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.     

Benchmark Dose Calculation for Ordered Categorical Responses

The use of benchmark dose (BMD) calculations for dichotomous or continuous responses is well established in the risk assessment of cancer and noncancer endpoints. In some cases, responses to exposure are categorized in terms of ordinal severity effects such as none, mild, adverse, and severe. Such responses can be assessed using categorical regression (CATREG) analysis. However, while CATREG has been employed to compare the benchmark approach and the no‐adverse‐effect‐level (NOAEL) approach in determining a reference dose, the utility of CATREG for risk assessment remains unclear. This study proposes a CATREG model to extend the BMD approach to ordered categorical responses by modeling severity levels as censored interval limits of a standard normal distribution. The BMD is calculated as a weighted average of the BMDs obtained at dichotomous cutoffs for each adverse severity level above the critical effect, with the weights being proportional to the reciprocal of the expected loss at the cutoff under the normal probability model. This approach provides a link between the current BMD procedures for dichotomous and continuous data. We estimate the CATREG parameters using a Markov chain Monte Carlo simulation procedure. The proposed method is demonstrated using examples of aldicarb and urethane, each with several categories of severity levels. Simulation studies comparing the BMD and BMDL (lower confidence bound on the BMD) using the proposed method to the correspondent estimates using the existing methods for dichotomous and continuous data are quite compatible; the difference is mainly dependent on the choice of cutoffs for the severity levels.     

Quantitative Risk Assessment for Developmental Neurotoxic Effects

Developmental neurotoxicity concerns the adverse health effects of exogenous agents acting on neurodevelopment. Because human brain development is a delicate process involving many cellular events, the developing fetus is rather susceptible to compounds that can alter the structure and function of the brain. Today, there is clear evidence that early exposure to many neurotoxicants can severely damage the developing nervous system. Although in recent years, there has been much attention given to model development and risk assessment procedures for developmental toxicants, the area of developmental neurotoxicity has been largely ignored. Here, we consider the problem of risk estimation for developmental neurotoxicants from animal bioassay data. Since most responses from developmental neurotoxicity experiments are nonquantal in nature, an adverse health effect will be defined as a response that occurs with very small probability in unexposed animals. Using a two-stage hierarchical normal dose-response model, upper confidence limits on the excess risk due to a given level of added exposure are derived. Equivalently, the model is used to obtain lower confidence limits on dose for a small negligible level of risk. Our method is based on the asymptotic distribution of the likelihood ratio statistic (cf. Crump, 1995). An example is used to provide further illustration.     

Calculation of Benchmark Doses from Continuous Data

A benchmark dose (BMD) is the dose of a substance that corresponds to a prescribed increase in the response (called the benchmark response or BMR) of a health effect. A statistical lower bound on the benchmark dose (BMDL) has been proposed as a replacement for the no-observed-adverse-effect-level (NOAEL) in setting acceptable human exposure levels. A method is developed in this paper for calculating BMDs and BMDLs from continuous data in a manner that is consistent with those calculated from quantal data. The method involves defining an abnormal response, either directly by specifying a cutoff x₀ that separates continuous responses into normal and abnormal categories, or indirectly by specifying the proportion P₀ of abnormal responses expected among unexposed subjects. The method does not involve actually dichotomizing individual continuous responses into quantal responses, and in certain cases can be applied to continuous data in summarized form (e.g., means and standard deviations of continuous responses among subjects in discrete dose groups). In addition to specifying the BMR and either x₀ or P₀ , the method requires specification of the distribution of continuous responses, including specification of the dose-response θ(d) for a measure of central tendency. A method is illustrated for selecting θ(d) to make the probability of an abnormal response any desired dose-response function. This enables the same dose-response model (Weibull, log-logistic, etc.) to be used for the probability of an abnormal response, regardless of whether the underlying data are continuous or quantal. Whenever the continuous responses are normally distributed with standard deviation σ (independent of dose), the method is equivalent to defining the BMD as the dose corresponding to a prescribed change in the mean response relative to σ.     

