Sensitivity Analysis of a Two‐Dimensional Quantitative Microbiological Risk Assessment: Keeping Variability and Uncertainty Separated

The aim of quantitative microbiological risk assessment is to estimate the risk of illness caused by the presence of a pathogen in a food type, and to study the impact of interventions. Because of inherent variability and uncertainty, risk assessments are generally conducted stochastically, and if possible it is advised to characterize variability separately from uncertainty. Sensitivity analysis allows to indicate to which of the input variables the outcome of a quantitative microbiological risk assessment is most sensitive. Although a number of methods exist to apply sensitivity analysis to a risk assessment with probabilistic input variables (such as contamination, storage temperature, storage duration, etc.), it is challenging to perform sensitivity analysis in the case where a risk assessment includes a separate characterization of variability and uncertainty of input variables. A procedure is proposed that focuses on the relation between risk estimates obtained by Monte Carlo simulation and the location of pseudo‐randomly sampled input variables within the uncertainty and variability distributions. Within this procedure, two methods are used—that is, an ANOVA‐like model and Sobol sensitivity indices—to obtain and compare the impact of variability and of uncertainty of all input variables, and of model uncertainty and scenario uncertainty. As a case study, this methodology is applied to a risk assessment to estimate the risk of contracting listeriosis due to consumption of deli meats.     

Variability and Uncertainty Meet Risk Management and Risk Communication

In the past decade, the use of probabilistic risk analysis techniques to quantitatively address variability and uncertainty in risks increased in popularity as recommended by the 1994 National Research Council that wrote Science and Judgment in Risk Assessment. Under the 1996 Food Quality Protection Act, for example, the U.S. EPA supported the development of tools that produce distributions of risk demonstrating the variability and/or uncertainty in the results. This paradigm shift away from the use of point estimates creates new challenges for risk managers, who now struggle with decisions about how to use distributions in decision making. The challenges for risk communication, however, have only been minimally explored. This presentation 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.     

Bayesian Hierarchical Structure for Quantifying Population Variability to Inform Probabilistic Health Risk Assessments

Human variability is a very important factor considered in human health risk assessment for protecting sensitive populations from chemical exposure. Traditionally, to account for this variability, an interhuman uncertainty factor is applied to lower the exposure limit. However, using a fixed uncertainty factor rather than probabilistically accounting for human variability can hardly support probabilistic risk assessment advocated by a number of researchers; new methods are needed to probabilistically quantify human population variability. We propose a Bayesian hierarchical model to quantify variability among different populations. This approach jointly characterizes the distribution of risk at background exposure and the sensitivity of response to exposure, which are commonly represented by model parameters. We demonstrate, through both an application to real data and a simulation study, that using the proposed hierarchical structure adequately characterizes variability across different populations.     

The Effect of Forest Management Strategy on Carbon Storage and Revenue in Western Washington: A Probabilistic Simulation of Tradeoffs

The objectives of this study are to understand tradeoffs between forest carbon and timber values, and evaluate the impact of uncertainty in improved forest management (IFM) carbon offset projects to improve forest management decisions. The study uses probabilistic simulation of uncertainty in financial risk for three management scenarios (clearcutting in 45‐ and 65‐year rotations and no harvest) under three carbon price schemes (historic voluntary market prices, cap and trade, and carbon prices set to equal net present value (NPV) from timber‐oriented management). Uncertainty is modeled for value and amount of carbon credits and wood products, the accuracy of forest growth model forecasts, and four other variables relevant to American Carbon Registry methodology. Calculations use forest inventory data from a 1,740 ha forest in western Washington State, using the Forest Vegetation Simulator (FVS) growth model. Sensitivity analysis shows that FVS model uncertainty contributes more than 70% to overall NPV variance, followed in importance by variability in inventory sample (3–14%), and short‐term prices for timber products (8%), while variability in carbon credit price has little influence (1.1%). At regional average land‐holding costs, a no‐harvest management scenario would become revenue‐positive at a carbon credit break‐point price of $14.17/Mg carbon dioxide equivalent (CO₂e). IFM carbon projects are associated with a greater chance of both large payouts and large losses to landowners. These results inform policymakers and forest owners of the carbon credit price necessary for IFM approaches to equal or better the business‐as‐usual strategy, while highlighting the magnitude of financial risk and reward through probabilistic simulation.     

