Estimation and testing with overdispersed proportions using the beta-logistic regression model of Heckman and Willis

Methods are presented for modeling dose-related effects in proportion data when extra-binomial variability is a concern. Motivation is taken from experiments in developmental toxicology, where similarity among conceptuses within a litter leads to intralitter correlations and to overdispersion in the observed proportions. Appeal is made to the well-known beta-binomial distribution to represent the overdispersion. From this, an exponential function of the linear predictor is used to model the dose-response relationship. The specification was introduced previously for econometric applications by Heckman and Willis; it induces a form of logistic regression for the mean response, together with a reciprocal biexponential model for the intralitter correlation. Large-sample, likelihood-based methods for estimating and testing the joint proportion-correlation response are studied. A developmental toxicity data set illustrates the methods.     

Mechanistic Modeling of Salmonellosis

The serious limitation of the available human data contributes to the need for making simplifying assumptions for dose-response modeling which has led to frequent use of a single function, the beta-Poisson function, as a default dose-response model form. This function is a concave, low-dose linear function. Sub-linear or convex curves may be more appropriate for some host-pathogen interactions due to the series of highly regulated innate and acquired defense systems of the healthy human body that protect against most microbial challenges. A systematic investigation of the steps of non-typhoid salmonellosis in humans leads to biological motivations for sub-linear, or non-concave, dose-response curves in microbial risk assessment. Three phenomena were identified that might contribute to sub-linear, or non-concave, dose-response curves: (1) clumping of bacterial cells in microcolonies in a food matrix; (2) quorum sensing, or density-dependency in expression of virulence genes or other metabolic actions; and (3) need, at least in some circumstances, for multiple lesions for progression to symptomatic illness. This investigation suggests that microbial risk assessors should routinely employ a variety of model forms in addition to the commonly used beta-Poisson model to depict more fully the uncertainty of the true dose-response model.     

A shared response model for clustered binary data in developmental toxicity studies

Existing distributions for modeling fetal response data in developmental toxicology such as the beta-binomial distribution have a tendency of inflating the probability of no malformed fetuses, and hence understating the risk of having at least one malformed fetus within a litter. As opposed to a shared probability extra-binomial model, we advocate a shared response model that allows a random number of fetuses within the same litter to share a common response. An explicit formula is given for the probability function and graphical plots suggest that it does not suffer from the problem of assigning too much probability to the event of no malformed fetuses. The EM algorithm can be used to estimate the model parameters. Results of a simulation study show that the EM estimates are nearly unbiased and the associated confidence intervals based on the usual standard error estimates have coverage close to the nominal level. Simulation results also suggest that the shared response model estimates of the marginal malformation probabilities are robust to misspecification of the distributional form, but not so for the estimates of intralitter correlation and the litter-level probability of having at least one malformed fetus. The proposed model is fitted to a set of data from the U.S. National Toxicology Program. For the same dose-response relationship, the fit based on the shared response distribution is superior to that based on the beta-binomial, and comparable to that based on the recently proposed q-power distribution (Kuk, 2004, Applied Statistics53, 369-386). An advantage of the shared response model over the q-power distribution is that it is more interpretable and can be extended more easily to the multivariate case. To illustrate this, a bivariate shared response model is fitted to fetal response data involving visceral and skeletal malformation.     

Influence of thyroid function upon "Substance P" induced secretion of saliva by submaxillary glands

The effects of changes in thyroid function on the action of "Substance P" upon the secretion of saliva by the submaxillary glands was studied in male Wistar rats, with parasympathetic decentralization on the left side. The dose-response curves to increasing doses of "Substance P" showed in hyperthyroid animals increased salivary secretion while in hypothyroid animals the dose-response curve to the drug was decreased. Every animal showed supersentivity to "Substance P" in the decentralized gland. The influence of changes in thyroid function in the denervated glands was the same as that in the unoperated side, increased salivary secretion in hyperthyroidism and decreased in hypothyroidism.     

Fish consumption and risk of rheumatoid arthritis: a dose-response meta-analysis

Introduction

The association between fish consumption and rheumatoid arthritis (RA) is unclear. The aim of this paper was to summarize the available evidence on the association between fish consumption and risk of RA using a dose-response meta-analysis.

