A Bayesian approach for determining the no effect concentration and hazardous concentration in ecotoxicology
Introduction
The species sensitivity distribution (SSD) is a cornerstone of modern ecotoxicology and provides a basis for establishing guidelines, trigger values, and limits on concentrations of hazardous chemicals in animals and the receiving environment. In the context of water quality, use of the SSD is underpinned by the well-validated belief that, at the community/assemblage level, aquatic species generally have different although predictable responses to increasing concentrations of physical–chemical toxicants. The familiar dose–response curves generated by laboratory experiments are used to estimate a variety of measures such as the LC50 (the concentration which is lethal to 50% of some defined population), the NEC (the maximum concentration which causes no adverse effect in a target organism), and ECx (the concentration which affects x% of organisms in a dose–response experiment). Classical tools of statistical inference such as t-tests, ANOVA, and multiple comparison techniques are also widely used to estimate related statistical measures such as the no observed effect concentration (NOEC) and the lowest observed effect concentration (LOEC). One of the difficulties with conventional practice is that many of these statistics are being used interchangeably which, as argued by Fox (2009) not only creates problems of interpretation but obfuscates what is really being protected. A number of authors have denounced the ad-hoc procedures for setting safe environmental concentrations that are based on NOECs and LOECs and have argued for a more rigorous, model-based approach (Fox, 2009; Jager et al., 2006; Kooijman, 2006).
A vast literature has accumulated over the last 20 years in which the theoretical, computational, statistical, and socio-economic aspects associated with the identification of ‘safe’ concentrations have been discussed. Readers requiring more detailed background information on the use of SSDs and their application in ecotoxicology and within a regulatory framework will find the collection of papers in Posthuma et al. (2002) a useful starting point. A good review of the statistical issues associated with ecotoxicological risk assessment is provided by Van der Hoeven (2004) while more recently Fox, 2006, Fox, 2008 reviewed the use of statistical methods in ecological risk assessment more generally.
The general problem addressed by this paper is as follows: how does one set a realistic threshold concentration on some contaminant or toxicant such that some arbitrary high fraction of all species will be protected provided environmental concentrations do not exceed the threshold? This is indeed not a new problem and has been studied extensively by many researchers (Wagner and Løkke, 1991; Kooijman et al., 1996; Aldenberg and Jaworska, 2000; Shao, 2000). Despite a plethora of models and incremental refinements, the numerous concerns with the NOEC-based procedures (Fox, 1999, Fox, 2006; Newman et al., 2000; Isnard et al., 2001; Pires et al., 2002; Verdonck et al., 2003) that underpin current practice in Australia, New Zealand, the United States, the Netherlands, and Denmark have not been extinguished and as recently noted by Newman (2008), the current ecotoxicological landscape is dominated by classical (i.e. frequentist) statistical methods.
Although a number of Bayesian papers have recently appeared in the ecotoxicological literature (Aldenberg and Jaworska, 2000; Verdonck et al., 2001; Grist et al., 2006; Billoir et al., 2008; Hickey et al., 2008) the framework by and large remains outside the realm of conventional ecotoxicological practice. Readers wanting to learn more about Bayesian statistics should consult any of a number of good introductory texts such as Lee (2004) or McCarthy (2007). The statistical profession spent many years and devoted many journal pages to the debate over the legitimacy of the Bayesian paradigm. Early objectors strenuously refuted the notion of a ‘prior’ distribution, the incorporation of subjective assessment, and treatment of parameters as random quantities. Thankfully the old divisions between the ‘Frequentist’ and ‘Bayesian’ schools of thought have largely given way to a more pragmatic approach that accommodates multiple modes of statistical inference with the choice increasingly based on the notion of ‘fit-for-purpose’ rather than ideological or pedagogical constructs. Furthermore, the advent of high-powered desk-top computers and associated software such as WinBUGS (Lunn et al., 2000) has removed any lingering impediments to the Bayesian analysis of complex, real-world problems.
