Benchmark dose profiles for joint‐action continuous data in quantitative risk assessment

Benchmark analysis is a widely used tool in biomedical and environmental risk assessment. Therein, estimation of minimum exposure levels, called benchmark doses (BMDs), that induce a prespecified benchmark response (BMR) is well understood for the case of an adverse response to a single stimulus. For cases where two agents are studied in tandem, however, the benchmark approach is far less developed. This paper demonstrates how the benchmark modeling paradigm can be expanded from the single‐agent setting to joint‐action, two‐agent studies. Focus is on continuous response outcomes. Extending the single‐exposure setting, representations of risk are based on a joint‐action dose–response model involving both agents. Based on such a model, the concept of a benchmark profile—a two‐dimensional analog of the single‐dose BMD at which both agents achieve the specified BMR—is defined for use in quantitative risk characterization and assessment.     

Multiplicity-adjusted inferences in risk assessment: benchmark analysis with quantal response data

A primary objective in quantitative risk or safety assessment is characterization of the severity and likelihood of an adverse effect caused by a chemical toxin or pharmaceutical agent. In many cases data are not available at low doses or low exposures to the agent, and inferences at those doses must be based on the high-dose data. A modern method for making low-dose inferences is known as benchmark analysis, where attention centers on the dose at which a fixed benchmark level of risk is achieved. Both upper confidence limits on the risk and lower confidence limits on the "benchmark dose" are of interest. In practice, a number of possible benchmark risks may be under study; if so, corrections must be applied to adjust the limits for multiplicity. In this short note, we discuss approaches for doing so with quantal response data.     

Benchmark Dose Profiles for Joint‐Action Quantal Data in Quantitative Risk Assessment

Summary Benchmark analysis is a widely used tool in public health risk analysis. Therein, estimation of minimum exposure levels, called Benchmark Doses (BMDs), that induce a prespecified Benchmark Response (BMR) is well understood for the case of an adverse response to a single stimulus. For cases where two agents are studied in tandem, however, the benchmark approach is far less developed. This article demonstrates how the benchmark modeling paradigm can be expanded from the single‐dose setting to joint‐action, two‐agent studies. Focus is on response outcomes expressed as proportions. Extending the single‐exposure setting, representations of risk are based on a joint‐action dose–response model involving both agents. Based on such a model, the concept of a benchmark profile (BMP) – a two‐dimensional analog of the single‐dose BMD at which both agents achieve the specified BMR – is defined for use in quantitative risk characterization and assessment. The resulting, joint, low‐dose guidelines can improve public health planning and risk regulation when dealing with low‐level exposures to combinations of hazardous agents.     

Quantitative Risk Assessment in Epidemiological Studies Investigating Threshold Effects

In this paper a method for quantitative risk assessment in epidemiological studies investigating threshold effects is proposed. The simple logistic regression model is used to describe the association between a binary response variable and a continuous risk factor. By defining acceptable levels for the absolute risk and the risk gradient the corresponding benchmark values of the risk factor can be calculated by means of nonlinear functions of the logistic regression coefficients. Standard errors and confidence intervals of the benchmark values are derived by means of the multivariate delta method. The proposed approach is compared with the threshold model of Ulm (1991) for assessing threshold values in epidemiological studies.     

Quantitative risk assessment for developmental toxicity.

Pharmaceutical companies and governmental regulatory agencies are becoming increasingly aware of the need for improved statistical methods for developmental toxicity experiments. Although a number of statisticians have become interested in this area, activity has centered mostly on the development of methods to analyze binary outcomes, such as malformations among live pups, while accounting appropriately for the correlation induced by the litter effect. In contrast, the topic of quantitative risk assessment has received relatively little attention. This paper addresses the specific question of how to assess risk appropriately when exposure causes a variety of adverse effects, including resorption and fetal death, in addition to malformations. It will be seen that risk assessments based on a single developmental outcome, such as malformation, may be conservative. A method is proposed for estimating an exposure level at which the overall risk of any adverse effect is acceptably low. The method is based on a continuation ratio formulation of a multinomial distribution, with an additional scale parameter to account for overdispersion. Comparisons are made with binary models on prenatal death and malformation, as well as a binary model that makes no distinction between death and malformation, but simply classifies each fetus as normal or abnormal. Data from several developmental toxicity studies illustrate the results and findings.     

Likelihood models for clustered binary and continuous outcomes: application to developmental toxicology

In developmental toxicology, methods based on dose response modeling and quantitative risk assessment are being actively pursued. Among live fetuses, the presence of malformations and reduction in fetal weight are of primary interest, but ordinarily, the dose-response relationships are characterized in each of the outcomes separately while appropriately accounting for clustering within litters. Jointly modeling the outcomes, allowing different relationships with dose while incorporating the correlation between the fetuses and the outcomes, may be more appropriate. We propose a likelihood-based model that is an extension of a correlated probit model to incorporate continuous outcomes. Our model maintains a marginal dose-response interpretation for the individual outcomes while taking into account both the correlations between outcomes on an individual fetus and those due to clustering. The joint risk of malformation and low birth weight can then be estimated directly. This approach is particularly well suited to estimating safe dose levels as part of quantitative risk assessment.     

Simultaneous confidence bands for log‐logistic regression with applications in risk assessment

In risk assessment, it is often desired to make inferences on the low dose levels at which a specific benchmark risk is attained. Applications of simultaneous hyperbolic confidence bands for low‐dose risk estimation with quantal data under different dose‐response models (multistage, Abbott‐adjusted Weibull, and Abbott‐adjusted log‐logistic models) have appeared in the literature. The use of simultaneous three‐segment bands under the multistage model has also been proposed recently. In this article, we present explicit formulas for constructing asymptotic one‐sided simultaneous hyperbolic and three‐segment bands for the simple log‐logistic regression model. We use the simultaneous construction to estimate upper hyperbolic and three‐segment confidence bands on extra risk and to obtain lower limits on the benchmark dose by inverting the upper bands on risk under the Abbott‐adjusted log‐logistic model. Monte Carlo simulations evaluate the characteristics of the simultaneous limits. An example is given to illustrate the use of the proposed methods and to compare the two types of simultaneous limits at very low dose levels.     

A Guide to Bivariate Ideas of Quantal Response Analysis,as Applied in Biometrics and Econometrics

A review is provided of the several bivariate generalisations of quantal response analysis that have appeared in the biometric and econometric literatures. There are three main types: (i) where a binary outcome is the result of two stimulants, and thus the bivariate distribution of the thresholds for response is relevant; (ii) where three or more alternative outcomes may arise from a single stimulant; and (iii) where the response itself is bivariate (i.e., two types of response may simultaneously be observed).     

A unification of multivariate methods for meta-analysis of genetic association studies

Methods for multivariate meta-analysis of genetic association studies are reviewed, summarized and presented in a unified framework. Modifications of standard models are described in detail in order to be applied in genetic association studies. The model based on summary data is uniformly defined for both discrete and continuous outcomes and analytical expressions for the covariance of the two jointly modeled outcomes are derived for both cases. The models based on the binary nature of the data are fitted using both prospective and retrospective likelihood. Furthermore, formal tests for assessing the genetic model of inheritance are developed based on standard normal theory. The general model is compared to the recently proposed genetic model-free bivariate approach (either using summary or binary data), and it is clearly shown that the estimates provided by this approach are nearly identical to the estimates derived by the general bivariate model using the aforementioned tests for the genetic model. The methods developed here as well as the tests, are easily implemented in all major statistical packages, escaping the need of self written software. The methods are applied in several already published meta-analyses of genetic association studies (with both discrete and continuous outcomes) and the results are compared against the widely used univariate approach as well as against the genetic model free approaches. Illustrative examples of code in Stata are given in the appendix. It is anticipated that the methods developed in this work will be widely applied in the meta-analysis of genetic association studies.     

Confidence bands for low-dose risk estimation with quantal response data

We study the use of simultaneous confidence bands for low-dose risk estimation with quantal response data, and derive methods for estimating simultaneous upper confidence limits on predicted extra risk under a multistage model. By inverting the upper bands on extra risk, we obtain simultaneous lower bounds on the benchmark dose (BMD). Monte Carlo evaluations explore characteristics of the simultaneous limits under this setting, and a suite of actual data sets are used to compare existing methods for placing lower limits on the BMD.     

Estimation of the benchmark dose by structural equation models

While epidemiological data typically contain a multivariate response and often also multiple exposure parameters, current methods for safe dose calculations, including the widely used benchmark approach, rely on standard regression techniques. In practice, dose-response modeling and calculation of the exposure limit are often based on the seemingly most sensitive outcome. However, this procedure ignores other available data, is inefficient, and fails to account for multiple testing. Instead, risk assessment could be based on structural equation models, which can accommodate both a multivariate exposure and a multivariate response function. Furthermore, such models will allow for measurement error in the observed variables, which is a requirement for unbiased estimation of the benchmark dose. This methodology is illustrated with the data on neurobehavioral effects in children prenatally exposed to methylmercury, where results based on standard regression models cause an underestimation of the true risk.     

