So far, we’ve learned that the equation for a logistic regression model looks like this:

$ln(\frac{p}{1-p}) = b_{0} + b_{1}x_{1} + b_{2}x_{2} +\cdots + b_{n}x_{n}$

Note that we’ve replaced y with the letter p because we are going to interpret it as a probability (eg., the probability of a student passing the exam). The whole left-hand side of this equation is called log-odds because it is the natural logarithm (ln) of odds (p/(1-p)). The right-hand side of this equation looks exactly like regular linear regression!

In order to understand how this link function works, let’s dig into the interpretation of log-odds a little more. The odds of an event occurring is:

$Odds = \frac{p}{1-p} = \frac{P(event\ occurring)}{P(event\ not\ occurring)}$

For example, suppose that the probability a student passes an exam is 0.7. That means the probability of failing is 1 - 0.7 = 0.3. Thus, the odds of passing are:

$Odds\ of\ passing = \frac{0.7}{0.3} = 2.\overline{33}$

This means that students are 2.33 times more likely to pass than to fail.

Odds can only be a positive number. When we take the natural log of odds (the log odds), we transform the odds from a positive value to a number between negative and positive infinity — which is exactly what we need! The logit function (log odds) transforms a probability (which is a number between 0 and 1) into a continuous value that can be positive or negative.