In Response: Some species sensitivity distribution statistics revisited—A governmental perspective

Estimating toxicity thresholds for aquatic, sedimentary, and terrestrial environments present insurmountable challenges, given the complexity of the systems involved. Since Kooijman 1 and Van Straalen and Denneman 2, the statistical species sensitivity distribution (SSD) method has provided a major shortcut in deriving hazardous concentrations (HCx, e.g., HC5) from a set of comparable laboratory toxicity endpoints. A 2002 volume on SSDs 3 documents overviews, applications, and extensions. More recently, an European Centre for Ecotoxicology and Toxicology of Chemicals workshop on estimating toxicity thresholds for SSDs was held in Amsterdam 4, where current SSD–related issues were discussed and summarized. Identified research areas include ecological validation, species selection, statistical problems, and mode of action of the chemicals. Among these, statistical challenges are manifold as well. In the present Response, it is shown that some statistical issues can be definitely addressed with present-day statistical methods. Three statistical topics are selected. The problem of data error is discussed firstly. Secondly, I will revisit predictive SSD fitting and extrapolation constants. Thirdly, the problem of censored data will be touched on. My coworkers and I have been concerned with the problem of how to account for data points within a SSD, some of which may derive from quantitative structure–activity relationship (QSAR) estimates 5. But as QSAR-based estimates bring in additional uncertainty, we investigated the effect of data error. But data error is not typical for model-injected data points: it is characteristic for experimental species data as well. The European Chemicals Agency's technical guidance says that SSD data points may be best conceived as species means, so it is advisable to average multiple sensitivity data points for the same species. This carries over to QSAR-based points as expected model predictions. We briefly explored a Bayesian hierarchical model with 2 variance components: within-species error and between-species error 5. Classical analysis of variance teaches us that high within-species error may mask the assessment of the SSD error of species means. If the SSD is mere species noise, the purported distribution of species means vanishes and becomes undetectable. In a numerical experiment for n = 5, we found that the within-species noise—up to one-half of the SSD between-species mean variation—did not have much effect on the estimate of the SSD standard deviation (SD). Neglecting data error in fact leads to a conservative estimate because the SSD SD decreases slightly for increasing data noise. So, we tentatively concluded that neglecting moderate data noise may often be acceptable. In the nonhierarchical Bayesian treatment 6, we have used the so-called noninformative prior, assuming the logarithm of the SSD sigma to be uniform. This is standard Bayesian practice. However, over the years, we have found this prior on sigma to be too optimistic (small). Why would the prior express that bigger SSD SDs are to be discounted? There is another argument: for the hierarchical model, the noninformative prior for the SSD SD does not work, however small the data noise 5. Hence, it seems unreasonable for an assumed value to be exactly 0. Consequently, we decided to calculate revised extrapolation constants for a uniform prior. We call this “cautionary extrapolation,” which we think amounts to more realistic extrapolation, especially for small sample sizes. In the Bayesian hierarchical model, we also looked at the so-called predictive distribution of the SSD. This can be understood as the vertical average of SSD spaghetti plot threads. This holds for both the SSD density curves as well as the cumulative distribution plots. The predictive distribution is single-threaded (1 curve) and, hence, a nice summary of the spaghetti plot and associated percentile curves. In our 2013 study 5, we compared lower quantiles (HCx) of the predictive distribution to former extrapolation values found in our 2000 study 6. Statistical theory reveals that the predictive distribution of a normal SSD is a Student t distribution with n − 1 degree of freedom. Cautionary extrapolation makes this n − 2, so the more realistic prior on the SSD SD causes the loss of 1 more degree of freedom. It turned out that predictive HCx values are more sensitive to low sample size than the former median estimates, doing more justice to small sample sizes. In Table 2, the classical HC5 estimates 6 are compared with the predictive HC5 (all based on the noninformative prior for the SSD SD) and the cautionary predictive estimate for the uniform prior SSD SD 5. Note that at the European Chemicals Agency–recommended sample sizes (Table 2, rows 10 and 15, printed in bold), the extrapolation constants for both priors converge to roughly –2. For sample size over 30, differences between median and both predictive estimates gradually disappear. The statistics of censored data modeling are well developed, see Helsel 7. We have applied this to SSD fitting of 14 acute antimony freshwater data 8. Note that 4 species data “points” have a right-censored value (Figure 5). The same analysis applies to left-censored values (e.g., detection limits), mixtures of both, as well as interval data. When the 4 “greater thans” are removed, the median HC5 equals 1.7 mg Sb/L. When included, the median HC5 is equal to 2.0 mg Sb/L. Note the asymmetry of the normal SSD percentile curves. A similar approach to censored values using Bayesian concepts is followed by Kon Kam King et al. 9 with the R-based web-tool MOSAIC_SSD. Figure 5 shows the lower bound of the censored values. It does not convey how the method of maximum likelihood takes censoring into account. For that, Kon Kam King et al. 9 and Helsel 7 supply the details. We hope to have shown that the SSD method, although incapable of modeling the full complexity of environmental systems, is by no means “done” yet and is amenable to extensions addressing a range of statistical challenges.