Quantitative Analysis of Variability and Uncertainty with Known Measurement Error: Methodology and Case Study

The appearance of measurement error in exposure and risk factor data potentially affects any inferences regarding variability and uncertainty because the distribution representing the observed data set deviates from the distribution that represents an error-free data set. A methodology for improving the characterization of variability and uncertainty with known measurement errors in data is demonstrated in this article based on an observed data set, known measurement error, and a measurement-error model. A practical method for constructing an error-free data set is presented and a numerical method based upon bootstrap pairs, incorporating two-dimensional Monte Carlo simulation, is introduced to address uncertainty arising from measurement error in selected statistics. When measurement error is a large source of uncertainty, substantial differences between the distribution representing variability of the observed data set and the distribution representing variability of the error-free data set will occur. Furthermore, the shape and range of the probability bands for uncertainty differ between the observed and error-free data set. Failure to separately characterize contributions from random sampling error and measurement error will lead to bias in the variability and uncertainty estimates. However, a key finding is that total uncertainty in mean can be properly quantified even if measurement and random sampling errors cannot be separated. An empirical case study is used to illustrate the application of the methodology.     

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.     

Analysis of Affordance,Time, and Adaptation in the Assessment of Industrial Control System Cybersecurity Risk

Industrial control systems increasingly use standard communication protocols and are increasingly connected to public networks—creating substantial cybersecurity risks, especially when used in critical infrastructures such as electricity and water distribution systems. Methods of assessing risk in such systems have recognized for some time the way in which the strategies of potential adversaries and risk managers interact in defining the risk to which such systems are exposed. But it is also important to consider the adaptations of the systems’ operators and other legitimate users to risk controls, adaptations that often appear to undermine these controls, or shift the risk from one part of a system to another. Unlike the case with adversarial risk analysis, the adaptations of system users are typically orthogonal to the objective of minimizing or maximizing risk in the system. We argue that this need to analyze potential adaptations to risk controls is true for risk problems more generally, and we develop a framework for incorporating such adaptations into an assessment process. The method is based on the principle of affordances, and we show how this can be incorporated in an iterative procedure based on raising the minimum period of risk materialization above some threshold. We apply the method in a case study of a small European utility provider and discuss the observations arising from this.     

Statistical Distributions of Daily Breathing Rates for Narrow Age Groups of Infants and Children

Children may be more susceptible to toxicity from some environmental chemicals than adults. This susceptibility may occur during narrow age periods (windows), which can last from days to years depending on the toxicant. Breathing rates specific to narrow age periods are useful to assess inhalation dose during suspected windows of susceptibility. Because existing breathing rates used in risk assessment are typically for broad age ranges or are based on data not representative of the population, we derived daily breathing rates for narrow age ranges of children designed to be more representative of the current U.S. children's population. These rates were derived using the metabolic conversion method of Layton (1993) and energy intake data adjusted to represent the U.S. population from a relatively recent dietary survey (CSFII 1994–1996, 1998). We calculated conversion factors more specific to children than those previously used. Both nonnormalized (L/day) and normalized (L/kg-day) breathing rates were derived and found comparable to rates derived using energy estimates that are accurate for the individuals sampled but not representative of the population. Estimates of breathing rate variability within a population can be used with stochastic techniques to characterize the range of risk in the population from inhalation exposures. For each age and age-gender group, we present the mean, standard error of the mean, percentiles (50th, 90th, and 95th), geometric mean, standard deviation, 95th percentile, and best-fit parametric models of the breathing rate distributions. The standard errors characterize uncertainty in the parameter estimate, while the percentiles describe the combined interindividual and intra-individual variability of the sampled population. These breathing rates can be used for risk assessment of subchronic and chronic inhalation exposures of narrow age groups of children.     

