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.     

Comparing Experimental Designs for Benchmark Dose Calculations for Continuous Endpoints

The BMD (benchmark dose) method that is used in risk assessment of chemical compounds was introduced by Crump (1984) and is based on dose-response modeling. To take uncertainty in the data and model fitting into account, the lower confidence bound of the BMD estimate (BMDL) is suggested to be used as a point of departure in health risk assessments. In this article, we study how to design optimum experiments for applying the BMD method for continuous data. We exemplify our approach by considering the class of Hill models. The main aim is to study whether an increased number of dose groups and at the same time a decreased number of animals in each dose group improves conditions for estimating the benchmark dose. Since Hill models are nonlinear, the optimum design depends on the values of the unknown parameters. That is why we consider Bayesian designs and assume that the parameter vector has a prior distribution. A natural design criterion is to minimize the expected variance of the BMD estimator. We present an example where we calculate the value of the design criterion for several designs and try to find out how the number of dose groups, the number of animals in the dose groups, and the choice of doses affects this value for different Hill curves. It follows from our calculations that to avoid the risk of unfavorable dose placements, it is good to use designs with more than four dose groups. We can also conclude that any additional information about the expected dose-response curve, e.g., information obtained from studies made in the past, should be taken into account when planning a study because it can improve the design.     

Bootstrap Estimation of Benchmark Doses and Confidence Limits with Clustered Quantal Data

The benchmark dose (BMD) is an exposure level that would induce a small risk increase (BMR level) above the background. The BMD approach to deriving a reference dose for risk assessment of noncancer effects is advantageous in that the estimate of BMD is not restricted to experimental doses and utilizes most available dose-response information. To quantify statistical uncertainty of a BMD estimate, we often calculate and report its lower confidence limit (i.e., BMDL), and may even consider it as a more conservative alternative to BMD itself. Computation of BMDL may involve normal confidence limits to BMD in conjunction with the delta method. Therefore, factors, such as small sample size and nonlinearity in model parameters, can affect the performance of the delta method BMDL, and alternative methods are useful. In this article, we propose a bootstrap method to estimate BMDL utilizing a scheme that consists of a resampling of residuals after model fitting and a one-step formula for parameter estimation. We illustrate the method with clustered binary data from developmental toxicity experiments. Our analysis shows that with moderately elevated dose-response data, the distribution of BMD estimator tends to be left-skewed and bootstrap BMDL s are smaller than the delta method BMDL s on average, hence quantifying risk more conservatively. Statistically, the bootstrap BMDL quantifies the uncertainty of the true BMD more honestly than the delta method BMDL as its coverage probability is closer to the nominal level than that of delta method BMDL. We find that BMD and BMDL estimates are generally insensitive to model choices provided that the models fit the data comparably well near the region of BMD. Our analysis also suggests that, in the presence of a significant and moderately strong dose-response relationship, the developmental toxicity experiments under the standard protocol support dose-response assessment at 5% BMR for BMD and 95% confidence level for BMDL.     

Harnessing the Theoretical Foundations of the Exponential and Beta‐Poisson Dose‐Response Models to Quantify Parameter Uncertainty Using Markov Chain Monte Carlo

Dose‐response models are the essential link between exposure assessment and computed risk values in quantitative microbial risk assessment, yet the uncertainty that is inherent to computed risks because the dose‐response model parameters are estimated using limited epidemiological data is rarely quantified. Second‐order risk characterization approaches incorporating uncertainty in dose‐response model parameters can provide more complete information to decisionmakers by separating variability and uncertainty to quantify the uncertainty in computed risks. Therefore, the objective of this work is to develop procedures to sample from posterior distributions describing uncertainty in the parameters of exponential and beta‐Poisson dose‐response models using Bayes's theorem and Markov Chain Monte Carlo (in OpenBUGS). The theoretical origins of the beta‐Poisson dose‐response model are used to identify a decomposed version of the model that enables Bayesian analysis without the need to evaluate Kummer confluent hypergeometric functions. Herein, it is also established that the beta distribution in the beta‐Poisson dose‐response model cannot address variation among individual pathogens, criteria to validate use of the conventional approximation to the beta‐Poisson model are proposed, and simple algorithms to evaluate actual beta‐Poisson probabilities of infection are investigated. The developed MCMC procedures are applied to analysis of a case study data set, and it is demonstrated that an important region of the posterior distribution of the beta‐Poisson dose‐response model parameters is attributable to the absence of low‐dose data. This region includes beta‐Poisson models for which the conventional approximation is especially invalid and in which many beta distributions have an extreme shape with questionable plausibility.     

