Abstracts from Recent & Upcoming Presentations



	Abstracts from Recent & Upcoming Presentations
	Computing cleanup targets is complicated (but possible) in probabilistic risk assessments Abstract Ferson, Scott, Troy W. Tucker and David Myers, Applied Biomathematics, 100 North Country Rd., Setauket, NY 11733 USA The deterministic expressions traditionally used in risk assessments are being replaced with probabilistic expressions that represent natural heterogeneity in populations and variability among possible exposure scenarios. This important advance has led to certain complications. So long as only point estimates were used in quantitative risk assessments, it was straightforward to solve for the environmental concentration that, if not exceeded, would ensure that resulting doses were below tolerable limits. Now that the mathematical equations involve probability distributions, however, it is not possible to simply rearrange the equations to solve for these concentrations. Because determining cleanup goals for remediation is often the primary purpose of a risk assessment, convenient and reliable methods are needed to untangle the probabilistic calculations involved in modern risk assessments. We present simple and efficient methods to compute cleanup goals that satisfy multiple simultaneous criteria expressed in terms of limits on resultant exposures or risks in the context of a probabilistic assessment. The approach can be used with arbitrarily many constraints on percentiles of the target risk. The approach characterizes concentration distributions that satisfy constraints prescribed by the regulator and the calculations yield two kinds of bounds on concentration: a ‘core’ and ‘shell’. If the concentration distribution is entirely inside the core, then the result surely obeys the prescribed constraints. If the concentration distribution is anywhere outside the shell, then the result certainly fails to comply with the prescribed constraints. If the concentration distribution is outside the core but inside the shell, then compliance must be determined by a forward calculation. The core is essentially comparable to a screening level familiar from deterministic assessments, but the shell is not similarly analogous to an action level, but rather to a frontier delimiting clearly unacceptable distributions.
	Basic statistics for imprecise data Abstract Scott Ferson, Applied Biomathematics, and Vladik Kreinovich, University of Texas at El Paso Risk analysts conscientiously distinguish between epistemic and aleatory uncertainty when the assessment demands it. This conscientiousness about the distinction should extend to data handling and statistical methods as well, but this has generally not been the case. For instance, guidance promulgated by national and international standards agencies for handling uncertainty in empirical data does not adequately distinguish between the two forms of uncertainty and tends to confound them together. Likewise, because most of the statistical methods developed over the last century have focused on assessing and projecting sampling uncertainty (which arises from having observed only a subset of a population) and neglect measurement uncertainty (which arises from imperfect mensuration, censoring or missing values), almost all of the statistical methods widely employed in risk analysis presume that individual measurements are infinitely precise, even though this is rarely justifiable. One reason for the ubiquity of the assumption is the difficulty of making the necessary calculations that account for both-and distinguish between-variability and imprecision in empirical data. For instance, computing the variance of a collection of interval data is generally an NP-hard computational problem. Nevertheless, some progress has been made. Within the last five years, for example, workable algorithms for computing basic descriptive statistics for interval data have been developed. We describe software implementing these and other algorithms for computing basic descriptive and inferential statistics for interval data. We also review the growing literature on censoring, missing values, interval data, and robust statistical methods that carefully make the distinction between epistemic and aleatory uncertainty.
	Modeling correlation and dependence among intervals Abstract S. Ferson and V. Kreinovich This note introduces the notion of dependence among intervals to account for observed or theoretical constraints on the relationships among uncertain inputs in mathematical calculations. We define dependence as any restriction on the possible pairings of values within respective intervals and define nondependence as the degenerate case of no restrictions (which we carefully distinguish from independence in probability theory). Traditional interval calculations assume nondependence, but alternative assumptions are possible, including several which might be practical in engineering settings that would lead to tighter enclosures on arithmetic functions intervals. We give best possible formulas for addition of intervals under several of these dependencies. We also suggest some potentially useful models of correlation, which are single-parameter families of dependencies, often from the identity dependence (u=v) representing maximal correlation, through nondependence, to opposite dependence (1-u=v) representing maximally negative correlation.
	Calculating a soil EPC when sampling and exposures are non-random Abstract Harlee Strauss, Scott Ferson, John Lortie, Richard McGrath, Susan Svirsky The exposure point concentration (EPC) is intended to represent the average concentration of soil with which a receptor comes into contact. For EPA risk assessments, the EPC is the 95th upper confidence limit for the mean of data collected from random sampling. Important assumptions underlying the EPC are that sampling is random and that an individual comes into contact with the contaminated soil in a random way across the exposure area (or site) under evaluation. In practice, the random sampling assumption is often violated if no adjustment is made for samples that are collected in a non-random manner, such as programs intended to define areas of elevated concentrations. The random exposure assumption may be violated when considering recreational activities such as hunting, fishing, birdwatching, and hiking, where there may be preferential areas or paths used by individuals, or avoidance of areas that are difficult to access because of dense vegetation or obstructions. In order to maintain consistency with the assumptions of random sampling and random exposure within an exposure area, we developed a methodology that included area-weighting to account for non-random sampling patterns and use-weighting to account for preferential use of certain areas within an exposure area. This EPC methodology was applied to PCBs in the floodplain soil of the Housatonic River in Massachusetts. The use-weighting system was based on information about ecological habitat and thus potential accessibility and attractiveness of the area. The area-weighting system also required information about habitat types as they were indicative of the topographic and hydrologic factors that governed the deposition of PCBs in the floodplain. The 95th UCL of the mean was calculated from area- and used-weighted data by generalizing a bootstrap resampling procedure that accounts for skewness.
	Propagating Epistemic and Aleatory Uncertainty in Nonlinear Dynamic Models Abstract Youdong Lin, Mark A. Stadtherr, Scott Ferson and George F. Corliss Engineering analysis and design problems frequently involve uncertain parameters and inputs. Propagating these uncertainties through a complex model to determine their effects on system states and outputs can be a challenging problem, especially for dynamic models. Lin and Stadtherr recently described the implementation of a new validating solver "VSPODE" for parametric ordinary differential equations (ODEs). Using this software, it is possible to obtain a Taylor-model representation (i.e., a Taylor polynomial function and an interval remainder bound) for the state variables and outputs in terms of the uncertain quantities. We give numerical examples to illustrate how these Taylor models can be used to propagate uncertainty about inputs through a system of nonlinear ODEs. We show that the approach can handle cases in which the uncertainty is represented by interval ranges, by probability distributions, or even by a set of possible cumulative probability distribution functions bounded by a pair of monotonically increasing functions (a "p-box"). The last case is useful when only partial information is available about the probability distributions, as is often the case when measurement error is non-negligible or in the early phases of engineering design when system features and properties have not yet be selected.
	The tradeoff between measurement precision and sample size: should we get more or better data? Abstract Scott Ferson, Applied Biomathematics, and Vladik Kreinovich, University of Texas at El Paso One intuitively expects there to be a tradeoff between precision and sample size of measurements. For instance, one might be able to spend a unit of additional resources to be devoted to measurement either to increase the number of samples, or to improve the precision of the individual samples. Many practitioners apparently believe, however, that the tradeoff always favors increasing the number of samples over improving their precision. This belief is understandable given that most of the statistical literature of the last century has focused on assessing and projecting sampling uncertainty, and has in comparison neglected the problem of assessing and projecting measurement uncertainty. The belief is nevertheless mistaken, as can easily be shown by straightforward numerical examples. Consider, for example, the problem of conservatively estimating an exposure point concentration (EPC) from sparse and imprecise data. We might use an upper confidence limit on the mean to account for the sampling uncertainty associated with having made only a few measurements. This value is affected by the sample size, but, if the calculation also accounts for the imprecision of the values in a reasonable way, it is also affected by the measurement precision. Using recent algorithms to compute basic statistics for interval data sets, we consider the EPC and describe a nonlinear tradeoff between precision and sample size. This nonlinearity means the optimal investment of empirical resources between increasing sampling and improving precision depends on the quantitative details of the problem. We describe how an analyst can plan an optimal empirical design.
	Uncertainty, Sensitivity and Validation Abstract S. Ferson, Applied Biomathematics. An honest assessment of the uncertainty in calculations and model predictions may be the only difference between prudent analysis and mere wishful thinking. Although traditional methods of error analysis and uncertainty assessments are useful, they typically require untenable or unjustified assumptions. Methods are needed that can relax these assumptions to reflect what is actually known and what is not known about the underlying system. Analysts in many fields draw a careful distinction between epistemic uncertainty and aleatory uncertainty. The latter comes from variability across time or space, heterogeneity within a population, and other sources of stochasticity, and it is commonly modeled with the methods of probability theory. The former arises from measurement imprecision, residual scientific ignorance about model structure, and other forms of incertitude. Many analysts are coming to believe that alternative methods must be used to fully account for epistemic uncertainty and to properly distinguish it from aleatory uncertainty. Considerations of these two kinds of uncertainty have suggested new approaches to some of the fundamental tasks in model building, including uncertainty propagation and especially the treatment of model uncertainty, but also sensitivity analysis and validation exercises. There are some strategies that can be used even for extremely complex models that have high-dimensional inputs and require long calculation times. For example, Monte Carlo techniques and the Cauchy deviate method have errors determined only by the number of replications, rather than the dimensionality of the problem. The former can project probabilistic uncertainty and the latter projects interval-like incertitude. For models that are so complex that very few runs can be computed, Kolmogorov-Smirnov confidence procedures can assess the sampling uncertainty associated with having few replications. Neglect of model uncertainty, which is often the elephant in the living room, is especially egregious in modeling. Analysts usually construct a model and then act as though it correctly represents the world. This understates the uncertainty associated with the model’s predictions, because it fails to express that the model might be in error. Standard methods recommended to account for model uncertainty have serious deficiencies, and some tend to erase the uncertainty rather than truly propagate it through calculations. Alternative strategies will be discussed. (Invited Tutorial for Center for Turbulence Research (CTR) Summer Program 2006, Friday, July 28, 2006 at 4PM, Bldg. 200, Room 002)
	Quasi-extinction risk in a wood frog (Rana sylvatica) metapopulation under environmental contamination by PCBs Abstract Tucker, W. Troy, Michael E. Thompson, John P. Lortie, Douglas J. Fort, Susan Svirsky, and Scott Ferson A stochastic population model projecting wood frog population trends into the future and computing the risk of population decline was constructed using vital rate information from the literature and abundances derived from studies of 27 vernal pools in western Massachusetts. The model was age- and sex-structured with yearly time steps, and both demographic and environmental stochasticity were incorporated. The model was spatially explicit and frogs were allowed to disperse between ponds as a function of distance. The impact of PCBs on this wood frog population was assessed by comparing population projections from a base population model, i.e., a wood frog population not impacted by PCBs, with projections from population models that included the effect of PCBs on population vital rates. Both a non-declining and a declining base population were simulated. Parameterizations included the effect of PCBs on initial population size and combinations of low and high estimates of the proportion of malformed frogs that subsequently died or became reproductively unfit due to PCB exposure. The impacts of PCBs were derived from vernal pool and laboratory studies in the study area and from literature sources. Based upon this modeling effort, PCBs appear to increase the risk of population decline and quasi-extinction at all levels for wood frog at the site.