Methodology for Determining the Appropriateness of a Linear Dose‐Response Function

Microbial food safety risk assessment models can often at times be simplified by eliminating the need to integrate a complex dose‐response relationship across a distribution of exposure doses. This is possible if exposure pathways lead to pathogens at exposure that consistently have a small probability of causing illness. In this situation, the probability of illness will follow an approximately linear function of dose. Consequently, the predicted probability of illness per serving across all exposures is linear with respect to the expected value of dose. The majority of dose‐response functions are approximately linear when the dose is low. Nevertheless, what constitutes “low” is dependent on the parameters of the dose‐response function for a particular pathogen. In this study, a method is proposed to determine an upper bound of the exposure distribution for which the use of a linear dose‐response function is acceptable. If this upper bound is substantially larger than the expected value of exposure doses, then a linear approximation for probability of illness is reasonable. If conditions are appropriate for using the linear dose‐response approximation, for example, the expected value for exposure doses is two to three logs₁₀ smaller than the upper bound of the linear portion of the dose‐response function, then predicting the risk‐reducing effectiveness of a proposed policy is trivial. Simple examples illustrate how this approximation can be used to inform policy decisions and improve an analyst's understanding of risk.     

Dose‐Response Modeling with Summary Data from Developmental Toxicity Studies

Dose‐response analysis of binary developmental data (e.g., implant loss, fetal abnormalities) is best done using individual fetus data (identified to litter) or litter‐specific statistics such as number of offspring per litter and proportion abnormal. However, such data are not often available to risk assessors. Scientific articles usually present only dose‐group summaries for the number or average proportion abnormal and the total number of fetuses. Without litter‐specific data, it is not possible to estimate variances correctly (often characterized as a problem of overdispersion, intralitter correlation, or “litter effect”). However, it is possible to use group summary data when the design effect has been estimated for each dose group. Previous studies have demonstrated useful dose‐response and trend test analyses based on design effect estimates using litter‐specific data from the same study. This simplifies the analysis but does not help when litter‐specific data are unavailable. In the present study, we show that summary data on fetal malformations can be adjusted satisfactorily using estimates of the design effect based on historical data. When adjusted data are then analyzed with models designed for binomial responses, the resulting benchmark doses are similar to those obtained from analyzing litter‐level data with nested dichotomous models.     

Bayesian Quantile Impairment Threshold Benchmark Dose Estimation for Continuous Endpoints

Quantitative risk assessment often begins with an estimate of the exposure or dose associated with a particular risk level from which exposure levels posing low risk to populations can be extrapolated. For continuous exposures, this value, the benchmark dose, is often defined by a specified increase (or decrease) from the median or mean response at no exposure. This method of calculating the benchmark dose does not take into account the response distribution and, consequently, cannot be interpreted based upon probability statements of the target population. We investigate quantile regression as an alternative to the use of the median or mean regression. By defining the dose–response quantile relationship and an impairment threshold, we specify a benchmark dose as the dose associated with a specified probability that the population will have a response equal to or more extreme than the specified impairment threshold. In addition, in an effort to minimize model uncertainty, we use Bayesian monotonic semiparametric regression to define the exposure–response quantile relationship, which gives the model flexibility to estimate the quantal dose–response function. We describe this methodology and apply it to both epidemiology and toxicology data.     

A Health Risk Benchmark for the Neurologic Effects of Styrene: Comparison with NOAEL/LOAEL Approach

Benchmark dose (BMD) analysis was used to estimate an inhalation benchmark concentration for styrene neurotoxicity. Quantal data on neuropsychologic test results from styrene-exposed workers [Mutti et al. (1984). American Journal of Industrial Medicine, 5, 275-286] were used to quantify neurotoxicity, defined as the percent of tested workers who responded abnormally to > or = 1, > or = 2, or > or = 3 out of a battery of eight tests. Exposure was based on previously published results on mean urinary mandelic- and phenylglyoxylic acid levels in the workers, converted to air styrene levels (15, 44, 74, or 115 ppm). Nonstyrene-exposed workers from the same region served as a control group. Maximum-likelihood estimates (MLEs) and BMDs at 5 and 10% response levels of the exposed population were obtained from log-normal analysis of the quantal data. The highest MLE was 9 ppm (BMD = 4 ppm) styrene and represents abnormal responses to > or = 3 tests by 10% of the exposed population. The most health-protective MLE was 2 ppm styrene (BMD = 0.3 ppm) and represents abnormal responses to > or = 1 test by 5% of the exposed population. A no observed adverse effect level/lowest observed adverse effect level (NOAEL/LOAEL) analysis of the same quantal data showed workers in all styrene exposure groups responded abnormally to > or = 1, > or = 2, or > or = 3 tests, compared to controls, and the LOAEL was 15 ppm. A comparison of the BMD and NOAEL/LOAEL analyses suggests that at air styrene levels below the LOAEL, a segment of the worker population may be adversely affected. The benchmark approach will be useful for styrene noncancer risk assessment purposes by providing a more accurate estimate of potential risk that should, in turn, help to reduce the uncertainty that is a common problem in setting exposure levels.     

