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.     

Uncertainty Analysis Based on Probability Bounds (P-Box) Approach in Probabilistic Safety Assessment

A wide range of uncertainties will be introduced inevitably during the process of performing a safety assessment of engineering systems. The impact of all these uncertainties must be addressed if the analysis is to serve as a tool in the decision-making process. Uncertainties present in the components (input parameters of model or basic events) of model output are propagated to quantify its impact in the final results. There are several methods available in the literature, namely, method of moments, discrete probability analysis, Monte Carlo simulation, fuzzy arithmetic, and Dempster-Shafer theory. All the methods are different in terms of characterizing at the component level and also in propagating to the system level. All these methods have different desirable and undesirable features, making them more or less useful in different situations. In the probabilistic framework, which is most widely used, probability distribution is used to characterize uncertainty. However, in situations in which one cannot specify (1) parameter values for input distributions, (2) precise probability distributions (shape), and (3) dependencies between input parameters, these methods have limitations and are found to be not effective. In order to address some of these limitations, the article presents uncertainty analysis in the context of level-1 probabilistic safety assessment (PSA) based on a probability bounds (PB) approach. PB analysis combines probability theory and interval arithmetic to produce probability boxes (p-boxes), structures that allow the comprehensive propagation through calculation in a rigorous way. A practical case study is also carried out with the developed code based on the PB approach and compared with the two-phase Monte Carlo simulation results.     

Exposure Assessment: Then, Now, and Quantum Leaps in the Future

Health risk assessments have become so widely accepted in the United States that their conclusions are a major factor in many environmental decisions. Although the risk assessment paradigm is 10 years old, the basic risk assessment process has been used by certain regulatory agencies for nearly 40 years. Each of the four components of the paradigm has undergone significant refinements, particularly during the last 5 years. A recent step in the development of the exposure assessment component can be found in the 1992 EPA Guidelines for Exposure Assessment. Rather than assuming worst-case or hypothetical maximum exposures, these guidelines are designed to lead to an accurate characterization, making use of a number of scientific advances. Many exposure parameters have become better defined, and more sensitive techniques now exist for measuring concentrations of contaminants in the environnment. Statistical procedures for characterizing variability, using Monte Carlo or similar approaches, eliminate the need to select point estimates for all individual exposure parameters. These probabilistic models can more accurately characterize the full range of exposures that may potentially be encountered by a given population at a particular site, reducing the need to select highly conservative values to account for this form of uncertainty in the exposure estimate. Lastly, our awareness of the uncertainties in the exposure assessment as well as our knowledge as to how best to characterize them will almost certainly provide evaluations that will be more credible and, therein, more useful to risk managers. If these refinements are incorporated into future exposure assessments, it is likely that our resources will be devoted to problems that, when resolved, will yield the largest improvement in public health.     

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.     

A Probabilistic Arsenic Exposure Assessment for Children Who Contact Chromated Copper Arsenate (CCA)-Treated Playsets and Decks, Part 2: Sensitivity and Uncertainty Analyses

A probabilistic model (SHEDS-Wood) was developed to examine children's exposure and dose to chromated copper arsenate (CCA)-treated wood, as described in Part 1 of this two-part article. This Part 2 article discusses sensitivity and uncertainty analyses conducted to assess the key model inputs and areas of needed research for children's exposure to CCA-treated playsets and decks. The following types of analyses were conducted: (1) sensitivity analyses using a percentile scaling approach and multiple stepwise regression; and (2) uncertainty analyses using the bootstrap and two-stage Monte Carlo techniques. The five most important variables, based on both sensitivity and uncertainty analyses, were: wood surface residue-to-skin transfer efficiency; wood surface residue levels; fraction of hand surface area mouthed per mouthing event; average fraction of nonresidential outdoor time a child plays on/around CCA-treated public playsets; and frequency of hand washing. In general, there was a factor of 8 for the 5th and 95th percentiles and a factor of 4 for the 50th percentile in the uncertainty of predicted population dose estimates due to parameter uncertainty. Data were available for most of the key model inputs identified with sensitivity and uncertainty analyses; however, there were few or no data for some key inputs. To evaluate and improve the accuracy of model results, future measurement studies should obtain longitudinal time-activity diary information on children, spatial and temporal measurements of residue and soil concentrations on or near CCA-treated playsets and decks, and key exposure factors. Future studies should also address other sources of uncertainty in addition to parameter uncertainty, such as scenario and model uncertainty.     

