Incorporating model uncertainties along with data uncertainties in microbial risk assessment

A Bayesian benchmark concentration analysis for urinary fluoride and intelligence in adults in Guizhou, China

Environmental fluoride exposure has been linked to numerous cases of fluorosis worldwide. Previous studies have indicated that long-term exposure to fluoride can result in intellectual damage among children. However, a comprehensive health risk assessment of fluorosis-induced intellectual damage is still pending. In this research, we utilized the Bayesian Benchmark Dose Analysis System (BBMD) to investigate the dose-response relationship between urinary fluoride (U-F) concentration and Raven scores in adults from Nayong, Guizhou, China. Our research findings indecate a dose-response relationship between the concentration of U-F and intelligence scores in adults. As the benchmark response (BMR) increased, both the benchmark concentration (BMCs) and the lower bound of the credible interval (BMCLs) increased. Specifically, BMCs for the association between U-F and IQ score were determined to be 0.18 mg/L (BMCL1 = 0.08 mg/L), 0.91 mg/L (BMCL5 = 0.40 mg/L), 1.83 mg/L (BMCL10 = 0.83 mg/L) when using BMRs of 1 %, 5 %, and 10 %. These results indicate that U-F can serve as an effective biomarker for monitoring the loss of IQ in population. We propose three interim targets for public policy in preventing interllectual harm from fluoride exposure.

Evidence of Phosphite Tolerance in Phytophthora cinnamomi from New Zealand Avocado Orchards

There is a limited number of chemical control agents for managing Phytophthora root and collar rot diseases of avocado internationally; of these, phosphite is one of the most effective. To determine whether prolonged phosphite use in New Zealand avocado orchards has led to decreased sensitivity of Phytophthora cinnamomi to phosphite, 57 isolates were collected from phosphite-treated and -untreated avocado orchards and screened for tolerance using a mycelial growth inhibition assay. The inhibitory effect of phosphite on mycelial growth was tested in vitro using six concentrations of phosphite. Based on changes in mycelial growth using optical density measurements to calculate the effective concentration to reduce growth by 50% (EC₅₀) estimates, three phosphite-susceptible (EC₅₀ range = 18.71 to 29.26 µg/ml) and three tolerant (EC₅₀ range = 81.85 to 123.89 µg/ml) isolates were selected. The effects of phosphite on the colonization of lupin (Lupinus angustifolius) seedling roots and sporangia and zoospore production of three susceptible and three tolerant isolates were determined. The three tolerant isolates colonized lupin roots more extensively than the three susceptible isolates in the presence of phosphite at 5 and 10 g/liter. The tolerant isolates were able to asymptomatically colonize further above the lesion margin in the lupin treated with phosphite at 5 g/liter relative to the phosphite-susceptible isolates but no isolates were completely resistant to phosphite. The tolerant isolates produced more sporangia and, consequently, zoospores in the presence of phosphite than the susceptible isolates. The detection of phosphite tolerance by P. cinnamomi in planta and in vivo is concerning for the future efficacy of phosphite to manage Phytophthora diseases.

Continuous Model Averaging for Benchmark Dose Analysis: Averaging Over Distributional Forms

When estimating a benchmark dose (BMD) from chemical toxicity experiments, model averaging is recommended by the National Institute for Occupational Safety and Health, World Health Organization and European Food Safety Authority. Though numerous studies exist for Model Average BMD estimation using dichotomous responses, fewer studies investigate it for BMD estimation using continuous response. In this setting, model averaging a BMD poses additional problems as the assumed distribution is essential to many BMD definitions, and distributional uncertainty is underestimated when one error distribution is chosen a priori. As model averaging combines full models, there is no reason one cannot include multiple error distributions. Consequently, we define a continuous model averaging approach over distributional models and show that it is superior to single distribution model averaging. To show the superiority of the approach, we apply the method to simulated and experimental response data.

Non-linear model analysis of the relationship between cholinesterase activity in rats exposed to 2, 2-dichlorovinyl dimethylphosphate (dichlorvos) and its metabolite concentrations in urine

