# Quantitative Ecotoxicology, Page 147, Example 4.3, LC50

This is about example 4.3 on page 147 of Quantitative Ecotoxicology. This example is about Calculations of $LC_{50}$ values. In this post I won’t reproduce the SAS-Code since I do not have any experience with SAS PROC PROBIT and I do not fully understand whats happening there.

Instead I will fit Dose-Response-Models using the `drc`

-package to this data.

Get the data from here and read it into R:

So the data consists of number of dead animals (DEAD) from all animals (TOTAL) exposed to a concentration (CONC). First we create a new column with the proportion of dead animals:

Lets have a look at the raw data (note that I use a logarithmic scale for the x-axis):

I will use the drc-package of Christian Ritz and Jens Strebig to fit dose-response-curves to this data. The main function of this package is `drm`

:

Here I fit a two-parameter log-logistic model to the data (see Ritz (2010) for a review of dose-response-models used in ecotoxicology):

So the usage is similar to `lm()`

or `nls()`

with some addtions:
The `fct`

argument defines the model that is fitted to the data here a 2-parameter log-logistic.
`weights`

specifies the sample size for each group and `type = 'binomial'`

specifies that we have a binomial response (default is `continuous`

for normally distributed errors).

We can compare this model with other models using the AIC (the smaller the better).

Here I compare the 2-parameter log-logistic model with a two-parameter Weibull and a 2-parameter Gompertz model.

`drc`

has for this purpose the `mselect()`

function that shows some diagnostics for every model:

- likelihood (the higher the better; however use this only when your models are nested)
- AIC (the lower the better)
- residual variance (the lower the better)

The LL.2-model has the lowest AIC so I will keep this. Lets see how the model looks like:

We can get the $LC_{50}$ with confidence interval from the model using the `ED()`

function:

Code and data is available on my github-repo under file name ‘p147’.

#### References

[1] C. Ritz. “Toward a unified approach to dose-response modeling
in ecotoxicology”. In: *Environmental Toxicology and Chemistry*
29.1 (Jan. 2010), pp. 220-229. DOI: 10.1002/etc.7. <URL:
http://dx.doi.org/10.1002/etc.7>.