Jens C. Streibig

Utilizing R Software Package for Dose-Response Studies: The Concept and Data Analysis

Advances in statistical software allow statistical methods for nonlinear regression analysis of dose-response curves to be carried out conveniently by non-statisticians. One such statistical software is the program R with the drc extension package. The drc package can: (1) simultaneously fit multiple dose-response curves; (2) compare curve parameters for significant differences; (3) calculate any point along the curve at the response level of interest, commonly known as an effective dose (e.g., ED30, ED50, ED90), and determine its significance; and (4) generate graphs for publications or presentations. We believe that the drc package has advantages that include: the ability to relatively simply and quickly compare multiple curves and select ED-levels easily along the curve with relevant statistics; the package is free of charge and does not require licensing fees, and the size of the package is only 70 MB. Therefore, our objectives are to: (1) provide a review of a few common issues in dose-response-curve fitting, and (2) facilitate the use of up-to-date statistical techniques for analysis of dose-response curves with this software. The methods described can be utilized to evaluate chemical and non-chemical weed control options. Benefits to the practitioners and academics are also presented.

Dose-Response Analysis Using R

Dose-response analysis can be carried out using multi-purpose commercial statistical software, but except for a few special cases the analysis easily becomes cumbersome as relevant, non-standard output requires manual programming. The extension package drc for the statistical environment R provides a flexible and versatile infrastructure for dose-response analyses in general. The present version of the package, reflecting extensions and modifications over the last decade, provides a user-friendly interface to specify the model assumptions about the dose-response relationship and comes with a number of extractors for summarizing fitted models and carrying out inference on derived parameters. The aim of the present paper is to provide an overview of state-of-the-art dose-response analysis, both in terms of general concepts that have evolved and matured over the years and by means of concrete examples.

A review of independent action compared to concentration addition as reference models for mixtures of compounds with different molecular target sites

From a theoretical point of view, it has often been argued that the model of independent action (IA) is the most correct reference model to use for predicting the joint effect of mixtures of chemicals with different molecular target sites. The theory of IA, however, relies on a number of assumptions that are rarely fulfilled in practice. It has even been argued that, theoretically, the concentration addition (CA) model could be just as correct. In the present study, we tested the accuracy of both IA and CA in describing binary dose-response surfaces of chemicals with different molecular targets using statistical software. We compared the two models to determine which best describes data for 158 data sets. The data sets represented 98 different mixtures of, primarily, pesticides and pharmaceuticals tested on one or several of seven test systems containing one of the following: Vibrio fischeri, activated sludge microorganisms, Daphnia magna, Pseudokirchneriella subcapitata, Lemna minor, Tripleurospermum inodorum, or Stellaria media. The analyses showed that approximately 20% of the mixtures were adequately predicted only by IA, 10% were adequately predicted only by CA, and both models could predict the outcome of another 20% of the experiment. Half of the experiments could not be correctly described with either of the two models. When quantifying the maximal difference between modeled synergy or antagonism and the reference model predictions at a 50% effect concentration, neither of the models proved significantly better than the other. Thus, neither model can be selected over the other on the basis of accuracy alone.

Improved empirical models describing hormesis

During the past two decades, the phenomenon of hormesis has gained increased recognition. To promote research in hormesis, a sound statistical quantification of important parameters, such as the level and significance of the increase in response and the range of concentration where it occurs, is strongly needed. Here, we present an improved statistical model to describe hormetic dose-response curves and test for the presence of hormesis. Using the delta method and freely available software, any percentage effect dose or concentration can be derived with its associated standard errors. Likewise, the maximal response can be extracted and the growth stimulation calculated. The new model was tested on macrophyte data from multiple-species experiments and on laboratory data of Lemna minor. For the 51 curves tested, significant hormesis was detected in 18 curves, and for another 17 curves, the hormesis model described that data better than the logistic model did. The increase in response ranged from 5 to 109%. The growth stimulation occurred at an average dose somewhere between zero and concentrations corresponding to approximately 20 to 25% of the median effective concentration (EC50). Testing the same data with the hormesis model proposed by Brain and Cousens in 1989, we found no significant hormesis. Consequently, the new model is shown to be far more robust than previous models, both in terms of variation in data and in terms of describing hormetic effects ranging from small effects of a 10% increase in response up to effects of an almost 100% increase in response.

The Occurrence of Hormesis in Plants and Algae

This paper evaluated the frequency, magnitude and dose/concentration range of hormesis in four species: The aquatic plant Lemna minor, the micro-alga Pseudokirchneriella subcapitata and the two terrestrial plants Tripleurospermum inodorum and Stellaria media exposed to nine herbicides and one fungicide and binary mixtures thereof. In total 687 dose-response curves were included in the database. The study showed that both the frequency and the magnitude of the hormetic response depended on the endpoint being measured. Dry weight at harvest showed a higher frequency and a larger hormetic response compared to relative growth rates. Evaluating hormesis for relative growth rates for all species showed that 25% to 76% of the curves for each species had treatments above 105% of the control. Fitting the data with a dose-response model including a parameter for hormesis showed that the average growth increase ranged from 9±1% to 16±16% of the control growth rate, while if measured on a dry weight basis the response increase was 38±13% and 43±23% for the two terrestrial species. Hormesis was found in >70% of the curves with the herbicides glyphosate and metsulfuron-methyl, and in >50% of the curves for acifluorfen and terbuthylazine. The concentration ranges of the hormetic part of the dose-response curves corresponded well with literature values.

