Uncertainty is pervasive in risk assessment. In ecotoxicological risk assessments, it arises from such sources as a lack of data, the simplification and abstraction of complex situations, and ambiguities in assessment endpoints (Burgman 2005; Suter 1993). When evaluating and managing risks, uncertainty needs to be explicitly considered in order to avoid erroneous decisions and to be able to make statements about the confidence that we can place in risk estimates. Although informative, previous approaches to dealing with uncertainty in ecotoxicological modelling have been found to be limited, inconsistent and often based on assumptions that may be false (Ferson & Ginzburg 1996; Suter 1998; Suter et al. 2002; van der Hoeven 2004; van Straalen 2002a; Verdonck et al. 2003a). In this thesis a Generalised Linear Modelling approach is proposed as an alternative, congruous framework for the analysis and prediction of a wide range of ecotoxicological effects. This approach was used to investigate the results of toxicity experiments on the effect of 2,4-Dichlorophenoxyacetic Acid (2,4-D) formulations and 4-Chlorophenol (4-CP, an associated breakdown product) on Daphnia carinata. Differences between frequentist Maximum Likelihood (ML) and Bayesian Markov-Chain Monte-Carlo (MCMC) approaches to statistical reasoning and model estimation were also investigated. These approaches are inferentially disparate and place different emphasis on aleatory and epistemic uncertainty (O'Hagan 2004). Bayesian MCMC and Probability Bounds Analysis methods for propagating uncertainty in risk models are also compared for the first time. For simple models, Bayesian and frequentist approaches to Generalised Linear Model (GLM) estimation were found to produce very similar results when non-informative prior distributions were used for the Bayesian models. Potency estimates and regression parameters were found to be similar for identical models, signifying that Bayesian MCMC techniques are at least a suitable and objective replacement for frequentist ML for the analysis of exposureresponse data. Applications of these techniques demonstrated that Amicide formulations of 2,4-D are more toxic to Daphnia than their unformulated, Technical Acid parent. Different results were obtained from Bayesian MCMC and ML methods when more complex models and data structures were considered. In the analysis of 4-CP toxicity, the treatment of 2 different factors as fixed or random in standard and Mixed-Effect models was found to affect variance estimates to the degree that different conclusions would be drawn from the same model, fit to the same data. Associated discrepancies in the treatment of overdispersion between ML and Bayesian MCMC analyses were also found to affect results. Bayesian MCMC techniques were found to be superior to the ML ones employed for the analysis of complex models because they enabled the correct formulation of hierarchical (nested) datastructures within a binomial logistic GLM. Application of these techniques to the analysis of results from 4-CP toxicity testing on two strains of Daphnia carinata found that between-experiment variability was greater than that within-experiments or between-strains. Perhaps surprisingly, this indicated that long-term laboratory culture had not significantly affected the sensitivity of one strain when compared to cultures of another strain that had recently been established from field populations. The results from this analysis highlighted the need for repetition of experiments, proper model formulation in complex analyses and careful consideration of the effects of pooling data on characterising variability and uncertainty. The GLM framework was used to develop three dimensional surface models of the effects of different length pulse exposures, and subsequent delayed toxicity, of 4-CP on Daphnia. These models described the relationship between exposure duration and intensity (concentration) on toxicity, and were constructed for both pulse and delayed effects. Statistical analysis of these models found that significant delayed effects occurred following the full range of pulse exposure durations, and that both exposure duration and intensity interacted significantly and concurrently with the delayed effect. These results indicated that failure to consider delayed toxicity could lead to significant underestimation of the effects of pulse exposure, and therefore increase uncertainty in risk assessments. A number of new approaches to modelling ecotoxicological risk and to propagating uncertainty were also developed and applied in this thesis. In the first of these, a method for describing and propagating uncertainty in conventional Species Sensitivity Distribution (SSD) models was described. This utilised Probability Bounds Analysis to construct a nonparametric 'probability box' on an SSD based on EC05 estimates and their confidence intervals. Predictions from this uncertain SSD and the confidence interval extrapolation methods described by Aldenberg and colleagues (2000; 2002a) were compared. It was found that the extrapolation techniques underestimated the width of uncertainty (confidence) intervals by 63% and the upper bound by 65%, when compared to the Probability Bounds (P3 Bounds) approach, which was based on actual confidence estimates derived from the original data. An alternative approach to formulating ecotoxicological risk modelling was also proposed and was based on a Binomial GLM. In this formulation, the model is first fit to the available data in order to derive mean and uncertainty estimates for the parameters. This 'uncertain' GLM model is then used to predict the risk of effect from possible or observed exposure distributions. This risk is described as a whole distribution, with a central tendency and uncertainty bounds derived from the original data and the exposure distribution (if this is also 'uncertain'). Bayesian and P-Bounds approaches to propagating uncertainty in this model were compared using an example of the risk of exposure to a hypothetical (uncertain) distribution of 4-CP for the two Daphnia strains studied. This comparison found that the Bayesian and P-Bounds approaches produced very similar mean and uncertainty estimates, with the P-bounds intervals always being wider than the Bayesian ones. This difference is due to the different methods for dealing with dependencies between model parameters by the two approaches, and is confirmation that the P-bounds approach is better suited to situations where data and knowledge are scarce. The advantages of the Bayesian risk assessment and uncertainty propagation method developed are that it allows calculation of the likelihood of any effect occurring, not just the (probability)bounds, and that the same software (WinBugs) and model construction may be used to fit regression models and predict risks simultaneously. The GLM risk modelling approaches developed here are able to explain a wide range of response shapes (including hormesis) and underlying (non-normal) distributions, and do not involve expression of the exposure-response as a probability distribution, hence solving a number of problems found with previous formulations of ecotoxicological risk. The approaches developed can also be easily extended to describe communities, include modifying factors, mixed-effects, population growth, carrying capacity and a range of other variables of interest in ecotoxicological risk assessments. While the lack of data on the toxicological effects of chemicals is the most significant source of uncertainty in ecotoxicological risk assessments today, methods such as those described here can assist by quantifying that uncertainty so that it can be communicated to stakeholders and decision makers. As new information becomes available, these techniques can be used to develop more complex models that will help to bridge the gap between the bioassay and the ecosystem.