Methods to Approximate Joint Uncertainty and Variability in Risk

As interest in quantitative analysis of joint uncertainty and interindividual variability (JUV) in risk grows, so does the need for related computational shortcuts. To quantify JUV in risk, Monte Carlo methods typically require nested sampling of JUV in distributed inputs, which is cumbersome and time-consuming. Two approximation methods proposed here allow simpler and more rapid analysis. The first consists of new upper-bound JUV estimators that involve only uncertainty or variability, not both, and so never require nested sampling to calculate. The second is a discrete-probability-calculus procedure that uses only the mean and one upper-tail mean for each input in order to estimate mean and upper-bound risk, which procedure is simpler and more intuitive than similar ones in use. Application of these methods is illustrated in an assessment of cancer risk from residential exposures to chloroform in Kanawah Valley, West Virginia. Because each of the multiple exposure pathways considered in this assessment had separate modeled sources of uncertainty and variability, the assessment illustrates a realistic case where a standard Monte Carlo approach to JUV analysis requires nested sampling. In the illustration, the first proposed method quantified JUV in cancer risk much more efficiently than corresponding nested Monte Carlo calculations. The second proposed method also nearly duplicated JUV-related and other estimates of risk obtained using Monte Carlo methods. Both methods were thus found adequate to obtain basic risk estimates accounting for JUV in a realistically complex risk assessment. These methods make routine JUV analysis more convenient and practical.     

COUNTRY PORTFOLIOS

Capital flows to developing countries are small and mostly take the form of loans rather than direct foreign investment. We build a simple model of North–South capital flows that highlights the interplay between diminishing returns, production risk, and sovereign risk. This model generates a set of country portfolios and a world distribution of capital stocks that resemble those in the data. (JEL: F32, F34)     

Parametric Inference and Dynamic State Recovery From Option Panels

We develop a new parametric estimation procedure for option panels observed with error. We exploit asymptotic approximations assuming an ever increasing set of option prices in the moneyness (cross‐sectional) dimension, but with a fixed time span. We develop consistent estimators for the parameters and the dynamic realization of the state vector governing the option price dynamics. The estimators converge stably to a mixed‐Gaussian law and we develop feasible estimators for the limiting variance. We also provide semiparametric tests for the option price dynamics based on the distance between the spot volatility extracted from the options and one constructed nonparametrically from high‐frequency data on the underlying asset. Furthermore, we develop new tests for the day‐by‐day model fit over specific regions of the volatility surface and for the stability of the risk‐neutral dynamics over time. A comprehensive Monte Carlo study indicates that the inference procedures work well in empirically realistic settings. In an empirical application to S&P 500 index options, guided by the new diagnostic tests, we extend existing asset pricing models by allowing for a flexible dynamic relation between volatility and priced jump tail risk. Importantly, we document that the priced jump tail risk typically responds in a more pronounced and persistent manner than volatility to large negative market shocks.     

Human Salmonellosis: Estimation of Dose-Illness from Outbreak Data

The quantification of the relationship between the amount of microbial organisms ingested and a specific outcome such as infection, illness, or mortality is a key aspect of quantitative risk assessment. A main problem in determining such dose-response models is the availability of appropriate data. Human feeding trials have been criticized because only young healthy volunteers are selected to participate and low doses, as often occurring in real life, are typically not considered. Epidemiological outbreak data are considered to be more valuable, but are more subject to data uncertainty. In this article, we model the dose-illness relationship based on data of 20 Salmonella outbreaks, as discussed by the World Health Organization. In particular, we model the dose-illness relationship using generalized linear mixed models and fractional polynomials of dose. The fractional polynomial models are modified to satisfy the properties of different types of dose-illness models as proposed by Teunis et al . Within these models, differences in host susceptibility (susceptible versus normal population) are modeled as fixed effects whereas differences in serovar type and food matrix are modeled as random effects. In addition, two bootstrap procedures are presented. A first procedure accounts for stochastic variability whereas a second procedure accounts for both stochastic variability and data uncertainty. The analyses indicate that the susceptible population has a higher probability of illness at low dose levels when the combination pathogen-food matrix is extremely virulent and at high dose levels when the combination is less virulent. Furthermore, the analyses suggest that immunity exists in the normal population but not in the susceptible population.     