Methods

Relevant studies were identified by a search of MEDLINE and EMBASE through December 2013, with no restrictions. A random-effects dose-response meta-analysis was conducted to combine study specific relative risks. Potential non-linear relation was investigated using restricted cubic splines. A stratified analysis was conducted by study design.

Results

Seven studies (four case-controls and three prospective cohorts) involving a total of 174 701 participants and 3346 cases were included in the meta-analysis. For each one serving per week increment in fish consumption, the relative risk (RR) of RA was 0.96 (95% confidence interval (CI) 0.91 to 1.01). Results did not change when stratifying by study design. No heterogeneity or publication bias was observed. When fish consumption was modeled using restricted cubic splines, the risk of RA was 20 to 24% lower for 1 up to 3 servings per week of fish (RR =0.76, 95% CI: 0.57 to 1.02) as compared to never consumption.

Conclusions

Results from this dose-response meta-analysis showed a non-statistically significant inverse association between fish consumption and RA.     

More Accurate Dose-Response Estimation Using Monte-Carlo Uncertainty Analysis: The Data Cube Approach

The traditional q₁ ^* methodology for constructing upper confidence limits (UCLs) for the low-dose slopes of quantal dose-response functions has two limitations: (i) it is based on an asymptotic statistical result that has been shown via Monte Carlo simulation not to hold in practice for small, real bioassay experiments (Portier and Hoel, 1983); and (ii) it assumes that the multistage model (which represents cumulative hazard as a polynomial function of dose) is correct. This paper presents an uncertainty analysis approach for fitting dose-response functions to data that does not require specific parametric assumptions or depend on asymptotic results. It has the advantage that the resulting estimates of the dose-response function (and uncertainties about it) no longer depend on the validity of an assumed parametric family nor on the accuracy of the asymptotic approximation. The method derives posterior densities for the true response rates in the dose groups, rather than deriving posterior densities for model parameters, as in other Bayesian approaches (Sielken, 1991), or resampling the observed data points, as in the bootstrap and other resampling methods. It does so by conditioning constrained maximum-entropy priors on the observed data. Monte Carlo sampling of the posterior (constrained, conditioned) probability distributions generate values of response probabilities that might be observed if the experiment were repeated with very large sample sizes. A dose-response curve is fit to each such simulated dataset. If no parametric model has been specified, then a generalized representation (e.g., a power-series or orthonormal polynomial expansion) of the unknown dose-response function is fit to each simulated dataset using “model-free” methods. The simulation-based frequency distribution of all the dose-response curves fit to the simulated datasets yields a posterior distribution function for the low-dose slope of the dose-response curve. An upper confidence limit on the low-dose slope is obtained directly from this posterior distribution. This “Data Cube” procedure is illustrated with a real dataset for benzene, and is seen to produce more policy-relevant insights than does the traditional q₁ ^* methodology. For example, it shows how far apart are the 90%, 95%, and 99% limits and reveals how uncertainty about total and incremental risk vary with dose level (typically being dominated at low doses by uncertainty about the response of the control group, and being dominated at high doses by sampling variability). Strengths and limitations of the Data Cube approach are summarized, and potential decision-analytic applications to making better informed risk management decisions are briefly discussed.     

Uncertainties in the risk assessment of three mycotoxins: aflatoxin, ochratoxin, and zearalenone

The risk assessment of mycotoxins is made up of two major components: an exposure assessment and a hazard assessment. There are many uncertainties in both of these components. This paper will briefly discuss the various aspects of the risk assessment process as it applies to mycotoxins and will then focus mainly on some of the uncertainties in the hazard assessment component of several carcinogenic mycotoxins. To arrive at an estimated "safe dose" (end point of the hazard assessment), we have previously used two major approaches: the no observed effect level (NOEL) divided by a safety factor approach and a mathematical (robust linear) extrapolation to a "virtual safe dose." Both of these approaches use only points from the no observed effect region of the dose-response curve and ignore valuable data from the response region. It is proposed to use the dose at which 50% of the animals would have developed tumors (the TD50) divided by a large safety factor of 50,000 as an additional estimate of "safe dose". For many studies, the TD50 lies within the observed response region of the dose-response curve and may have more validity. It is also suggested in certain cases that some of the uncertainties regarding the NOEL can be reduced if one uses a statistically derived no effect level (NEL).     