In light of these developments it is timely to revisit the role and place of Bayesian statistics in the context of determining hazardous concentrations for ecosystem protection. Some of the more serious limitations associated with conventional NOEC-based analyses center on the following: (i) the unknown (and perhaps unknowable) underlying distributional form for NOECs; (ii) the statistical method by which a NOEC is determined; (iii) the inability to represent uncertainty in the estimated NOEC; and (iv) the non-random selection of a small number of species (van der Hoeven, 1997; Crane and Newman, 2000). As will be demonstrated later in this paper, these issues are addressed through the use of posterior distributions to represent and describe uncertainty in the estimated no effect concentration (NEC). Uncertainty in the collection of NECs can be incorporated into a final estimate of a hazardous concentration to which a statement of ‘confidence’ (or the Bayesian analog credibility) can be attached.
Our starting point is a flexible and realistic model for the raw data generated by a dose–response experiment – this is consistent with the recommendations of Kooijman et al. (1996), Van der Hoeven (1997) and Van der Hoeven (2004). In the remainder of the paper we describe the procedure and illustrate its implementation with the use of previously published data sets.
Section snippets
A Bayesian model for the NEC
A number of models have been proposed to describe the dose–response relationship in ecotoxicological studies. We have adopted the model used by Pires et al. (2002) which relates the response (Y) to concentration (x) such that Y is constant from x=0 up to a threshold, γ and thereafter exhibits an exponential decay. It is important to note that the incorporation of γ in our model does not presuppose the existence of a threshold – it simply allows for one to be estimated if that is what the data
Example – estimation of a NEC for Daphnia magna
Biesinger et al. (1982) reported on a study into the chronic toxicity of mercury (Hg) to D. magna. The compounds of mercury tested were mercuric chloride (HgCl2), methyl mercuric chloride (MMC), and phenyl mercuric acetate (PMA). A range of Hg concentrations was prepared and for each toxicant the number, yi out of an initial sample of ni individuals surviving after 21 days was recorded against the ith concentration. A summary of the data is shown in Table 1.
The response variable here is
Example – estimation of the HCx for pond water at a uranium mine
The Ranger mine located 230 km east of Darwin, Australia is one of the world's largest open-pit uranium mines. The mine site is located within the environmentally sensitive Kakadu National Park. On March 9, 2007 tropical cyclone ‘George’ impacted the Pilbara mining region in the far north-west of Western Australia. The Ranger mine was also affected by this system with nearly 850 mm of rain falling in the 7 days to March 4, including 750 mm in one 72-h period causing flooding of the mine and the
Estimation of the no effect concentration
The response variables (Yi) for each of the four species used by Hogan et al. (2008) are assumed to be normally distributed as with μI given by Eq. (2) and non-informative gamma priors for the precision terms The prior distributions assigned to the model terms in Eq. (2) are listed in Table 3. We have chosen informative priors that reflect an ‘educated’ guess (informed by an inspection of a plot of the data) as to the likely position of the NEC. In essence, this corresponds
Estimation of the HC1
In Australia, the recommended procedure for estimating the HCx is embedded in the software tool known as BurrliOz which is distributed with the national water quality guidelines document (ANZECC/ARMCANZ, 2000). The procedure is a generalization of the method described by Aldenberg and Slob (1993). Using the BurrliOz software with the data in Table 5 we examined the differences in the estimated HC1. Following Hogan et al. (2008) we used a concentration of 50 (%) for the fifth species (Mogurnda)
Discussion
Over the past decade deterministic methods of risk assessment such as the hazard quotient (HQ) approach have given way to probabilistic methods. Thus for example, in Australia and elsewhere, the use of species sensitivity distributions (SSDs) is the preferred method for establishing concentration thresholds (or ‘trigger values’) for chemical contaminants in water bodies (ANZECC/ARMCANZ, 2000). While probabilistic ecological risk assessments (PERAs) are underpinned by a more comprehensive
Conclusions
In this paper we have described a general Bayesian framework for identifying critical threshold concentrations for ecosystem protection. We have developed the models and provided sufficient mathematical and computational detail with the use of realistic examples to help facilitate the integration of Bayesian methods into the environmental toxicologists’ toolkit of statistical techniques. We believe that our approach for estimating a NEC is superior to current NOEC-based methods by virtue of the
Acknowledgments
The author is most grateful to the following individuals for their encouragement and helpful comments on an earlier draft of this paper: Graeme Batley (CSIRO, Land and Water), Chris Humphrey, Rick van Dam, Alicia Hogan, and Andrew Harford (Environmental Research Institute of the Supervising Scientist, Supervising Scientist Division, Department of the Environment, Water, Heritage and Arts). The feedback and constructive comments from three anonymous reviewers significantly improved the final
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