GLM for Binomial Counts

The binomial GLM

The GLM of the binomial counts analyzes the number of each batch of beetles killed while taking into account the size of each group (the binomial denominator, the sum of the number killed, and the number alive). Some statistical software packages require the number of successes and the binomial denominator, while others— such as R—need the numbers of successes and failures.

The numbers of dead and alive beetles (successes and failures) must be bound together using the cbind() function so that they can be jointly supplied to the response variable argument for the binomial GLM. We use the family argument to specify the binomial distribution. The logistic function logit is the default link function.

The logistic link function and the binomial distribution are chosen to take account of the properties of and constraints on the pattern of the mean and variance for the binomial count data, as we saw in the section “Logits and the logistic curve” from the previous lesson.Box_15_2 Since we’re interested in the mortality rate, we put the number of beetles killed as the successes and the number of beetles still alive as the failures, as shown below:

m1_logit <- glm(cbind(killed, alive) ~ Dose, data = beetle,
    family = binomial(link = "logit"))

The logistic curve is linear on the logit scale, and the coefficients are the regression intercept $(−14.6)$ and the slope $(0.25)$ of this line: