Accounting for Uncertainty in Heteroscedasticity in Nonlinear Regression

Toxicologists and pharmacologists often describe toxicity of a chemical using parameters of a nonlinear regression model. Thus estimation of parameters of a nonlinear regression model is an important problem. The estimates of the parameters and their uncertainty estimates depend upon the underlying error variance structure in the model. Typically, a priori the researcher would not know if the error variances are homoscedastic (i.e., constant across dose) or if they are heteroscedastic (i.e., the variance is a function of dose). Motivated by this concern, in this paper we introduce an estimation procedure based on preliminary test which selects an appropriate estimation procedure accounting for the underlying error variance structure. Since outliers and influential observations are common in toxicological data, the proposed methodology uses M-estimators. The asymptotic properties of the preliminary test estimator are investigated; in particular its asymptotic covariance matrix is derived. The performance of the proposed estimator is compared with several standard estimators using simulation studies. The proposed methodology is also illustrated using a data set obtained from the National Toxicology Program.     

Optimal designs for response functions with a downturn

In many toxicological assays, interactions between primary and secondary effects may cause a downturn in mean responses at high doses. In this situation, the typical monotonicity assumption is invalid and may be quite misleading. Prior literature addresses the analysis of response functions with a downturn, but so far as we know, this paper initiates the study of experimental design for this situation. A growth model is combined with a death model to allow for the downturn in mean doses. Several different objective functions are studied. When the number of treatments equals the number of parameters, Fisher information is found to be independent of the model of the treatment means and on the magnitudes of the treatments. In general, A- and D_A-optimal weights for estimating adjacent mean differences are found analytically for a simple model and numerically for a biologically motivated model. Results on c-optimality are also obtained for estimating the peak dose and the EC₅₀ (the treatment with response half way between the control and the peak response on the increasing portion of the response function). Finally, when interest lies only in the increasing portion of the response function, we propose composite D-optimal designs.     

Dose-Response Analysis Using Spreadsheets

The task of fitting dose-response models to experimental data can be performed using a spreadsheet with a built-in optimization engine. This paper shows how the task of point and interval estimation can be performed using Microsoft EXCEL. A case study is presented on the carcinogenic dose-response behavior of chloroform.     

Biological Basis of Chemical Carcinogenesis: Insights from Benzene

Benzene is one of the best studied of the known human carcinogens. It causes leukemia in humans and a variety of solid tumors in rats and mice. Decades of research on benzene metabolism, pharmacokinetics, cytotoxicity, genotoxicity, and carcinogenicity in vivo and in vitro are starting to converge on a small set of overlapping hypotheses about the most probable biological mechanisms of benzene toxicity and carcinogenicity. Although there is still room for surprises, it seems likely that the ultimate answer to the mystery of how benzene exerts its multiple effects will consist of elaborations and extensions of one or more of the current hypotheses. This paper reviews benzene health effects and biology, showing how various aspects of metabolism and cytotoxicity fit together with genotoxic and nongenotoxic effects to help explain how benzene may cause cancer. Its goals are: (i) to introduce the qualitative biological background needed for detailed quantitative dose-response modeling of benzene cancer risks; and (ii) to survey a rapidly evolving area of research that shows promise of producing fundamental insights into the mechanisms of toxicity and carcinogenesis for several chemicals--benzene and perhaps phenols, catechols, and other hydroxylated ring hydrocarbons--in the decade ahead.     

Characterizing Dose-Response Relationships in Multiple Cancer Bioassays

In the evaluation of chemical compounds for carcinogenic risk, regulatory agencies such as the U.S. Environmental Protection Agency and National Toxicology Program (NTP) have traditionally fit a dose-response model to data from rodent bioassays, and then used the fitted model to estimate a Virtually Safe Dose or the dose corresponding to a very small increase (usually 10(-6)) in risk over background. Much recent interest has been directed at incorporating additional scientific information regarding the properties of the specific chemical under investigation into the risk assessment process, including biological mechanisms of cancer induction, metabolic pathways, and chemical structure and activity. Despite the fact that regulatory agencies are currently poised to allow use of nonlinear dose-response models based on the concept of an underlying threshold for nongenotoxic chemicals, there have been few attempts to investigate the overall relationship between the shape of dose-response curves and mutagenicity. Using data from an historical database of NTP cancer bioassays, the authors conducted a repeated-measures Analysis of the estimated shape from fitting extended Weibull dose-response curves. It was concluded that genotoxic chemicals have dose-response curves that are closer to linear than those for nongenotoxic chemicals, though on average, both types of compounds have dose-response curves that are convex and the effect of genotoxicity is small.     