The Impact of Portfolio Location Uncertainty on Probabilistic Seismic Risk Analysis

Probabilistic seismic risk analysis is a well‐established method in the insurance industry for modeling portfolio losses from earthquake events. In this context, precise exposure locations are often unknown. However, so far, location uncertainty has not been in the focus of a large amount of research. In this article, we propose a novel framework for treatment of location uncertainty. As a case study, a large number of synthetic portfolios resembling typical real‐world cases were created. We investigate the effect of portfolio characteristics such as value distribution, portfolio size, or proportion of risk items with unknown coordinates on the variability of loss frequency estimations. The results indicate that due to loss aggregation effects and spatial hazard variability, location uncertainty in isolation and in conjunction with ground motion uncertainty can induce significant variability to probabilistic loss results, especially for portfolios with a small number of risks. After quantifying its effect, we conclude that location uncertainty should not be neglected when assessing probabilistic seismic risk, but should be treated stochastically and the resulting variability should be visualized and interpreted carefully.     

Exposure to Persistent Organochlorines in Canadian Breast Milk: A Probabilistic Assessment

Exposure to persistent organochlorines in breast milk was estimated probabilistically for Canadian infants. Noncancer health effects were evaluated by comparing the predicted exposure distributions to published guidance values. For chemicals identified as potential human carcinogens, cancer risks were evaluated using standard methodology typically applied in Canada, as well as an alternative method developed under the Canadian Environmental Protection Act. Potential health risks associated with exposure to persistent organochlorines were quantitatively and qualitatively weighed against the benefits of breast-feeding. Current levels of the majority of contaminants identified in Canadian breast milk do not pose unacceptable risks to infants. Benefits of breast-feeding are well documented and qualitatively appear to outweigh potential health concerns associated with organochlorine exposure. Furthermore, the risks of mortality from not breast-feeding estimated by Rogan and colleagues exceed the theoretical cancer risks estimated for infant exposure to potential carcinogens in Canadian breast milk. Although levels of persistent compounds have been declining in Canadian breast milk, potentially significant risks were estimated for exposure to polychlorinated biphenyls, dibenzo-p-dioxins, and dibenzofurans. Follow-up work is suggested that would involve the use of a physiologically based toxicokinetic model with probabilistic inputs to predict dioxin exposure to the infant. A more detailed risk analysis could be carried out by coupling the exposure estimates with a dose–response analysis that accounts for uncertainty.     

A Bayesian Approach to Integrated Ecological and Human Health Risk Assessment for the South River,Virginia Mercury‐Contaminated Site

We conducted a regional‐scale integrated ecological and human health risk assessment by applying the relative risk model with Bayesian networks (BN‐RRM) to a case study of the South River, Virginia mercury‐contaminated site. Risk to four ecological services of the South River (human health, water quality, recreation, and the recreational fishery) was evaluated using a multiple stressor–multiple endpoint approach. These four ecological services were selected as endpoints based on stakeholder feedback and prioritized management goals for the river. The BN‐RRM approach allowed for the calculation of relative risk to 14 biotic, human health, recreation, and water quality endpoints from chemical and ecological stressors in five risk regions of the South River. Results indicated that water quality and the recreational fishery were the ecological services at highest risk in the South River. Human health risk for users of the South River was low relative to the risk to other endpoints. Risk to recreation in the South River was moderate with little spatial variability among the five risk regions. Sensitivity and uncertainty analysis identified stressors and other parameters that influence risk for each endpoint in each risk region. This research demonstrates a probabilistic approach to integrated ecological and human health risk assessment that considers the effects of chemical and ecological stressors across the landscape.     

