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Notes: Risk assessors often use different probability plots as a way to assessthe fit of a particular distribution or model by comparing the plotted points to a straight line and to obtain estimates of the parameters in parametric distributions or models. When empirical data do not fall in a sufficiently straight line on a probability plot, and when no other single parametricdistribution provides an acceptable (graphical) fit to the data, the risk assessor may consider a mixture model with two component distributions. Animated probability plots are a way to visualize the possible behaviors of mixture models with two component distributions. When no single parametric distribution provides an adequate fit to an empirical dataset, animated probability plots can help an analyst pick some plausible mixture models for the data based on their qualitative fit. After using animations during exploratory data analysis, the analyst must then use other statistical tools, including but not limited to: Maximum Likelihood Estimation (MLE) to find the optimal parameters, Goodness of Fit (GoF) tests, and a variety of diagnostic plots to check the adequacy of the fit. Using a specific example with two LogNormal components, we illustrate the use of animated probability plots asa tool for exploring the suitability of a mixture model with two component distributions. Animations work well with other types of probability plots, and they may be extended to analyze mixture models with three or more component distributions.

Notes: Using exploratory data analysis, probability plots, scatterplots, and computer animations to rotate and visualize the data, we fit a trivariate Normal distribution to data for the height, the natural logarithm of body weight, and the body fat for 646 men between the ages of 50 and 80 years as reported by the medical staff of the U.S. Veterans Administration's “Normative Aging Study” in Boston, MA. Although these data do not include any children, women, or young men, the measurements represent the best data that we could find through a 4-year search. We believe that these data are well measured and reliable for men in the specified age range and that these data reveal an interesting statistical pattern for use in probabilistic PBPK models.

Notes: This article develops and fits probability distributions for the variability in projected (total) job tenure for adult men and women in 31 industries and 22 occupations based on data reported by the U.S. Department of Labor's Bureau of Labor Statistics. It extends previously published results and updates those results from January 1987 to February 1996. The model provides probability distributions for the variability in projected (total) job tenures within the time range of the data, and it extrapolates the distributions beyond the time range of the data, i.e., beyond 25 years.

Notes: Finite mixture models, that is, weighted averages of parametric distributions, provide a powerful way to extend parametric families of distributions to fit data sets not adequately fit by a single parametric distribution. First-order finite mixture models have been widely used in the physical, chemical, biological, and social sciences for over 100 years. Using maximum likelihood estimation, we demonstrate how a first-order finite mixture model can represent the large variability in data collected by the U.S. Environmental Protection Agency for the concentration of Radon 222 in drinking water supplied from ground water, even when 28% of the data fall at or below the minimum reporting level. Extending the use of maximum likelihood, we also illustrate how a second-order finite mixture model can separate and represent both the variability and the uncertainty in the data set.

Notes: Variability arises due to differences in the value of a quantity among different members of a population. Uncertainty arises due to lack of knowledge regarding the true value of a quantity for a given member of a population. We describe and evaluate two methods for quantifying both variability and uncertainty. These methods, bootstrapsimulation and a likelihood-based method, are applied to three datasets. The datasetsinclude a synthetic sample of 19 values from a Lognormal distribution, a sample of nine values obtained from measurements of the PCB concentration in leafy produce, and asample of five values for the partitioning of chromium in the flue gas desulfurization system of coal-fired power plants. For each of these datasets, we employ the two methods to characterize uncertainty in the arithmetic mean and standard deviation, cumulative distribution functions based upon fitted parametric distributions, the 95th percentile of variability, and the 63rd percentile of uncertainty for the 81st percentile of variability. The latter is intended to show that it is possible to describe anypoint within the uncertain frequency distribution by specifying an uncertainty percentile and a Variability percentile. Using the bootstrap method, we compare results based upon use of the method of matching moments and the method of maximum likelihood for fitting distributions to data. Our results indicate that with only 5-19 data pointsas in the datasets we have evaluated, there is substantial uncertainty based upon random sampling error. Both the boostrap and likelihood-based approaches yield comparable uncertainty estimates in most cases.