SEGMENTED DOSE-RESPONSE MODELS FOR REPEATED MEASURES DATA

In a dose-response analysis, logit-transformed responses are modelled as a function of log-transformed doses. The linear trend is commonly observed. The comparison among treatment groups can be made based on the linear trend. An example in this paper came from a study to estimate the effect of aminophylline on dose-response curve of atracurium. Unlike the usual dose-response curve, this example has repeated measures and seems to have two slopes to which the usual dose-response model is not adequate to fit. We propose segmented regression models that allow two different slopes. The proposed model is an extension of the segmented regression model with a univariate response per subject. We illustrate the proposed model fits data better than the usual dose-response model.     

A Quantitative Microbial Risk Assessment Model for Legionnaires'' Disease: Animal Model Selection and Dose-Response Modeling

Legionnaires' disease (LD), first reported in 1976, is an atypical pneumonia caused by bacteria of the genus Legionella, and most frequently by L. pneumophila (Lp). Subsequent research on exposure to the organism employed various animal models, and with quantitative microbial risk assessment (QMRA) techniques, the animal model data may provide insights on human dose-response for LD. This article focuses on the rationale for selection of the guinea pig model, comparison of the dose-response model results, comparison of projected low-dose responses for guinea pigs, and risk estimates for humans. Based on both in vivo and in vitro comparisons, the guinea pig (Cavia porcellus) dose-response data were selected for modeling human risk. We completed dose-response modeling for the beta-Poisson (approximate and exact), exponential, probit, logistic, and Weibull models for Lp inhalation, mortality, and infection (end point elevated body temperature) in guinea pigs. For mechanistic reasons, including low-dose exposure probability, further work on human risk estimates for LD employed the exponential and beta-Poisson models. With an exposure of 10 colony-forming units (CFU) (retained dose), the QMRA model predicted a mild infection risk of 0.4 (as evaluated by seroprevalence) and a clinical severity LD case (e.g., hospitalization and supportive care) risk of 0.0009. The calculated rates based on estimated human exposures for outbreaks used for the QMRA model validation are within an order of magnitude of the reported LD rates. These validation results suggest the LD QMRA animal model selection, dose-response modeling, and extension to human risk projections were appropriate.     

Food Safety Objectives Should Integrate the Variability of the Concentration of Pathogen

The World Trade Organization introduced the concept of appropriate level of protection (ALOP) as a public health target. For this public health objective to be interpretable by the actors in the food chain, the concept of food safety objective (FSO) was proposed by the International Commission on Microbiological Specifications for Foods and adopted later by the Codex Alimentarius Food Hygiene Committee. The way to translate an ALOP into a FSO is still in debate. The purpose of this article is to develop a methodological tool to derive a FSO from an ALOP being expressed as a maximal annual marginal risk. We explore the different models relating the annual marginal risk to the parameters of the FSO depending on whether the variability in the survival probability and in the concentration of the pathogen are considered or not. If they are not, determination of the FSO is straightforward. If they are, we propose to use stochastic Monte Carlo simulation models and logistic discriminant analysis in order to determine which sets of parameters are compatible with the ALOP. The logistic discriminant function was chosen such that the kappa coefficient is maximized. We illustrate this method by the example of the risks of listeriosis and salmonellosis in one type of soft cheese. We conclude that the definition of the FSO should integrate three dimensions: the prevalence of contamination, the average concentration per contaminated typical serving, and the dispersion of the concentration among those servings.     

A Hierarchical Model for the Skew-normal Distribution with Application in Developmental Neurotoxicology

The distribution of the mean of a random sample drawn from a skew-normal population was derived by Chen et al. (2004). Here, we consider a hierarchical structure and derive the distribution of the sample mean when the location parameter itself is a random variable with a normal distribution. In neurotoxicological bioassay experiments with laboratory animals, often the response of interest is continuous in nature and the mean of responses is used for inferential purposes (Chen, 2006). However, in developmental neurotoxicity experiments where the neurological effect of a compound on the developing fetus is of interest, because of the intra-litter correlation, the mean of the response distribution may vary from one litter to another. The unconditional distribution of the litter sample mean is derived and its application in the analysis of data from developmental neurotoxicology is described. An example with real experimental data is used to provide further illustration.