Social play in the development of sibling relations in Eurasian lynx (Lynx lynx)

Social play fulfills an important function in creating and maintaining relations between siblings. However, its relationship with the intralitter social processes is poorly understood. It was noticed that, in Eurasian lynx (Lynx lynx) litters, sex differences in social play are absent in the first 2–3 months of life. It was found that the most intense periods of play behavior (at an age of 9 and 12 weeks) coincide with periods of aggression. Gradual change in play interactions, which require close physical contact by play elements with increased motor activity, are described. This reflects the changes in the relevance of certain skills of lynx cubs as they grow older.     

Alanine Aminotransferase and Risk of the Metabolic Syndrome: A Linear Dose-Response Relationship

Background

Elevated baseline circulating alanine aminotransferase (ALT) level has been demonstrated to be associated with an increased risk of the metabolic syndrome (MetS), but the nature of the dose-response relationship is uncertain.

Methods

We performed a systematic review and meta-analysis of published prospective cohort studies to characterize in detail the nature of the dose-response relationship between baseline ALT level and risk of incident MetS in the general population. Relevant studies were identified in a literature search of MEDLINE, EMBASE, and Web of Science up to December 2013. Prospective studies in which investigators reported relative risks (RRs) of MetS for 3 or more categories of ALT levels were eligible. A potential nonlinear relationship between ALT levels and MetS was examined using restricted cubic splines.

Results

Of the 489 studies reviewed, relevant data were available on 29,815 non-overlapping participants comprising 2,125 incident MetS events from five prospective cohort studies. There was evidence of a linear association (P for nonlinearity = 0.38) between ALT level and risk of MetS, characterised by a graded increase in MetS risk at ALT levels 6–40 U/L. The risk of MetS increased by 14% for every 5 U/L increment in circulating ALT level (95% CI: 12–17%). Evidence was lacking of heterogeneity and publication bias among the contributing studies.

Conclusions

Baseline ALT level is associated with risk of the MetS in a linear dose-response manner. Studies are needed to determine whether the association represents a causal relationship.     

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.     

Univariate Dose-Response Models in Case-Control Studies

For modelling dose-response relationships in case-control studies the multiplicative logistic regression model, assuming the relative risk to be an exponential function of the dose, is widely known. If the relative risk is assumed to be a linear function of the dose, several authors (see e.g. BERRY (1980)) have proposed an additive (linear) model. This model has a better fit with the data if such a linear relation holds. Confidence limits for the relative risk derived from the information matrix, however, appear to be rather inaccurate. Therefore, use of the ‘standard’ logistic model in two different ways was studied: extension with a quadratic term or a logarithmic transformation of the dose. By applying the methods both to an empirical data set and in a simulation experiment, it is shown that appropriate transformation (often logarithmic) of the dosage and then applying the ‘standard’ logistic model is an useful approach if a linear dose-response relationship holds.     

A systems-based mathematical modelling framework for investigating the effect of drugs on solid tumours

Background and Purpose

Most information on the dose-response of radiation-induced cancer is derived from data on the A-bomb survivors. Since, for radiation protection purposes, the dose span of main interest is between zero and one Gy, the analysis of the A-bomb survivors is usually focused on this range. However, estimates of cancer risk for doses larger than one Gy are becoming more important for radiotherapy patients. Therefore in this work, emphasis is placed on doses relevant for radiotherapy with respect to radiation induced solid cancer.

Materials and methods

For various organs and tissues the analysis of cancer induction was extended by an attempted combination of the linear-no-threshold model from the A-bomb survivors in the low dose range and the cancer risk data of patients receiving radiotherapy for Hodgkin's disease in the high dose range. The data were fitted using organ equivalent dose (OED) calculated for a group of different dose-response models including a linear model, a model including fractionation, a bell-shaped model and a plateau-dose-response relationship.

Results

The quality of the applied fits shows that the linear model fits best colon, cervix and skin. All other organs are best fitted by the model including fractionation indicating that the repopulation/repair ability of tissue is neither 0 nor 100% but somewhere in between. Bone and soft tissue sarcoma were fitted well by all the models. In the low dose range beyond 1 Gy sarcoma risk is negligible. For increasing dose, sarcoma risk increases rapidly and reaches a plateau at around 30 Gy.

