# Using a generalised linear model for tennis winning probability

I have a dataset of historical tennis matches, one row per match. Each row has the ELO points of each player, and a calculated ELO difference. And of course a column to indicate if Player 1 won as a 1 or 0.

There are other predictor variables involved but for the sake of this example I just want to know how would I go building this kind of model where there can be both numerical and categorical independent variables, and a 1 or 0 dependent variable. (I would be using SAS).

What kind of distribution function would I be using, and what kind of link function would I be using?

Ideally the result would give the probability of winning as a function of ELO difference. Is this possible using a GLM?

You should be using a Binomial GLM, a.k.a. Binomial/Binary logistic regression. Categorical independent variables will be converted to dummies without affecting the generic model structure. For $$N$$ Binomial input samples and $$M$$ independent variables (including dummified factors), your GLM components should be the following...

1. Stochastic component:

$$y_i \sim Binomial(n_i, p_i), ~~i=1,...,N$$

(NB: For binary response, $$n_i=1, \forall i$$)

2. Systematic component:

$$\eta_i = \beta_0 + \beta_1 x_1 + ... + \beta_Mx_M$$

$$g(p_i) = logit(p_i) = \eta_i$$
$$logit(p_i) = \beta_0 + \beta_1 x_1 + ... + \beta_kx_k$$