Threshold Models for Combination Data from Reproductive and Developmental Experiments

Logistic Regression Models with Unspecified Low Dose-Response Relationships and Experimental Designs for Hormesis Studies

Hormesis refers to a nonmonotonic (biphasic) dose-response relationship in toxicology, environmental science, and related fields. In the presence of hormesis, a low dose of a toxic agent may have a lower risk than the risk at the control dose, and the risk may increase at high doses. When the sample size is small due to practical, logistic, and ethical considerations, a parametric model may provide an efficient approach to hypothesis testing at the cost of adopting a strong assumption, which is not guaranteed to be true. In this article, we first consider alternative parameterizations based on the traditional three-parameter logistic regression. The new parameterizations attempt to provide robustness to model misspecification by allowing an unspecified dose-response relationship between the control dose and the first nonzero experimental dose. We then consider experimental designs including the uniform design (the same sample size per dose group) and the c -optimal design (minimizing the standard error of an estimator for a parameter of interest). Our simulation studies showed that (1) the c -optimal design under the traditional three-parameter logistic regression does not help reducing an inflated Type I error rate due to model misspecification, (2) it is helpful under the new parameterization with three parameters (Type I error rate is close to a fixed significance level), and (3) the new parameterization with four parameters and the c -optimal design does not reduce statistical power much while preserving the Type I error rate at a fixed significance level.

Incorporating regulatory guideline values in analysis of epidemiology data

Fundamental to regulatory guidelines is to identify chemicals that are implicated with adverse human health effects and inform public health risk assessors about "acceptable ranges" of such environmental exposures (e.g., from consumer products and pesticides). The process is made more difficult when accounting for complex human exposures to multiple environmental chemicals. Herein we propose a new class of nonlinear statistical models for human data that incorporate and evaluate regulatory guideline values into analyses of health effects of exposure to chemical mixtures using so-called 'desirability functions' (DFs). The DFs are incorporated into nonlinear regression models to allow for the simultaneous estimation of points of departure for risk assessment of combinations of individual substances that are parts of chemical mixtures detected in humans. These are, in contrast to published so-called biomonitoring equivalent (BE) values and human biomonitoring (HBM) values that link regulatory guideline values from in vivo studies of single chemicals to internal concentrations monitored in humans. We illustrate the strategy through the analysis of prenatal concentrations of mixtures of 11 chemicals with suspected endocrine disrupting properties and two health effects: birth weight and language delay at 2.5 years. The strategy allows for the creation of a Mixture Desirability Function i.e., MDF, which is a uni-dimensional construct of the set of single chemical DFs; thus, it focuses the resulting inference to a single dimension for a more powerful one degree-of-freedom test of significance. Based on the application of this new method we conclude that the guideline values need to be lower than those for single chemicals when the chemicals are observed in combination to achieve a similar level of protection as was aimed for the individual chemicals. The proposed modeling may thus suggest data-driven uncertainty factors for single chemical risk assessment that takes environmental mixtures into account.

Detecting Departure From Additivity Along a Fixed-Ratio Mixture Ray With a Piecewise Model for Dose and Interaction Thresholds

For mixtures of many chemicals, a ray design based on a relevant, fixed mixing ratio is useful for detecting departure from additivity. Methods for detecting departure involve modeling the response as a function of total dose along the ray. For mixtures with many components, the interaction may be dose dependent. Therefore, we have developed the use of a three-segment model containing both a dose threshold and an interaction threshold. Prior to the dose threshold, the response is that of background; between the dose threshold and the interaction threshold, an additive relationship exists; the model allows for departure from additivity beyond the interaction threshold. With such a model, we can conduct a hypothesis test of additivity, as well as a test for a region of additivity. The methods are illustrated with cytotoxicity data that arise when Chinese hamster ovary cells are exposed to a mixture of nine haloacetic acids.