Urinary dialkylphosphates (DAPs) are measured to assess exposure to organophosphorus pesticides (OPs), but they are common metabolites of OPs and not specific indices for individual agents. Biomonitoring (BM) of urinary DAPs has been widely adopted as an assessment of individual exposure in general environments, however, guidance values for DAPs based on health effects have yet to be established. The present study aimed to clarify the relationship between the amount of urinary dimethylphosphate (DMP), a metabolite of dichlorvos (DDVP), and the inhibition of cholinesterase (ChE) activity in rats exposed to DDVP. The relationship was analyzed using a nonlinear model analysis, and the excretion level of urinary DMP equivalent to ChE 20 % inhibition (EL20) and the lower limit of the 95 % confidence interval of EL20 (ELL20) were estimated. EL20 and ELL20 (mg/24 h urine) of brain, erythrocyte, and plasma ChE activities after 10-day administration of DDVP were 0.21 and 0.15, 0.11 and 0.06, and 0.23 and 0.09, respectively. Extrapolating ELL20 of the brain ChE to humans, the range of 24 h urinary DMP concentration according to the 20 % inhibition of cholinesterase activity was estimated to be 20.5-30.8 mg/l. In conclusion, the amount of urinary DMP as ELL20 for DDVP exposure was identified and could probably be used as a novel index for the assessment of risk from OP exposure. Further studies are needed to clarify the ELL20 s derived from OPs other than DDVP, for informing efforts to establish guidance values of urinary OP metabolites that should prevent neurotoxicity.

bmd: an R package for benchmark dose estimation

The benchmark dose (BMD) methodology is used to derive a hazard characterization measure for risk assessment in toxicology or ecotoxicology. The present paper's objective is to introduce the R extension package bmd, which facilitates the estimation of BMD and the benchmark dose lower limit for a wide range of dose-response models via the popular package drc. It allows using the most current statistical methods for BMD estimation, including model averaging. The package bmd can be used for BMD estimation for binomial, continuous, and count data in a simple set up or from complex hierarchical designs and is introduced using four examples. While there are other stand-alone software solutions available to estimate BMDs, the package bmd facilitates easy estimation within the established and flexible statistical environment R. It allows the rapid implementation of available, novel, and future statistical methods and the integration of other statistical analyses.

Simulation-based assessment of model selection criteria during the application of benchmark dose method to quantal response data

Background

To employ the benchmark dose (BMD) method in toxicological risk assessment, it is critical to understand how the BMD lower bound for reference dose calculation is selected following statistical fitting procedures of multiple mathematical models. The purpose of this study was to compare the performances of various combinations of model exclusion and selection criteria for quantal response data.

Methods

Simulation-based evaluation of model exclusion and selection processes was conducted by comparing validity, reliability, and other model performance parameters. Three different empirical datasets for different chemical substances were analyzed for the assessment, each having different characteristics of the dose-response pattern (i.e. datasets with rich information in high or low response rates, or approximately linear dose-response patterns).

Results

The best performing criteria of model exclusion and selection were different across the different datasets. Model averaging over the three models with the lowest three AIC (Akaike information criteria) values (MA-3) did not produce the worst performance, and MA-3 without model exclusion produced the best results among the model averaging. Model exclusion including the use of the Kolmogorov-Smirnov test in advance of model selection did not necessarily improve the validity and reliability of the models.

Conclusions

If a uniform methodological suggestion for the guideline is required to choose the best performing model for exclusion and selection, our results indicate that using MA-3 is the recommended option whenever applicable.

A Review of Recent Advances in Benchmark Dose Methodology

In this review, recent methodological developments for the benchmark dose (BMD) methodology are summarized. Specifically, we introduce the advances for the main steps in BMD derivation: selecting the procedure for defining a BMD from a predefined benchmark response (BMR), setting a BMR, selecting a dose-response model, and estimating the corresponding BMD lower limit (BMDL). Although the last decade has shown major progress in the development of BMD methodology, there is still room for improvement. Remaining challenges are the implementation of new statistical methods in user-friendly software and the lack of consensus about how to derive the BMDL.

Genetic and phenotypic variation of Phytophthora crassamura isolates from California nurseries and restoration sites

Phenotypic and sequence data were used to characterize 28 isolates resembling Phytophthora megasperma from 14 host species in 2 plant production facilities and 10 restoration sites across the San Francisco Bay Area (California; USA). Size of the oogonia and DNA sequences (nuclear internal transcribed spacer (ITS) and mitochondrial cytochrome c oxidase subunit 1 (COX 1)) were compared, and sensitivity to mefenoxam and pathogenicity were measured. Based on ITS 61 % of isolates matched ex-type sequences of Phytophthora crassamura from Italy, and the remainder matched or were close to the P. megasperma ex-type. However, all California P. crassamura genotypes belonged to four unique COX 1 haplotype lineages isolated from both nurseries and restoration sites. Although lineages were sensitive to mefenoxam, a significant difference in sensitivity was identified, and all continued growth in-vitro. These results suggested previous mefenoxam exposure in plant production facilities resulting in tolerance. In conclusion, all evidence pointed to a nursery origin of novel P. crassamura lineages found in California restoration sites. In this study, COX 1 sequences and oogonia size provided information relevant to identify geographic and evolutionary intraspecific variation within P. crassamura, and was additionally used to track the spread of this species from nurseries into wildlands.

