More on log odds and odds ratio

Recall that the odds refer to the ratio of the probability of an event happening over its complement:

odds =  probability of event happening   ________________________   probability of event not happening  =  p _ 1 − p  =  P(y = 1) _ P(y = 0) 

Here, the probability is calculated as follows:

p = P(y = 1) =  1 _ 1 + e −z 

1 − p = 1 −  1 _ 1 + e −z  =  e −z _ 1 + e −z 

Plugging in the definition of p and 1 − p gives us the following:

odds =  p _ 1 − p  = e z

Instead of directly working with the odds, we often use the log odds or logit. This term is typically modeled as a linear combination of predictors in a logistic regression model via the following:

log P(y = 1) _ P(y...