Conclusions

In this work OED for various organs was calculated for a linear, a bell-shaped, a plateau and a mixture between a bell-shaped and plateau dose-response relationship for typical treatment plans of Hodgkin's disease patients. The model parameters (α and R) were obtained by a fit of the dose-response relationships to these OED data and to the A-bomb survivors. For any three-dimensional inhomogenous dose distribution, cancer risk can be compared by computing OED using the coefficients obtained in this work.     

Dose-response function for Painter's SCE-model

Summary The replication model for sister chromatid exchange (SCE), when introduced in 1980 by Painter, was claimed to be consistent with the one hit property of SCE. However, the argument offered in favour of the one hit property was based on a defective dose-response function, as shown in this paper, since dose as the independent parameter of any dose-response function was not included in the considerations. This missing part of the model's dose-response function is added and, by using Bessel functions, a formula for the complete dose-response function is presented. A re-examination of the newly derived function shows that, in the model, linearity holds only under certain restricted circumstances.     

Excess Body Mass Index and Risk of Liver Cancer: A Nonlinear Dose-Response Meta-Analysis of Prospective Studies

Background

Excess body weight measured as body mass index (BMI) has a positive association with risk of common cancers. However, previous meta-analyses related to BMI and liver cancer had inconsistent results. The purpose of the current study is to establish a nonlinear dose-response relationship between BMI and incidence risk of liver cancer.

Methods

A systematic literature search for relevant articles published from 1966 to November 2011 was conducted in PUBMED and EMBASE digital databases. Additional articles were manually searched by using the reference lists of identified papers. Restricted cubic splines and generalized least-squares regression methods were used to model a potential curvilinear relationship and to make a dose-response meta-analysis. Stratified analysis, sensitivity analysis and assessment of bias were performed in our meta-analysis.

Results

8 articles including 1,779,471 cohort individuals were brought into meta-analysis. A non-linear dose-response association between BMI and risk of liver cancer was visually significant (P for nonlinearity<0.001), besides, the point value of BMI also enhanced the results quantitatively, where relative risks were 1.02 (95%CI = 1.02–1.03), 1.35 (95%CI = 1.24–1.47) and 2.22-fold (95%CI = 1.74–2.83) when BMI was at the point of 25, 30 and 35 kg/m² compared with reference (the median value of the lowest category), respectively. The ethnicity of the population was found as the main source of heterogeneity. In subsequent stratified analysis, no evidence of heterogeneity was showed in Asian and White populations (P for heterogeneity>0.1), and all value of BMI still presented significantly increased risk of cancer.

Conclusions

The findings from meta-analysis provided that excess BMI had significant increased association with risk of liver cancer, although the biological mechanisms underlying the obesity-cancer link still need to be clarified.     

Comparative effects of insecticides with different mechanisms of action on Chrysoperla externa （Neuroptera＂ Chrysopidae）＂ Lethal,sublethal and dose-response effects

The comprehensive knowledge that the delayed systemic and reproduction side effects can be even more deleterious than acute toxicity, has caused a shift in focus toward sublethal effects assessment on physiology and behavior of beneficial insects. In this study, we assessed the risks posed by some insecticides with different mode of action through lethal and delayed systemic sublethal effects on the pupation, adult emergence, and repro- duction of the chrysopid Chrysoperla externa （Hagen, 1861; Neuroptera： Chrysopidae）, an important predator in pest biological control. The maximum field recommended dose （MFRD） and twice （2xMFRD） for chlorantraniliprole, tebufenozide, and pyriproxyfen were harmless to C. externa. In contrast, all the tested chitin synthesis inhibitors （CSIs） were highly detrimental to the predator, despite of their lack of acute lethal toxicity. There- fore, the safety assumed by using IGRs toward beneficial insects is not valid for chrysopids. Dose-response data showed that although all CSIs have a similar mechanism of action, the relative extent of toxicity may differ （novaluron 〉 lufenuron 〉 teflubenzuron）. For CSIs, the delayed systemic effects became obvious at adult emergence, where the predicted no observable effect dose （NOED） was 1/2 048 of the MFRD for novaluron （0.085 ng/insect）, and 1/256 of the MFRD for both lufenuron （0.25 ng/insect） and teflubenzuron （0.6 ng/insect）. Finally, this work emphasized the significance of performing toxicity risk assessments with an adequate posttreatment period to avoid underestimating the toxicities of insecticides, as the acute lethal toxicity assays may not provide accurate information regarding the long-range effects of hazardous compounds.     