Application of a Biologically-Based RFD Estimation Method to Tetrachlorodibenzo-P-Dioxin (TCDD) Mediated Immune Suppression and Enzyme Induction

The current methods for a reference dose (RfD) determination can be enhanced through the use of biologically-based dose-response analysis. Methods developed here utilizes information from tetrachlorodibenzo- p -dioxin (TCDD) to focus on noncancer endpoints, specifically TCDD mediated immune system alterations and enzyme induction. Dose-response analysis, using the Sigmoid-Emax (EMAX) function, is applied to multiple studies to determine consistency of response. Through the use of multiple studies and statistical comparison of parameter estimates, it was demonstrated that the slope estimates across studies were very consistent. This adds confidence to the subsequent effect dose estimates. This study also compares traditional methods of risk assessment such as the NOAEL/safety factor to a modified benchmark dose approach which is introduced here. Confidence in the estimation of an effect dose (ED₁₀) was improved through the use of multiple datasets. This is key to adding confidence to the benchmark dose estimates. In addition, the Sigmoid-Emax function when applied to dose-response data using nonlinear regression analysis provides a significantly improved fit to data increasing confidence in parameter estimates which subsequently improve effect dose estimates.     

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.     

An Approach for Modeling Noncancer Dose Responses with an Emphasis on Uncertainty

This paper presents an approach for characterizing the probability of adverse effects occurring in a population exposed to dose rates in excess of the Reference Dose (RfD). The approach uses a linear threshold (hockey stick) model of response and is based on the current system of uncertainty factors used in setting RfDs. The approach requires generally available toxicological estimates such as No-Observed-Adverse-Effect Levels (NOAELs) or Benchmark Doses and doses at which adverse effects are observed in 50% of the test animals (ED₅₀s). In this approach, Monte Carlo analysis is used to characterize the uncertainty in the dose response slope based on the range and magnitude of the key sources of uncertainty in setting protective doses. The method does not require information on the shape of the dose response curve for specific chemicals, but is amenable to the inclusion of such data. The approach is applied to four compounds to produce estimates of response rates for dose rates greater than the RfD     

Comparison of Risk Predicted by Multiple Norovirus Dose–Response Models and Implications for Quantitative Microbial Risk Assessment

The application of quantitative microbial risk assessments (QMRAs) to understand and mitigate risks associated with norovirus is increasingly common as there is a high frequency of outbreaks worldwide. A key component of QMRA is the dose–response analysis, which is the mathematical characterization of the association between dose and outcome. For Norovirus, multiple dose–response models are available that assume either a disaggregated or an aggregated intake dose. This work reviewed the dose–response models currently used in QMRA, and compared predicted risks from waterborne exposures (recreational and drinking) using all available dose–response models. The results found that the majority of published QMRAs of norovirus use the ₁F₁ hypergeometric dose–response model with α = 0.04, β = 0.055. This dose–response model predicted relatively high risk estimates compared to other dose–response models for doses in the range of 1–1,000 genomic equivalent copies. The difference in predicted risk among dose–response models was largest for small doses, which has implications for drinking water QMRAs where the concentration of norovirus is low. Based on the review, a set of best practices was proposed to encourage the careful consideration and reporting of important assumptions in the selection and use of dose–response models in QMRA of norovirus. Finally, in the absence of one best norovirus dose–response model, multiple models should be used to provide a range of predicted outcomes for probability of infection.     

Impact of Acquired Immunity and Dose‐Dependent Probability of Illness on Quantitative Microbial Risk Assessment

Dose‐response models in microbial risk assessment consider two steps in the process ultimately leading to illness: from exposure to (asymptomatic) infection, and from infection to (symptomatic) illness. Most data and theoretical approaches are available for the exposure‐infection step; the infection‐illness step has received less attention. Furthermore, current microbial risk assessment models do not account for acquired immunity. These limitations may lead to biased risk estimates. We consider effects of both dose dependency of the conditional probability of illness given infection, and acquired immunity to risk estimates, and demonstrate their effects in a case study on exposure to Campylobacter jejuni. To account for acquired immunity in risk estimates, an inflation factor is proposed. The inflation factor depends on the relative rates of loss of protection over exposure. The conditional probability of illness given infection is based on a previously published model, accounting for the within‐host dynamics of illness. We find that at low (average) doses, the infection‐illness model has the greatest impact on risk estimates, whereas at higher (average) doses and/or increased exposure frequencies, the acquired immunity model has the greatest impact. The proposed models are strongly nonlinear, and reducing exposure is not expected to lead to a proportional decrease in risk and, under certain conditions, may even lead to an increase in risk. The impact of different dose‐response models on risk estimates is particularly pronounced when introducing heterogeneity in the population exposure distribution.     