Model Averaging with AIC Weights for Hypothesis Testing of Hormesis at Low Doses

For many dose-response studies, large samples are not available. Particularly, when the outcome of interest is binary rather than continuous, a large sample size is required to provide evidence for hormesis at low doses. In a small or moderate sample, we can gain statistical power by the use of a parametric model. It is an efficient approach when it is correctly specified, but it can be misleading otherwise. This research is motivated by the fact that data points at high experimental doses have too much contribution in the hypothesis testing when a parametric model is misspecified. In dose-response analyses, to account for model uncertainty and to reduce the impact of model misspecification, averaging multiple models have been widely discussed in the literature. In this article, we propose to average semiparametric models when we test for hormesis at low doses. We show the different characteristics of averaging parametric models and averaging semiparametric models by simulation. We apply the proposed method to real data, and we show that P values from averaged semiparametric models are more credible than P values from averaged parametric methods. When the true dose-response relationship does not follow a parametric assumption, the proposed method can be an alternative robust approach.

Dose-Response Analysis Using R

Dose-response analysis can be carried out using multi-purpose commercial statistical software, but except for a few special cases the analysis easily becomes cumbersome as relevant, non-standard output requires manual programming. The extension package drc for the statistical environment R provides a flexible and versatile infrastructure for dose-response analyses in general. The present version of the package, reflecting extensions and modifications over the last decade, provides a user-friendly interface to specify the model assumptions about the dose-response relationship and comes with a number of extractors for summarizing fitted models and carrying out inference on derived parameters. The aim of the present paper is to provide an overview of state-of-the-art dose-response analysis, both in terms of general concepts that have evolved and matured over the years and by means of concrete examples.

Bayesian model-averaged benchmark dose analysis via reparameterized quantal-response models

An important objective in biomedical and environmental risk assessment is estimation of minimum exposure levels that induce a pre-specified adverse response in a target population. The exposure points in such settings are typically referred to as benchmark doses (BMDs). Parametric Bayesian estimation for finding BMDs has grown in popularity, and a large variety of candidate dose-response models is available for applying these methods. Each model can possess potentially different parametric interpretation(s), however. We present reparameterized dose-response models that allow for explicit use of prior information on the target parameter of interest, the BMD. We also enhance our Bayesian estimation technique for BMD analysis by applying Bayesian model averaging to produce point estimates and (lower) credible bounds, overcoming associated questions of model adequacy when multimodel uncertainty is present. An example from carcinogenicity testing illustrates the calculations.

Disability adjusted life year (DALY): a useful tool for quantitative assessment of environmental pollution

Disability adjusted life year (DALY) has been widely used since 1990s for evaluating global and/or regional burden of diseases. As many environmental pollutants are hazardous to human health, DALY is also recognized as an indicator to quantify the health impact of environmental pollution related to disease burden. Based on literature reviews, this article aims to give an overview of the applicable methodologies and research directions for using DALY as a tool for quantitative assessment of environmental pollution. With an introduction of the methodological framework of DALY, the requirements on data collection and manipulation for quantifying disease burdens are summarized. Regarding environmental pollutants hazardous to human beings, health effect/risk evaluation is indispensable for transforming pollution data into disease data through exposure and dose-response analyses which need careful selection of models and determination of parameters. Following the methodological discussions, real cases are analyzed with attention paid to chemical pollutants and pathogens usually encountered in environmental pollution. It can be seen from existing studies that DALY is advantageous over conventional environmental impact assessment for quantification and comparison of the risks resulted from environmental pollution. However, further studies are still required to standardize the methods of health effect evaluation regarding varied pollutants under varied circumstances before DALY calculation.

Historical Context and Recent Advances in Exposure-Response Estimation for Deriving Occupational Exposure Limits

Virtually no occupational exposure standards specify the level of risk for the prescribed exposure, and most occupational exposure limits are not based on quantitative risk assessment (QRA) at all. Wider use of QRA could improve understanding of occupational risks while increasing focus on identifying exposure concentrations conferring acceptably low levels of risk to workers. Exposure-response modeling between a defined hazard and the biological response of interest is necessary to provide a quantitative foundation for risk-based occupational exposure limits; and there has been considerable work devoted to establishing reliable methods quantifying the exposure-response relationship including methods of extrapolation below the observed responses. We review several exposure-response modeling methods available for QRA, and demonstrate their utility with simulated data sets.

A diversity index for model space selection in the estimation of benchmark and infectious doses via model averaging

In chemical and microbial risk assessments, risk assessors fit dose-response models to high-dose data and extrapolate downward to risk levels in the range of 1-10%. Although multiple dose-response models may be able to fit the data adequately in the experimental range, the estimated effective dose (ED) corresponding to an extremely small risk can be substantially different from model to model. In this respect, model averaging (MA) provides more robustness than a single dose-response model in the point and interval estimation of an ED. In MA, accounting for both data uncertainty and model uncertainty is crucial, but addressing model uncertainty is not achieved simply by increasing the number of models in a model space. A plausible set of models for MA can be characterized by goodness of fit and diversity surrounding the truth. We propose a diversity index (DI) to balance between these two characteristics in model space selection. It addresses a collective property of a model space rather than individual performance of each model. Tuning parameters in the DI control the size of the model space for MA.

