I'm trying to understand how logistic regression uses the binomial distribution.
Let's say I'm studying nest success in birds. The probability of a nest being successful is 0.6. Using the binomial distribution, I can calculate the probability of r successes given n trials (numbers of nests studied).
But how is the binomial distribution used in a modelling context? Let's say I want to know how mean daily temperature affects nest success and I use logistic regression to explore this question.
Within the context I've described, how does the logistic regression use the binomial distribution?
I'm looking for an intuitive answer, therefore an answer without equations! I think equations are only useful once understanding has been achieved at an intuitive level.
Best Answer
Suppose you observe several nests at different mean daily temperatures $t$. How does the probability $\pi(t)$ of nest success depend on the temperature $t$? (If nests are independent, the number of nests with success at temperature $t$ is then binomially distributed with $n$ equal to the number of nests observed and success probability $\pi(t)$.)
Logistic regression is one approach (using the logistic function) of specifying the success probability as a function of temperature via stretching and shifting the logistic curve, with the amount of stretching and shifting required to be estimated from the data.