	Model Validation, Calibration and Predictive Capability Abstract Scott Ferson (Applied Biomathematics) and William L. Oberkampf (Sandia National Laboratories) The theoretical and mathematical issues involved in model validation, calibration, and predictive capability touch the foundations of science. As the capability of computational simulation continues to dramatically increase, these issues must be addressed more directly and clearly in science and engineering. Sandia National Laboratories’ Validation Challenge Problems were designed to require most of the key issues be addressed. We describe an analysis of the thermal challenge problem that does not assume that the experimental data given in the problem are obtained without experimental measurement uncertainty. Instead, we assume a model for the experimental measurement uncertainty, along with the appropriate model constants and use the material characterization data provided to calibrate estimates for thermal conductivity and volumetric heat capacity as a regression function of temperature. The calibration includes the unit-to-unit variability of the samples tested, and reflects the limited number of samples. We then use the onedimensional, unsteady, thermal model provided to make predictions for both the ensemble and accreditation experiments. As part of these predictions, we address the uncertainty introduced from the incompatibility between the assumption of constant thermal properties in the thermal model as compared to the temperature-dependent trend exposed in the material characterization data. In comparing the predictions and experimental data for the ensemble and accreditation cases, we construct validation metrics to quantify the disagreement between the two sets of data. Predictions are then made for the regulatory compliance condition, taking into account the temperaturedependent properties, the validation metric results, and the extrapolative nature of the prediction. Separate predictions are made for each of the three quantities of experimental data (low, medium, and high), as well as with and without experimental measurement uncertainty. Each of the predictions includes an assessment of satisfying the probabilistic requirement for the regulatory condition.
	Environmental contamination in ecological systems Abstract S. Ferson, Applied Biomathematics (NAC SETAC Keynote Address Abstract). Many human activities introduce chemical contaminants into the natural environment. Manufacturing by-products, agricultural fertilizers and pesticides, leachates from mine tailings, combustion residues, waste and effluent streams deliver anthropogenic toxicants and other chemicals into aquatic and terrestrial ecosystems. Planning mitigation and remediation strategies and designing systems for minimum environmental impact require clear assessment of the nature, magnitude and consequence of the impacts of these contaminants. Risk assessors are beginning to appreciate the need to include ecological processes in their assessment models. The need arises because ecological systems have an inherent complexity that can completely erase the effects of an impact or greatly magnify it, depending on the life histories of the biological species involved. This complexity can also delay the consequence of an impact or alter its expression in other ways. Three central themes have emerged in ecological risk assessment: 1) Variability versus incertitude. Natural biological systems fluctuate in time and space, partially due to interactions we understand, but substantially due to various factors that we cannot foresee. The variability of ecological patterns and processes, and our incertitude about them, prevent us from making precise, deterministic estimates of the effects of environmental impacts. Because of this, comprehensive impact assessment requires a probabilistic language of risk that recognizes variability and incertitude, yet permits quantitative statements of what can be predicted. The emergence of this risk language has been an important development in applied ecology over the last decade. A risk-analytic endpoint is a natural summary that can integrate disparate impacts on a biological system. 2) Population-level assessment. In the past, assessments were conducted at the level of the individual organism, or, in the case of toxicity impacts, even at the level of tissues or enzyme function. To justify costly decisions about remediation and mitigation, biologists are often asked “So what?” questions that demand predictions about the consequences of impacts on higher levels of biological organization. Management plans require predictions of the consequent effects on biological populations and ecological communities. Our scientific understanding of community and ecosystem ecology is very limited, however, and quantitative predictions, even in terms of risks, for complex systems would require vastly more data and mechanistic knowledge than are usually available. Extrapolating the results of individual-level impacts to potential effects on the ecosystem may simply be beyond the current scientific capacity of ecology, which still lacks wide agreement about even fundamental equations governing predator-prey interactions. How can we satisfy the desire for ecological relevance when we are limited by our understanding of how ecosystems actually work? As a practical matter, focusing on populations, meta-populations (assemblages of distinct local populations), and short food chains may be a workable compromise between the organism and ecosystem levels. Risk assessment at the population level requires the combination of several technical tools including demographic models, potentially with explicit age, stage or geographic structure, and methods for probabilistic uncertainty propagation, which are usually implemented with Monte Carlo simulation. Meta-populations and short food chains are likely to be at the frontier of what we can address with scientifically credible models over the next decade. 3) Cumulative attributable risk. Assessments should focus on the change in risk due to a particular impact. The risk that a population declines to, say, 50% of its current abundance in the next 50 years is sometimes substantial whether it is impacted by anthropogenic activity or not. Only the potential change in risk, not the risk itself, should be attributed to impact. On the other hand, for environmental protection to be effective, remediation and mitigation must be designed with reference to the cumulative risks suffered by an ecological system from impacts and from all the various stresses present cumulated through time.
	The elephant in the living room: What to do about model uncertainty Abstract S. Ferson, Applied Biomathematics Analysts usually construct a model and then act as though it correctly represents the state of the world. This understates the uncertainty associated with the model’s predictions, because it fails to express the analyst’s uncertainty that the model itself might be in error. Several approaches have been proposed to account for model uncertainty within a probabilistic assessment, including what-if analyses, stochastic mixtures, Bayesian model averaging, probability bounds analysis, robust Bayes analyses, and imprecise probabilities. Although each approach has advantages that make it attractive in some situations, each also has serious limitations. For example, several approaches require the analyst to explicitly enumerate all the possible competing models. This might sometimes be reasonable, but the uncertainty will often be more profound and there might be possible models of which the analyst is not even aware. Although Bayesian model averaging and stochastic mixture strategies are considered by many to be the state of the art in accounting for model uncertainty in probabilistic assessments, numerical examples show that both approaches actually tend to erase model uncertainty rather than truly propagate it through calculations. In contrast, probability bounds analysis, robust Bayes methods, and imprecise probability methods can be used even if the possible models cannot be explicitly enumerated and, moreover, they do not average away the uncertainty but propagate it fully through calculations.
	Overdriving the headlights: empirical data limit risk analyses Abstract Scott Ferson, W. Troy Tucker, and Lev R. Ginzburg, Applied Biomathematics Even though there may be little relevant empirical information, Monte Carlo simulation requires an analyst to select a precise statistical distribution for every variable in an assessment. Moreover, even when there is no information about correlations among the variables, analysts must make some assumptions about their dependencies. Typically, analysts assume independence even between variables that are mechanistically related. By making assumptions merely for the sake of mathematical convenience that do not have empirical justification, risk assessments based on Monte Carlo simulation yield results that cannot be considered reliable. Although many have argued that two-dimensional or second-order probabilistic risk assessments can account for uncertainties about distribution shapes and parameters, and dependencies and model structure, it is easy to show that the results obtained from such analyses can be grossly misleading. Probability bounds analysis, on the other hand, allows an analyst to relax inappropriately precise statements about statistical distributions as well as untenable independence assumptions. It bounds the probabilistic results in a rigorous way and characterizes their reliability. It can even comprehensively account for many kinds of model uncertainty that may attend a risk calculation. Several numerical examples involving the real-world ecological and human health risk calculations are described and graphically contrasted with results from precise Monte Carlo simulations and second-order simulations.
	Risk perception and the problems we make for ourselves Abstract W. Troy Tucker, Scott Ferson, and Lev R. Ginzburg, Applied Biomathematics The failures of risk communication are often blamed on public ignorance of technical issues or mistrust of industry or government. We suggest that often neither ignorance nor mistrust is fundamentally responsible for the difficulty. Instead, humans seem wired by natural evolution to use a mental calculus for reckoning risk and making decisions that can be substantially different from probability theory. We suggest that several important biases of risk perception recognized by psychometricians can be interpreted as adaptive strategies for responding to incertitude, variation, and multiple dimensions of risk. In particular, we deduce evolutionary reasons why (i) people routinely misestimate risks, (ii) people are insensitive to prior probabilities, (iii) the notion of independence is so difficult to correctly interpret, and (iv) people concentrate on the worst case (and ignore how unlikely it is). If these biases are fundamental to human perception and not removable by general education or specific training, perhaps risk analysts should make their calculations and arguments more natural, interesting, and compelling to humans. We describe such an approach to risk assessment and communication based on a practical review of recent findings in evolutionary psychology and neurobiology. Implications for medical decision making in the context of uncertainty are explored.
	Reliable calculation of probabilities Abstract S. Ferson, Applied Biomathematics. A variety of practical computational problems arise in risk and safety assessments, forensic statistics and decision analyses in which the probability of some event or proposition E is to be estimated from the probabilities of a finite list of related subevents or propositions F, G, H, ... . When the probabilities of the subevents are known only poorly from crude estimations, or the dependencies among the subevents are unknown, one cannot use traditional methods of fault/event tree analysis to estimate the probability of the top event. Representing probability estimates as interval ranges on [0,1] has been suggested as a way to address these sources of uncertainty. Interval operations are given that can be used to compute rigorous bounds on the the probability of the top event which, it turns out, account for both imprecision about subevent probabilities and uncertainty about their dependencies at the same time.
	Why Laplace was wrong: using copulas to bound convolutions when marginals are known Abstract S. Ferson, Applied Biomathematics. Williamson's algorithms that compute bounds on the distributions of arithmetic operations on random numbers when only marginal distributions are known also work when the marginals can only be bounded. They therefore constitute a probability bounds analysis that renders obsolete the maximum entropy criterion for selection of input distributions.
	Total exposure does not equal average daily exposure times days exposed Abstract S. Ferson, Applied Biomathematics. Suppose we are interested in estimating the probability distribution among exposed individuals of their total exposures over some time period. Using simple convolution (i.e., what @Risk or Crystal Ball does) with the distribution of toxicant concentrations and the distribution of individuals' bodyweights leads to an answer whose variance can be grossly overestimated if exposures are iterated over the time period. The reason is that this calculation assumes that the magnitude of every exposure event through time is the same for an individual. In other words, if a person is given a small exposure once, then he will always experience exposures of the same size over the entire time period. This approach fails to appreciate that the toxicant concentration encountered may be different for different exposure events. When time periods are long, as they are for lifetime exposures needed in cancer r isk assessments, the resulting error can be substantial. Even if the temporal autocorrelation among sequential exposures for an individual is exceedingly high (e.g., 0.99), sufficiently many iterations will eventually overwhelm the autocorrelation. The simplistic calculation is appropriate only if exposure events are perfectly correlated (which seems unlikely in most practical cases). Nevertheless, this approach has been almost universally used since exposure assessments have been conducted within the pr obabilistic framework. Several examples of recent assessments that have made this mistake will be reviewed. In most cases, the effect of the error is to very strongly overestimate the chance of large lifetime exposures. Simple approximations are described that can be used to improve the estimate for the distribution of total exposure but do not require a full description of the autocorrelation function or an elaborate simulation of event-to-event variation in exposures.
	Combining toxicant kinetics, population dynamics and trophic interactions Abstract S. Ferson, M. Spencer, W.-X. Wang, N. Fisher and L. Ginzburg, Applied Biomathematics and Stony Brook University. The chemistry and kinetics of an environmental toxicant and the population dynamics and food chain relationships among the species exposed to the toxicant are all fundamental phenomena for the questions we pose in ecological risk analysis. Yet the disciplines that address these phenomena have developed almost completely separately from each other. As a consequence it is not at all clear how we should combine models of toxicant kinetics, population dynamics and food chain relations together. Models of toxicant kinetics usually assume that trophic interactions and populations dynamics are so slow as to be constant. Models of population dynamics usually assume trophic interactions and especially toxicant kinetics to be so fast as to be equilibrial. In real-world systems, such assumptions are not always reasonable. When the assumptions must be abandoned and we have to consider all three phenomena simultaneously, what compromises are necessary and practical? We describe case studies involving heavy metal accumulation in marine and freshwater food chain systems of zooplankton, copepods and bivalves using the new software RAMAS Ecotoxicology and RAMAS Ecosystem.
	Detecting rare event clusters when data are extremely sparse Abstract S. Ferson and K. Hwang, Applied Biomathematics. Detection of clustering among rare events can be very important in recognizing engineering design flaws and cryptic common-mode or common-cause depen- dencies among rare events such as component failures. However, traditional statistical tests for clustering assume asymptotically large sample size. Simulation studies show that with small data sets the Type I error rates for traditional test s such as chi-square or the likelihood ratio can be much larger than nominal levels. Moreover, these tests are sensitive to a specific kind of deviation from randomness and may not provide the most appropriate measure of clustering in a particular circumstance. We describe five new statistical tests, implemented in a convenient software package, that can be used to detect clustering of rare events in structured environments. Because the formulations employ combinatorial expressions, they yield exact P-va lues and can therefore be used even when data sets are extremely small. These new statistical methods allow risk and safety analysts to detect clustering of rare events in data sets of the size usually encountered in practice. We characterize the relative statistical power of the tests under different kinds of clustering mechanisms and data set configurations. This work was supported by SBIR grant R44GM49521 from the National Institutes of Health.
	Ecotoxicology: how to bring ecology and toxicology together Abstract M. Spencer and S. Ferson, Applied Biomathematics, L. Ginzburg, State University of New York. Models of the kinetics of environmental toxicants generally assume that ecological processes such as population growth and predation can be ignored. Models that concentrate instead on the ecological processes typically assume that toxicant concentrations are constant through time and ignore toxicant kinetics. Clearly, the discipline of ecotoxicology will require us to consider both kinds of phenomena simultaneously, and to focus on their interaction through time. We describe a software shell running under Windows operating systems in which users can build and evaluate their own models to make probabilistic forecasts of the impacts of environmental toxicants on ecosystems consisting of food chains with several species, or single-species systems with age or stage structure. The shell of RAMAS Ecosystem uses state-of-the-art second-order Monte Carlo simulation and supports a rich array of risk-analytic summaries of the results. Users can choose among several different models of dose-reponse functions (such as Weibull, probit, logit, etc.), toxicant kinetics (first-order, constant, constant'environment), density dependence (ceiling, logistic, Ricker, etc.), and trophic interactions (Lotka-Volterra, ratio-dependent, Holling type II). The user can specify parameters as scalar numbers, intervals to represent measurement error (e.g., p10,15] mg per liter), or one of twenty name statistical distributions to represent temporal variability (e.g., lognormal(10,1) mg per liter). The software performs automatic unit conversions and checking of dimensional consistency. We illustrate its use with a study of the effect of an organophosphate on a food chain consisting of worms, sparrows and hawks. The results of the simulation appear to mirror what actually happens in such systems and would not have been expected from study of either the toxicant kinetics or the ecological dynamics separately.
	Checking the computational integrity of probabilistic risk analyses Abstract S. Ferson, Applied Biomathematics. Many have argued that probabilistic risk analyses should not be widely required because they would simply be too difficult for regulators to review on a routine basis. I describe a series of elementary checks that can be employed with modest effort by a reviewer to test (1) dimensional soundness of the expressions used in the calculations, (2) feasibility of the correlation structure among the input variables, and (3) quantitative plausibility of the computed risk estimates. The first check uses dimension and units analysis such as is currently available in several software implementations. The second check tests for positive semi-definiteness of the matrix of correlations among the input distributions of random numbers. The third check uses both interval analysis on the ranges of the input variables and moment constraint propagation on their first and second moments (i.e. their means and variances) to determine whether the reported risks are possible within the stated assumptions of the model. With software to perform the checks themselves, a reviewer need only transcribe the input variables (both the characterized distributions and their units) and the mathematic expression(s). The automated analyses can reveal many if not most of the serious qualitative and quantitative errors likely to occur in probabilistic risk analyses, and should therefore relieve the reviewer of much of his or her burden. Several examples taken from the recent risk analysis literature will be used to illustrate how errors, ranging from the simple to the profound, could have been caught by applying these straightforward checks.
	Cancer clusters: how to be sure you have one Abstract K. Hwang, S. Ferson, Applied Biomathematics, and R. Grimson, Stony Brook University. How much should we worry about ten more cases of childhood cancers in a city than would be expected from its population size and the background incidence rate? Until now, it has been difficult or impossible to compute P-values for putative diseases clusters when the data set is small (as it usually is for cancer). Traditional statistical tests for clustering assume asymptotically large sample sizes and are therefore not strictly applicable when data are sparse. Numerical studies show, in fact, that widel y used tests such as chi-square routinely and strongly overestimate the evidence for clustering. Thus, they can cause more alarm than is warranted. We describe several new statistical methods, implemented in a convenient software package, that can be used to compute exact P-values for clustering. These new methods can be used whatever the size of the data set, and are especially useful when data sets are extremely small. They provide tools to public health researchers and epidemiologists that, for the f irst time, have wide applicability for detecting clustering and other epidemiologic patterns in data sets of the size usually encountered in practice. We describe the relative statistical power of these tests under different kinds of clustering mechanisms and data set configurations. This work was supported by SBIR grant (R44GM49521) from the National Institutes of Health.
	Reliable probabilities when sample sizes are small and the model is uncertain Abstract S. Ferson, Applied Biomathematics. A risk analyst's knowledge can be incomplete in two ways. First, the probabilities of the inputs may be imprecisely known from statistical estimations, perhaps based on very small sample sizes. Second, dependency relationships among the inputs may be known imprecisely. Representing probability estimates as interval ranges on [0,1] has been suggested as a way to address the first source of imprecision. A suite of logical operators (AND, OR and NOT) defined by the classical Fréchet inequalities permit these probability intervals to be used in calculations that address the second source of imprecision, in many cases, in a best possible way. One can employ statistical confidence intervals to estimate the inputs, but doing so introduces the question of which confidence level should be used, and unravels the closure properties of this approach. A solution to this problem is to characterize each probability as a nested stack of intervals for all possible levels of statistical confidence, from a point estimate (0% confidence) to the entire unit interval (100% confidence). The implied calculation reduces by symmetry to a manageably simple computational problem. The resulting logical calculus can be implemented in software that allows users to compute comprehensive and often best-possible bounds on probabilities for logical functions of events. It yields reliable computations in the sense that answers are sure to enclose the true probabilities. The approach is illustrated with an example problem from forensic statistics. This work was supported by SBIR grant (R43ES06857) from the National Institutes of Health.
	Six methods to use when information is limited (as it always is) Abstract S. Ferson, Applied Biomathematics and D.R.J. Moore, The Cadmus Group. Several different methods for uncertainty propagation might be used in a risk analysis when there is little empirical information. Because the methods have different underlying axioms and assumptions about their input, they may be appropriate in different analytical situations. We compared six methods: 1) Monte Carlo analysis using conventional assumptions, 2) probabilistic analysis based on the maximum entropy criterion, 3) second-order (Monte Carlo) analysis, 4) probability bounds analysis, 5) fuzzy arithmetic, and 6) interval or worst case analysis. In applying these six methods to case studies, we found that they can yield substantially different results and that it may often be instructive to use more than one method in an analysis. Each approach has its theoretical advantages, but which is best in a particular circumstance depends on what and how much empirical information is available. Standard probabilistic approaches (methods 1, 2 and 3) cannot handle non-statistical uncertainty such as doubt about the proper form of the model. The bounding ap proaches (methods 4, 5 and 6) are not most powerful when very detailed information is available. We rank the six methods in terms of their conservativism, ease of use, data requirements, necessary assumptions, and appropriateness for use in screening and higher-tier assessments. Although Monte Carlo analysis is often touted as the only practical approach for uncertainty propagation, we conclude that other methods may sometimes be more appropriate. This work was supported by in part by SBIR grant R43ES06857 from the National Institutes of Health.
	Quality assurance for an environmental risk assessment Abstract S. Donald and S. Ferson, Applied Biomathematics. Although a probabilistic risk assessment using Monte Carlo techniques is widely regarded as the most comprehensive kind of uncertainty analysis, in current practice there are usually several assumptions that accompany an assessment that have not been justified by empirical information but are made for the sake of mathematical convenience. For example, an analyst typically specifies precise statistical distributions to be used as inputs to the model even though evidence supporting the particular choices ma y be fairly sparse. In some cases, there is controversy about the form of the model itself, including the nature of the dependencies among the variables and even the mathematical functions that tie them together. Simple methods based on the notion of interval probabilities have recently been described that can be used to incorporate these kinds of uncertainty into the analysis in a way that does not confound an analyst's subjective uncertainty with the true stochasticity of the system. These methods allow one to directly compute bounds on the probabilities of adverse events and therefore provide estimates of the reliability of a probabilistic risk assessment. We performed a quality assurance assessment using these methods on a previously reported human health risk analysis from multiple-pathway exposures to chemical contamination at an industrial facility by a complex mixture of PAHs and dioxins. The results quantify the reliability of conclusions drawn from the analysis about exposures and the need for remediation.
	Probability bounds analysis Abstract S. Ferson, Applied Biomathematics, and S. Donald, University of New Mexico. Probabilistic risk assessments almost always seem to demand more information about the statistical distributions and dependencies of input variables than is available empirically. For instance, to use Monte Carlo simulation, one generally needs to specify the particular shapes and precise parameters for all the input variables even if relevant data are very sparse. Moreover, imperfect understanding of how common-cause or common-mode mechanisms induce correlations or complicated dependencies among the variables typically forces analysts to assume independence even if they suspect otherwise. Most practitioners acknowledge the limitations induced by these problems, yet rarely employ sensitivity studies or second-order simulations to assess their possible significance because they would be computationally prohibitive. Probability bounds analysis allows assessors to sidestep both uncertainty about the precise specifications of input variables and imperfect information about the correlation and dependency structure among the variables. However, this approach has not been accessible to risk analysts because of a lack of convenient software. We describe probability bounds analysis software for Windows operating systems. The software has a graphical interface that is convenient and natural for quantitative risk assessors and permits calculations that are vastly less expensive computationally than alternative approaches. For instance, a second-order Monte Carlo simulation that required several days to compute on a microcomputer can be replaced by a calculation with probability bounds analysis that takes only seconds. A probability distribution or bounds on a probability distribution (p-bounds) can be specified interactively according to what empirical information is available. For instance, the parameters of the distributions can be given as precise numbers or as intervals. If the shape of the underlying distribution is not known, but some parameters such as the mean, mode, variance, etc. can be specified (or given as intervals), the software will construct bounds that are guaranteed to enclose the distribution subject to whatever constraints are specified. The software supports the full complement of standard operators and functions, including +, -, *, /, <, <=",">, >=, equality comparison, variable assignment, maximum, minimum, power, exponential, natural and common logarithm, square root, integer part, sine, cosine, arc tangent, and the basic logical operations (and, or, not). Binary operations are computed according to what can be assumed about the dependence between the operands. The software supports operations (i) under an assumption of independence, (ii) assuming the operands are positively dependent, (iii) assuming the operands are negatively dependent, or (iv) without any assumption whatever about the dependence between the operands. All operations and functions are transparently supported for pure or mixed expressions involving scalars, intervals, probability distributions and p-bounds. Expressions are evaluated as they are entered and the resulting values automatically displayed graphically. The software also accepts and propagates units embedded with the numbers in expressions. It checks that dimensions conform in additive and comparison operations so that trying, for instance, to add meters and seconds generates an error message. The software will also perform conversions automatically when needed to interpret input and complete calculations. For instance, it will correctly interpret inputs such as an interval with one endpoint given in feet and the other in meters, or a probability distribution with its mean given in kilograms and its standard deviation in grams, and it will produce the correct answer when a distribution in units of days is convolved with a distribution in units of hours. The software also supports basic programming constructs for conditional execution (if), looping (while), blocking (begin-end), string operations, user-defined functions and procedures, script execution, user-defined display windows, hypertext help, and an extensive library of examples that illustrate the properties of uncertainty arithmetic. This work was supported by funding from the National Institutes of Health, Southern California Edison and the Electric Power Research Institute.