Comparison of multi-stage dose-response mixture models, with applications

This article concerns the analysis of a stochastic model that we propose for the population that generates a response (response measure) to the dose with the multi-stage model. The parameter uncertainty is dealt with via random dose and random size of the population at risk. The response measure is modeled by a random sum of mixed Bernoulli random variables with arbitrary distribution for the mixing parameters. Some extensions of the model are defined by functionals of the infection probability, fulfilling some convexity properties. We analyze the response by stochastic comparisons under different stochastic relations on the random dosages and the random sizes of the population at risk; or on the random infection rates. We provide stochastic exact bounds of the mixture model for the response, using inequalities and the positive quadrant dependence. Numerical bounds of the response by a dose having a scalar value or having an exponential or uniform distributions are obtained. Some conclusions are derived: the lower estimation of the response measure in the increasing convex order sense by replacing the dosages by their means; effects of the variation of the dose on the magnitude of the probability distribution of the response; effects of parameter correlation on the degree of variability of the response to any random dose; the low-dose region assessment; and also, the classical multi-stage model is compared versus the mixture model featuring independence and versus that with positive quadrant dependence.

Benchmark Dose Analysis via Nonparametric Regression Modeling

Estimation of benchmark doses (BMDs) in quantitative risk assessment traditionally is based upon parametric dose-response modeling. It is a well-known concern, however, that if the chosen parametric model is uncertain and/or misspecified, inaccurate and possibly unsafe low-dose inferences can result. We describe a nonparametric approach for estimating BMDs with quantal-response data based on an isotonic regression method, and also study use of corresponding, nonparametric, bootstrap-based confidence limits for the BMD. We explore the confidence limits' small-sample properties via a simulation study, and illustrate the calculations with an example from cancer risk assessment. It is seen that this nonparametric approach can provide a useful alternative for BMD estimation when faced with the problem of parametric model uncertainty.

Model Uncertainty and Bayesian Model Averaged Benchmark Dose Estimation for Continuous Data

The benchmark dose (BMD) approach has gained acceptance as a valuable risk assessment tool, but risk assessors still face significant challenges associated with selecting an appropriate BMD/BMDL estimate from the results of a set of acceptable dose-response models. Current approaches do not explicitly address model uncertainty, and there is an existing need to more fully inform health risk assessors in this regard. In this study, a Bayesian model averaging (BMA) BMD estimation method taking model uncertainty into account is proposed as an alternative to current BMD estimation approaches for continuous data. Using the "hybrid" method proposed by Crump, two strategies of BMA, including both "maximum likelihood estimation based" and "Markov Chain Monte Carlo based" methods, are first applied as a demonstration to calculate model averaged BMD estimates from real continuous dose-response data. The outcomes from the example data sets examined suggest that the BMA BMD estimates have higher reliability than the estimates from the individual models with highest posterior weight in terms of higher BMDL and smaller 90th percentile intervals. In addition, a simulation study is performed to evaluate the accuracy of the BMA BMD estimator. The results from the simulation study recommend that the BMA BMD estimates have smaller bias than the BMDs selected using other criteria. To further validate the BMA method, some technical issues, including the selection of models and the use of bootstrap methods for BMDL derivation, need further investigation over a more extensive, representative set of dose-response data.

Information-theoretic model-averaged benchmark dose analysis in environmental risk assessment

An important objective in environmental risk assessment is estimation of minimum exposure levels, called Benchmark Doses (BMDs), that induce a pre-specified Benchmark Response (BMR) in a dose-response experiment. In such settings, representations of the risk are traditionally based on a specified parametric model. It is a well-known concern, however, that existing parametric estimation techniques are sensitive to the form employed for modeling the dose response. If the chosen parametric model is in fact misspecified, this can lead to inaccurate low-dose inferences. Indeed, avoiding the impact of model selection was one early motivating issue behind development of the BMD technology. Here, we apply a frequentist model averaging approach for estimating benchmark doses, based on information-theoretic weights. We explore how the strategy can be used to build one-sided lower confidence limits on the BMD, and we study the confidence limits' small-sample properties via a simulation study. An example from environmental carcinogenicity testing illustrates the calculations. It is seen that application of this information-theoretic, model averaging methodology to benchmark analysis can improve environmental health planning and risk regulation when dealing with low-level exposures to hazardous agents.