Reflections on the implications of thresholds of mutagenic activity for the labelling of chemicals by the European Union

During the course of the safety evaluation and regulatory control of chemicals it is important to distinguish between "potential hazard" and "actual risk" of exposure to toxins. In the case of DNA reactive chemicals, it has been prudent to assume that hazard is expressed as risk at low exposure concentrations. However, analysis of the dose-response relationships of both DNA reactive and non-DNA reactive genotoxins (e.g., aneugens) indicate that there are exposure concentrations below which protective mechanisms such as DNA repair activity and the presence of multiple targets may lead to the prediction of no risk until threshold concentrations are achieved. Current European Union management procedures for mutagenic chemicals are based predominantly upon hazard assessment rather than assessment of actual risk under likely exposure scenarios. As our knowledge of protective mechanisms increases, the time is now appropriate to undertake a re-evaluation of European Union criteria and to base the clarification mutagenic chemical more firmly upon the basis of actual risks to the human population and to the environment.     

The quantitative patterns of the modified one-center bioreceptor model

A modified one site model of the bioreceptor have been used to estimate quantitatively the phenomenon of the full agonism. Threshold phenomenon, spare receptors and linear dependence of the biological effect on concentration of complex agonist-receptor has been determinate by the general correlation equations. The equations of one site model provides a good fit to the experimental curves "dose-response" for the full agonists and allows to calculate the value of space receptors. The model includes occupancy Clark's theory and law "all or nothing". The interaction of acetylcholine and aklyltrimethylammonium salts with muscarinic acetylholine receptors is analysed as an example of use of this equation.     

Statistical models for low dose exposure

Extrapolation of health risks from high to low doses has received a considerable amount of attention in carcinogenic risk assessment over decades. Fitting statistical dose-response models to experimental data collected at high doses and use of the fitted model for estimating effects at low doses lead to quite different risk predictions. Dissatisfaction with this procedure was formulated both by toxicologists who saw a deficit of biological knowledge in the models as well as by risk modelers who saw the need of mechanistically-based stochastic modeling. This contribution summarizes the present status of low dose modeling and the determination of the shape of dose-response curves. We will address the controversial issues of the appropriateness of threshold models, the estimation of no observed adverse effect levels (NOAEL), and their relevance for low dose modeling. We will distinguish between quantal dose-response models for tumor incidence and models of the more informative age/time dependent tumor incidence. The multistage model and the two-stage model of clonal expansion are considered as dose-response models accounting for biological mechanisms. Problems of the identifiability of mechanisms are addressed, the relation between administered dose and effective target dose is illustrated by examples, and the recently proposed Benchmark Dose concept for risk assessment is presented with its consequences for mechanistic modeling and statistical estimation.     

Optimal Designs for Estimating the Percentiles of the Risk in Multistage Models of Carcinogenesis

The multistage carcinogenesis models describe a process by which a normal cell becomes malignant and gives rise to a tumor. This paper aims at evaluating the percentiles of the risk function derived as dose-response relationship in a multi-stage model. These percentiles have been known as “virtual safe dose” levels or risk specific dose levels. The optimal design theory is applied to estimate the appropriate percentile and the sequential approach of design is adopted through a stochastic approximation scheme. If the initial design is D-optimal the limit design is D-optimal as well and it is the one with the minimum entropy.     

The risk of chronic myeloid leukemia: can the dose-response curve be U-shaped?

Chronic myeloid leukemia (CML) is caused by a BCR-ABL chromosome translocation in a primitive hematopoietic stem cell. The number of hematopoietic stem cells in the body is thus a major factor in CML risk. Evidence suggests that the number of hematopoietic stem cells in the body is only loosely regulated, having a broad "dead-band" of physiologically acceptable values. The existence of a dead-band is important, because it would imply that low levels of hematopoietic stem cell killing can be permanent; i.e., it would imply that low doses of ionizing radiation can cause permanent reductions in the total number of CML target cells and thus permanent reductions in the subsequent risk of spontaneous CML. Such reductions in risk could be substantial if hematopoietic stem cells are also hypersensitive to radiation killing at low dose. Our calculations indicate that, due to dead-band hematopoietic stem cell control, if hematopoietic stem cells are as hypersensitive to killing at low doses as epithelial cells, reductions in the spontaneous CML risk could exceed the low-dose risks of induced CML; i.e., the net lifetime CML risk could have a U-shaped dose-response curve.