	Statistically detecting clustering in very sparse data sets Abstract K. Hwang, S. Ferson, Applied Biomathematics, and R. Grimson, Stony Brook University. Cluster detection is considered to be essential in many environmental and epidemiological studies. Likewise, it can be very important in engineering studies for recognizing design flaws and cryptic common-mode or common-cause dependencies among rare events such as component failures. How can we tell whether a town's incidence of childhood cancers is significantly greater than the national average? How can we tell whether an airline's crash history is significantly worse than what one would expect given the industry's performance? The answers to these questions require some statistical method to detect of clustering among rare events. However, traditional statistical tests for detecting clustering assume asymptotically large sample sizes and are therefore not applicable when data are sparse-as they generally are for rare events. In fact, simulation studies show that the Type I error rates for traditional tests such as chi-square or the log-likelihood ratio (G-test) are routinely much larger than their no minal levels when applied to small data sets. As a result these tests can seriously overestimate the evidence for clustering and thus cause more alarm than is warranted. In other cases, traditional cluster tests can fail to detect clusters that can be shown by other methods to be statistically significant. Thus, the traditional approaches will provide an inefficient review of the available data. It is difficult to anticipate whether the traditional test will overestimate or underestimate the probability of clustering for a particular data set. Moreover, these tests are sensitive to a specific kind of deviation from randomness and may not provide the most appropriate measure of clustering from a specific mechanism. We describe eight new statistical methods, implemented in a convenient software package, that can be used to detect clustering of rare events in structured environments. Because the new tests employ exact methods based on combinatorial formulations, they yield exact P-values and cannot violate their nominal Type I error rates like the traditional tests do. As a result, the new tests are reliable whatever the size of the data set, and are especially useful when data sets are extremely small. By design, these tests are sensitive to different aspects of clusters and should be useful in discerning not only the fact of clustering but also something about the processes that generated the clustering. We characterize the relative statistical power of the new tests under different kinds of clustering mechanisms and data set configurations. The new statistical tests, along with several traditional and Monte Carlo tests for clustering, have been implemented in convenient graphical software for Windows operating systems which should be useful to risk and safety analysts for detect clustering of rare events in data sets even when the available data are sparse. This work was supported by SBIR grant R44GM49521 from the National Institute of General Medical Sciences of the National Institutes of Health.
	Why probability is insufficient for handling uncertainty in risk analysis Abstract S. Ferson, Applied Biomathematics. Risk analysts commonly distinguish variability resulting from heterogeneity or stochasticity from incertitude (partial ignorance) resulting from systematic measurement error or subjective uncertainty. In almost all risk assessments, both forms of uncertainty are present, although their respective magnitudes can vary widely. Most analysts now agree that variability and incertitude should be treated separately in risk analyses so that planners can design effective risk management strategies and future empricial efforts. My claim is much stronger: the two kinds of uncertainty require entirely difference propagation calculi. Some approach that does not make unjustified assumptions about randomness and independence should be used to propagate incertitude, and probability theory (with appropriate assumptions) should be used to propagate variability. Using conventional probabilistic methods for incertitude leads to results that are clearly erroneous, at least in terms of the goals of risk analysis. As the number of variables increases, or the extrapolation time grows, the problem becomes more and more severe. It is possible to represent and manipulate both incertitude and variability simultaneously in a coherent analysis that does not confound the two forms of uncertainty and distinguishes what is known from what is assumed, although it does not appear that two-dimensional Monte Carlo is sufficient to accomplish this.
	Probability bounds analysis (Why Laplace was wrong) Abstract S. Ferson, Applied Biomathematics. Whenever probability theory has been used to make calculations, analysts have routinely assumed (i) probabilities and probability distributions can be precisely specified, (ii) variables are all independent of one another, and (iii) model structure is known without error. For the most part, these assumptions have been made for the sake of mathematical convenience, rather than with any empirical justification. And, until now, these assumptions were pretty much necessary in order to get any answer at all. New methods now allow us to compute bounds on estimates of probabilities and probability distributions that are guaranteed to be correct even when one or more of the assumptions is relaxed or removed. In many cases, the results obtained are the best possible bounds, which means that tightening them would require additional empirical information.
	Why humans are so bad at interpreting probabilities Abstract S. Ferson and L.R. Ginzburg, Applied Biomathematics. It is widely recognized that the lay public has difficulty in grasping the meaning of risk analyses. The sometimes spectacular failures of risk communication strategies are often blamed on the public’s ignorance of technical issues or its mistrust of industry or government. We suggest, however, that it is neither ignorance nor mistrust that is fundamentally responsible for the difficulty. Instead, humans seem to have been wired by natural evolution to use a mental calculus for reckoning uncertainty and making decisions that is substantially different from probability theory. We suggest that several of the most important biases of probability perception that have been recognized by psychometricians can be interpreted as highly adaptive strategies for responding to variation and risk. Given that humans evolved in a strongly autocorrelated natural environment that heavily rewarded pattern recognition skills and often punished indecision more sternly than it did a suboptimal decision, it is easy to deduce evolutionary reasons why it is that (i) people routinely underestimate risks, (ii) people are insensitive to prior probabilities, (iii) the notion of independence is so difficult to correctly interpret, and (iv) people always ask how bad it could be (and ignore how unlikely that outcome is). If these biases and misconceptions are fundamental in human perception and not removable by general education or even specific training, perhaps it is incumbent on risk analysts to make calculations and arguments in a way that is more natural for humans, and that yields results that are interesting and compelling to them. We describe some of the properties and features of such an approach to risk assessment and communication.
	Symposium at the Society for Risk Analysis annual meeting The Balance of Nature: Can Toxicologists and Ecologists Come to Consensus? Abstract Chair: S. Ferson, Applied Biomathematics. The subject of the proposed symposium is the difference in perspective of toxicologists and ecologists and how it affects the evaluation of risks from contaminant effects to natural communities. The consequence of homeostasis of organisms and other stability mechanisms in natural communities are interpreted differently by the disciplines. Toxicologists often view these mechanisms as a cushion against toxicological insults. Ecologists, on the other hand, are more likely to view the balance of nature as an unstable one, which can be destroyed by toxicant effects. The goals of the symposium are to explore the differences between two communities and to illustrate ways to resolve disagreements to achieve a synthetic assessment of ecotoxicological risks. The speakers and their titles are: Todd S. Bridges, USACE Waterways Experiment Station, "Straining the gnat and swallowing the camel: The importance of scale and uncertainty in assessing ecological risk" Beth McGee, University of Maryland, "Gleaning ecological risks from sediment toxicity tests" Sara Hoover, Golder Associates, "Competing issues in contaminated site risk assessment: a case study" Bruce K. Hope, Oregon Department of Environmental Quality, "A new diet of worms: ecologists, toxicologists, and regulators"
	Symposium at the Society for Risk Analysis annual meeting Backcalculating cleanup goals Abstract Chair: S. Ferson, Applied Biomathematics. Risks from environmental contaminants are estimated with a risk equation involving contaminant concentration and other factors. Because the intent of environmental remediation is to ensure that these risks are not intolerably large, some way is needed to backcalculate from tolerance limits on risk mandated by regulators to the allowable environmental concentration for the contaminant. It is now well known that the approach used in deterministic assessments of simply inverting the risk equation to compute the cleanup goal does not work in a probabilistic assessment. Several approaches sidestepping the underlying mathematical problems have been proposed by various authors. The goal of this symposium is to review the needs of the regulatory community and examine whether and how the recent methodological work satisfies these needs. The speakers and their titles are: Ted Simon, U.S. Environmental Protection Agency, Region 4, "Calculating cleanup levels with monte carlo: regulatory concerns and perspective" Matthew Butcher and Rob Pastorok, Exponent Environmental, and Brad Sample, CH2M Hill, "Setting soil screening levels for wildlife at superfund sites" Robert Fares and K.G. Symms, Environmental Standards, Inc., "Back-calculation of soil cleanup levels utilizing monte carlo techniques" David Myers and Scott Ferson, Applied Biomathematics, "How to satisfy multiple constraints on cleanup goals in a probabilistic assessment"
	Surprising dynamics during ecological recovery after heavy metal contamination Abstract S. Ferson, Applied Biomathematics, and J. Crutchfield, Carolina Power & Light. The population of bluegill sunfish "Lepomis macrochirus" in part of a lake in North Carolina was decimated by toxicological and developmental effects of selenium leached from ash settling ponds. To forecast the potential recovery after cessation of heavy metal contamination, a demographic model was created for the bluegill population based on data collected from on-going biological monitoring at the lake. The model included density dependence which is known to be an important aspect of the life history of this species and used Monte Carlo methods to analyze the effect of natural environmental variability. The life history of the species revealed by analysis of the population model suggests that, if selenium poisoning were stopped, the population could recover to pre-impact abundances within two years. The increased abundan ce would be unevenly distributed among age groups, however. Following this increase in abundance, the biology predicts a population crash, especially among older year classes (which are prized by sportfishermen). This crash is due to the time-delayed effects of selenium on the population resulting from the strong non-linearity of density dependence in this species. The sharp increase in population size itself precipitates the crash. If this crash were not forecast in advance, its unanticipated occurrenc e could cause considerable consternation among managers, regulators and the interested public. This example shows that it can be important to predict ecological consequences to understand the nature and duration of biological recovery from toxicological insults. Without the understanding provided by the ecological analysis, the population decline would probably be completely misinterpreted as the failure of the mitigation program.
	Ecological risk assessment based on extinction distributions Abstract S. Ferson, Applied Biomathematics. Many researchers now agree that an ecological risk assessment should be a probabilistic forecast of effects at the level of the population. The emerging consensus has two essential themes: (i) individual-level effects are less important for ecological management, and (ii) deterministic models cannot adequately portray the environmental stochasticity that is ubiquitous in nature. It is important to resist the temptation to reduce a probabilistic analysis to a scalar summary based on the mean. An assessment of the full distribution of risks will be the most comprehensive and flexible endpoint. There are two ways to visualize a distributional risk assessment of a chemical's impact on a population. The first is to display, side by side, the two risk distributions arising from separate simulations with and without the impact but alike in every other respect. Alternatively, one can display the risk of differences between population trajectories with and without impact but alike in every other respect. Like all scientific forecasts, an ecological risk assessment requires appropriate uncertainty propagation. This can be accomplished by using a mixture of interval analysis and Monte Carlo simulation techniques.
	Robust risk analysis: What can we be sure about? Abstract S. Ferson, Applied Biomathematics. Four major problems inhibit the routine use of Monte Carlo methods in risk and uncertainty analyses: correlations and dependencies are often ignored, input distributions are usually not available, mathematical structure of the model is questionable, and there are no practical methods to back-calculate solutions for constraints on probabilistic equations. Most practitioners acknowledge the limitations induced by these problems, yet rarely employ sensitivity studies or other methods to assess their consequences. Recently developed techniques employing probability bounds analysis can be used to obtain (often optimal) bounds on risk calculations that do not require false or unjustified assumptions. The techniques therefore yield a comprehensive risk assessment without the need for precise information about inter-variable dependencies, and even without selection of specific input distributions for those variables. While Monte Carlo techniques for propagating model uncertainty are useful when it can be reduced to parametric uncertainty, a probability bounds approach is far more general and is useful whenever model uncertainty can be reduced to distributional uncertainty. Finally, the new techniques provide convenient solutions to probabilistic equations involving constraints such as are needed to compute the distribution of environmental concentrations that are sure to yield a distribution of doses specified by multiple constraints.
	Biological invasion Abstract S. Ferson, Applied Biomathematics. Biological invasion consists of dispersal and establishment of a species into new habitats. Invasions include diverse processes such as range expansion, spread of epidemics, and introductions of exotics, all of which can induce irreversible changes in an ecological community. The pattern of invasion is determined by how fast the species can disperse and the extent of the invadable habitat. Dispersal is often modeled with percolation theory, as exemplified by the well known "Game of Life" often played on computer screens. We learn from such models that when offspring invade only adjacent areas, the invasion is rather slow because the advance is limited by the perimeter length of the range. When offspring can invade distant areas, however, then the invasion can be much faster because it is limited by the total abundance of the population. Despite their extreme complexity, the frontiers of invasions are not fractals as some have suggested. However, measuring the complexity of the frontier may let us directly compute the relative chance for establishment of an invasion. Invasion patterns are reviewed for laboratory examples and several famous cases, including the starlings in North America, killer bees in South and Central America, and muskrat in Europe. For exotics and pest species, we are especially concerned with the chances of the population becoming large. We call the probability that a population grows to large abundances the "risk of population explosion". Estimating this risk is the symmetrically opposite problem to estimating the risk of extinction, and similar mathematical methods can be employed for both problems.