Integration of Probabilistic Exposure Assessment and Probabilistic Hazard Characterization

A method is proposed for integrated probabilistic risk assessment where exposure assessment and hazard characterization are both included in a probabilistic way. The aim is to specify the probability that a random individual from a defined (sub)population will have an exposure high enough to cause a particular health effect of a predefined magnitude, the critical effect size ( CES ). The exposure level that results in exactly that CES in a particular person is that person's individual critical effect dose ( ICED ). Individuals in a population typically show variation, both in their individual exposure ( IEXP ) and in their ICED . Both the variation in IEXP and the variation in ICED are quantified in the form of probability distributions. Assuming independence between both distributions, they are combined (by Monte Carlo) into a distribution of the individual margin of exposure ( IMoE ). The proportion of the IMoE distribution below unity is the probability of critical exposure ( PoCE ) in the particular (sub)population. Uncertainties involved in the overall risk assessment (i.e., both regarding exposure and effect assessment) are quantified using Monte Carlo and bootstrap methods. This results in an uncertainty distribution for any statistic of interest, such as the probability of critical exposure ( PoCE ). The method is illustrated based on data for the case of dietary exposure to the organophosphate acephate. We present plots that concisely summarize the probabilistic results, retaining the distinction between variability and uncertainty. We show how the relative contributions from the various sources of uncertainty involved may be quantified.     

Exploring the Uncertainties in Cancer Risk Assessment Using the Integrated Probabilistic Risk Assessment (IPRA) Approach

Current methods for cancer risk assessment result in single values, without any quantitative information on the uncertainties in these values. Therefore, single risk values could easily be overinterpreted. In this study, we discuss a full probabilistic cancer risk assessment approach in which all the generally recognized uncertainties in both exposure and hazard assessment are quantitatively characterized and probabilistically evaluated, resulting in a confidence interval for the final risk estimate. The methodology is applied to three example chemicals (aflatoxin, N‐nitrosodimethylamine, and methyleugenol). These examples illustrate that the uncertainty in a cancer risk estimate may be huge, making single value estimates of cancer risk meaningless. Further, a risk based on linear extrapolation tends to be lower than the upper 95% confidence limit of a probabilistic risk estimate, and in that sense it is not conservative. Our conceptual analysis showed that there are two possible basic approaches for cancer risk assessment, depending on the interpretation of the dose‐incidence data measured in animals. However, it remains unclear which of the two interpretations is the more adequate one, adding an additional uncertainty to the already huge confidence intervals for cancer risk estimates.     

Robustness of Risk Maps and Survey Networks to Knowledge Gaps About a New Invasive Pest

In pest risk assessment it is frequently necessary to make management decisions regarding emerging threats under severe uncertainty. Although risk maps provide useful decision support for invasive alien species, they rarely address knowledge gaps associated with the underlying risk model or how they may change the risk estimates. Failure to recognize uncertainty leads to risk‐ignorant decisions and miscalculation of expected impacts as well as the costs required to minimize these impacts. Here we use the information gap concept to evaluate the robustness of risk maps to uncertainties in key assumptions about an invading organism. We generate risk maps with a spatial model of invasion that simulates potential entries of an invasive pest via international marine shipments, their spread through a landscape, and establishment on a susceptible host. In particular, we focus on the question of how much uncertainty in risk model assumptions can be tolerated before the risk map loses its value. We outline this approach with an example of a forest pest recently detected in North America, Sirex noctilio Fabricius. The results provide a spatial representation of the robustness of predictions of S. noctilio invasion risk to uncertainty and show major geographic hotspots where the consideration of uncertainty in model parameters may change management decisions about a new invasive pest. We then illustrate how the dependency between the extent of uncertainties and the degree of robustness of a risk map can be used to select a surveillance network design that is most robust to knowledge gaps about the pest.     

Sensitivity Analysis and Estimation of Extreme Tail Behavior in Two‐Dimensional Monte Carlo Simulation

Two‐dimensional Monte Carlo simulation is frequently used to implement probabilistic risk models, as it allows for uncertainty and variability to be quantified separately. In many cases, we are interested in the proportion of individuals from a variable population exceeding a critical threshold, together with uncertainty about this proportion. In this article we introduce a new method that can accurately estimate these quantities much more efficiently than conventional algorithms. We also show how those model parameters having the greatest impact on the probabilities of rare events can be quickly identified via this method. The algorithm combines elements from well‐established statistical techniques in extreme value theory and Bayesian analysis of computer models. We demonstrate the practical application of these methods with a simple example, in which the true distributions are known exactly, and also with a more realistic model of microbial contamination of milk with seven parameters. For the latter, sensitivity analysis (SA) is shown to identify the two inputs explaining the majority of variation in distribution tail behavior. In the subsequent prediction of probabilities of large contamination events, similar results are obtained using the new approach taking 43 seconds or the conventional simulation that requires more than 3 days.     