Review: Confirmation of Resistance to Herbicides and Evaluation of Resistance Levels

Abstract As cases of resistance to herbicides escalate worldwide, there is increasing demand from growers to test for weed resistance and learn how to manage it. Scientists have developed resistance-testing protocols for numerous herbicides and weed species. Growers need immediate answers and scientists are faced with the daunting task of testing an increasingly large number of samples across a variety of species and herbicides. Quick tests have been, and continue to be, developed to address this need, although classical tests are still the norm. Newer methods involve molecular techniques. Whereas the classical whole-plant assay tests for resistance regardless of the mechanism, many quick tests are limited by specificity to an herbicide, mode of action, or mechanism of resistance. Advancing knowledge in weed biology and genomics allows for refinements in sampling and testing protocols. Thus, approaches in resistance testing continue to diversify, which can confound the less experienced. We aim to help weed science practitioners resolve questions pertaining to the testing of herbicide resistance, starting with field surveys and sampling methods, herbicide screening methods, data analysis, and, finally, interpretation. More specifically, this article discusses approaches for sampling plants for resistance confirmation assays, provides brief overviews on the biological and statistical basis for designing and analyzing dose–response tests, and discusses alternative procedures for rapid resistance confirmation, including molecular-based assays. Resistance confirmation procedures often need to be slightly modified to suit a specific situation; thus, the general requirements as well as pros and cons of quick assays and DNA-based assays are contrasted. Ultimately, weed resistance testing research, as well as resistance management decisions arising from research, needs to be practical, feasible, and grounded in science-based methods.

Random-Effect Models in Nonlinear Regression with Applications to Bioassay

SUMMARY Transformation and weighting techniques are applied to dose-response curve models. In particular, weighting methods derived from a controlled-variable, random-effect model and a closely related random-coefficient model are studied. These two models correspond to additive and multiplicative effects of variations in the dose, and both lead to variance components proportional to the square of the derivative of the response function with respect to dose. When the dose-response curve is nonlinear in dose, the variance components are typically identifiable even without replicate measurements of dose. In a bioassay example the fit of a logistic model is studied. The transform-both-sides technique with a power transformation is shown to give a vast improvement in fit, compared to the analysis with no transformation and no weighting, and it also gives considerably better estimates of the parameters in the logistic function. For the data set studied, a significant further improvement in the fit is possible by use of the random-effect models.

Reproducibility of binary‐mixture toxicity studies

Binary-mixture studies often are conducted with the aim of elucidating the effect of one specific chemical on the biological action of another. The results can be interpreted in relation to reference models by the use of response-surface analyses and isobolograms. The amount of data needed for these analyses is, however, extensive, and the experiments therefore rarely are repeated. In the present study, we investigate the reproducibility of isobole shapes of binary-mixture toxicity experiments in terms of deviation from the reference model of concentration addition (CA), dose-level dependence, and isobole asymmetry. We use data from four herbicide mixtures tested in three to five independent experiments on the aquatic test plant Lemna minor and the terrestrial plant Tripleurospermum inodorum. The results showed that the variation both within and among experiments was approximately half the size for the aquatic test system compared to the terrestrial system. As a consequence, a consistent deviation from CA could be obtained in three of four herbicide mixtures for L. minor, whereas this was only the case for one or two of the herbicide mixtures tested on T. inodorum. For one mixture on T. inodorum, both CA synergism and antagonism were detected. Dose-dependent effects could not be repeated consistently, just as the asymmetry found in some isoboles could not. The study emphasizes the importance of repeating mixture toxicity experiments, especially for test systems with large variability, and using caution when drawing biological conclusions from the test results.

Can the choice of endpoint lead to contradictory results of mixture‐toxicity experiments?

Theoretically, the effect of two independently acting compounds in a mixture will depend on the slope of the dose-response curves of the individual compounds if evaluated in relation to the model of concentration addition (CA). In the present study, we explored development of the shape of the dose-response relationship for four different recommended endpoints (surface area, frond number, fresh weight-specific, and dry weight-specific relative growth rates [RGRA, RGRF, RGRFW, and RGRDW, respectively]) and for two differently acting herbicides (metsulfuron-methyl and terbuthylazine) over time (3-15 d) on the standard test plant Lemna minor to identify endpoints and experiment times for which predictions of independent action (IA) would depart the most from those of CA. After a test time of 6 d, predictions of IA based on RGRA and RGRFW showed antagonism in relation to CA. Based on RGRDW, synergy was predicted, whereas IA based on RGRF was indistinguishable from CA. To test the prediction of choice of endpoint giving different results in mixture-toxicity experiments, three endpoints and six combinations of independently acting herbicides were evaluated using isobolograms. The experiments showed that in four of six herbicide combinations, different conclusions were reached depending on endpoint. The contradictory isoboles did not follow the theory of IA and, therefore, are more likely to be related to differences in susceptibility of the physiological processes affecting each endpoint than to the shape of the dose-response curve.

An isobole-based statistical model and test for synergism/antagonism in binary mixture toxicity experiments

Synergism and antagonism are often defined in relation to the model of Concentration Addition (CA). Hence, it is vital for the conclusion of mixture toxicity studies to be able to test whether an observed deviation from CA reflects a true deviation or whether it is simply due to random variation. In this paper we consider a non-linear regression model for the classical ray designs for binary mixture experiments. The model combines dose–response curves for each mixture in the experiment with an isobole model, describing possible deviations from CA. The method allows us to test whether the chosen isobole model is reasonable for the data and to test the hypothesis of CA. Furthermore, it provides us with a measure of the degree of synergism/antagonism. The method is flexible since both the dose–response relationships and the isobole model can be chosen arbitrarily. We demonstrate the use of the method on datasets where combinations of pesticides are tested on a floating plant, Lemna minor, and an algae, Pseudokirchneriella subcapitata. Furthermore, we conduct a simulation study in order to explore the power with which a specific deviation from CA can be distinguished in different test-systems.