	Probabilistic screening assessment of ecological risks Abstract S. Ferson, Applied Biomathematics. To be practically useful in regulatory settings, screening assessments must have two properties: they must be conservative and they must be easy to perform. Most analysts presume that making an assessment probabilistic makes it relatively more complex and costly and will therefore force it into a higher-tier assessment. But the requirements of conservativism and ease do not in themselves preclude an assessment from being probabilistic. We describe an example of a screening assessment for ecological risks to a biological population that is fully probabilistic. The Malthusian model of population growth (the simplest model possible) can be written as a first-passage problem and solved for the probability that the population declines to any given level within a given time horizon. The formulation uses five quantities, including three describing the population (i) growth rate, (ii) variation in the growth rate, (iii) initial population size, and two quantities determined by the interests of the analyst, (iv) crossing threshold and (v) the time horizon. This formulation permits the evaluation of risks to a population from toxicological or other impacts whenever they can be expressed as changes in any of these quantities. There are only three quantities for which empirical estimates are needed and these estimates can be made conservative to temporal trending and uncertainty in a completely straightforward way. Therefore, a fully probabilistic assessment of ecological risks at the population level may be conducted in screening assessments rather than necessarily being relegated to higher tiers.
	What to do about model uncertainty Abstract M. Butcher, Exponent Environmental, and S. Ferson, Applied Biomathematics. When model uncertainty is ignored in a risk assessment, analysts may be overly confident in and thus misled by the results obtained. Probabilistic risk assessments based on Monte Carlo methods typically propagate model uncertainty by randomly choosing among possible models, which treats it in the same fashion as parameter uncertainty. Like the duck-hunting statisticians who shot above and below a bird and declared a hit, this Monte Carlo approach averages the available models, and can produce an aggregate model supported by no theory whatever. The approach represents this uncertainty in the choice of models by their mixture and the resulting answers can be dramatically wrong. We propose an alternative method that can comprehensively represent and propagate model uncertainty based on the idea of the envelope of distributions corresponding to different models. The central feature of this strategy is that it does not average together mutually incompatible models. What it provides are bounds on the resulting distribution from the risk assessment. This method is comprehensive in that it can handle the uncertainty associated with all identifiable models. It cannot, however, anticipate the true surprise of completely unrecognized mechanisms, although it may be more forgiving in such circumstances. We describe software that implements this strategy and illustrate its use in numerical examples. We also contrast the strategy with other possible approaches to the problem, including Bayesian and other kinds of model averaging.
	Cleanup Goals in a Probabilistic Assessment Abstract S. Ferson, Applied Biomathematics, D.S. Myers, Applied Biomathematics, and M. Butcher, Exponent Environmental. The ecological risk from a contaminant is estimated with a risk equation involving the contaminant's environmental concentration and other factors. Because the intent of environmental remediation is to ensure that these risks are not intolerably large, we need some way to backcalculate from constraints on risk mandated by regulation or prudence to the allowable environmental concentration for the contaminant. It is now well known that the approach used in deterministic assessments of simply inverting the risk equation to compute the cleanup goal does not work in a probabilistic assessment. Several approaches that sidestep the underlying mathematical problems have been proposed by various authors, but all are strongly limited in their generality. We present for the first time simple and efficient methods to compute cleanup goals that satisfy multiple simultaneous criteria in the context of a probabilistic assessment. The approach can be used with multiple receptors and with arbitrarily many constraints on percentiles or moments of the target risk. This approach uses probability bounds analysis to characterize concentration distributions that satisfy the constraints. The calculations yield two kinds of bounds on concentration: a 'core' and 'shell'. If the concentration distribution is entirely inside the core, then the result surely obeys the prescribed constraints. If the concentration distribution is anywhere outside the shell, then the result certainly fails to comply with the prescribed constraints. If the concentration distribution is outside the core but inside the shell, then compliance must be determined by a forward calculation. Although the core is essentially comparable to the screening level familiar from deterministic assessments, the shell cannot be similarly analogized with an action level.
	Population-level assessments with almost no data Abstract S. Ferson, Applied Biomathematics. Risks to biological populations are among the most relevant endpoints in environmental assessment. However, in most tiered assessment systems, ecological risks are usually considered only at the highest tiers, where monitoring costs are greatest and analyses are most complex. We show how a probabilistic population-level assessment can be conducted at the screening level where answers must be both conservative and cheap to obtain. The Malthusian model of population growth can be expressed as a first-passage problem and solved for the probability that the population declines to any given level within a given time horizon. The formulation uses five quantities, three of which describe the population: (i) average rate of population growth, (ii) variation in the growth rate, and (iii) initial population size. Two quantities represent interests of the analyst: (v) crossing threshold and (vi) the time horizon. This formulation permits the evaluation of additional risks to a population from toxicological impacts whenever they can be expressed as changes in any of these quantities. This means that evaluating ecological risks at the population level may be conducted in screening assessments rather than necessarily being relegated to higher tiers. There are of course many ecological phenomena that are ignored in this simple formulation, including age or spatial structure, density dependence, dispersal, windows of recruitment, trophic interactions, demographic stochasticity, and autocorrelation in environmental conditions. However, it is generally possible to bound the effects of such phenomena and make conservative calculations as needed for a screening assessment. Comparisons with more comprehensive population assessments that use extensive empirical information demonstrate that the results of the screening-level assessments bound the best estimates of population risks.
	Can we know the iceberg from its tip? Censored distributions in risk assessment Abstract M. Butcher, Exponent Environmental, T. Barry, U.S. Environmental Protection Agency, and S. Ferson, Applied Biomathematics. Data censoring occurs when empirical information about a quantity is limited to knowing only that its value is less than (or greater than) some threshold. Censored distributions commonly arise when observed chemical concentrations contain results that are reported as non-detects. There are several methods available for estimating the underlying distribution from censored data sets. These methods approximate the underlying distribution based on assumptions about the underlying distribution, but their performance degrades as the proportion of non-detects grows to dominate the sample. We describe a method for incorporating all the available data from a censored data set in a risk assessment, without imposing any assumptions on them. We show that even in the cases of highly censored distributions, the information contained in the detected values is still useful, because it is they that contribute to highest dose or exposure. In conjunction with the information known from the censored observations (number of samples and their detection limit), these data may be used to define a probability region, in place of an estimate of a single distribution. The method is sufficiently general to be applied to any source of measurement error, and therefore may be extended to the more difficult cases of multiple detection limits within a data set and the inclusion of uncertainty provided by the analytical laboratory for each concentration. Software for this method is described, and numerical examples involving radon exposures are provided. a name="orleans">
	Ecosystem models for ecological risk analysis: From single species to communities Abstract S. Ferson, Applied Biomathematics. To justify regulatory and mitigation decisions, toxicologist are often asked the "so what?" questions that demand predictions about the population or even ecosystem response to contamination. RAMAS Ecotoxicology and RAMAS Ecosystem are microcomputer software specifically created to assist toxicologists answer such questions by extrapolating effects on organisms observed in bioassays to their eventual population-level consequences. It provides a software shell from which users can construct their own models for projecting toxi city effects through the complex filters of demography, density dependence and ecological interactions in foodchains. It allows various standard choices about low-dose response models (probit, etc.), which vital parameters are affected by the toxicant, the magnitudes and variabilities of these impacts, and species-specific life history descriptions. During the calculations, the software distinguishes between measurement error and stochastic variability. It forecasts the expected risks of population decli nes resulting from toxicity of the contaminant and provides estimates of the reliability of these expectations in the face of empirical uncertainty. This risk-analytic endpoint is a natural summary that integrates disparate impacts on biological functions over many organization levels. Where applicable, the software automatically performs consistency tests to check the at the input conforms to statistical assumptions and is dimensionally coherent. Parameterizations have already been prepared for several vertebrate and invertebrate species for use in assessments of soil or sediment contamination.
	Additional Abstracts Regan, H.M., B.E. Sample and S. Ferson. Comparison of deterministic and probabilistic calculation of ecological soil screening levels. Environmental Toxicology and Chemistry [accepted for publication]. The U.S. EPA is sponsoring development of Ecological Soil Screening Levels (Eco-SSLs) for terrestrial wildlife. These are intended to be used to identify chemicals of potential ecological concern at Superfund sites. Eco-SSLs represent concentrations of contaminants in soils that are believed to be protective of ecological receptors. An exposure model, based on soil and food ingestion rates, and the relationship between the concentrations of contaminants in soil and food, has been developed for estimation of wildlife Eco-SSLs. It is important to understand how conservative and protective these values are, how parameterization of the model influences the resulting Eco-SSL, and how the treatment of uncertainty impacts results. Eco-SSLs were calculated for meadow voles (Microtus pennsylvanicus) and northern short-tailed shrews (Blarina brevicauda) for lead and DDT using deterministic and probabilistic methods. Conclusions obtained include: use of central-tendency point estimates may result in hazard quotients much larger than one; a Monte Carlo approach also leads to hazard quotients that can be substantially larger than one; if no hazard quotients larger than one are allowed, any probabilistic approach is identical to a worse-case approach; the larger the uncertainty about inputs, the smaller the Eco-SSL must be. This is the inherent cost of uncertainty. Ferson, S., L.R. Ginzburg and H.R. Akçakaya. Whereof one cannot speak: when input distributions are unknown. Risk Analysis [accepted for publication]. One of the major criticisms of probabilistic risk assessment is that the requisite input distributions are often not available. Several approaches to this problem have been suggested, including creating a library of standard empirically fitted distributions, employing maximum entropy criteria to synthesize distributions from a priori constraints, and even using 'default' inputs such as the triangular distribution. Since empirical information is often sparse, analysts commonly must make assumptions to select the input distributions without empirical justification. This practice diminishes the credibility of the assessment and any decisions based on it. There is no absolute necessity, however, of assuming particular shapes for input distributions in probabilistic risk assessments. It is possible to make the needed calculations using inputs specified only as bounds on probability distributions. We describe such bounds for a variety of circumstances where empirical information is extremely limited, and illustrate how these bounds can be used in computations to represent uncertainty about input distributions far more comprehensively than is possible with current approaches. Ferson, S. Checking for errors in calculations and software: dimensional balance and conformance of units. Accountability in Research: Policies and Quality Assurance [accepted for publication]. Although there has always been a general awareness that mathematical expressions must make dimensional sense in terms of the units involved, it is very easy to make simple mistakes in quantitative work that result in profound and potentially dangerous errors. Such errors are ubiquitous in modern research, as can be seen by reviewing government publications where dimensional errors persist despite peer and public review. Software methods have recently become available for checking calculations, equations, algorithms and programs for dimensional soundness. Correctness depends on conformance at two levels: balance of dimensions and agreement among units. Error at either level can now be detected automatically by software. Disagreement among units can even be automatically corrected by software procedures. These software tools can be used to check for errors in calculations and software source code in a way that is similar to using a spelling or grammar checker for text. Spencer, M., N.S. Fisher, W.