A literature review of metal mixture interaction analyses identified that there was not a consistent method to determine the impact of metal mixtures on an aquatic organism. The review also revealed that a majority of the research on mixtures made use of water concentrations only. Therefore research was conducted to determine the relationship between exposure, bioaccumulation and chronic effects of the four elements As, Co, Cr and Mn individually. Mechanistically based saturation models of bioaccumulation and toxicity were determined for the benthic invertebrate Hyalella azteca, from which lethal water concentrations and body concentrations were also determined. These models were then combined with those previously done for the metals Cd, Cu, Ni, Pb, Tl and Zn to model the impact of 10 metal mixtures on bioaccumulation in short term (1-week) exposures and on bioaccumulation and toxicity in chronic (4-week) exposures at “equi-toxic” concentrations. Interactions between the metals were identified in which; Cd, Co and Ni bioaccumulations were significantly inhibited, Tl and Zn bioaccumulations were marginally inhibited, there was no impact on Cr, Cu or Mn bioaccumulation, and both As and Pb bioaccumulation were enhanced by some mixtures of metals. It was determined that strict competitive inhibition may be a plausible mechanism of interaction affecting Co, Cd and Ni bioaccumulation but not for any of the other metals. However, it is possible that other interactions such as non-competitive or anti-competitive inhibition may have been responsible. A metal effects addition model (MEAM) was developed for Hyalella azteca based on both the bioaccumulation (body concentrations) to effects and the exposure (water concentration) to effects relationships developed from the single metal only studies The MEAM was used to predict the impact of metal mixture exposures on mortality. Toxicity was under-estimated when based on measured water or body concentrations, however, its best prediction was based on body concentrations. The MEAM, when based on measured body concentrations, takes bioavailability into account, which is important since the chemical characteristics of water can greatly alter the bioavailability and therefore toxicity of metals. The MEAM was compared to the traditional Concentration Addition Model (CAM), which calculates toxic units based on water concentrations and LC50s or body concentrations and LBC50s. The CAM overestimated toxicity, but had its best prediction when based on water concentrations. Over all, the best fit to observed mortality was the prediction by the MEAM, based on body concentrations. The measurement of bioaccumulated metals and the use of the MEAM could be important in field site assessments since it takes into account changes in bioavailability due to different site water chemistries whereas the traditional CAM based on water concentration does not.

A literature review of metal mixture interaction analyses identified that there was not a consistent method to determine the impact of metal mixtures on an aquatic organism. The review also revealed that a majority of the research on mixtures made use of water concentrations only. Therefore research was conducted to determine the relationship between exposure, bioaccumulation and chronic effects of the four elements As, Co, Cr and Mn individually. Mechanistically based saturation models of bioaccumulation and toxicity were determined for the benthic invertebrate Hyalella azteca, from which lethal water concentrations and body concentrations were also determined. These models were then combined with those previously done for the metals Cd, Cu, Ni, Pb, Tl and Zn to model the impact of 10 metal mixtures on bioaccumulation in short term (1-week) exposures and on bioaccumulation and toxicity in chronic (4-week) exposures at “equi-toxic” concentrations. Interactions between the metals were identified in which; Cd, Co and Ni bioaccumulations were significantly inhibited, Tl and Zn bioaccumulations were marginally inhibited, there was no impact on Cr, Cu or Mn bioaccumulation, and both As and Pb bioaccumulation were enhanced by some mixtures of metals. It was determined that strict competitive inhibition may be a plausible mechanism of interaction affecting Co, Cd and Ni bioaccumulation but not for any of the other metals. However, it is possible that other interactions such as non-competitive or anti-competitive inhibition may have been responsible. A metal effects addition model (MEAM) was developed for Hyalella azteca based on both the bioaccumulation (body concentrations) to effects and the exposure (water concentration) to effects relationships developed from the single metal only studies The MEAM was used to predict the impact of metal mixture exposures on mortality. Toxicity was under-estimated when based on measured water or body concentrations, however, its best prediction was based on body concentrations. The MEAM, when based on measured body concentrations, takes bioavailability into account, which is important since the chemical characteristics of water can greatly alter the bioavailability and therefore toxicity of metals. The MEAM was compared to the traditional Concentration Addition Model (CAM), which calculates toxic units based on water concentrations and LC50s or body concentrations and LBC50s. The CAM overestimated toxicity, but had its best prediction when based on water concentrations. Over all, the best fit to observed mortality was the prediction by the MEAM, based on body concentrations. The measurement of bioaccumulated metals and the use of the MEAM could be important in field site assessments since it takes into account changes in bioavailability due to different site water chemistries whereas the traditional CAM based on water concentration does not.