Influence of Distribution of Animals between Dose Groups on Estimated Benchmark Dose and Animal Distress for Quantal Responses

Increasingly, dose‐response data are being evaluated with the benchmark dose (BMD) approach rather than by the less precise no‐observed‐adverse‐effect‐level (NOAEL) approach. However, the basis for designing animal experiments, using equally sized dose groups, is still primed for the NOAEL approach. The major objective here was to assess the impact of using dose groups of unequal size on both the quality of the BMD and overall animal distress. We examined study designs with a total number of 200 animals distributed in four dose groups employing quantal data generated by Monte Carlo simulations. Placing more animals at doses close to the targeted BMD provided an estimate of BMD that was slightly better than the standard design with equally sized dose groups. In situations involving a clear dose‐response, this translates into fewer animals receiving high doses and thus less overall animal distress. Accordingly, in connection with risk and safety assessment, animal distress can potentially be reduced by distributing the animals appropriately between dose groups without decreasing the quality of the information obtained.     

Inferring an Augmented Bayesian Network to Confront a Complex Quantitative Microbial Risk Assessment Model with Durability Studies: Application to Bacillus Cereus on a Courgette Purée Production Chain

The Monte Carlo (MC) simulation approach is traditionally used in food safety risk assessment to study quantitative microbial risk assessment (QMRA) models. When experimental data are available, performing Bayesian inference is a good alternative approach that allows backward calculation in a stochastic QMRA model to update the experts’ knowledge about the microbial dynamics of a given food‐borne pathogen. In this article, we propose a complex example where Bayesian inference is applied to a high‐dimensional second‐order QMRA model. The case study is a farm‐to‐fork QMRA model considering genetic diversity of Bacillus cereus in a cooked, pasteurized, and chilled courgette purée. Experimental data are Bacillus cereus concentrations measured in packages of courgette purées stored at different time‐temperature profiles after pasteurization. To perform a Bayesian inference, we first built an augmented Bayesian network by linking a second‐order QMRA model to the available contamination data. We then ran a Markov chain Monte Carlo (MCMC) algorithm to update all the unknown concentrations and unknown quantities of the augmented model. About 25% of the prior beliefs are strongly updated, leading to a reduction in uncertainty. Some updates interestingly question the QMRA model.     

Quantitative Risk Assessment: Developing a Bayesian Approach to Dichotomous Dose–Response Uncertainty

Model averaging for dichotomous dose–response estimation is preferred to estimate the benchmark dose (BMD) from a single model, but challenges remain regarding implementing these methods for general analyses before model averaging is feasible to use in many risk assessment applications, and there is little work on Bayesian methods that include informative prior information for both the models and the parameters of the constituent models. This article introduces a novel approach that addresses many of the challenges seen while providing a fully Bayesian framework. Furthermore, in contrast to methods that use Monte Carlo Markov Chain, we approximate the posterior density using maximum a posteriori estimation. The approximation allows for an accurate and reproducible estimate while maintaining the speed of maximum likelihood, which is crucial in many applications such as processing massive high throughput data sets. We assess this method by applying it to empirical laboratory dose–response data and measuring the coverage of confidence limits for the BMD. We compare the coverage of this method to that of other approaches using the same set of models. Through the simulation study, the method is shown to be markedly superior to the traditional approach of selecting a single preferred model (e.g., from the U.S. EPA BMD software) for the analysis of dichotomous data and is comparable or superior to the other approaches.     

