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.     

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.     

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.     

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.     

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.     

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.     

A Simulation Study of the Influence of Study Design on the Estimation of Benchmark Doses for Developmental Toxicity1

The benchmark dose (BMD)⁴ approach is emerging as replacement to determination of the No Observed Adverse Effect Level (NOAEL) in noncancer risk assessment. This possibility raises the issue as to whether current study designs for endpoints such as developmental toxicity, optimized for detecting pair wise comparisons, could be improved for the purpose of calculating BMDs. In this paper, we examine various aspects of study design (number of dose groups, dose spacing, dose placement, and sample size per dose group) on BMDs for two endpoints of developmental toxicity (the incidence of abnormalities and of reduced fetal weight). Design performance was judged by the mean-squared error (reflective of the variance and bias) of the maximum likelihood estimate (MLE) from the log-logistic model of the 5% added risk level (the likely target risk for a benchmark calculation), as well as by the length of its 95% confidence interval (the lower value of which is the BMD). We found that of the designs evaluated, the best results were obtained when two dose levels had response rates above the background level, one of which was near the ED₀₅, were present. This situation is more likely to occur with more, rather than fewer dose levels per experiment. In this instance, there was virtually no advantage in increasing the sample size from 10 to 20 litters per dose group. If neither of the two dose groups with response rates above the background level was near the ED₀₅, satisfactory results were also obtained, but the BMDs tended to be more conservative (i.e., lower). If only one dose level with a response rate above the background level was present, and it was near the ED₀₅, reasonable results for the MLE and BMD were obtained, but here we observed benefits of larger dose group sizes. The poorest results were obtained when only a single group with an elevated response rate was present, and the response rate was much greater than the ED₀₅. The results indicate that while the benchmark dose approach is readily applicable to the standard study designs and generally observed dose-responses in developmental assays, some minor design modifications would increase the accuracy and precision of the BMD.     

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.     

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.     

Models of Neurotoxicity: Extrapolation of Benchmark Doses in Vitro

In risk assessment, no observed exposure level (NOAEL) and benchmark dose (BMD) are usually derived either from epidemiological studies in humans or from animal experiments. In many in vitro studies, concentration-effect/response curves have been analyzed using different mathematical models finalized to the identification of EC50. In the present article, we propose a model to fit dose-response curves in vitro. The BMD approach has been used to compare the cell viability (MIT assay) of different rat (C6 and PC12, glial and neuronal, respectively) and human cell lines (D384 and SK-N-MC, glial and neuronal, respectively) after 24-hour exposure to the following neurotoxic substances: manganese chloride (MnCl2), methyl-mercury (Me-Hg), and the enantiomers of styrene oxide (SO). For all rat and human cell lines, the potency of the examined compounds was: MnCl2 < S-SO < R-SO < Me-Hg. A preliminary comparison with in vivo toxicity data for these substances gave rise to consistent results. Whereas a reasonable agreement between in vitro and in vivo data has been found for Mn and styrene oxide, a wide scatter of LOAEL has been reported for Me-Hg and these appear to be either much higher or lower than the BMD for the MIT assay we observed in vitro.     

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.     

Correlation of Noncancer Benchmark Doses in Short‐ and Long‐Term Rodent Bioassays

This study investigated whether, in the absence of chronic noncancer toxicity data, short‐term noncancer toxicity data can be used to predict chronic toxicity effect levels by focusing on the dose–response relationship instead of a critical effect. Data from National Toxicology Program (NTP) technical reports have been extracted and modeled using the Environmental Protection Agency's Benchmark Dose Software. Best‐fit, minimum benchmark dose (BMD), and benchmark dose lower limits (BMDLs) have been modeled for all NTP pathologist identified significant nonneoplastic lesions, final mean body weight, and mean organ weight of 41 chemicals tested by NTP between 2000 and 2012. Models were then developed at the chemical level using orthogonal regression techniques to predict chronic (two years) noncancer health effect levels using the results of the short‐term (three months) toxicity data. The findings indicate that short‐term animal studies may reasonably provide a quantitative estimate of a chronic BMD or BMDL. This can allow for faster development of human health toxicity values for risk assessment for chemicals that lack chronic toxicity data.     

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.     

Characterizing Dose-Response I: Critical Assessment of the Benchmark Dose Concept

We present a critical assessment of the benchmark dose (BMD) method introduced by Crump⁽¹⁾ as an alternative method for setting a characteristic dose level for toxicant risk assessment. The no-observed-adverse-effect-level (NOAEL) method has been criticized because it does not use all of the data and because the characteristic dose level obtained depends on the dose levels and the statistical precision (sample sizes) of the study design. Defining the BMD in terms of a confidence bound on a point estimate results in a characteristic dose that also varies with the statistical precision and still depends on the study dose levels.⁽²⁾ Indiscriminate choice of benchmark response level may result in a BMD that reflects little about the dose-response behavior available from using all of the data. Another concern is that the definition of the BMD for the quantal response case is different for the continuous response case. Specifically, defining the BMD for continuous data using a ratio of increased effect divided by the background response results in an arbitrary dependence on the natural background for the endpoint being studied, making comparison among endpoints less meaningful and standards more arbitrary. We define a modified benchmark dose as a point estimate using the ratio of increased effect divided by the full adverse response range which enables consistent placement of the benchmark response level and provides a BMD with a more consistent relationship to the dose-response curve shape.     

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.     