Model uncertainty and model averaging in the estimation of infectious doses for microbial pathogens

Food-borne infection is caused by intake of foods or beverages contaminated with microbial pathogens. Dose-response modeling is used to estimate exposure levels of pathogens associated with specific risks of infection or illness. When a single dose-response model is used and confidence limits on infectious doses are calculated, only data uncertainty is captured. We propose a method to estimate the lower confidence limit on an infectious dose by including model uncertainty and separating it from data uncertainty. The infectious dose is estimated by a weighted average of effective dose estimates from a set of dose-response models via a Kullback information criterion. The confidence interval for the infectious dose is constructed by the delta method, where data uncertainty is addressed by a bootstrap method. To evaluate the actual coverage probabilities of the lower confidence limit, a Monte Carlo simulation study is conducted under sublinear, linear, and superlinear dose-response shapes that can be commonly found in real data sets. Our model-averaging method achieves coverage close to nominal in almost all cases, thus providing a useful and efficient tool for accurate calculation of lower confidence limits on infectious doses.

Nonparametric estimation of benchmark doses in environmental risk assessment

An important statistical objective in environmental risk analysis is estimation of minimum exposure levels, called benchmark doses (BMDs), that induce a pre-specified benchmark response in a dose-response experiment. In such settings, representations of the risk are traditionally based on a parametric dose-response model. It is a well-known concern, however, that if the chosen parametric form is misspecified, inaccurate and possibly unsafe low-dose inferences can result. We apply a nonparametric approach for calculating benchmark doses, based on an isotonic regression method for dose-response estimation with quantal-response data (Bhattacharya and Kong, 2007). We determine the large-sample properties of the estimator, develop bootstrap-based confidence limits on the BMDs, and explore the confidence limits' small-sample properties via a short simulation study. An example from cancer risk assessment illustrates the calculations.

The Impact of Model Uncertainty on Benchmark Dose Estimation

We study the popular benchmark dose (BMD) approach for estimation of low exposure levels in toxicological risk assessment, focusing on dose-response experiments with quantal data. In such settings, representations of the risk are traditionally based on a specified, parametric, dose-response model. It is a well-known concern, however, that uncertainty can exist in specification and selection of the model. If the chosen parametric form is in fact misspecified, this can lead to inaccurate, and possibly unsafe, lowdose inferences. We study the effects of model selection and possible misspecification on the BMD, on its corresponding lower confidence limit (BMDL), and on the associated extra risks achieved at these values, via large-scale Monte Carlo simulation. It is seen that an uncomfortably high percentage of instances can occur where the true extra risk at the BMDL under a misspecified or incorrectly selected model can surpass the target BMR, exposing potential dangers of traditional strategies for model selection when calculating BMDs and BMDLs.

Monotonic Bayesian semiparametric benchmark dose analysis

Quantitative risk assessment proceeds by first estimating a dose-response model and then inverting this model to estimate the dose that corresponds to some prespecified level of response. The parametric form of the dose-response model often plays a large role in determining this dose. Consequently, the choice of the proper model is a major source of uncertainty when estimating such endpoints. While methods exist that attempt to incorporate the uncertainty by forming an estimate based upon all models considered, such methods may fail when the true model is on the edge of the space of models considered and cannot be formed from a weighted sum of constituent models. We propose a semiparametric model for dose-response data as well as deriving a dose estimate associated with a particular response. In this model formulation, the only restriction on the model form is that it is monotonic. We use this model to estimate the dose-response curve from a long-term cancer bioassay, as well as compare this to methods currently used to account for model uncertainty. A small simulation study is conducted showing that the method is superior to model averaging when estimating exposure that arises from a quantal-linear dose-response mechanism, and is similar to these methods when investigating nonlinear dose-response patterns.

Application of the Margin of Exposure (MOE) approach to substances in food that are genotoxic and carcinogenic

This paper presents the work of an expert group established by the International Life Sciences Institute - European branch (ILSI Europe) to follow up the recommendations of an international conference on "Risk Assessment of Compounds that are both Genotoxic and Carcinogenic: New Approaches". Twelve genotoxic and carcinogenic chemicals that can be present in food were selected for calculation of a Margin of Exposure (MOE) between a point of departure on the dose-response for oral carcinogenicity in animal studies and estimates of human dietary exposure. The MOE can be used to support prioritisation of risk management action and, if the MOE is very large, on communication of a low level of human health concern. Depending on the approaches taken in determining the point of departure and the estimation of exposure, it is possible to derive very different values for the MOE. It is therefore essential that the selection of the cancer endpoint and mathematical treatment of the data are clearly described and justified if the results of the MOE approach are to be trusted and of value to risk managers. An outline framework for calculating an MOE is proposed in order to help to ensure transparency in the results.