Model-based experimental design for assessing effects of mixtures of chemicals

We exposed flour beetles (Tribolium castaneum) to a mixture of four poly aromatic hydrocarbons (PAHs). The experimental setup was chosen such that the emphasis was on assessing partial effects. We interpreted the effects of the mixture by a process-based model, with a threshold concentration for effects on survival. The behavior of the threshold concentration was one of the key features of this research. We showed that the threshold concentration is shared by toxicants with the same mode of action, which gives a mechanistic explanation for the observation that toxic effects in mixtures may occur in concentration ranges where the individual components do not show effects. Our approach gives reliable predictions of partial effects on survival and allows for a reduction of experimental effort in assessing effects of mixtures, extrapolations to other mixtures, other points in time, or in a wider perspective to other organisms.

Summary of dose-response modeling for developmental toxicity studies

Developmental toxicity studies are an important area in the field of toxicology. Endpoints measured on fetuses include weight and indicators of death and malformation. Binary indicator measures are typically summed over the litter and a discrete distribution is assumed to model the number of adversely affected fetuses. Additionally, there is noticeable variation in the litter responses within dose groups that should be taken into account when modeling. Finally, the dose-response pattern in these studies exhibits a threshold effect. The threshold dose-response model is the default model for non-carcinogenic risk assessment, according to the USEPA, and is encouraged by the agency for the use in the risk assessment process. Two statistical models are proposed to estimate dose-response pattern of data from the developmental toxicity study: the threshold model and the spline model. The models were applied to two data sets. The advantages and disadvantages of these models, potential other models, and future research possibilities will be summarized.

Statistical power considerations show the endocrine disruptor low-dose issue in a new light

BACKGROUND

The endocrine disruptor field has been vexed by difficulties in reproducing various claims of effects at unusually low doses. In previous analyses, variations in control responses from experiment to experiment and problems with observing effects in positive controls have been identified as possible explanations of the resulting impasse.

OBJECTIVE

In this article, we argue that both of these viewpoints fail to take sufficient account of the problems that exist in estimating low effects and low-effect doses. We have carried out post hoc power analyses on selected published data to illustrate that claims of low-dose effects (or their absence) are often compromised by insufficient statistical power of the chosen experimental design.

CONCLUSIONS

We demonstrate that low-dose estimates such as the no observed adverse effect levels derived from statistical hypothesis-testing procedures are dependent on the specific experimental conditions used for testing. Thus, below the statistical detection limit of the experiment, the presence of effects can neither be proven nor ruled out. Common practice is to attempt to establish "doses without effect." However, low-dose estimations in the endocrine-disruptor field could be improved if decisions regarding the toxicologic effect size of relevance formed the starting point of testing procedures. Statistical power considerations could then reveal the resources necessary to demonstrate effect magnitudes of concern.

A Regression Spline Model for Developmental Toxicity Data

Observed dose-response patterns of data from several developmental toxicity experiments appear to be nonlinear and should be characterized by an appropriate model to adequately fit this observed pattern. Information from these animal studies of ambient substances that are noncarcinogenic, yet potentially toxic, to humans is used by federal protection agencies (Environmental Protection Agency, Occupational Safety and Health Administration, Food and Drug Administration) to determine safe exposure levels, such as no observed adverse effects level and benchmark dose. We have developed a flexible regression linear B-spline model for application to developmental toxicity dose-response data from animal studies of these noncarcinogens. We apply our model to data from two CD-1 mice studies of the National Toxicology Program; the observed dose-response pattern from both appears nonlinear: (1) experiment of 131 pregnant mice allocated over five exposure levels (0, 0.025, 0.05, 0.10, and 0.15% diet) of diethylhexyl phthalate and (2) experiment of 111 pregnant mice exposed to five levels (0, 62.5, 125, 250, and 500 mg/kg/day) of diethylene glycol dimethyl ether. In each study, we measure litter response as the proportion of adversely affected fetuses. Upon applying our B-spline model to the data from both studies, we predict nonlinear dose-response, with improvement over the more typical logistic dose-response model in each of the two studies.

