## INTRODUCTION

The so-called species sensitivity distribution (SSD) methodology has been used since 1996 in the European Community to determine quality criteria such as the predicted no-effect concentration of a chemical substance or for deriving probable impacts in exposed systems. This approach (which was developed by Kooijman 1 and expanded by Van Straalen and Denneman 2, Wagner and Løkke 3, and Aldenberg and Slob 4) is based on the assumption that the species for which ecotoxicological tests results are available are representative of the sensitivity of the rest of the species in an ecosystem. A likely SSD is then estimated from these results, and a concentration that is likely to protect a given percentage of the species can be extrapolated. For the first use of SSDs—that is, for setting quality criteria—the agreed European concentration is HC5_{50%}, the hazardous concentration affecting 5% of species with 50% confidence, which is equivalent to 95% of the species being protected with a confidence limit of 50% (in this article, this value will be denoted by HC5). The SSD approach is now widely used for operational risk assessments of chemical compounds, even if it raises a number of fundamental challenges 5, 6. First, the assumption that the data arise from randomly sampled species that represent the actual ecosystem has major flaws in practice 6, 7 because species are usually selected for laboratory purposes. Second, because of the scarcity of available tests, the data used to generate SSDs often represent a mixture of end points (e.g., growth, reproduction, and mortality), each of them being more or less relevant for protecting the target 6. Third, the trophic composition of taxa is heterogeneous within the data for a substance, which limits the relevance of comparative analyses 8. The goal of this article is not to discuss these fundamental limitations of the SSD approach (even if they must be considered), especially those related to the quality of the ecotoxicological data, but to focus on the statistical method used to build SSDs.

Since the SSD concept was first developed, several frequentist statistical methods for estimating distributions and calculating HC5s have been suggested and compared 4, 8–16. These methods differ in the choice of the underlying statistical distribution (empirical distribution, log-normal or log-logistic distribution, and others) and in the method used to estimate the confidence interval (bootstrap, asymptotic theory, and nonparametric statistics). The results of these methods can vary significantly when applied to the same data 17. Although Aldenberg and Slob 4 have derived extrapolation factors for small data sets, frequentist approaches inherently require a minimum quantity of data to guarantee goodness-of-fit. For example, Forbes and Calow 6 proposed that a minimum of 19 and 9 data points are required to estimate the 5th and 10th percentiles, respectively. Thus, the SSD methodology can only be used if there are sufficient chronic ecotoxicological test results for the substance (a minimum of approximately 15). In addition, three taxonomic groups (algae, invertebrates, and vertebrates) need to be represented in aquatic ecosystems. As a consequence, the SSD approach can only be used in practice for a few substances, most of which are metals. When the quantity of chronic data is insufficient, the predicted no-effect concentration estimate is based on an alternative approach that uses assessment factors. Only the lowest no observed effect concentration (NOEC) or effect concentration (ECx) value divided by predefined assessment factors (10, 100, or 1,000, depending on the case) is used to estimate the predicted no-effect concentration, and no confidence intervals can be derived for it.

Alternative approaches that use Bayesian theory have also been proposed for creating SSDs 12, 14, 17, 18. In the Bayesian paradigm, the data sample is fixed and unique, and the distribution parameters themselves are uncertain. For example, if the SSD is assumed to be log-normally distributed, the Bayesian method aims at calculating the joint distribution of the mean and standard deviation of the SSD; in contrast to the frequentist approaches, the mean and standard deviation of the SSD are then probabilistically described. Aldenberg and Jaworska 12 used a Bayesian method to derive extrapolation factors for calculating an HC5 from the sample mean and standard deviation of small data sets. This approach was compared with frequentist methods (maximum likelihood, parametric bootstrapping, and nonparametric bootstrapping) by Verdonck et al. 14. Hickey and Craig 18 proposed modifying these extrapolation factors by introducing an asymmetric LINEX (linear-exponential) loss function for deriving asymmetric extrapolation factors for over- and underpredictions. However, all of these authors used noninformative priors for the means and standard deviations of the SSD. Grist et al. 17 introduced expert judgment to define priors; these expert judgments were used to define an a priori sensitivity index for several of the taxa used to construct the SSD. A panel of experts consulted by the European Food Safety Authority 19 recommended using informative Bayesian approaches to derive HC5s from small data sets, but their approach was theoretical and was not tested on real data sets.

Nevertheless, the experience gained from constructing SSDs for well-known substances allows collecting prior information. For example, Duboudin et al. 8, 20 calculated the chronic and acute SSDs of several substances for which sufficient data were available. If the standard deviation is relatively homogeneous among substances, prior information about the expected variance of the SSD can be defined for a new substance.

Therefore, our objective was to develop and test a Bayesian approach to deriving SSDs and HC5s using informative priors derived from past experiences related to species sensitivity variance. This method was tested on a set of 21 substances; the cross-validation allowed comparing the “true” HC5 (i.e., the HC5 calculated from the complete data set) and the simulated Bayesian HC5 derived from truncated samples.