Health Risk Characterization of Chlorpyrifos Using Epidemiological Dose‐Response Data and Probabilistic Techniques: A Case Study with Rice Farmers in Vietnam

Various methods for risk characterization have been developed using probabilistic approaches. Data on Vietnamese farmers are available for the comparison of outcomes for risk characterization using different probabilistic methods. This article addresses the health risk characterization of chlorpyrifos using epidemiological dose‐response data and probabilistic techniques obtained from a case study with rice farmers in Vietnam. Urine samples were collected from farmers and analyzed for trichloropyridinol (TCP), which was converted into absorbed daily dose of chlorpyrifos. Adverse health response doses due to chlorpyrifos exposure were collected from epidemiological studies to develop dose‐adverse health response relationships. The health risk of chlorpyrifos was quantified using hazard quotient (HQ), Monte Carlo simulation (MCS), and overall risk probability (ORP) methods. With baseline (prior to pesticide spraying) and lifetime exposure levels (over a lifetime of pesticide spraying events), the HQ ranged from 0.06 to 7.1. The MCS method indicated less than 0.05% of the population would be affected while the ORP method indicated that less than 1.5% of the population would be adversely affected. With postapplication exposure levels, the HQ ranged from 1 to 32.5. The risk calculated by the MCS method was that 29% of the population would be affected, and the risk calculated by ORP method was 33%. The MCS and ORP methods have advantages in risk characterization due to use of the full distribution of data exposure as well as dose response, whereas HQ methods only used the exposure data distribution. These evaluations indicated that single‐event spraying is likely to have adverse effects on Vietnamese rice farmers.     

Propagation of Uncertainty in Risk Assessments: The Need to Distinguish Between Uncertainty Due to Lack of Knowledge and Uncertainty Due to Variability

In quantitative uncertainty analysis, it is essential to define rigorously the endpoint or target of the assessment. Two distinctly different approaches using Monte Carlo methods are discussed: (1) the end point is a fixed but unknown value (e.g., the maximally exposed individual, the average individual, or a specific individual) or (2) the end point is an unknown distribution of values (e.g., the variability of exposures among unspecified individuals in the population). In the first case, values are sampled at random from distributions representing various "degrees of belief" about the unknown "fixed" values of the parameters to produce a distribution of model results. The distribution of model results represents a subjective confidence statement about the true but unknown assessment end point. The important input parameters are those that contribute most to the spread in the distribution of the model results. In the second case, Monte Carlo calculations are performed in two dimensions producing numerous alternative representations of the true but unknown distribution. These alternative distributions permit subject confidence statements to be made from two perspectives: (1) for the individual exposure occurring at a specified fractile of the distribution or (2) for the fractile of the distribution associated with a specified level of individual exposure. The relative importance of input parameters will depend on the fractile or exposure level of interest. The quantification of uncertainty for the simulation of a true but unknown distribution of values represents the state-of-the-art in assessment modeling.     

Comparative Probabilistic Assessment of Occupational Pesticide Exposures Based on Regulatory Assessments

Implementation of probabilistic analyses in exposure assessment can provide valuable insight into the risks of those at the extremes of population distributions, including more vulnerable or sensitive subgroups. Incorporation of these analyses into current regulatory methods for occupational pesticide exposure is enabled by the exposure data sets and associated data currently used in the risk assessment approach of the Environmental Protection Agency (EPA). Monte Carlo simulations were performed on exposure measurements from the Agricultural Handler Exposure Database and the Pesticide Handler Exposure Database along with data from the Exposure Factors Handbook and other sources to calculate exposure rates for three different neurotoxic compounds (azinphos methyl, acetamiprid, emamectin benzoate) across four pesticide‐handling scenarios. Probabilistic estimates of doses were compared with the no observable effect levels used in the EPA occupational risk assessments. Some percentage of workers were predicted to exceed the level of concern for all three compounds: 54% for azinphos methyl, 5% for acetamiprid, and 20% for emamectin benzoate. This finding has implications for pesticide risk assessment and offers an alternative procedure that may be more protective of those at the extremes of exposure than the current approach.     

Feasibility Study on the Use of Probabilistic Migration Modeling in Support of Exposure Assessment from Food Contact Materials

The use of probabilistic approaches in exposure assessments of contaminants migrating from food packages is of increasing interest but the lack of concentration or migration data is often referred as a limitation. Data accounting for the variability and uncertainty that can be expected in migration, for example, due to heterogeneity in the packaging system, variation of the temperature along the distribution chain, and different time of consumption of each individual package, are required for probabilistic analysis. The objective of this work was to characterize quantitatively the uncertainty and variability in estimates of migration. A Monte Carlo simulation was applied to a typical solution of the Fick's law with given variability in the input parameters. The analysis was performed based on experimental data of a model system (migration of Irgafos 168 from polyethylene into isooctane) and illustrates how important sources of variability and uncertainty can be identified in order to refine analyses. For long migration times and controlled conditions of temperature the affinity of the migrant to the food can be the major factor determining the variability in the migration values (more than 70% of variance). In situations where both the time of consumption and temperature can vary, these factors can be responsible, respectively, for more than 60% and 20% of the variance in the migration estimates. The approach presented can be used with databases from consumption surveys to yield a true probabilistic estimate of exposure.     