Importance of Uncertainty and Variability to Predicted Risks from Trophic Transfer of PCBs in Dredged Sediments

Biomagnification of organochlorine and other persistent organic contaminants by higher trophic level organisms represents one of the most significant sources of uncertainty and variability in evaluating potential risks associated with disposal of dredged materials. While it is important to distinguish between population variability (e.g., true population heterogeneity in fish weight, and lipid content) and uncertainty (e.g., measurement error), they can be operationally difficult to define separately in probabilistic estimates of human health and ecological risk. We propose a disaggregation of uncertain and variable parameters based on: (1) availability of supporting data; (2) the specific management and regulatory context (in this case, of the U.S. Army Corps of Engineers/U.S. Environmental Protection Agency tiered approach to dredged material management); and (3) professional judgment and experience in conducting probabilistic risk assessments. We describe and quantitatively evaluate several sources of uncertainty and variability in estimating risk to human health from trophic transfer of polychlorinated biphenyls (PCBs) using a case study of sediments obtained from the New York-New Jersey Harbor and being evaluated for disposal at an open water off-shore disposal site within the northeast region. The estimates of PCB concentrations in fish and dietary doses of PCBs to humans ingesting fish are expressed as distributions of values, of which the arithmetic mean or mode represents a particular fractile. The distribution of risk values is obtained using a food chain biomagnification model developed by Gobas by specifying distributions for input parameters disaggregated to represent either uncertainty or variability. Only those sources of uncertainty that could be quantified were included in the analysis. Results for several different two-dimensional Latin Hypercube analyses are provided to evaluate the influence of the uncertain versus variable disaggregation of model parameters. The analysis suggests that variability in human exposure parameters is greater than the uncertainty bounds on any particular fractile, given the described assumptions.     

Harnessing the Theoretical Foundations of the Exponential and Beta‐Poisson Dose‐Response Models to Quantify Parameter Uncertainty Using Markov Chain Monte Carlo

Dose‐response models are the essential link between exposure assessment and computed risk values in quantitative microbial risk assessment, yet the uncertainty that is inherent to computed risks because the dose‐response model parameters are estimated using limited epidemiological data is rarely quantified. Second‐order risk characterization approaches incorporating uncertainty in dose‐response model parameters can provide more complete information to decisionmakers by separating variability and uncertainty to quantify the uncertainty in computed risks. Therefore, the objective of this work is to develop procedures to sample from posterior distributions describing uncertainty in the parameters of exponential and beta‐Poisson dose‐response models using Bayes's theorem and Markov Chain Monte Carlo (in OpenBUGS). The theoretical origins of the beta‐Poisson dose‐response model are used to identify a decomposed version of the model that enables Bayesian analysis without the need to evaluate Kummer confluent hypergeometric functions. Herein, it is also established that the beta distribution in the beta‐Poisson dose‐response model cannot address variation among individual pathogens, criteria to validate use of the conventional approximation to the beta‐Poisson model are proposed, and simple algorithms to evaluate actual beta‐Poisson probabilities of infection are investigated. The developed MCMC procedures are applied to analysis of a case study data set, and it is demonstrated that an important region of the posterior distribution of the beta‐Poisson dose‐response model parameters is attributable to the absence of low‐dose data. This region includes beta‐Poisson models for which the conventional approximation is especially invalid and in which many beta distributions have an extreme shape with questionable plausibility.     

Confronting Deep Uncertainties in Risk Analysis

How can risk analysts help to improve policy and decision making when the correct probabilistic relation between alternative acts and their probable consequences is unknown? This practical challenge of risk management with model uncertainty arises in problems from preparing for climate change to managing emerging diseases to operating complex and hazardous facilities safely. We review constructive methods for robust and adaptive risk analysis under deep uncertainty. These methods are not yet as familiar to many risk analysts as older statistical and model‐based methods, such as the paradigm of identifying a single “best‐fitting” model and performing sensitivity analyses for its conclusions. They provide genuine breakthroughs for improving predictions and decisions when the correct model is highly uncertain. We demonstrate their potential by summarizing a variety of practical risk management applications.     