A Distributional Approach to Characterizing Low-Dose Cancer Risk

Since cancer risk at very low doses cannot be directly measured in humans or animals, mathematical extrapolation models and scientific judgment are required. This article demonstrates a probabilistic approach to carcinogen risk assessment that employs probability trees, subjective probabilities, and standard bootstrapping procedures. The probabilistic approach is applied to the carcinogenic risk of formaldehyde in environmental and occupational settings. Sensitivity analyses illustrate conditional estimates of risk for each path in the probability tree. Fundamental mechanistic uncertainties are characterized. A strength of the analysis is the explicit treatment of alternative beliefs about pharmacokinetics and pharmacodynamics. The resulting probability distributions on cancer risk are compared with the point estimates reported by federal agencies. Limitations of the approach are discussed as well as future research directions.     

Methods for Assessing Uncertainty in Fundamental Assumptions and Associated Models for Cancer Risk Assessment

The distributional approach for uncertainty analysis in cancer risk assessment is reviewed and extended. The method considers a combination of bioassay study results, targeted experiments, and expert judgment regarding biological mechanisms to predict a probability distribution for uncertain cancer risks. Probabilities are assigned to alternative model components, including the determination of human carcinogenicity, mode of action, the dosimetry measure for exposure, the mathematical form of the dose‐response relationship, the experimental data set(s) used to fit the relationship, and the formula used for interspecies extrapolation. Alternative software platforms for implementing the method are considered, including Bayesian belief networks (BBNs) that facilitate assignment of prior probabilities, specification of relationships among model components, and identification of all output nodes on the probability tree. The method is demonstrated using the application of Evans, Sielken, and co‐workers for predicting cancer risk from formaldehyde inhalation exposure. Uncertainty distributions are derived for maximum likelihood estimate (MLE) and 95th percentile upper confidence limit (UCL) unit cancer risk estimates, and the effects of resolving selected model uncertainties on these distributions are demonstrated, considering both perfect and partial information for these model components. A method for synthesizing the results of multiple mechanistic studies is introduced, considering the assessed sensitivities and selectivities of the studies for their targeted effects. A highly simplified example is presented illustrating assessment of genotoxicity based on studies of DNA damage response caused by naphthalene and its metabolites. The approach can provide a formal mechanism for synthesizing multiple sources of information using a transparent and replicable weight‐of‐evidence procedure.     

Issues in Qualitative and Quantitative Risk Analysis for Developmental Toxicology1

The qualitative and quantitative evaluation of risk in developmental toxicology has been discussed in several recent publications.(^1–3) A number of issues still are to be resolved in this area. The qualitative evaluation and interpretation of end points in developmental toxicology depends on an understanding of the biological events leading to the end points observed, the relationships among end points, and their relationship to dose and to maternal toxicity. The interpretation of these end points is also affected by the statistical power of the experiments used for detecting the various end points observed. The quantitative risk assessment attempts to estimate human risk for developmental toxicity as a function of dose. The current approach is to apply safety (uncertainty) factors to die no observed effect level (NOEL). An alternative presented and discussed here is to model the experimental data and apply a safety factor to an estimated risk level to achieve an “acceptable” level of risk. In cases where the dose-response curves upward, this approach provides a conservative estimate of risk. This procedure does not preclude the existence of a threshold dose. More research is needed to develop appropriate dose-response models that can provide better estimates for low-dose extrapolation of developmental effects.     

