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.
 A: Both models are binomial generalized linear models (GLMs). The "binomial" part of that phrase means that we think the outcome $Y_i$ for each observation $i$ is generated as the result of a single weighted coin flip, 0 or 1, governed by the parameter $p_i$, which is equal to the probability of getting a 1 (i.e., $p_i = P(Y_i=1)$).
Where the models differ is in the form of the relationship between the predictors $X_i$ and $p_i$. In logistic regression, the relationship is
specified as $$
p_i=\text{expit}(X_i \beta)=\frac{1}{1+\exp(-X_i\beta)}
$$
or, equivalently,$$
\text{logit}(p_i)=\frac{p_i}{1-p_i}=X_i\beta
$$
In log-binomial regression, the relationship is specified as $$
p_i=\exp(X_i \beta)
$$
or, equivalently,
$$
\log(p_i)=X_i \beta
$$
The coefficients $\beta$ are interpreted differently between the two models. For both, it makes sense to exponentiate them. In logistic regression, $\exp(\beta_k)$ is the odds ratio corresponding to a 1-unit change in $X_k$, holding all other variables in the model constant. In log-binomial regression, $\exp(\beta_k)$ is the risk ratio corresponding to a 1-unit change in $X_k$, holding all other variables in the model constant.
Logistic regression always produces estimates of $p_i$ that are between 0 and 1. This is not true for log-binomial regression; when a model is not saturated, it is generally possible, given a set of estimated coefficients, to compute an estimate of $p_i$ that is greater than 1 (but it will always be positive). There can be numerical problems when using log-binomial regression for this reason (i.e., the model can be hard for computer programs to fit). Which model fits your data better is an empirical question, but logistic regression will almost always perform better unless there are very few 1s.
For saturated models, the two models will fit exactly the same and produce the same predicted probabilities. The log-binomial model is therefore most useful when you have a saturated model and you want to interpret a coefficient as a risk ratio, e.g., when the only predictor in your model is treatment status in a randomized trial.
See here and here for more discussions of these models.
A: 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.