Risk assessment for quantitative responses using a mixture model

A problem that frequently occurs in biological experiments with laboratory animals is that some subjects are less susceptible to the treatment than others. A mixture model has traditionally been proposed to describe the distribution of responses in treatment groups for such experiments. Using a mixture dose-response model, we derive an upper confidence limit on additional risk, defined as the excess risk over the background risk due to an added dose. Our focus will be on experiments with continuous responses for which risk is the probability of an adverse effect defined as an event that is extremely rare in controls. The asymptotic distribution of the likelihood ratio statistic is used to obtain the upper confidence limit on additional risk. The method can also be used to derive a benchmark dose corresponding to a specified level of increased risk. The EM algorithm is utilized to find the maximum likelihood estimates of model parameters and an extension of the algorithm is proposed to derive the estimates when the model is subject to a specified level of added risk. An example is used to demonstrate the results, and it is shown that by using the mixture model a more accurate measure of added risk is obtained.     

A general noninteractive multiple toxicity model including probit, logit, and Weibull transformations

A multiple toxicity model for the quantal response of organisms is constructed based on an existing bivariate theory. The main assumption is that the tolerances follow a multivariate normal distribution function. However, any monotone tolerance distribution can be applied by mapping the integration region in the n-dimensional space of transforms on the n-dimensional space of normal equivalent deviates. General requirements to noninteractive bivariate tolerance distributions are discussed, and it is shown that bivariate logit and Weibull distributions, constructed according to the mapping procedure, meet these criteria. The univariate Weibull dose-response model is given a novel interpretation in terms of reactions between toxicant molecules and a hypothetical key receptor of the organism. The application of the multiple toxicity model is demonstrated using literature data for the action of gamma-benzene hexachloride and pyrethrins on flour beetles (Tribolium castaneum). Nonnormal tolerance distributions are needed when the mortality data include extreme response probabilities.     

Association Models for Clustered Data with Binary and Continuous Responses

Summary .  We consider analysis of clustered data with mixed bivariate responses, i.e., where each member of the cluster has a binary and a continuous outcome. We propose a new bivariate random effects model that induces associations among the binary outcomes within a cluster, among the continuous outcomes within a cluster, between a binary outcome and a continuous outcome from different subjects within a cluster, as well as the direct association between the binary and continuous outcomes within the same subject. For the ease of interpretations of the regression effects, the marginal model of the binary response probability integrated over the random effects preserves the logistic form and the marginal expectation of the continuous response preserves the linear form. We implement maximum likelihood estimation of our model parameters using standard software such as PROC NLMIXED of SAS . Our simulation study demonstrates the robustness of our method with respect to the misspecification of the regression model as well as the random effects model. We illustrate our methodology by analyzing a developmental toxicity study of ethylene glycol in mice.     

Bayesian dose-finding in phase I/II clinical trials using toxicity and efficacy odds ratios

A Bayesian adaptive design is proposed for dose-finding in phase I/II clinical trials to incorporate the bivariate outcomes, toxicity and efficacy, of a new treatment. Without specifying any parametric functional form for the drug dose-response curve, we jointly model the bivariate binary data to account for the correlation between toxicity and efficacy. After observing all the responses of each cohort of patients, the dosage for the next cohort is escalated, deescalated, or unchanged according to the proposed odds ratio criteria constructed from the posterior toxicity and efficacy probabilities. A novel class of prior distributions is proposed through logit transformations which implicitly imposes a monotonic constraint on dose toxicity probabilities and correlates the probabilities of the bivariate outcomes. We conduct simulation studies to evaluate the operating characteristics of the proposed method. Under various scenarios, the new Bayesian design based on the toxicity-efficacy odds ratio trade-offs exhibits good properties and treats most patients at the desirable dose levels. The method is illustrated with a real trial design for a breast medical oncology study.     