-X. Wang, S. Ferson. Temporal variability and ignorance in Monte Carlo contaminant bioaccumulation models: a case study with selenium in Mytilus edulis. Risk Analysis [accepted for publication]. Although the parameters for contaminant bioaccumulation models most likely vary over time, lack of data makes it impossible to quantify this variability As a consequence,Monte Carlo models of contaminant bioaccumulation often treat all parameters as having .xed true values that are unknown.This can lead to biased distributions of predicted contaminant concentrations This article demonstrates this phenomenon with a case study of selenium accumulation in the mussel Mytilus edulis in San Francisco Bay "Ignorance-only " simulations (in which phytoplankton and bioavailable selenium concentrations are constant over time, but sampled from distributions of field measurements taken at different times),which an analyst might be forced to use due to lack of data,were compared with "variability and ignorance " simulations (sampling phytoplankton and bioavailable selenium concentrations each month).Ignorance-only simulations may underestimate or overestimate the median predicted contaminant concentration at any time,relative to variability and ignorance simulations However,over a long enough time period (such as the complete seasonal cycle in a seasonal model), treating temporal variability as if it were ignorance would at least give a range of predicted concentrations that encloses the range predicted by explicit treatment of temporal variability Comparing the temporal variability in field data with that predicted by simulations may indicate whether the right amount of temporal variability is being included in input variables Sensitivity analysis combined with biological knowledge may suggest which parameters might make important contributions to temporal variability Temporal variability is potentially more complicated to deal with than other types of stochastic variability,because of the range of time scales over which parameters may vary. Burgman, M.A., D.R. Breininger, B.W. Duncan and S. Ferson. Setting reliability bounds on habitat suitability indicies. Ecological Applications [in press]. We expressed quantitative and qualitative uncertainties in suitability index functions as triangular distributions and used the mechanics of fuzzy numbers to solve for the distribution of uncertainty around the habitat suitability indices derived from them. We applied this approach to a habitat model for the Florida Scrub-Jay. The results demonstrate that priorities and decisions associated with management and assessment of ecological systems may be influenced by an explicit consideration of the reliability of the indices. Crutchfield, J. and S. Ferson. 2000. Predicting recovery of a fish population after heavy metal impacts. Environmental Science and Policy 3: S183-S189. Bluegill sunfish, Lepomis macrochirus, in part of Hyco Reservoir (North Carolina) were decimated by toxicological and developmental effects of selenium leached from coal ash settling ponds during 1970-1980. Bluegill are especially sensitive to elevated concentrations of the heavy metal, and near-complete recruitment failure of zero-year olds was observed. To predict the potential recovery after cessation of heavy metal contamination, a demographic model was created for the bluegill population based on data collected from on-going biological monitoring at the lake. The model included density dependence and used Monte Carlo methods to analyze the effects of natural environmental variability. The life history of the species suggests that once selenium poisoning stopped, the population could recover to pre-impact abundances within two years, although the increased abundance would be unevenly distributed among age groups. However, following this increase in abundance, we predicted a population crash due to the time-delayed effects of selenium on the population resulting from the strong nonlinearity of density dependence in this species. The sharp increase in population size itself precipitates the crash which, if not forecast in advance, could cause considerable concern among managers, regulators and the interested public. This example shows that it can be important to predict ecological consequences to understand the nature and duration of biological recovery of toxicological insults. Without the understanding provided by the ecological analysis, the population decline would probably be completely misinterpreted as the failure of the mitigation program. Akçakaya, H.R., S. Ferson, M. Burgman, D. Keith, G. Mace and C. Todd. 2000. Making consistent IUCN classifications under uncertainty. Conservation Biology 14: 1001-1013. The World Conservation Union (IUCN) defined a set of categories for conservation status supported by decision rules based on thresholds of parameters such as distributional range, population size, population history, and risk of extinction. These rules have received international acceptance and have become one of the most important decision tools in conservation biology because of their wide applicability, objectivity, and simplicity of use. The input data for these rules are often estimated with considerable uncertainty due to measurement error, natural variation, and vagueness in definitions of parameters used in the rules. Currently, no specific guidelines exist for dealing with uncertainty. Interpretation of uncertain data by different assessors may lead to inconsistent classifications because attitudes toward uncertainty and risk may have an important influence on the classification of threatened species. We propose a method of dealing with uncertainty that can be applied to the current IUCN criteria without altering the rules, thresholds, or intent of these criteria. Our method propagates the uncertainty in the input parameters and assigns the evaluated species either to a single category (as the current criteria do) or to a range of plausible of categories, depending on the nature and extent of uncertainties. Fortin, M.-J., R.J. Olson, S. Ferson. L. Iverson, D. Levine, K. Buteras, V. Klemas and C. Hunsaker. 2000. Detecting boundaries and gradients associated with ecotones. Landscape Ecology 15: 453-466. Ecotones are inherent features of landscapes, transitional zones, and play more than one function role in ecosystem dynamics. The delineation of ecotones and environmental boundaries is therefore an important step in land-use management planning. The delineation of ecotones depends on the phenomenon of interest and the statistical methods used as well as the associated spatial and temporal resolution of the data available. In the context of delineating wetland and riparian ecosystems, various data types (field data, remotely sensed data) can be used to delineate ecotones. Methodological issues related to their detection need to be addressed, however, so that their management and monitoring can yield useful information about their dynamics and functional roles in ecosystems. The aim of this paper is to review boundary detection methods. Because the most appropriate methods to detect and characterize boundaries depend of the spatial resolution and the measurement type of the data, a wide range of approaches are presented: GIS, remote sensing and statistical ones. Goldwasser, L., L. Ginzburg and S. Ferson. 2000 Variability and measurement error in extinction risk analysis: the northern spotted owl on the Olympic Peninsula. Quantitative Methods for Conservation Biology, S. Ferson and M. Burgman (eds.), Springer-Verlag, New York. We distinguish two sources of uncertainty in analyzing risks of extinction. The first is stochastic variability caused by fluctuations in the environment and variation among individuals. The second is measurement error induced by incomplete sampling and empirical imprecision. We argue that the two kinds of uncertainty require different treatments when uncertainty is propagated through models of population dynamics. We use published data on the Olympic Peninsula population of the northern spotted owl (Strix occidentalis caurina) to illustrate the importance of this difference. Using recently proposed quantitative criteria for extinction threat, this owl population appears either to be threatened if the current number of owl breeding territories is 200, or to be non-threatened if this number is closer to 300. Taking measurement error into account strongly increases the uncertainty in this determination. Although we can be reasonably confident that this population will persist at least for the next 30 40 years, reliable risk evaluations over a 100-year time scale are probably impossible. Ferson, S., H.R. Akçakaya and A. Dunham. 1999. Using fuzzy intervals to represent measurement error and scientific uncertainty in endangered species classification. Real World Application of Fuzzy Logic and Soft Computing, R.N. Davé and T. Sudkamp (eds.), Proceedings of the 18th International Conference of NAFIPS, IEEE, Piscataway, New Jersey. Although fuzzy numbers (including fuzzy intervals) are often used to capture semantic ambiguity, they are also useful to represent and propagate measurement error. In this application, a classification scheme used by international authorities for assigning biological species into categories of relative endangerment is generalized to accept intervals and triangular or trapezoidal fuzzy numbers as inputs representing empirical estimates of unknown quantities. Non-traditional definitions for fuzzy magnitude comparisons and logical operations were required but, otherwise, standard fuzzy arithmetic was used. A defuzzification step, which explicitly reveals the analyst's attitudes regarding evidence, can condense the result from the fuzzified classification scheme to a single category. But this step is not required and may be counterproductive. Ferson, S. 1999. Ecological risk assessment based on extinction probabilities of populations. Proceedings of the Second International Workshop on Risk Evaluation and Management of Chemicals, J. Nakanishi (ed.), Japan Science and Technology Corporation, Yokohama. Many researchers now agree that an ecological risk assessment should be a probabilistic forecast of effects at the level of the population. The emerging consensus has two essential themes: (i) individual-level effects are less important for ecological management, and (ii) deterministic models cannot adequately portray the environmental stochasticity that is ubiquitous in nature. It is important to resist the temptation to reduce a probabilistic analysis to a scalar summary based on the mean. An assessment of the full distribution of risks will be the most comprehensive and flexible endpoint. There are two ways to visualize a distributional risk assessment of a chemical's impact on a population. The first is to display, side by side, the two risk distributions arising from separate simulations with and without the impact but alike in every other respect. Alternatively, one can display the risk of differences between population trajectories with and without impact but alike in every other respect. Like all scientific forecasts, an ecological risk assessment requires appropriate uncertainty propagation. This can be accomplished by using a mixture of interval analysis and Monte Carlo simulation techniques. Ferson, S. and S. Donald. 1998. Probability bounds analysis, pp. 1203-1208 in Probabilistic Safety Assessment and Management, A. Mosleh and R.A. Bari (eds.), Springer-Verlag, New York. Probabilistic risk assessments almost always demand more information about the statistical distributions and dependencies of input variables than is available empirically. For instance, to use Monte Carlo simulation, one generally needs to specify the particular shapes and exact parameters for all the input variables. Imperfect understanding of how common-cause or common-mode mechanisms induce correlations or complicated dependencies among the variables typically forces analysts to assume independence even if they suspect otherwise. Probability bounds analysis allows assessors to sidestep both uncertainty about the precise specifications of input variables and imperfect information about the correlation and dependency structure among the variables to compute rigorous bounds on the resulting risks. Hwang, K. and S. Ferson. 1998. Detecting rare event clustering in very sparse data sets, pp. 1235-1240 in Probabilistic Safety Assessment and Management, A. Mosleh and R.A. Bari (eds.), Springer-Verlag, New York. Traditional statistical methods to detect clustering are inappropriate for the sparse data available for rare events because they assume large sample sizes. These methods can either underestimate or overestimate the evidence for clustering, and it is hard to predict the direction of the error. Moreover, the traditional statistics are very coarse indicators of non-random pattern and are not especially designed to detect clustering per se. We describe eight new statistical tests, implemented in a convenient software package, that can be used to detect clustering of rare events in structured environments. Because the new tests employ combinatorial formulations, they are exact methods. As a result, the new tests are reliable whatever the size of the data set, and are especially useful when data sets are extremely small. By design, these tests are sensitive to different aspects of clusters and should be useful in discerning not only the fact of clustering but also something about the processes that generated the clustering. Ferson, S. and T.F. Long. 1997. Deconvolution can reduce uncertainty in risk analyses. Risk Assessment: Measurement and Logic, M. Newman and C. Strojan (eds.), Ann Arbor Press. The operations implemented in familiar software packages such as @Risk, Crystal Ball and Analytica routinely estimate convolutions of random variables. In comprehensive risk analyses, however, deconvolutions should be used whenever justified because they will often yield narrower distributions containing less overall uncertainty. Furthermore, in some circumstances, failing to use deconvolutions when appropriate can lead to conclusions that are insufficiently protective. Deconvolutions reduce uncertainty by taking into account knowledge about the underlying relationship among the variables. Although they are almost never used in current risk analyses, situations which justify their use arise frequently in practical situations. Both probabilistic and traditional worst-case or bounding-estimate risk analyses can benefit from appropriate use of deconvolution. We outline when deconvolutions can be used and how they are computed in both analytical settings. Ferson, S. 1997. Probability bounds analysis software. Computing in Environmental Resource Management. Proceedings of the Conference, A. Gertler (ed.), Air and Waste Management Association and the U.S. Environmental Protection Agency, Pittsburgh, Pennsylvania. pp. 669-678. Probabilistic risk assessments almost always require more information about the statistical distributions and dependencies of input variables than is empirically available. For instance, to use Monte Carlo simulation, one generally needs to specify the particular shapes and precise parameters for all the input variables even if relevant data are very sparse. Moreover, imperfect understanding of how common-cause or common-mode mechanisms induce correlations or complicated dependencies among the variables typically forces analysts to assume independence even if they suspect otherwise. Most practitioners acknowledge the limitations induced by these problems, yet rarely employ sensitivity studies or second-order simulations to assess their possible significance because they would be computationally prohibitive. Probability bounds analysis allows assessors to sidestep both uncertainty about the precise specifications of input variables and imperfect information about the correlation and dependency structure among the variables. Probability bounds analysis is therefore an important tool for establishing quality assurance of probabilistic risk assessments, yet until now this tool has not been accessible to the practicing risk analyst because of a lack of convenient software. The package Risk Calc implements probability bounds analysis under the Windows operating system. The interface is convenient and natural for quantitative risk assessors. It offers a powerful array of standard functions, which are extensible by means of user-written programs. The software permits calculations that are vastly less expensive computationally than alternative approaches. For instance, a second-order Monte Carlo simulation that required two weeks to compute on a microcomputer can be replaced by a calculation with probability bounds analysis in Risk Calc that takes only seconds. The features and uses of Risk Calc are outlined. Ferson, S. 1996. What Monte Carlo methods cannot do. Human and Ecological Risk Assessment 2:990-1007. Although extremely flexible and obviously useful for many risk assessment problems, Monte Carlo methods have four significant limitations that risk analysts should keep in mind. (1) Like most methods based on probability theory, Monte Carlo methods are data-intensive. Consequently, they usually cannot produce results unless a considerable body of empirical information has been collected, or unless the analyst is willing to make several assumptions in the place of such empirical information. (2) Although appropriate for handling variability and stochasticity, Monte Carlo methods cannot be used to propagate partial ignorance under any frequentist interpretation of probability. (3) Monte Carlo methods cannot be used to conclude that exceedance risks are no larger than a particular level. (4) Finally, Monte Carlo methods cannot be used to effect deconvolutions to solve backcalculation problems such as often arise in remediation planning. This paper reviews a series of ten exemplar problems in risk analysis for which classical Monte Carlo methods yield an incorrect answer. Ferson, S. and L.R. Ginzburg. 