Testing the robustness of GLM, Bayesian GLM and Maximum Likelihood for the estimation of LC50 in different experimental setups using simulations

LC50, the concentration of a substance that kills 50% of a population of an organism, is a standard measurement in toxicology and its estimation is a common task in fields dealing with chemical substances, like pesticides. There are different strategies to estimate LC50, depending on the kind of data available and the kind of response the tested organism displays. One of the most commonly used methodologies is to assume that the mortality produced by the chemical follows a Log-Logistic curve and calculate the relevant parameters using different mathematical approaches. For the simulations we assumed that the organisms came from a population LC50 = 5. Each population was sampled and exposed to 10 different concentrations of a theoretic chemical with a toxicity modeled on the Log-Logistic curve, with LC50 = 5 and slope = 10. This curve defined the probability of dying for each dose of the chemical and each sample was simulated with a binomial distribution. To perform a power analysis, I ran simulations creating experiments differing in the number of replicates (2, 3, 4, 5, 10, 50) and the number of tested individuals per replicate (5, 10, 20, 30, 50, 100, 1000) for a total of 42 total experiments. For each experiment I ran 10.000 simulations and estimated the LC50 and the slope using three different methods: regular logistic regression, Bayesian logistic regression and maximum likelihood estimation from a 2-parameter Log-Logistic model (as performed in the R package “drm”). All three methods produced similar estimates, except with low numbers, where the Bayesian approach seems more robust. And number of insects appears to be the important variable when planning an experiment. We ran another round of simulations this time sampling from a population that had a normally distributed LC50, with mean = 5 and standard deviation = 1. In this case the methods estimated a lower LC50 than expected, suggesting that variability in the tolerance to the pesticide in the tested population can produce lower estimates for this important parameter. Further research is underway to estimate how the level of variation in the population affects LC50.