Default Distributions
To avoid such model redundancy (Burnham and Anderson 2002), ssdtools and the accompanying Shiny app (Dalgarno 2018) fit three different shape distributions by default: the log-normal, gamma and the Burr Type-III 2-parameter distributions.
The three distributions are plotted below with a mean of 2 and standard deviation of 2 on the (natural) log concentration scale or around 7.4 on the concentration scale.
library(ssdtools)
library(ggplot2)
set.seed(7)
ssd_plot_cdf(ssd_match_moments(meanlog = 2, sdlog = 2))
Log-normal distribution
The log-normal distribution was selected as the starting distribution given the data are for effect concentrations. The log-logistic distribution is often used as a candidate SSD primarily because of its analytic tractability (Aldenderg and Slob 1993). However, we did not include the log-logistic distribution in the default set of distribution because 1) it has a very similar shape to the log-normal distribution and 2) it is a specific case of the more general Burr family of distributions (Shao 2000).
The log-normal distribution does have a couple limitations to consider when fitting species sensitivity data. First, on the logarithmic scale, the normal distribution is symmetrical. Second, the log-normal distribution decays too quickly in the tails giving narrow tails which may not adequately fit the data. Additional distributions were selected to compensate for these deficiencies.
Gamma distribution
The gamma distribution is a two-parameter distribution commonly used to model failure times or time to events. For use in modeling species sensitivity data, the gamma distribution has two key features that provide additional flexibility relative to the log-normal distribution: 1) it is non-symmetrical on the logarithmic scale; and 2) it has wider tails. The Weibull distribution was considered as an alternative but the Gamma distribution is generally more flexible.
Burr Type III
Burr (1942) developed a family of 12 distributions with a flexible shape. The original Burr Type III distribution (also known as the inverse Burr distribution or the Dagum distribution) is a three parameter distribution that can take many shapes as dictated by the data and has a heavy tail which is useful in modeling extreme concentrations (Domma et al 2011).
The original Burr III distribution has three parameters (two shape and one scale parameter). However, the estimated parameters are very highly correlated for small samples and the distribution is essentially non-identifiable (i.e. different sets of parameter values give rise to essentially the same fit). For this reason, we have selected as a default distribution, the two-parameter (one shape and one scale parameter) version of the Burr III distribution that has better properties in small samples.
