs due to sampling error.  In this article, sensitivity analysis results identified inputs related to the storage of ground beef servings at home as the most important. Risk management recommendations are suggested in the form of a consumer advisory for better handling and storage practices.

This article describes uncertainty importance measures as quantitative tools aiming at identifying the contribution of uncertain inputs to output uncertainty. Uncertainty importance application ranges from food safety to hurricane losses. Results and indications an analyst derives depend on the method selected for the study. By means of an example the author shows that output variance does not always reflect a decisionmaker state of knowledge of the inputs. In this article, numerical results demonstrate that both moment-independent and variance-based indicators agree in identifying noninfluential parameters. However, differences in the ranking of the most relevant factors show that inputs that influence variance the most are not necessarily the ones that influence the output uncertainty distribution the most.

This article illustrates that uncertainty in the detection and evaluation of chemical hazards to health leads to challenges when conducting risk assessments. Some of the uncertainty has to do with data, some with incomplete understanding of processes, and some with the most fundamental ways of viewing the questions.

True variability--across space, in time, or among individuals--complicates the search for understanding many important aspects of risk. A few statistical and toxicologic tools are available to assess uncertainty. This article briefly discusses three methods of classifying uncertainty. 

This article articulates the usefulness of modeling and evaluating risk uncertainty and variability separately.  This paper proposes a Bayesian procedure for growth parameter estimation which makes it possible to separate these two components by means of hyperparameters.

This model incorporates in a single step the logistic equation with delay as a primary growth model and the cardinal temperature equation as a secondary growth model.

The estimation of Listeria monocytogenes growth parameters in milk using literature data is proposed as a detailed application.

While this model should be applied on genuine data, it is highlighted that the proposed approach may be convenient for estimating the variability and uncertainty of growth parameters separately, using a complete predictive microbiology model.

The authors of this letter agreed with Vicari et al. that second order modeling is highly recommended in microbial risk assessment, but disagree with the choice of dose-response model by Vicari et al. They also pointed out that, in contrast to what is stated in the introduction, this approach has already been demonstrated in the literature.

This study describes an analytical framework that permits quantitative consideration of variability and uncertainty in microbial hazard characterization. Second-order modeling that used two-dimensional Monte Carlo simulation and stratification into homogeneous population subgroups was applied to integrate uncertainty and variability.  

The results indicate that uncertainty associated with dose-response modeling has a dominating influence on the analytical outcome. In contrast, inclusion of the age factor has a limited impact. While the advocacy of more closely modeling variability in hazard characterization is warranted, the characterization of key sources of uncertainties and their consistent propagation throughout a microbial risk assessment actually appear of greater importance.

Predictive models are now commonly used for exposure assessment, with growth parameters defined for each microbial species. In this study, we tried to take into account microbial growth variability among strains of a single species. Bacillus cereus in pasteurized milk was chosen to illustrate the influence of the biological variability on the outcome of exposure assessment. Each parameter of the exposure assessment (growth parameters, shelf-life conditions) was characterized by a probability distribution describing variability and/or uncertainty. The impact of the intra-species variability on the result of the exposure assessment was then quantified and discussed. Two simple domestic shelf life conditions were tested. The results confirm that the biological variability has a great impact on the accuracy of the result and should not be systematically neglected.

Quantitative risk assessment (QRA) second-order modeling, involving the separation of uncertainty and variability of model parameters, is considered of increasing importance in several fields of risk analysis. In this paper the relevance of second-order modeling in microbial risk assessment is illustrated by a simple example of a risk assessment of growth of B. cereus in pasteurised milk. It shows that the prediction of the outbreak size may depend on the way that uncertainty and variability are separated, and that a major outbreak may be overlooked if the distinction between uncertainty and variability is neglected.

This article introduces the use of two new quantitative risk assessment computing resources (R packages) specifically developed to help risk assessors in their projects. The first package, “fitdistrplus”, gathers tools for choosing and fitting a parametric univariate distribution to a given dataset. The second package, “mc2d”, helps to build and study two dimensional (or second-order) Monte-Carlo simulations in which the estimation of variability and uncertainty in the risk estimates is separated. The usefulness of these packages is illustrated through a risk assessment of hemolytic and uremic syndrome in children linked to the presence of Escherichia coli O157:H7 in ground beef. These R packages are freely available at the Comprehensive R Archive Network (cran.r-project.org).

This software tool, developed for the Office of Research and Development, Environmental Protection Agency, is for use in quantifying variability and uncertainty in quantitative analysis. The tool was presented at the Society for Risk Analysis Annual Meeting in New Orleans, LA, in a workshop entitled "Bootstrap Simulation and Two- Dimensional Monte Carlo Simulation: Dealing with Variability and Uncertainty, Mixture Distributions, Measurement Error, and Censored Data." The software is available to download, as is a user's manual, technical documentation, workshop handouts and slides, and a paper entitled "Quantification of variability and uncertainty using mixture distributions: evaluation of sample size, mixing weights and separation between components"

Report on methods for evaluating uncertainty in risk assessment mathematical equations and computer models. Includes discussion of analytical and numerical methods for error propogation and of methods for identifying the most important contributors to uncertainty

This report was presented at the Conference on International Food Trade Beyond 2000: Science-Based Decisions, Harmonization, Equivalence, and Mutual Recognition, in Melbourne, Australia, October 11-15, 1999. It discusses the importance of scientific advice in the risk analysis process, and includes suggestions on strengthening that process