Uncertain Numbers and Uncertainty in the Selection of Input Distributions—Consequences for a Probabilistic Risk Assessment of Contaminated Land

Risks from exposure to contaminated land are often assessed with the aid of mathematical models. The current probabilistic approach is a considerable improvement on previous deterministic risk assessment practices, in that it attempts to characterize uncertainty and variability. However, some inputs continue to be assigned as precise numbers, while others are characterized as precise probability distributions. Such precision is hard to justify, and we show in this article how rounding errors and distribution assumptions can affect an exposure assessment. The outcome of traditional deterministic point estimates and Monte Carlo simulations were compared to probability bounds analyses. Assigning all scalars as imprecise numbers (intervals prescribed by significant digits) added uncertainty to the deterministic point estimate of about one order of magnitude. Similarly, representing probability distributions as probability boxes added several orders of magnitude to the uncertainty of the probabilistic estimate. This indicates that the size of the uncertainty in such assessments is actually much greater than currently reported. The article suggests that full disclosure of the uncertainty may facilitate decision making in opening up a negotiation window. In the risk analysis process, it is also an ethical obligation to clarify the boundary between the scientific and social domains.     

A Critical Discussion and Practical Recommendations on Some Issues Relevant to the Nonprobabilistic Treatment of Uncertainty in Engineering Risk Assessment

Models for the assessment of the risk of complex engineering systems are affected by uncertainties due to the randomness of several phenomena involved and the incomplete knowledge about some of the characteristics of the system. The objective of this article is to provide operative guidelines to handle some conceptual and technical issues related to the treatment of uncertainty in risk assessment for engineering practice. In particular, the following issues are addressed: (1) quantitative modeling and representation of uncertainty coherently with the information available on the system of interest; (2) propagation of the uncertainty from the input(s) to the output(s) of the system model; (3) (Bayesian) updating as new information on the system becomes available; and (4) modeling and representation of dependences among the input variables and parameters of the system model. Different approaches and methods are recommended for efficiently tackling each of issues (1)?(4) above; the tools considered are derived from both classical probability theory as well as alternative, nonfully probabilistic uncertainty representation frameworks (e.g., possibility theory). The recommendations drawn are supported by the results obtained in illustrative applications of literature.     

A Risk Assessment Model for Enterotoxigenic Staphylococcus aureus in Pasteurized Milk: A Potential Route to Source‐Level Inference

This article describes a probabilistic model that quantifies hazards that arise from Staphylococcus aureus in milk that is sold as pasteurized in the United Kingdom. The model is centered on coupled dynamics for S. aureus populations, staphylococcal enterotoxins, and the concentration of alkaline phosphatase throughout the milk chain. The chain includes farm collection and storage of pooled milk, further pooling for off‐farm processing, high temperature short time thermal processing, and possible postprocess contamination. The model is implemented as a Bayesian belief network. The results indicate that milk sold as pasteurized is relatively safe with respect to the hazards associated with S. aureus and that most risk is associated with small scale on‐farm processing. An additional analysis of likelihood ratios shows that alkaline phosphatase concentrations in filler tank milk are a good indicator of potential hazards and that these concentrations, in conjunction with other measurements, can be used effectively to discriminate over possible failure modes. The ability to discriminate over potential failure modes can support preemptive actions, such as maintenance or hygiene, which assist with milk chain management and, over extended periods, accumulate to drive improved safety, efficiency, and security.     

Science and Decisions: Advancing Risk Assessment

At the request of the U.S. Environmental Protection Agency (EPA), the National Research Council (NRC) recently completed a major report, Science and Decisions: Advancing Risk Assessment, that is intended to strengthen the scientific basis, credibility, and effectiveness of risk assessment practices and subsequent risk management decisions. The report describes the challenges faced by risk assessment and the need to consider improvements in both the technical analyses of risk assessments (i.e., the development and use of scientific information to improve risk characterization) and the utility of risk assessments (i.e., making assessments more relevant and useful for risk management decisions). The report tackles a number of topics relating to improvements in the process, including the design and framing of risk assessments, uncertainty and variability characterization, selection and use of defaults, unification of cancer and noncancer dose‐response assessment, cumulative risk assessment, and the need to increase EPA's capacity to address these improvements. This article describes and summarizes the NRC report, with an eye toward its implications for risk assessment practices at EPA.     