Conundrums with uncertainty factors

The practice of uncertainty factors as applied to noncancer endpoints in the IRIS database harkens back to traditional safety factors. In the era before risk quantification, these were used to build in a "margin of safety." As risk quantification takes hold, the safety factor methods yield to quantitative risk calculations to guarantee safety. Many authors believe that uncertainty factors can be given a probabilistic interpretation as ratios of response rates, and that the reference values computed according to the IRIS methodology can thus be converted to random variables whose distributions can be computed with Monte Carlo methods, based on the distributions of the uncertainty factors. Recent proposals from the National Research Council echo this view. Based on probabilistic arguments, several authors claim that the current practice of uncertainty factors is overprotective. When interpreted probabilistically, uncertainty factors entail very strong assumptions on the underlying response rates. For example, the factor for extrapolating from animal to human is the same whether the dosage is chronic or subchronic. Together with independence assumptions, these assumptions entail that the covariance matrix of the logged response rates is singular. In other words, the accumulated assumptions entail a log-linear dependence between the response rates. This in turn means that any uncertainty analysis based on these assumptions is ill-conditioned; it effectively computes uncertainty conditional on a set of zero probability. The practice of uncertainty factors is due for a thorough review. Two directions are briefly sketched, one based on standard regression models, and one based on nonparametric continuous Bayesian belief nets.

Model-averaged benchmark concentration estimates for continuous response data arising from epidemiological studies

Worker populations often provide data on adverse responses associated with exposure to potential hazards. The relationship between hazard exposure levels and adverse response can be modeled and then inverted to estimate the exposure associated with some specified response level. One concern is that this endpoint may be sensitive to the concentration metric and other variables included in the model. Further, it may be that the models yielding different risk endpoints are all providing relatively similar fits. We focus on evaluating the impact of exposure on a continuous response by constructing a model-averaged benchmark concentration from a weighted average of model-specific benchmark concentrations. A method for combining the estimates based on different models is applied to lung function in a cohort of miners exposed to coal dust. In this analysis, we see that a small number of the thousands of models considered survive a filtering criterion for use in averaging. Even after filtering, the models considered yield benchmark concentrations that differ by a factor of 2 to 9 depending on the concentration metric and covariates. The model-average BMC captures this uncertainty, and provides a useful strategy for addressing model uncertainty.

Model averaging in microbial risk assessment using fractional polynomials

The alleviation of food-borne diseases caused by microbial pathogen remains a great concern in order to ensure the well-being of the general public. The relation between the ingested dose of organisms and the associated infection risk can be studied using dose-response models. Traditionally, a model selected according to a goodness-of-fit criterion has been used for making inferences. In this article, we propose a modified set of fractional polynomials as competitive dose-response models in risk assessment. The article not only shows instances where it is not obvious to single out one best model but also illustrates that model averaging can best circumvent this dilemma. The set of candidate models is chosen based on biological plausibility and rationale and the risk at a dose common to all these models estimated using the selected models and by averaging over all models using Akaike's weights. In addition to including parameter estimation inaccuracy, like in the case of a single selected model, model averaging accounts for the uncertainty arising from other competitive models. This leads to a better and more honest estimation of standard errors and construction of confidence intervals for risk estimates. The approach is illustrated for risk estimation at low dose levels based on Salmonella typhi and Campylobacter jejuni data sets in humans. Simulation studies indicate that model averaging has reduced bias, better precision, and also attains coverage probabilities that are closer to the 95% nominal level compared to best-fitting models according to Akaike information criterion.

Towards quantitative uncertainty assessment for cancer risks: Central estimates and probability distributions of risk in dose–response modeling

Regulatory agencies and the scientific community have been engaged in a long-term effort to strengthen health risk assessment procedures. Recently the momentum of this effort has accelerated to increasing biological information for a variety of toxic compounds and emphasis on the policy goal of broader characterization of scientific uncertainty (in contrast to providing only a single risk estimate). For example, the OMB Regulatory Analysis Guidelines [OMB, 2003. Office of Management and Budget. Circular A-4. Available from: <http://www.whitehouse.gov/omb/circulars/a004/a-4.html/>] suggest that a formal quantitative uncertainty analysis be performed for economic assessments in support of major regulatory analyses, a process that can utilize both expected values and probability distributions for risk estimates. Some efforts have been made in the past to provide probability distributions of risk estimates. In this article, we examine a procedure for constructing probability distributions and expected values of risk estimates using a Bayesian framework. This approach has the advantage of mathematical soundness and computational feasibility, given the Markov chain Monte Carlo software tools that are available today. Importantly, the Bayesian framework can serve as a unifying platform for uncertainty analysis in cancer risk assessment. This paper provides some initial applications of Bayesian methods in quantitative analysis of uncertainty in cancer risk assessment, including implementation with cancer dose-response data sets for two chemicals. The Bayesian expected risk calculations provide an approach to generating a central estimate of risk that does not have the instability problems that have often limited utility of MLE risk estimates.