1996. Different methods are needed to propagate ignorance and variability. Reliability Engineering and Systems Safety 54:133-144. There are two kinds of uncertainty. One kind arises as variability resulting from heterogeneity or stochasticity. The other arises as partial ignorance resulting from systematic measurement error or subjective (epistemic) uncertainty. As most researchers recognize, variability and ignorance should be treated separately in risk analyses. Although a second-order Monte Carlo simulation is commonly employed for this task, this approach often requires unjustified assumptions may be inappropriate in some circumstances. We argue the two kinds of uncertainty should be propagated through mathematical expressions with different calculation methods. Basically, interval analysis should be used to propagate ignorance, and probability theory should be used to propagate variability. We demonstrate how using an inappropriate method can yield erroneous results. We also show how ignorance and variability can be represented simultaneously and manipulated in a coherent analysis that does not confound the two forms of uncertainty and distinguishes what is known from what is assumed. Cooper, J.A., S. Ferson and L.R. Ginzburg. 1996. Hybrid processing of stochastic and subjective uncertainty data. Risk Analysis 16: 785-791. Uncertainty analyses typically recognize separate stochastic and subjective sources of uncertainty, but do not systematically combine the two, although a large amount of data used in analyses is partly stochastic and partly subjective. We have developed methodology for mathematically combining stochastic and subjective data uncertainty, based on new "hybrid number" approaches. The methodology can be utilized in conjunction with various traditional techniques, such as PRA (probabilistic risk assessment) and risk analysis decision support. Hybrid numbers have been previously examined as a potential method to represent combinations of stochastic and subjective information, but mathematical processing has been impeded by the requirements inherent in the structure of the numbers, e.g., there was no known way to multiply hybrids. In this paper, we will demonstrate methods for calculating with hybrid numbers that avoid the difficulties. By formulating a hybrid number as a probability distribution that is only fuzzily known, or alternatively as a random distribution of fuzzy numbers, methods are demonstrated for the full suite of arithmetic operations, permitting complex mathematical calculations. It will be shown how information about relative subjectivity (the ratio of subjective to stochastic knowledge about a particular datum) can be incorporated. Techniques are also developed for conveying uncertainty information visually, so that the stochastic and subjective constituents of the uncertainty, as well as the ratio of knowledge about the two, are readily apparent. The techniques demonstrated have the capability to process uncertainty information for independent, uncorrelated data, and for some types of dependent and correlated data. Example applications are suggested, illustrative problems are worked, and graphical results are given. Ginzburg, L.R., C. Janson, and S. Ferson. 1996. Judgment under uncertainty: evolution may not favor a probabilistic calculus. Behavioral and Brain Sciences 19: 24f. The environment in which humans evolved is strongly positively autocorrelated in space and time. Probabilistic judgements based on the assumption of independence may not yield evolutionarily adaptive behavior. A number of 'faults' of human reasoning are not faulty under fuzzy arithmetic, a non-probabilistic calculus of reasoning under uncertainty that may be closer to that underlying human decision making. Ferson, S. 1996. Reliable calculation in probabilistic logic: accounting for small sample size and model uncertainty. Intelligent Systems: A Semiotic Perspective, NIST, Gaithersburg, Maryland. 115-121. A variety of practical computational problems arise in risk and safety assessments, forensic statistics and decision analyses in which the probability of some event or proposition E is to be estimated from the probabilities of a finite list of related subevents or propositions F,G,H,.... In practice, the analyst's knowledge may be incomplete in two ways. First, the probabilities of the subevents may be imprecisely known from statistical estimations, perhaps based on very small sample sizes. Second, relationships among the subevents may be known imprecisely. For instance, there may be only limited information about their stochastic dependencies. Representing probability estimates as interval ranges on [0,1] has been suggested as a way to address the first source of imprecision. A suite of AND, OR and NOT operators defined with reference to the classical Fréchet inequalities permit these probability intervals to be used in calculations that address the second source of imprecision, in many cases, in a best possible way. Using statistical confidence intervals as inputs unravels the closure properties of this approach however. One solution is to characterize each probability as a nested stack of intervals for all possible levels of statistical confidence, from a point estimate (0% confidence) to the entire unit interval (100% confi-dence). The corresponding logical operations implied by convolutive application of the logical operators for every possible pair of confidence intervals reduces by symmetry to a manageably simple level-wise iteration. The resulting logical calculus can be implemented in software that allows users to compute comprehensive and often level-wise best possible bounds on probabilities for logical functions of events. Ferson, S., L.R. Ginzburg and R.A. Goldstein. 1995. Inferring ecotoxicological risk from toxicity bioassays. Water, Air and Soil Pollution 90:71-82. Results from toxicological bioassays can express the likely impact of environmental contamination on biochemical function, histopathology, development, reproduction and survivorship. However, justifying environmental regulatory decisions and management plans requires predictions of the consequent effects on ecological populations and communities. Although extrapolating the results of toxicity bioassays to potential effects on the ecosystem may be beyond the current scientific capacity of ecology, it is possible to make detailed forecasts at the level of a population. We give examples in which toxicological impacts are either magnified or diminished by population-dynamic phenomena and argue that ecological risk assessments should be conducted at a level no lower than the population. Although methods recently proposed by EPA acknowledge that ecological risk evaluations should reflect population-level effects, they adopt approaches from human health risk analysis that focus on individuals. Ferson, S. 1995. Quality assurance for Monte Carlo risk assessments. Proceedings of the 1995 Joint ISUMA/NAFIPS Symposium on Uncertainty Modeling and Analysis, IEEE Computer Society Press, Los Alamitos, California, pp. 14-19. Three major problems inhibit the routine use of Monte Carlo methods in risk and uncertainty analyses: (1) correlations and dependencies are often ignored, (2) input distributions are usually not available, and (3) mathematical structure of the model is questionable. Most practitioners acknowledge the limitations induced by these problems, yet rarely employ sensitivity studies or other methods to assess their consequences. This paper reviews several computational methods that can be used to check a risk assessment for the presence of certain kinds of fundamental modeling mistakes, and to assess the possible error that could arise when variables are incorrectly assumed to be independent or when input distributions are incompletely specified. decisions. Ferson, S. and L. Ginzburg. 1995. Hybrid arithmetic. Proceedings of the 1995 Joint ISUMA/NAFIPS Symposium on Uncertainty Modeling and Analysis, IEEE Computer Society Press, Los Alamitos, California, pp. 619-623. Kaufmann's formulation of hybrid numbers, which simultaneously express fuzzy and probabilistic uncertainty, allows addition and subtraction, but offers no obvious way to do multiplication, division or other operations. We describe another, more comprehensive formulation for hybrid numbers that allows the full suite of arithmetic operations, permitting them to be incorporated into complex mathematical calculations. There are two complementary approaches to computing with these hybrid numbers. The first is extremely efficient and yields theoretically optimal results in many circumstances. The second more general approach is based on Monte Carlo simulation using intervals or fuzzy numbers rather than scalar numbers. Ferson, S. and M. Burgman. 1995. Correlations, dependency bounds and extinction risks. Biological Conservation 73:101-105. Methods for estimating bounds on extinction risks are described for cases where correlations among the parameters in a quantitative population viability analysis are unknown, and more generally where the forms of statistical dependence among the model's parameters are unknown. An example using Leadbeater's possum shows that making incorrect assumptions about correlations and dependencies may lead to severe over-estimation or under-estimation of extinction risks. Ferson, S. 1995. Using approximate deconvolution to estimate cleanup targets in probabilistic risk analyses, pages 239-248 in Hydrocarbon Contaminated Soils, P. Kostecki (ed). Amherst Scientific Press, Amherst, Massachusetts. Although deconvolution seems to be virtually unknown among practicing risk analysts, it will likely play an important role in backcalculating the soil cleanup goals necessary to satisfy remediation requirements. Many algorithms for computing deconvolutions have been described, but most are numerically unstable and extremely sensitive to noise. Even modern nonlinear methods which have considerably improved performance characteristics can be unwieldy in the kinds of uses required in risk analysis. A simple but robust approximate method to estimate deconvolutions is described and illustrated with a numerical example. Ferson, S. and T.F. Long. 1995. Conservative uncertainty propagation in environmental risk assessments. Environmental Toxicology and Risk Assessment, Third Volume, ASTM STP 1218, J.S. Hughes, G.R. Biddinger and E. Mones (eds.), American Society for Testing and Materials, Philadelphia, pp. 97-110. Toxicological risk analysis is at a crossroads. The traditional approach using worst case analysis is widely regarded as fundamentally flawed since it yields conclusions that are often strongly biased and presumed hyperconservative. Probabilistic analysis using Monte Carlo simulation can yield overly optimistic conclusions when used without information about the correlation structure among variables. What is needed is a conservative methodology that makes no assumptions unwarranted by empirical evidence. To be conservative means both that the estimated risk is not systematically lower than the actual impact, but also that the uncertainty around the estimate is not narrower than justified by the available data. An appropriate methodology is dependency bounds analysis which computes bounds on the distributions for arithmetic operations on random variables when only their marginal distributions are known. Used in a risk analysis, it yields conservative results because it does not depend on knowledge about the correlation structure among all of the variables used in the analysis. Ferson, S. 1994. Naive Monte Carlo methods yield dangerous underestimates of tail probabilities. Proceedings of the High Consequence Operations Safety Symposium, Sandia National Laboratories, SAND94-2364, J.A. Cooper (ed.), pp. 507-514. Extreme-event probabilities (i.e., the tails) of a statistical distribution resulting from probabilistic risk analysis can depend strongly on dependencies among the variables involved in the calculation. Although well known techniques exist for incorporating correlation into analyses, in practice they are often neglected on account of a paucity of information about joint distributions. Furthermore, certain forms of dependency that are not adequately measured by simple correlation must perforce be omitted from such assessments. Two general techniques may be used to compute conservative estimates of the tails in the context of ignorance about correlation and dependency among the variables. The first is based on a variance maximization/minimization trick. It is compatible with existing Monte Carlo methods, but it accounts only for linear dependencies. The second is based on Fréchet inequalities, and, although incompatible with Monte Carlo methods, it guarantees conservative estimates of tail probabilities no matter what dependence structure exists among the variables in the analysis. Ferson, S. and R. Kuhn. 1994. Interactive microcomputer software for fuzzy arithmetic. Proceedings of the High Consequence Operations Safety Symposium, Sandia National Laboratories, SAND94-2364, J.A. Cooper (ed.). We describe a new microcomputer implementation of fuzzy arithmetic for the Microsoft Windows operating system based on the strict definition of fuzzy numbers (i.e., fuzzy sets that are both convex and normal). Fuzzy numbers can be specified by a user as intervals in the form [a,b] or [a±b], as triangular numbers in the form [a,b,c], as trapezoidal numbers in the form [a,b,c,d], or as an arbitrary list of points describing the fuzzy membership function. A fuzzy number can also be synthesized as the consensus of a list of intervals. The software supports the full complement of standard operators and functions, including +, -, ´, ¸¸ >, ³, <, £, equality comparison, variable assignment, additive and multiplicative deconvolution, maximum, minimum, power, exponential, natural and common logarithm, square root, integer part, sine, cosine, and arc tangent. The basic logical operations (and, or, not) are interpreted in both traditional boolean and fuzzy-logical contexts. Also several new functions for fuzzy operands are introduced, including least and greatest possible value, breadth of uncertainty, most possible interval, interval of specified possibility, and possibility level of a scalar. All operations and functions are transparently supported for pure or mixed expressions involving scalars, intervals, and fuzzy numbers. Expressions are evaluated as they are entered and the resulting values automatically displayed graphically. The software also accepts and propagates (arbitrarily named) units embedded with the numbers in expressions and checks that their dimensions conform in additive and comparison operations. The software features fundamental programming constructs for conditional execution and looping, strings and basic string operators, hypertext help, script execution, user-defined display windows, and an extensive library of examples that illustrate the properties of fuzzy arithmetic. Ferson, S. 1994. Using fuzzy arithmetic in Monte Carlo simulation of fishery populations. Management Strategies for Exploited Fish Populations, G. Kruse et al. (eds.)