Model uncertainty and choices made by modelers: lessons learned from the International Atomic Energy Agency model intercomparisons.

The treatment of uncertainties associated with modeling and risk assessment has recently attracted significant attention. The methodology and guidance for dealing with parameter uncertainty have been fairly well developed and quantitative tools such as Monte Carlo modeling are often recommended. However, the issue of model uncertainty is still rarely addressed in practical applications of risk assessment. The use of several alternative models to derive a range of model outputs or risks is one of a few available techniques. This article addresses the often-overlooked issue of what we call "modeler uncertainty," i.e., difference in problem formulation, model implementation, and parameter selection originating from subjective interpretation of the problem at hand. This study uses results from the Fruit Working Group, which was created under the International Atomic Energy Agency (IAEA) BIOMASS program (BIOsphere Modeling and ASSessment). Model-model and model-data intercomparisons reviewed in this study were conducted by the working group for a total of three different scenarios. The greatest uncertainty was found to result from modelers' interpretation of scenarios and approximations made by modelers. In scenarios that were unclear for modelers, the initial differences in model predictions were as high as seven orders of magnitude. Only after several meetings and discussions about specific assumptions did the differences in predictions by various models merge. Our study shows that parameter uncertainty (as evaluated by a probabilistic Monte Carlo assessment) may have contributed over one order of magnitude to the overall modeling uncertainty. The final model predictions ranged between one and three orders of magnitude, depending on the specific scenario. This study illustrates the importance of problem formulation and implementation of an analytic-deliberative process in risk characterization.     

Probabilistic Framework for the Estimation of the Adult and Child Toxicokinetic Intraspecies Uncertainty Factors

Human health risk assessments use point values to develop risk estimates and thus impart a deterministic character to risk, which, by definition, is a probability phenomenon. The risk estimates are calculated based on individuals and then, using uncertainty factors (UFs), are extrapolated to the population that is characterized by variability. Regulatory agencies have recommended the quantification of the impact of variability in risk assessments through the application of probabilistic methods. In the present study, a framework that deals with the quantitative analysis of uncertainty (U) and variability (V) in target tissue dose in the population was developed by applying probabilistic analysis to physiologically-based toxicokinetic models. The mechanistic parameters that determine kinetics were described with probability density functions (PDFs). Since each PDF depicts the frequency of occurrence of all expected values of each parameter in the population, the combined effects of multiple sources of U/V were accounted for in the estimated distribution of tissue dose in the population, and a unified (adult and child) intraspecies toxicokinetic uncertainty factor UFH-TK was determined. The results show that the proposed framework accounts effectively for U/V in population toxicokinetics. The ratio of the 95th percentile to the 50th percentile of the annual average concentration of the chemical at the target tissue organ (i.e., the UFH-TK) varies with age. The ratio is equivalent to a unified intraspecies toxicokinetic UF, and it is one of the UFs by which the NOAEL can be divided to obtain the RfC/RfD. The 10-fold intraspecies UF is intended to account for uncertainty and variability in toxicokinetics (3.2x) and toxicodynamics (3.2x). This article deals exclusively with toxicokinetic component of UF. The framework provides an alternative to the default methodology and is advantageous in that the evaluation of toxicokinetic variability is based on the distribution of the effective target tissue dose, rather than applied dose. It allows for the replacement of the default adult and children intraspecies UF with toxicokinetic data-derived values and provides accurate chemical-specific estimates for their magnitude. It shows that proper application of probability and toxicokinetic theories can reduce uncertainties when establishing exposure limits for specific compounds and provide better assurance that established limits are adequately protective. It contributes to the development of a probabilistic noncancer risk assessment framework and will ultimately lead to the unification of cancer and noncancer risk assessment methodologies.