A gradient Markov chain Monte Carlo algorithm for computing multivariate maximum likelihood estimates and posterior distributions: mixture dose-response assessment

Multivariate probability distributions, such as may be used for mixture dose‐response assessment, are typically highly parameterized and difficult to fit to available data. However, such distributions may be useful in analyzing the large electronic data sets becoming available, such as dose‐response biomarker and genetic information. In this article, a new two‐stage computational approach is introduced for estimating multivariate distributions and addressing parameter uncertainty. The proposed first stage comprises a gradient Markov chain Monte Carlo (GMCMC) technique to find Bayesian posterior mode estimates (PMEs) of parameters, equivalent to maximum likelihood estimates (MLEs) in the absence of subjective information. In the second stage, these estimates are used to initialize a Markov chain Monte Carlo (MCMC) simulation, replacing the conventional burn‐in period to allow convergent simulation of the full joint Bayesian posterior distribution and the corresponding unconditional multivariate distribution (not conditional on uncertain parameter values). When the distribution of parameter uncertainty is such a Bayesian posterior, the unconditional distribution is termed predictive. The method is demonstrated by finding conditional and unconditional versions of the recently proposed emergent dose‐response function (DRF). Results are shown for the five‐parameter common‐mode and seven‐parameter dissimilar‐mode models, based on published data for eight benzene–toluene dose pairs. The common mode conditional DRF is obtained with a 21‐fold reduction in data requirement versus MCMC. Example common‐mode unconditional DRFs are then found using synthetic data, showing a 71% reduction in required data. The approach is further demonstrated for a PCB 126‐PCB 153 mixture. Applicability is analyzed and discussed. Matlab^® computer programs are provided.     

A Simulation Study of Quantitative Risk Assessment for Bivariate Continuous Outcomes

The neurotoxic effects of chemical agents are often investigated in controlled studies on rodents, with binary and continuous multiple endpoints routinely collected. One goal is to conduct quantitative risk assessment to determine safe dose levels. Yu and Catalano (2005) describe a method for quantitative risk assessment for bivariate continuous outcomes by extending a univariate method of percentile regression. The model is likelihood based and allows for separate dose‐response models for each outcome while accounting for the bivariate correlation. The approach to benchmark dose (BMD) estimation is analogous to that for quantal data without having to specify arbitrary cutoff values. In this article, we evaluate the behavior of the BMD relative to background rates, sample size, level of bivariate correlation, dose‐response trend, and distributional assumptions. Using simulations, we explore the effects of these factors on the resulting BMD and BMDL distributions. In addition, we illustrate our method with data from a neurotoxicity study of parathion exposure in rats.     

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.     

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.     

Relation Between Benchmark Dose and No-Observed-Adverse-Effect Level in Clinical Research: Effects of Daily Alcohol Intake on Blood Pressure in Japanese Salesmen

The benchmark dose (BMD) is defined as the dose that corresponds to a specific change in an adverse response compared to the response in unexposed subjects, and the lower 95% confidence limit is termed the benchmark dose level (BMDL). In this study, the threshold of daily ethanol intake affecting blood pressure was calculated by both the BMD approach and multiple logistic regression analysis to clarify the relation between the BMDL and no-observed-adverse-effect level (NOAEL). Systolic and diastolic blood pressures (SBP and DBP) and daily ethanol intake were explored in 1,100 Japanese salesmen. The SBP and DBP were positively related to daily ethanol intake (p < 0.001) when adjusting for possible confounders such as age, body mass index, and smoking status. The adjusted risk for hypertension (SBP >or= 140 mmHg or DBP >or= 90 mmHg) increased significantly when daily ethanol intake exceeded 60 g/day, and the categorical dose of interest was 60.1-90 g/day. The BMDL and BMD of ethanol intake for increased SBP and DBP were estimated to be approximately 60 and 75 g/day, respectively. These findings suggest that the BMDL and BMD correspond to the NOAEL and lowest-observed-adverse-effect level, respectively, if the sample number of clinical data is large enough to confirm the dose-response association.     

