Data in Environmental Science and Eco(toxico-)logy
Quantitative Ecotoxicology, Page 223, Example 5.1, Using GLM
Previously, I showed how to analyse the fish survival data using the arcsine square root transformation.
Warton & Hui (2011) demonstrated that the arcsine transform should not be used in either circumstance, but instead use Generalized Linear Models.
This post is about how to analyse example 5.1 on page 223 of Quantitative Ecotoxicology using Generalised Linear Models.
The binomial distribution is described by two parameters:
where N is the number of fish and $\pi$ is the probability of survival.
Here I plotted the binomial distribution for ten fish at different probabilities ($\pi = 0.5, 0.75, 0.95, 0.05$), to give you a feeling how this distribution looks like at difference $\pi$:
We can now build a model, where the probability of success is a function of treatment:
Let’s explain this a little bit:
, basically says: We assume that the number of dead fish ($y$) binomially distributed, where N = exposed animals and $\pi$ i is the probability of survival, which together give the expected number of surviving fish ($E(y) = N \times \pi$).
, basically says: We are modelling the probability of survival as function of treatment (x) [note the right-hand side of the formula is similar to linear regression]. However, we need to ensure that , and therefore the relationship is on the logit scale. The estimated parameters ($\beta) are directly interpretable as changes in log odds between treatments.
, says that variance of the binomial distribution
is a quadratic function of the mean. Note, that in linear regression we assume a constant variance.
How to do it in R?
First I read again the data into R and prepare it:
The compute a binomial GLM in R we can use the glm() function:
Note, that I use the weights argument to specify that in each tank we had 10 fish (N in the above formulas).
The summary gives the estimated parameters:
The estimates for (Intercept) are the log odds to survive in the treatment:
Similar the log odds for the highest treatment:
The estimate for conc512 (-3.675) gives you the difference in the log odds, as can be seen here:
Note, that this kind of interpretation is not possible with the arcsine transformation.
Similar to the previous post we can perform an F-Test:
Or do multiple comparisons:
Which one is better?
I would say GLMs!
They have greater power (Warton & Hui, 2011), are simpler to interpret and are readily available in most software packages.
Warton, D. I., & Hui, F. K. (2011). The arcsine is asinine: the analysis of proportions in ecology. Ecology, 92(1), 3-10.