Framework for Microbial Food‐Safety Risk Assessments Amenable to Bayesian Modeling

Regulatory agencies often perform microbial risk assessments to evaluate the change in the number of human illnesses as the result of a new policy that reduces the level of contamination in the food supply. These agencies generally have regulatory authority over the production and retail sectors of the farm‐to‐table continuum. Any predicted change in contamination that results from new policy that regulates production practices occurs many steps prior to consumption of the product. This study proposes a framework for conducting microbial food‐safety risk assessments; this framework can be used to quantitatively assess the annual effects of national regulatory policies. Advantages of the framework are that estimates of human illnesses are consistent with national disease surveillance data (which are usually summarized on an annual basis) and some of the modeling steps that occur between production and consumption can be collapsed or eliminated. The framework leads to probabilistic models that include uncertainty and variability in critical input parameters; these models can be solved using a number of different Bayesian methods. The Bayesian synthesis method performs well for this application and generates posterior distributions of parameters that are relevant to assessing the effect of implementing a new policy. An example, based on Campylobacter and chicken, estimates the annual number of illnesses avoided by a hypothetical policy; this output could be used to assess the economic benefits of a new policy. Empirical validation of the policy effect is also examined by estimating the annual change in the numbers of illnesses observed via disease surveillance systems.     

Quantitative Risk Assessment from Farm to Fork and Beyond: A Global Bayesian Approach Concerning Food-Borne Diseases

A novel approach to the quantitative assessment of food-borne risks is proposed. The basic idea is to use Bayesian techniques in two distinct steps: first by constructing a stochastic core model via a Bayesian network based on expert knowledge, and second, using the data available to improve this knowledge. Unlike the Monte Carlo simulation approach as commonly used in quantitative assessment of food-borne risks where data sets are used independently in each module, our consistent procedure incorporates information conveyed by data throughout the chain. It allows "back-calculation" in the food chain model, together with the use of data obtained "downstream" in the food chain. Moreover, the expert knowledge is introduced more simply and consistently than with classical statistical methods. Other advantages of this approach include the clear framework of an iterative learning process, considerable flexibility enabling the use of heterogeneous data, and a justified method to explore the effects of variability and uncertainty. As an illustration, we present an estimation of the probability of contracting a campylobacteriosis as a result of broiler contamination, from the standpoint of quantitative risk assessment. Although the model thus constructed is oversimplified, it clarifies the principles and properties of the method proposed, which demonstrates its ability to deal with quite complex situations and provides a useful basis for further discussions with different experts in the food chain.     

Application of Correspondence Analysis in the Assessment of Mine Tailings Dam Breakage Risk in the Mediterranean Region

A new statistical approach for preliminary risk evaluation of breakage in tailings dam is presented and illustrated by a case study regarding the Mediterranean region. The objective of the proposed method is to establish an empirical scale of risk, from which guidelines for prioritizing the collection of further specific information can be derived. The method relies on a historical database containing, in essence, two sets of qualitative data: the first set concerns the variables that are observable before the disaster (e.g., type and size of the dam, its location, and state of activity), and the second refers to the consequences of the disaster (e.g., failure type, sludge characteristics, fatalities categorization, and downstream range of damage). Based on a modified form of correspondence analysis, where the second set of attributes are projected as "supplementary variables" onto the axes provided by the eigenvalue decomposition of the matrix referring to the first set, a "qualitative regression" is performed, relating the variables to be predicted (contained in the second set) with the "predictors" (the observable variables). On the grounds of the previously derived relationship, the risk of breakage in a new case can be evaluated, given observable variables. The method was applied in a case study regarding a set of 13 test sites where the ranking of risk obtained was validated by expert knowledge. Once validated, the procedure was included in the final output of the e-EcoRisk UE project (A Regional Enterprise Network Decision-Support System for Environmental Risk and Disaster Management of Large-Scale Industrial Spills), allowing for a dynamic historical database updating and providing a prompt rough risk evaluation for a new case. The aim of this section of the global project is to provide a quantified context where failure cases occurred in the past for supporting analogue reasoning in preventing similar situations.     

The Effect of Changes in Risk Attitude on Strategic Behavior

We study families of normal‐form games with fixed preferences over pure action profiles but varied preferences over lotteries. That is, we subject players' utilities to monotone but nonlinear transformations and examine changes in the rationalizable set and set of equilibria. Among our results: The rationalizable set always grows under concave transformations (risk aversion) and shrinks under convex transformations (risk love). The rationalizable set reaches an upper bound under extreme risk aversion, and lower bound under risk love, and both of these bounds are characterized by elimination processes. For generic two‐player games, under extreme risk love or aversion, all Nash equilibria are close to pure and the limiting set of equilibria can be described using preferences over pure action profiles.     