Modeling U‐Shaped Exposure‐Response Relationships for Agents that Demonstrate Toxicity Due to Both Excess and Deficiency

Essential elements such as copper and manganese may demonstrate U‐shaped exposure‐response relationships due to toxic responses occurring as a result of both excess and deficiency. Previous work on a copper toxicity database employed CatReg, a software program for categorical regression developed by the U.S. Environmental Protection Agency, to model copper excess and deficiency exposure‐response relationships separately. This analysis involved the use of a severity scoring system to place diverse toxic responses on a common severity scale, thereby allowing their inclusion in the same CatReg model. In this article, we present methods for simultaneously fitting excess and deficiency data in the form of a single U‐shaped exposure‐response curve, the minimum of which occurs at the exposure level that minimizes the probability of an adverse outcome due to either excess or deficiency (or both). We also present a closed‐form expression for the point at which the exposure‐response curves for excess and deficiency cross, corresponding to the exposure level at which the risk of an adverse outcome due to excess is equal to that for deficiency. The application of these methods is illustrated using the same copper toxicity database noted above. The use of these methods permits the analysis of all available exposure‐response data from multiple studies expressing multiple endpoints due to both excess and deficiency. The exposure level corresponding to the minimum of this U‐shaped curve, and the confidence limits around this exposure level, may be useful in establishing an acceptable range of exposures that minimize the overall risk associated with the agent of interest.     

A Bayesian Semiparametric Model for Radiation Dose‐Response Estimation

In evaluating the risk of exposure to health hazards, characterizing the dose‐response relationship and estimating acceptable exposure levels are the primary goals. In analyses of health risks associated with exposure to ionizing radiation, while there is a clear agreement that moderate to high radiation doses cause harmful effects in humans, little has been known about the possible biological effects at low doses, for example, below 0.1 Gy, which is the dose range relevant to most radiation exposures of concern today. A conventional approach to radiation dose‐response estimation based on simple parametric forms, such as the linear nonthreshold model, can be misleading in evaluating the risk and, in particular, its uncertainty at low doses. As an alternative approach, we consider a Bayesian semiparametric model that has a connected piece‐wise‐linear dose‐response function with prior distributions having an autoregressive structure among the random slope coefficients defined over closely spaced dose categories. With a simulation study and application to analysis of cancer incidence data among Japanese atomic bomb survivors, we show that this approach can produce smooth and flexible dose‐response estimation while reasonably handling the risk uncertainty at low doses and elsewhere. With relatively few assumptions and modeling options to be made by the analyst, the method can be particularly useful in assessing risks associated with low‐dose radiation exposures.     

Heterogeneity: A Major Factor Influencing Microbial Exposure and Risk Assessment

Microbial risk assessment is dependent on several biological and environmental factors that affect both the exposure characteristics to the biological agents and the mechanisms of pathogenicity involved in the pathogen‐host relationship. Many exposure assessment studies still focus on the location parameters of the probability distribution representing the concentration of the pathogens and/or toxin. However, the mean or median by themselves are insufficient to evaluate the adverse effects that are associated with a given level of exposure. Therefore, the effects on the risk of disease of a number of factors, including the shape parameters characterizing the distribution patterns of the pathogen in their environment, were investigated. The statistical models, which were developed to provide a better understanding of the factors influencing the risk, highlight the role of heterogeneity and its consequences on the commonly used risk assessment paradigm. Indeed, the heterogeneity characterizing the spatial and temporal distribution of the pathogen and/or the toxin contained in the water or food consumed is shown to be a major factor that may influence the magnitude of the risk dramatically. In general, the risk diminishes with higher levels of heterogeneity. This scheme is totally inverted in the presence of a threshold in the dose‐response relationship, since heterogeneity will then have a tremendous impact, namely, by magnifying the risk when the mean concentration of pathogens is below the threshold. Moreover, the approach of this article may be useful for risk ranking analysis, regarding different exposure conditions, and may also lead to improved water and food quality guidelines.     

Biological Markers of Environmental and Ecological Contamination: An Overview

An approach, using biomarkers (biological responses) for assessing the biological and ecological significance of contaminants present in the environment is described. Living organisms integrate exposure to contaminants in their environment and respond in some measurable and predictable way. Responses are observed at several levels of biological organization from the biomolecular level, where pollutants can cause damage to critical cellular macromolecules and elicit defensive strategies such as detoxication and repair mechanisms, to the organismal level, where severe disturbances are manifested as impairment in growth, reproduction, developmental abnormalities, or decreased survival. Biomarkers can provide not only evidence of exposure to a broad spectrum of anthropogenic chemicals, but also a temporally integrated measure of bioavailable contaminant levels. A suite of biomarkers are evaluated over time to determine the magnitude of the problem and possible consequences. Relationships between biomarker response and adverse ecological effects are determined from estimates of animal health and population structure.