An approach to risk assessment for TiO2

Titanium dioxide (TiO2) is a poorly soluble, low-toxicity (PSLT) particle. Fine TiO2 (<2.5 microm) has been shown to produce lung tumors in rats exposed to 250 mg/m3, and ultrafine TiO2 (< 0.1 microm diameter) has been shown to produce lung tumors in rats at 10 mg/m3. We have evaluated the rat dose-response data and conducted a quantitative risk assessment for TiO2. Preliminary conclusions are: (1) Fine and ultrafine TiO2 and other PSLT particles show a consistent dose-response relationship when dose is expressed as particle surface area; (2) the mechanism of TiO2 tumor induction in rats appears to be a secondary genotoxic mechanism associated with persistent inflammation; and (3) the inflammatory response shows evidence of a nonzero threshold. Risk estimates for TiO2 depend on both the dosimetric approach and the statistical model that is used. Using 7 different dose-response models in the U.S. Environmental Protection Agency (EPA) benchmark dose software, the maximum likelihood estimate (MLE) rat lung dose associated with a 1 per 1000 excess risk ranges from 0.0076 to 0.28 m2/g-lung of particle surface area, with 95% lower confidence limits (LCL) of 0.0059 and 0.042, respectively. Using the ICRP particle deposition and clearance model, estimated human occupational exposures yielding equivalent lung burdens range from approximately 1 to 40 mg/m3 (MLE) for fine TiO2, with 95% LCL approximately 0.7-6 mg/m3. Estimates using an interstitial sequestration lung model are about one-half as large. Bayesian model averaging techniques are now being explored as a method for combining the various estimates into a single estimate, with a confidence interval expressing model uncertainty.

Development of a tolerable daily intake for N-nitrosodimethylamine using a modified benchmark dose methodology

N-Nitrosodimethylamine (NDMA) is an environmental contaminant that has recently been detected in Australian drinking-water supplies and that is principally generated in chloramination systems. NDMA is acutely toxic to humans at high doses, is genotoxic after cytochrome P-450 metabolism, and is carcinogenic in several animal species. An extremely large lifetime cancer dose-response study reported by Peto and colleagues (1984, 1991a, 1991b) of NDMA in drinking water given to rats is used in risk assessment by various jurisdictions. We have recently reported on use of an Australian modified benchmark dose (mBMD) methodology for developing tolerable daily intakes (TDIs) and guideline values for environmental carcinogens based on cancer dose response in the low-dose region, and have applied this to the NDMA rat liver tumor data. The application of a suite of mathematical models to the incidence data for hepatocellular carcinomas and hemangiosarcomas, followed by arithmetic and exponential-weight averaging of the 5% extra risk dose (mBMD(0.05)) for the various models, produced an mBMD(0.05) range of 0.020-0.028 mg/kg/d. This was then divided by a range of modifying factors to account for seriousness of the carcinogenic endpoint, adequacy of the database, and inter- and intraspecies differences, generating a TDI range of 4.0 to 9.3 ng/kg/d. This may be employed in developing guideline values for NDMA in environmental media.

Properties of model-averaged BMDLs: a study of model averaging in dichotomous response risk estimation

Model averaging (MA) has been proposed as a method of accounting for model uncertainty in benchmark dose (BMD) estimation. The technique has been used to average BMD dose estimates derived from dichotomous dose-response experiments, microbial dose-response experiments, as well as observational epidemiological studies. While MA is a promising tool for the risk assessor, a previous study suggested that the simple strategy of averaging individual models' BMD lower limits did not yield interval estimators that met nominal coverage levels in certain situations, and this performance was very sensitive to the underlying model space chosen. We present a different, more computationally intensive, approach in which the BMD is estimated using the average dose-response model and the corresponding benchmark dose lower bound (BMDL) is computed by bootstrapping. This method is illustrated with TiO(2) dose-response rat lung cancer data, and then systematically studied through an extensive Monte Carlo simulation. The results of this study suggest that the MA-BMD, estimated using this technique, performs better, in terms of bias and coverage, than the previous MA methodology. Further, the MA-BMDL achieves nominal coverage in most cases, and is superior to picking the "best fitting model" when estimating the benchmark dose. Although these results show utility of MA for benchmark dose risk estimation, they continue to highlight the importance of choosing an adequate model space as well as proper model fit diagnostics.

Second-order modeling of variability and uncertainty in microbial hazard characterization

This study describes an analytical framework that permits quantitative consideration of variability and uncertainty in microbial hazard characterization. Second-order modeling that used two-dimensional Monte Carlo simulation and stratification into homogeneous population subgroups was applied to integrate uncertainty and variability. Specifically, the bootstrap method was used to simulate sampling error due to the limited sample size in microbial dose-response modeling. A data set from human feeding trials with Campylobacter jejuni was fitted to the log-logistic dose-response model, and results from the analysis of FoodNet surveillance data provided further information on variability and uncertainty in Campylobacter susceptibility due to the effect of age. Results of our analyses indicate that uncertainty associated with dose-response modeling has a dominating influence on the analytical outcome. In contrast, inclusion of the age factor has a limited impact. While the advocacy of more closely modeling variability in hazard characterization is warranted, the characterization of key sources of uncertainties and their consistent propagation throughout a microbial risk assessment actually appear of greater importance.