A Semiparametric Approach to Risk Assessment for Quantitative Outcomes

Characterizing the dose-effect relationship and estimating acceptable exposure levels are the primary goals of quantitative risk assessment. A semiparametric approach is proposed for risk assessment with continuously measured or quantitative outcomes which has advantages over existing methods by requiring fewer assumptions. The approach is based on pairwise ranking between the response values in the control group and those in the exposed groups. The work generalizes the rank-based Wilcoxon-Mann-Whitney test, which for the two-group comparison is effectively a test of whether a response from the control group is different from (larger than) a response in an exposed group. We develop a regression framework that naturally extends this metric to model the dose effect in terms of a risk function. Parameters of the regression model can be estimated with standard software. However, inference requires an additional step to estimate the variance structure of the estimated parameters. An effective dose (ED) and associated lower confidence limit (LED) are easily calculated. The method is supported by a simulation study and is illustrated with a study on the effects of aconiazide. The method offers flexible modeling of the dose effect, and since it is rank-based, it is more resistant to outliers, nonconstant variance, and other departures from normality than previously described approaches.     

An Empirical,Nonparametric Simulator for Multivariate Random Variables with Differing Marginal Densities and Nonlinear Dependence with Hydroclimatic Applications

Multivariate simulations of a set of random variables are often needed for risk analysis. Given a historical data set, the goal is to develop simulations that reproduce the dependence structure in that data set so that the risk of potentially correlated factors can be evaluated. A nonparametric, copula‐based simulation approach is developed and exemplified. It can be applied to multiple variables or to spatial fields with arbitrary dependence structures and marginal densities. The nonparametric simulator uses logspline density estimation in the univariate setting, together with a sampling strategy to reproduce dependence across variables or spatial instances, through a nonparametric numerical approximation of the underlying copula function. The multivariate data vectors are assumed to be independent and identically distributed. A synthetic example is provided to illustrate the method, followed by an application to the risk of livestock losses in Mongolia.     

On the Need for Restricting the Probabilistic Analysis in Risk Assessments to Variability

It is common perspective in risk analysis that there are two kinds of uncertainties: i) variability as resulting from heterogeneity and stochasticity (aleatory uncertainty) and ii) partial ignorance or epistemic uncertainties resulting from systematic measurement error and lack of knowledge. Probability theory is recognized as the proper tool for treating the aleatory uncertainties, but there are different views on what is the best approach for describing partial ignorance and epistemic uncertainties. Subjective probabilities are often used for representing this type of ignorance and uncertainties, but several alternative approaches have been suggested, including interval analysis, probability bound analysis, and bounds based on evidence theory. It is argued that probability theory generates too precise results when the background knowledge of the probabilities is poor. In this article, we look more closely into this issue. We argue that this critique of probability theory is based on a conception of risk assessment being a tool to objectively report on the true risk and variabilities. If risk assessment is seen instead as a method for describing the analysts’ (and possibly other stakeholders’) uncertainties about unknown quantities, the alternative approaches (such as the interval analysis) often fail in providing the necessary decision support.     

Fitting Hierarchical Holographic Modeling into the Theory of Scenario Structuring and a Resulting Refinement to the Quantitative Definition of Risk

A point of view is suggested from which the Hierarchical Holographic Modeling (HHM) method can be seen as one more method within the Theory of Scenario Structuring (TSS), which is that part of Quantitative Risk Assessment having to do with the task of identifying the set of risk scenarios. Seen in this way, HHM brings strongly to our attention the fact that different methods within TSS can result in different sets of risk scenarios for the same underlying problem. Although this is not a problem practically, it is a bit awkward conceptually from the standpoint of the "set of triplets" definition of risk, in which the scenario set is part of the definition. Accordingly, the present article suggests a refinement to the set of triplets definition, which removes the specific set of scenarios, found by any of the TSS methods, from the definition of risk and casts it, instead, as an approximation to the "true" set of scenarios that is native to the problem at hand and not affected by the TSS method used.