What is the difference between a logistic regression and log binomial regression? I understand that the outcomes have different meanings such as in epidemiology but what is the difference between the two models? I am new to statistics and will greatly appreciate an easy to understand answer and preferably with an example.

They are both GLMs, and they both take the variance of the response to have a mean-variance relationship as p(1-p) with p = probability of response. The difference regards the transformation of the expected response in the GLMs. The logistic regression models the logit transformation of the p whereas log binomial models the log of the p. The exponentiated (non-intercept) coefficients for logistic regression model are interpretted as odds ratios whereas for log-binomial they are relative risk ratios. Logistic regression is valid in outcome dependent sampling like case control studies. Log-binomial models risk overfitting response probabilities to values greater than 1. They give approximately equal estimation and inference when the probaibility of response is very low.