Model averaging using the Kullback information criterion in estimating effective doses for microbial infection and illness

Since the National Food Safety Initiative of 1997, risk assessment has been an important issue in food safety areas. Microbial risk assessment is a systematic process for describing and quantifying a potential to cause adverse health effects associated with exposure to microorganisms. Various dose-response models for estimating microbial risks have been investigated. We have considered four two-parameter models and four three-parameter models in order to evaluate variability among the models for microbial risk assessment using infectivity and illness data from studies with human volunteers exposed to a variety of microbial pathogens. Model variability is measured in terms of estimated ED01s and ED10s, with the view that these effective dose levels correspond to the lower and upper limits of the 1% to 10% risk range generally recommended for establishing benchmark doses in risk assessment. Parameters of the statistical models are estimated using the maximum likelihood method. In this article a weighted average of effective dose estimates from eight two- and three-parameter dose-response models, with weights determined by the Kullback information criterion, is proposed to address model uncertainties in microbial risk assessment. The proposed procedures for incorporating model uncertainties and making inferences are illustrated with human infection/illness dose-response data sets.

Model uncertainty and risk estimation for experimental studies of quantal responses

Experimental animal studies often serve as the basis for predicting risk of adverse responses in humans exposed to occupational hazards. A statistical model is applied to exposure-response data and this fitted model may be used to obtain estimates of the exposure associated with a specified level of adverse response. Unfortunately, a number of different statistical models are candidates for fitting the data and may result in wide ranging estimates of risk. Bayesian model averaging (BMA) offers a strategy for addressing uncertainty in the selection of statistical models when generating risk estimates. This strategy is illustrated with two examples: applying the multistage model to cancer responses and a second example where different quantal models are fit to kidney lesion data. BMA provides excess risk estimates or benchmark dose estimates that reflects model uncertainty.

A comparison of microbial dose–response models fitted to human data

A study of eight mathematical dose-response models for microbial risk assessment was conducted using infectivity and illness data on a variety of microbial pathogens from published studies with human volunteers. The purpose was to evaluate variability among the models for human microbial dose-response data in order to determine whether two-parameter models might suffice for most microbial dose-response data or whether three-parameter models should generally be fitted. Model variability was measured in terms of estimated ED01s and ED10s, with the view that these effective dose levels correspond to the lower and upper limits of the 1-10% risk range generally recommended for establishing benchmark doses in risk assessment. An investigation of the ranks of the ED01 and ED10 values among the models led to the conclusion that the two-parameter models captured at least as much uncertainty as the three-parameter models for the data examined. A further evaluation of the two-parameter models did not result in the selection of one "best" model, but it did provide some insights into the models' relative behavior. The model uncertainty analysis proposed by Kang et al. [Regulat. Toxicol. Pharmacol. 32 (2000) 68] using four two-parameter models was reinforced.

Risk characterization: principles and practice

In the field of risk assessment, characterizing the nature and magnitude of human health or environmental risks is arguably the most important step in the analytical process. In this step, data on the dose-response relationship of an agent are integrated with estimates of the degree of exposure in a population to characterize the likelihood and severity of risk. Although the purpose of risk characterizations is to make sense of the available data and describe what they mean to a broad audience, this step is often given insufficient attention in health risk evaluations. Too often, characterizations fail to interpret or summarize risk information in a meaningful way, or they present single numerical estimates of risk without an adequate discussion of the uncertainties inherent in key exposure parameters or the dose-response assessment, model assumptions, or analytical limitations. Consequently, many users of risk information have misinterpreted the findings of a risk assessment or have false impressions about the degree of accuracy (or the confidence of the scientist) in reported risk estimates. In this article we collected and integrated the published literature on conducting and reporting risk characterizations to provide a broad, yet comprehensive, analysis of the risk characterization process as practiced in the United States and some other countries. Specifically, the following eight topics are addressed: (1) objective of risk characterization, (2) guidance documents on risk characterization, (3) key components of risk characterizations, (4) toxicity criteria for evaluating health risks, (5) descriptors used to characterize health risks, (6) methods for quantifying human health risks, (7) key uncertainties in risk characterizations, and (8) the risk decision-making process. A brief discussion is also provided on international aspects of risk characterization. A number of examples are presented that illustrate key concepts, and citations are provided for approximately 100 of the most relevant papers.