Probabilistic Reliability Modeling for Oil Exploration & Production (E&P) Facilities in the Tallgrass Prairie Preserve

The aging domestic oil production infrastructure represents a high risk to the environment because of the type of fluids being handled (oil and brine) and the potential for accidental release of these fluids into sensitive ecosystems. Currently, there is not a quantitative risk model directly applicable to onshore oil exploration and production (E&P) facilities. We report on a probabilistic reliability model created for onshore exploration and production (E&P) facilities. Reliability theory, failure modes and effects analysis (FMEA), and event trees were used to develop the model estimates of the failure probability of typical oil production equipment. Monte Carlo simulation was used to translate uncertainty in input parameter values to uncertainty in the model output. The predicted failure rates were calibrated to available failure rate information by adjusting probability density function parameters used as random variates in the Monte Carlo simulations. The mean and standard deviation of normal variate distributions from which the Weibull distribution characteristic life was chosen were used as adjustable parameters in the model calibration. The model was applied to oil production leases in the Tallgrass Prairie Preserve, Oklahoma. We present the estimated failure probability due to the combination of the most significant failure modes associated with each type of equipment (pumps, tanks, and pipes). The results show that the estimated probability of failure for tanks is about the same as that for pipes, but that pumps have much lower failure probability. The model can provide necessary equipment reliability information for proactive risk management at the lease level by providing quantitative information to base allocation of maintenance resources to high-risk equipment that will minimize both lost production and ecosystem damage.     

A Probabilistic Framework for the Reference Dose (Probabilistic RfD)

Determining the probabilistic limits for the uncertainty factors used in the derivation of the Reference Dose (RfD) is an important step toward the goal of characterizing the risk of noncarcinogenic effects from exposure to environmental pollutants. If uncertainty factors are seen, individually, as "upper bounds" on the dose-scaling factor for sources of uncertainty, then determining comparable upper bounds for combinations of uncertainty factors can be accomplished by treating uncertainty factors as distributions, which can be combined by probabilistic techniques. This paper presents a conceptual approach to probabilistic uncertainty factors based on the definition and use of RfDs by the US. EPA. The approach does not attempt to distinguish one uncertainty factor from another based on empirical data or biological mechanisms but rather uses a simple displaced lognormal distribution as a generic representation of all uncertainty factors. Monte Carlo analyses show that the upper bounds for combinations of this distribution can vary by factors of two to four when compared to the fixed-value uncertainty factor approach. The probabilistic approach is demonstrated in the comparison of Hazard Quotients based on RfDs with differing number of uncertainty factors.     

Uncertainty and Variation in Indirect Exposure Assessments: An Analysis of Exposure to Tetrachlorodibenzo-p-Dioxin from a Beef Consumption Pathway

Indirect exposures to 2,3,7,8-tetrachlorodibenzo- p -dioxin (TCDD) and other toxic materials released in incinerator emissions have been identified as a significant concern for human health. As a result, regulatory agencies and researchers have developed specific approaches for evaluating exposures from indirect pathways. This paper presents a quantitative assessment of the effect of uncertainty and variation in exposure parameters on the resulting estimates of TCDD dose rates received by individuals indirectly exposed to incinerator emissions through the consumption of home-grown beef. The assessment uses a nested Monte Carlo model that separately characterizes uncertainty and variation in dose rate estimates. Uncertainty resulting from limited data on the fate and transport of TCDD are evaluated, and variations in estimated dose rates in the exposed population that result from location-specific parameters and individuals'behaviors are characterized. The analysis indicates that lifetime average daily dose rates for individuals living within 10 km of a hypothetical incinerator range over three orders of magnitude. In contrast, the uncertainty in the dose rate distribution appears to vary by less than one order of magnitude, based on the sources of uncertainty included in this analysis. Current guidance for predicting exposures from indirect exposure pathways was found to overestimate the intakes for typical and high-end individuals.     

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.     

