In this final part of my series on Bradley-Terry models I will talk about how the simple concepts behind Bradley-Terry models link with and underpin some more well-known and advanced concepts.

### 1. Logistic Regression

Let’s start by making a substitution in the formula of .

With a bit of mathematical manipulation we can recast this into a more familiar form.

These look very similar to the form of a logistic regression. This is a regression where the dependent variable can only take two values – it is binary. In our case we only have the outcomes of team i winning or team i losing.

Then if we substitute our initial expression into the logistic transformation we obtain the following terms which can be furthered simplified using the fact that .

From here it is simple to invert the transformation to get the final result, which is that the probability of team i beating team j is just a logistic regression on .