Abstracts from Recent & Upcoming Presentations

Computing cleanup targets is complicated (but possible) in probabilistic risk assessments

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Basic statistics for imprecise data

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Modeling correlation and dependence among intervals

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Calculating a soil EPC when sampling and exposures are non-random

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Propagating Epistemic and Aleatory Uncertainty in Nonlinear Dynamic Models

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The tradeoff between measurement precision and sample size: should we get more or better data?

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Uncertainty, Sensitivity and Validation

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Quasi-extinction risk in a wood frog (Rana sylvatica) metapopulation under environmental contamination by PCBs

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Model Validation, Calibration and Predictive Capability

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Environmental contamination in ecological systems

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The elephant in the living room: What to do about model uncertainty

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Overdriving the headlights: empirical data limit risk analyses

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Risk perception and the problems we make for ourselves

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Reliable calculation of probabilities

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Why Laplace was wrong: using copulas to bound convolutions when marginals are known

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Total exposure does not equal average daily exposure times days exposed

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Combining toxicant kinetics, population dynamics and trophic interactions

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Detecting rare event clusters when data are extremely sparse

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Ecotoxicology: how to bring ecology and toxicology together

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Checking the computational integrity of probabilistic risk analyses

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Cancer clusters: how to be sure you have one

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Reliable probabilities when sample sizes are small and the model is uncertain

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Six methods to use when information is limited (as it always is)

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Quality assurance for an environmental risk assessment

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Probability bounds analysis

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Statistically detecting clustering in very sparse data sets

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Why probability is insufficient for handling uncertainty in risk analysis

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Probability bounds analysis (Why Laplace was wrong)

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Why humans are so bad at interpreting probabilities

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Symposium at the Society for Risk Analysis annual meeting The Balance of Nature: Can Toxicologists and Ecologists Come to Consensus?

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Symposium at the Society for Risk Analysis annual meeting Backcalculating cleanup goals

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Surprising dynamics during ecological recovery after heavy metal contamination

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Ecological risk assessment based on extinction distributions

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Robust risk analysis: What can we be sure about?

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Biological invasion

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Probabilistic screening assessment of ecological risks

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What to do about model uncertainty

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Cleanup Goals in a Probabilistic Assessment

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Population-level assessments with almost no data

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Can we know the iceberg from its tip? Censored distributions in risk assessment

Symposium at the Society for Risk Analysis annual meeting
The Balance of Nature: Can Toxicologists and Ecologists Come to Consensus?

Symposium at the Society for Risk Analysis annual meeting
Backcalculating cleanup goals