Discrete Probability Distributions for Probabilistic Fracture Mechanics

Recently, discrete probability distributions (DPDs) have been suggested for use in risk analysis calculations to simplify the numerical computations which must be performed to determine failure probabilities. Specifically, DPDs have been developed to investigate probabilistic functions, that is, functions whose exact form is uncertain. The analysis of defect growth in materials by probabilistic fracture mechanics (PFM) models provides an example in which probabilistic functions play an important role. This paper compares and contrasts Monte Carlo simulation and DPDs as tools for calculating material failure due to fatigue crack growth. For the problem studied, the DPD method takes approximately one third the computation time of the Monte Carlo approach for comparable accuracy. It is concluded that the DPD method has considerable promise in low-failure-probability calculations of importance in risk assessment. In contrast to Monte Carlo, the computation time for the DPD approach is relatively insensitive to the magnitude of the probability being estimated.     

Measures of Compounding Conservatism in Probabilistic Risk Assessment

Concern about the degree of uncertainty and potential conservatism in deterministic point estimates of risk has prompted researchers to turn increasingly to probabilistic methods for risk assessment. With Monte Carlo simulation techniques, distributions of risk reflecting uncertainty and/or variability are generated as an alternative. In this paper the compounding of conservatism⁽¹⁾ between the level associated with point estimate inputs selected from probability distributions and the level associated with the deterministic value of risk calculated using these inputs is explored. Two measures of compounded conservatism are compared and contrasted. The first measure considered, F , is defined as the ratio of the risk value, R _d, calculated deterministically as a function of n inputs each at the j th percentile of its probability distribution, and the risk value, R _j that falls at the j th percentile of the simulated risk distribution (i.e., F=R_d/R_j). The percentile of the simulated risk distribution which corresponds to the deterministic value, R_d , serves as a second measure of compounded conservatism. Analytical results for simple products of lognormal distributions are presented. In addition, a numerical treatment of several complex cases is presented using five simulation analyses from the literature to illustrate. Overall, there are cases in which conservatism compounds dramatically for deterministic point estimates of risk constructed from upper percentiles of input parameters, as well as those for which the effect is less notable. The analytical and numerical techniques discussed are intended to help analysts explore the factors that influence the magnitude of compounding conservatism in specific cases.     

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.     

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.     

Reproductive and Developmental Risks from Ethylene Oxide: A Probabilistic Characterization of Possible Regulatory Thresholds

Ethylene oxide is a gas produced in large quantities in the United States that is used primarily as a chemical intermediate in the production of ethylene glycol, propylene glycol, non-ionic surfactants, ethanolamines, glycol ethers, and other chemicals. It has been well established that ethylene oxide can induce cancer, genetic, reproductive and developmental, and acute health effects in animals. The U.S. Environmental Protection Agency is currently developing both a cancer potency factor and a reference concentration (RfC) for ethylene oxide. This study used the rich database on the reproductive and developmental effects of ethylene oxide to develop a probabilistic characterization of possible regulatory thresholds for ethylene oxide. This analysis was based on the standard regulatory approach for noncancer risk assessment, but involved several innovative elements, such as: (1) the use of advanced statistical methods to account for correlations in developmental outcomes among littermates and allow for simultaneous control of covariates (such as litter size); (2) the application of a probabilistic approach for characterizing the uncertainty in extrapolating the animal results to humans; and (3) the use of a quantitative approach to account for the variation in heterogeneity among the human population. This article presents several classes of results, including: (1) probabilistic characterizations of ED10s for two quantal reproductive outcomes-resorption and fetal death, (2) probabilistic characterizations of one developmental outcome-the dose expected to yield a 5% reduction in fetal (or pup) weight, (3) estimates of the RfCs that would result from using these values in the standard regulatory approach for noncancer risk assessment, and (4) a probabilistic characterization of the level of ethylene oxide exposure that would be expected to yield a 1/1,000 increase in the risk of reproductive or developmental outcomes in exposed human populations.     

Quantitative Risk Assessment of Listeria monocytogenes in French Cold-Smoked Salmon: I. Quantitative Exposure Assessment

A quantitative assessment of the exposure to Listeria monocytogenes from cold-smoked salmon (CSS) consumption in France is developed. The general framework is a second-order (or two-dimensional) Monte Carlo simulation, which characterizes the uncertainty and variability of the exposure estimate. The model takes into account the competitive bacterial growth between L. monocytogenes and the background competitive flora from the end of the production line to the consumer phase. An original algorithm is proposed to integrate this growth in conditions of varying temperature. As part of a more general project led by the French Food Safety Agency (Afssa), specific data were acquired and modeled for this quantitative exposure assessment model, particularly time-temperature profiles, prevalence data, and contamination-level data. The sensitivity analysis points out the main influence of the mean temperature in household refrigerators and the prevalence of contaminated CSS on the exposure level. The outputs of this model can be used as inputs for further risk assessment.