We know the Logit distribution is $\frac{exp(\beta'x_i)}{1+exp(\beta'x_i)}$
In R, if we want to execute a logit regression, we use:
glm.logit=glm(model,binomial(),Data.df)
This returns the relevant coefficients. How do we interpret these coefficients algebraically? I read another response which said that R returns the log odds, so each coefficient therefore has the interpretation of log$(\frac{p_i}{1-p_i})$? Where $p_i =\frac{exp(\beta'x_i)} {1+exp(\beta'x_i)}$ So p, the probability is equal to the distributional function? If this is correct, then all I need to do to get the odds ratio is take the exponential of, i.e. log$(\frac{p_i}{1-p_i})$ = $\beta'x_i$ $\implies$ $\frac{p_i}{1-p_i} = exp(\beta'x_i)$
in R:
exp(coefficients(glm.logit))
How would we calculate the logit model 'by hand'? Would you specify the logit distribution function, F(z) and then set z= linear model (OLS)?
Thank you