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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?

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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 $


3. Link function:

$g(p_i) = logit(p_i) = \eta_i $

With for final model being:

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

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