Issues in Exposure and Dose Assessment for Epidemiology and Risk Assessment

Epidemiologic studies have been effective in identifying human environmental and occupational hazards. However, most epidemiologic data has been difficult to use in quantitative risk assessments because of the vague specification of exposure and dose. Toxicologic animal studies have used applied doses (quantities administered, or exposures with fixed duration) and well characterized end points to determine effects. However, direct use of animal data in human risk assessment has been limited by uncertainties in the extrapolation. The applied dose paradigm of toxicology is not suited for cross species extrapolation, nor for use in epidemiology as a dose metric because of the complexity of human exposures. Physiologically based pharmacokinetic (PBPK) modeling can estimate the time course of tissue concentrations in humans, given an exposure-time profile, and it has been used for extrapolating findings from animals to humans. It is proposed that human PBPK modeling can be used in appropriately designed epidemiologic studies to estimate tissue concentrations. Secondly, tissue time courses can be used to form dose metrics based on the type and time course of adverse effects. These dose metrics will strengthen the determination of epidemiologic dose-response relationships by reducing misclassification. Findings from this approach can be readily integrated into quantitative risk assessment.     

Improving Risk Assessment: Priorities for Epidemiologic Research

The Epidemiology Work Group at the Workshop on Future Research for Improving Risk Assessment Methods, Of Mice, Men, and Models, held August 16 to 18, 2000, at Snowmass Village, Aspen, Colorado, concluded that in order to improve the utility of epidemiologic studies for risk assessment, methodologic research is needed in the following areas: (1) aspects of epidemiologic study designs that affect doseresponse estimation; (2) alternative methods for estimating dose in human studies; and (3) refined methods for dose-response modeling for epidemiologic data. Needed research in aspects of epidemiologic study design includes recognition and control of study biases, identification of susceptible subpopulations, choice of exposure metrics, and choice of epidemiologic risk parameters. Much of this research can be done with existing data. Research needed to improve determinants of dose in human studies includes additional individual-level data (e.g., diet, co-morbidity), development of more extensive human data for physiologically based pharmacokinetic (PBPK) dose modeling, tissue registries to increase the availability of tissue for studies of exposure/dose and susceptibility biomarkers, and biomarker data to assess exposures in humans and animals. Research needed on dose-response modeling of human studies includes more widespread application of flexible statistical methods (e.g., general additive models), development of methods to compensate for epidemiologic bias in dose-response models, improved biological models using human data, and evaluation of the benchmark dose using human data. There was consensus among the Work Group that, whereas most prior risk assessments have focused on cancer, there is a growing need for applications to other health outcomes. Developmental and reproductive effects, injuries, respiratory disease, and cardiovascular disease were identified as especially high priorities for research. It was also a consensus view that epidemiologists, industrial hygienists, and other scientists focusing on human data need to play a stronger role throughout the risk assessment process. Finally, the group agreed that there was a need to improve risk communication, particularly on uncertainty inherent in risk assessments that use epidemiologic data.     

Benchmark dose calculation from epidemiological data

A threshold for dose-dependent toxicity is crucial for standards setting but may not be possible to specify from empirical studies. Crump (1984) instead proposed calculating the lower statistical confidence bound of the benchmark dose, which he defined as the dose that causes a small excess risk. This concept has several advantages and has been adopted by regulatory agencies for establishing safe exposure limits for toxic substances such as mercury. We have examined the validity of this method as applied to an epidemiological study of continuous response data associated with mercury exposure. For models that are linear in the parameters, we derived an approximative expression for the lower confidence bound of the benchmark dose. We find that the benchmark calculations are highly dependent on the choice of the dose-effect function and the definition of the benchmark dose. We therefore recommend that several sets of biologically relevant default settings be used to illustrate the effect on the benchmark results and to stimulate research that will guide an a priori choice of proper default settings.     

Joint mapping of quantitative trait Loci for multiple binary characters

Joint mapping for multiple quantitative traits has shed new light on genetic mapping by pinpointing pleiotropic effects and close linkage. Joint mapping also can improve statistical power of QTL detection. However, such a joint mapping procedure has not been available for discrete traits. Most disease resistance traits are measured as one or more discrete characters. These discrete characters are often correlated. Joint mapping for multiple binary disease traits may provide an opportunity to explore pleiotropic effects and increase the statistical power of detecting disease loci. We develop a maximum-likelihood method for mapping multiple binary traits. We postulate a set of multivariate normal disease liabilities, each contributing to the phenotypic variance of one disease trait. The underlying liabilities are linked to the binary phenotypes through some underlying thresholds. The new method actually maps loci for the variation of multivariate normal liabilities. As a result, we are able to take advantage of existing methods of joint mapping for quantitative traits. We treat the multivariate liabilities as missing values so that an expectation-maximization (EM) algorithm can be applied here. We also extend the method to joint mapping for both discrete and continuous traits. Efficiency of the method is demonstrated using simulated data. We also apply the new method to a set of real data and detect several loci responsible for blast resistance in rice.