D-Optimal Experimental Designs to Test for Departure from Additivity in a Fixed-Ratio Mixture Ray

Traditional factorial designs for evaluating interactions among chemicals in a mixture may be prohibitive when the number of chemicals is large. Using a mixture of chemicals with a fixed ratio (mixture ray) results in an economical design that allows estimation of additivity or nonadditive interaction for a mixture of interest. This methodology is extended easily to a mixture with a large number of chemicals. Optimal experimental conditions can be chosen that result in increased power to detect departures from additivity. Although these designs are used widely for linear models, optimal designs for nonlinear threshold models are less well known. In the present work, the use of D-optimal designs is demonstrated for nonlinear threshold models applied to a fixed-ratio mixture ray. For a fixed sample size, this design criterion selects the experimental doses and number of subjects per dose level that result in minimum variance of the model parameters and thus increased power to detect departures from additivity. An optimal design is illustrated for a 2:1 ratio (chlorpyrifos:carbaryl) mixture experiment. For this example, and in general, the optimal designs for the nonlinear threshold model depend on prior specification of the slope and dose threshold parameters. Use of a D-optimal criterion produces experimental designs with increased power, whereas standard nonoptimal designs with equally spaced dose groups may result in low power if the active range or threshold is missed.

Testing Threshold and Hormesis in a Random Effects Dose-Response Model Applied to Developmental Toxicity Data

Here we describe a random effects threshold dose-response model for clustered binary-response data from developmental toxicity studies. For our model we assume that a hormetic effect occurs in addition to a threshold effect. Therefore, the dose-response curve is based on two components: relationships below the threshold (hormetic u-shaped model) and those above the threshold (logistic model). In the absence of hormesis and threshold effects, the estimation procedure is straightforward. We introduce score tests that are derived from a random effects hormetic-threshold dose-response model. The model and tests are applied to clustered binary data from developmental toxicity studies of animals to test for hormesis and threshold effects. We also compare the score test and likelihood ratio test to test for hormesis and threshold effects in a simulated study.

Analysis of an interaction threshold in a mixture of drugs and/or chemicals

Increasingly, humans are exposed to drug/chemical mixtures. These exposures can result from therapeutic interventions or environmental sources. Of interest is the interaction that may occur among the components of these mixtures. Since interaction can be dose-dependent, it is important to determine exposure levels to either exploit the benefits of the interaction in a therapeutic application or to avoid the effect of the interaction in the case of an environmental risk assessment. We propose generalized linear models that permit the estimation of interaction threshold boundaries. The methods developed are applied to the combination of ethanol and chloral hydrate.

A new threshold dose-response model including random effects for data from developmental toxicity studies

Usually, in teratological dose finding studies, there are not only threshold effects but also extra variations that cannot be accounted for by the beta-binomial model alone. The beta-binomial model assumes correlation between fetuses in the same litter. The general random effect threshold (RE) model allows the additional variability that arises due to correlation and between litter variability to be modeled, in combination with threshold in the model. The goal of this research was to investigate a threshold dose-response model with random effects (RE) to model the variability that exists between litters of animals in studies of toxic agents. Data from a developmental toxicity study of a toxic agent were analysed, using the proposed RE threshold dose-response model, which is an extension of logit in form. Also, an approximate likelihood function was used to derive parameter estimates from this model, and tests were performed to determine the significance of the model parameters, in particular, the RE parameter. A simulation study was conducted to assess the performance of the RE threshold model in estimating the model parameters.

Estimating regression models with unknown break-points

This paper deals with fitting piecewise terms in regression models where one or more break-points are true parameters of the model. For estimation, a simple linearization technique is called for, taking advantage of the linear formulation of the problem. As a result, the method is suitable for any regression model with linear predictor and so current software can be used; threshold modelling as function of explanatory variables is also allowed. Differences between the other procedures available are shown and relative merits discussed. Simulations and two examples are presented to illustrate the method.