Influence of Distribution of Animals between Dose Groups on Estimated Benchmark Dose and Animal Welfare for Continuous Effects

The benchmark dose (BMD) approach is increasingly used as a preferred approach for dose–effect analysis, but standard experimental designs are generally not optimized for BMD analysis. The aim of this study was to evaluate how the use of unequally sized dose groups affects the quality of BMD estimates in toxicity testing, with special consideration of the total burden of animal distress. We generated continuous dose–effect data by Monte Carlo simulation using two dose–effect curves based on endpoints with different shape parameters. Eighty‐five designs, each with four dose groups of unequal size, were examined in scenarios ranging from low‐ to high‐dose placements and with a total number of animals set to 40, 80, or 200. For each simulation, a BMD value was estimated and compared with the “true” BMD. In general, redistribution of animals from higher to lower dose groups resulted in an improved precision of the calculated BMD value as long as dose placements were high enough to detect a significant trend in the dose–effect data with sufficient power. The improved BMD precision and the associated reduction of the number of animals exposed to the highest dose, where chemically induced distress is most likely to occur, are favorable for the reduction and refinement principles. The result thereby strengthen BMD‐aligned design of experiments as a means for more accurate hazard characterization along with animal welfare improvements.     

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.     

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     

Estimation of Benchmark Dose as the Threshold Amount of Alcohol Consumption for Blood Pressure in Japanese Workers

In order to determine the threshold amount of alcohol consumption for blood pressure, we calculated the benchmark dose (BMD) of alcohol consumption and its 95% lower confidence interval (BMDL) in Japanese workers. The subjects consisted of 4,383 males and 387 females in a Japanese steel company. The target variables were systolic, diastolic, and mean arterial pressures. The effects of other potential covariates such as age and body mass index were adjusted by including these covariates in the multiple linear regression models. In male workers, BMD/BMDL for alcohol consumption (g/week) at which the probability of an adverse response was estimated to increase by 5% relative to no alcohol consumption, were 396/315 (systolic blood pressure), 321/265 (diastolic blood pressure), and 326/269 (mean arterial pressures). These values were based on significant regression coefficients of alcohol consumption. In female workers, BMD/BMDL for alcohol consumption based on insignificant regression coefficients were 693/134 (systolic blood pressure), 199/90 (diastolic blood pressure), and 267/77 (mean arterial pressure). Therefore, BMDs/BMDLs in males were more informative than those in females as there was no significant relationship between alcohol and blood pressure in females. The threshold amount of alcohol consumption determined in this study provides valuable information for preventing alcohol-induced hypertension.     

An International Pooled Analysis for Obtaining a Benchmark Dose for Environmental Lead Exposure in Children

Lead is a recognized neurotoxicant, but estimating effects at the lowest measurable levels is difficult. An international pooled analysis of data from seven cohort studies reported an inverse and supra‐linear relationship between blood lead concentrations and IQ scores in children. The lack of a clear threshold presents a challenge to the identification of an acceptable level of exposure. The benchmark dose (BMD) is defined as the dose that leads to a specific known loss. As an alternative to elusive thresholds, the BMD is being used increasingly by regulatory authorities. Using the pooled data, this article presents BMD results and applies different statistical techniques in the analysis of multistudy data. The calculations showed only a limited variation between studies in the steepness of the dose‐response functions. BMD results were quite robust to modeling assumptions with the best fitting models yielding lower confidence limits (BMDLs) of about 0.1–1.0 μ g/dL for the dose leading to a loss of one IQ point. We conclude that current allowable blood lead concentrations need to be lowered and further prevention efforts are needed to protect children from lead toxicity.     

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.     

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.     

Estimation of a Benchmark Dose in the Presence or Absence of Hormesis Using Posterior Averaging

U.S. Environment Protection Agency benchmark doses for dichotomous cancer responses are often estimated using a multistage model based on a monotonic dose‐response assumption. To account for model uncertainty in the estimation process, several model averaging methods have been proposed for risk assessment. In this article, we extend the usual parameter space in the multistage model for monotonicity to allow for the possibility of a hormetic dose‐response relationship. Bayesian model averaging is used to estimate the benchmark dose and to provide posterior probabilities for monotonicity versus hormesis. Simulation studies show that the newly proposed method provides robust point and interval estimation of a benchmark dose in the presence or absence of hormesis. We also apply the method to two data sets on carcinogenic response of rats to 2,3,7,8‐tetrachlorodibenzo‐p‐dioxin.