Uncertainty of the hazardous concentration and fraction affected for normal species sensitivity distributions

Species in the environment vary according to their sensitivity to a toxicant. Because these differences in sensitivity are unique to the toxicant at consideration and laboratory data sets to assess this variability are very small due to cost, it is important to provide uncertainty estimates of (1) environmental quality objectives (hazardous concentrations) derived from these laboratory data and (2) fraction of species affected at given, or predicted, laboratory or environmental concentrations. This article focuses on the normal (Gaussian) distribution of species sensitivity. It examines and compares results of Problems (1) and (2) from two opposing statistical philosophies, Bayesian and Classical, leading to vastly different numerical approaches. For the normal model, both approaches lead to identical answers, numerically. Extrapolation factors for the lower, median, and upper estimates of the hazardous concentration at six levels of protection are derived. Furthermore, upper, median, and lower estimates of the fraction affected at given, standardized, logarithmic concentrations have been tabulated. This table can be used directly for risk assessment without reference to protection levels or hazardous concentrations. The confidence limits for hazardous concentration and fraction affected depend heavily on the number of species tested and are independent of the toxic substance involved (provided the model is right), due to correction for the mean and standard deviation of the toxicity data. The equivalence of confidence limits for hazardous concentration and fraction affected is captured in the law of extrapolation: the upper (median, lower) confidence limit for the fraction affected at the lower (median, upper) confidence limit of the hazardous concentration is equal to the fraction affected (e.g., 5%) used to define the hazardous concentration. The upper confidence limit for the fraction affected at the median estimate of the hazardous concentration for 5% of the species is a fixed number depending on the sample size of the toxicity data only. It amounts to 46% at n=3, down to 20% at n=10, and still 12% at n 30.