Notes: This paper reviews existing data on the variability in parameters relevant for health risk analyses. We cover both exposure-related parameters andparameters related to individual susceptibility to toxicity. The toxicity/susceptibility data base under construction is part of a longer term research effort to lay the groundwork for quantitative distributional analyses of non-cancer toxic risks. These data are broken down into a variety of parameter types that encompass different portions of the pathway from external exposure to the production of biological responses. The discrete steps in this pathway, as we now conceive them, are:〈list xml:id="l2" style="custom"〉•Contact Rate (Breathing rates per body weight; fish consumption per bodyweight)•Uptake or Absorption as a Fraction of Intake or Contact Rate•General Systemic Availability Net of First Pass Elimination and Dilutionvia Distribution Volume (e.g., intial blood concentration per mg/kg of uptake)•Systemic Elimination (half life or clearance)•Active Site Concentration per Systemic Blood or Plasma Concentration•Physiological Parameter Change per Active Site Concentration (expressed as the dose required to make a given percentage change in different people, or the dose required to achieve some proportion of an individual's maximum response to the drug or toxicant)•Functional Reserve Capacity-Change in Baseline Physiological Parameter Needed to Produce a Biological Response or Pass a Criterion of Abnormal FunctionComparison of the amounts of variability observed for the different parameter types suggests that appreciable variability is associated with the final step in the process-differences among people in “functional reserve capacity.” This has the implication that relevant information for estimatingeffective toxic susceptibility distributions may be gleaned by direct studies of the population distributions of key physiological parameters in people that are not exposed to the environmental and occupational toxicants thatare thought to perturb those parameters. This is illustrated with some recent observations of the population distributions of Low Density Lipoprotein Cholesterol from the second and third National Health and Nutrition Examination Surveys.

Notes: The North American Free Trade Agreement (NAFTA) and the General Agreement on Tariffs and Trade (GATT) have focused attention on risk assessment of potential insect, weed, and animal pests and diseases of livestock. These risks have traditionally been addressed through quarantine protocols ranging from limits on the geographical areas from which a product may originate, postharvest disinfestation procedures like fumigation, and inspections at points of export and import, to outright bans. To ensure that plant and animal protection measures are not used as nontariff trade barriers, GATT and NAFTA require pest risk analysis (PRA) to support quarantine decisions. The increased emphasis on PRA has spurred multiple efforts at the national and international level to design frameworks for the conduct of these analyses. As approaches to pest risk analysis proliferate, and the importance of the analyses grows, concerns have arisen about the scientific and technical conduct of pest risk analysis. In January of 1997, the Harvard Center for Risk Analysis (HCRA) held an invitation-only workshop in Washington, D.C. to bring experts in risk analysis and pest characterization together to develop general principles for pest risk analysis. Workshop participants examined current frameworks for PRA, discussed strengths and weaknesses of the approaches, and formulated principles, based on years of experience with risk analysis in other setting and knowledge of the issues specific to analysis of pests. The principles developed highlight the both the similarities of pest risk analysis to other forms of risk analysis, and its unique attributes.

Notes: Most public health risk assessments assume and combine a series of average, conservative, and worst-case values to derive a conservative point estimate of risk. This procedure has major limitations. This paper demonstrates a new methodology for extended uncertainty analyses in public health risk assessments using Monte Carlo techniques. The extended method begins as do some conventional methods—with the preparation of a spreadsheet to estimate exposure and risk. This method, however, continues by modeling key inputs as random variables described by probability density functions (PDFs). Overall, the technique provides a quantitative way to estimate the probability distributions for exposure and health risks within the validity of the model used. As an example, this paper presents a simplified case study for children playing in soils contaminated with benzene and benzo(a)pyrene (BaP).

Notes: We fit lognormal distributions to data collected in a national survey for both total water intake and tap water intake by children and adults for these age groups in years: 0 〈 age 〈 1; 1 ≤ age 〈 11; 11 ≤ age 〈 20; 20 ≤ age 〈 65; 65 ≤ age; and all people in the survey taken as a single group. These distributions are suitable for use in public health risk assessments.

Notes: This note presents parameterized distributions of estimates of the amount of soil ingested by children based on data collected by Binder et al. (1986). Following discussions with Dr. Binder, we modified the Binder study data by using the actual stool weights instead of the 15 g value used in the original study. After testing the data for lognormality, we generated parameterized distributions for use in risk assessment uncertainty analyses such as Monte Carlo simulations.