Potential Uncertainty Reduction in Model‐Averaged Benchmark Dose Estimates Informed by an Additional Dose Study

A methodology is presented for assessing the information value of an additional dosage experiment in existing bioassay studies. The analysis demonstrates the potential reduction in the uncertainty of toxicity metrics derived from expanded studies, providing insights for future studies. Bayesian methods are used to fit alternative dose‐response models using Markov chain Monte Carlo (MCMC) simulation for parameter estimation and Bayesian model averaging (BMA) is used to compare and combine the alternative models. BMA predictions for benchmark dose (BMD) are developed, with uncertainty in these predictions used to derive the lower bound BMDL. The MCMC and BMA results provide a basis for a subsequent Monte Carlo analysis that backcasts the dosage where an additional test group would have been most beneficial in reducing the uncertainty in the BMD prediction, along with the magnitude of the expected uncertainty reduction. Uncertainty reductions are measured in terms of reduced interval widths of predicted BMD values and increases in BMDL values that occur as a result of this reduced uncertainty. The methodology is illustrated using two existing data sets for TCDD carcinogenicity, fitted with two alternative dose‐response models (logistic and quantal‐linear). The example shows that an additional dose at a relatively high value would have been most effective for reducing the uncertainty in BMA BMD estimates, with predicted reductions in the widths of uncertainty intervals of approximately 30%, and expected increases in BMDL values of 5–10%. The results demonstrate that dose selection for studies that subsequently inform dose‐response models can benefit from consideration of how these models will be fit, combined, and interpreted.     

Estimation of Benchmark Dose of Lifetime Cadmium Intake for Adverse Renal Effects Using Hybrid Approach in Inhabitants of an Environmentally Exposed River Basin in Japan

The aim of this study is to estimate the reference level of lifetime cadmium intake (LCd) as the benchmark doses (BMDs) and their 95% lower confidence limits (BMDLs) for various renal effects by applying a hybrid approach. The participants comprised 3,013 (1,362 men and 1,651 women) and 278 (129 men and 149 women) inhabitants of the Cd‐polluted and nonpolluted areas, respectively, in the environmentally exposed Kakehashi River basin. Glucose, protein, aminonitrogen, metallothionein, and β₂‐microglobulin in urine were measured as indicators of renal dysfunction. The BMD and BMDL that corresponded to an additional risk of 5% were calculated with background risk at zero exposure set at 5%. The obtained BMDLs of LCd were 3.7 g (glucose), 3.2 g (protein), 3.7 g (aminonitrogen), 1.7 g (metallothionein), and 1.8 g (β₂‐microglobulin) in men and 2.9 g (glucose), 2.5 g (protein), 2.0 g (aminonitrogen), 1.6 g (metallothionein), and 1.3 g (β₂‐microglobulin) in women. The lowest BMDL was 1.7 g (metallothionein) and 1.3 g (β₂‐microglobulin) in men and women, respectively. The lowest BMDL of LCd (1.3 g) was somewhat lower than the representative threshold LCd (2.0 g) calculated in the previous studies. The obtained BMDLs may contribute to further discussion on the health risk assessment of cadmium exposure.     

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.     

A Comparison of Methods for Estimating the Benchmark Dose Based on Overdispersed Data from Developmental Toxicity Studies

Developmental anomalies resulting from prenatal toxicity can be manifested in terms of both malformations among surviving offspring and prenatal death. Although these two endpoints have traditionally been analyzed separately in the assessment of risk, multivariate methods of risk characterization have recently been proposed. We examined this and other issues in developmental toxicity risk assessment by evaluating the accuracy and precision of estimates of the effective dose ( ED ₀₅) and the benchmark dose ( BMD ₀₅) using computer simulation. Our results indicated that different variance structures (Dirichlet-trinomial and generalized linear model) used to characterize overdispersion yielded comparable results when fitting joint dose response models based on generalized estimating equations. (The choice of variance structure in separate modeling was also not critical.) However, using the Rao-Scott transformation to eliminate overdispersion tended to produce estimates of the ED ₀₅ with reduced bias and mean squared error. Because joint modeling ensures that the ED ₀₅ for overall toxicity (based on both malformations and prenatal death) is always less than the ED ₀₅ for either malformations or prenatal death, joint modeling is preferred to separate modeling for risk assessment purposes.     

Parameters of a Dose‐Response Model Are on the Boundary: What Happens with BMDL?

It is well known that, under appropriate regularity conditions, the asymptotic distribution for the likelihood ratio statistic is χ². This result is used in EPA's benchmark dose software to obtain a lower confidence bound (BMDL) for the benchmark dose (BMD) by the profile likelihood method. Recently, based on work by Self and Liang, it has been demonstrated that the asymptotic distribution of the likelihood ratio remains the same if some of the regularity conditions are violated, that is, when true values of some nuisance parameters are on the boundary. That is often the situation for BMD analysis of cancer bioassay data. In this article, we study by simulation the coverage of one- and two-sided confidence intervals for BMD when some of the model parameters have true values on the boundary of a parameter space. Fortunately, because two-sided confidence intervals (size 1–2α) have coverage close to the nominal level when there are 50 animals in each group, the coverage of nominal 1−α one-sided intervals is bounded between roughly 1–2α and 1. In many of the simulation scenarios with a nominal one-sided confidence level of 95%, that is, α= 0.05, coverage of the BMDL was close to 1, but for some scenarios coverage was close to 90%, both for a group size of 50 animals and asymptotically (group size 100,000). Another important observation is that when the true parameter is below the boundary, as with the shape parameter of a log-logistic model, the coverage of BMDL in a constrained model (a case of model misspecification not uncommon in BMDS analyses) may be very small and even approach 0 asymptotically. We also discuss that whenever profile likelihood is used for one-sided tests, the Self and Liang methodology is needed to derive the correct asymptotic distribution.