Generalised linear models (GLMs) explained for betting

In one sentence

A GLM keeps the simple weighted-sum idea of linear regression but lets the outcome follow a distribution that suits it, such as Poisson for goals.

How it works

Linear regression assumes outcomes are bell-shaped around a straight line. Goals, cards and corners are whole numbers that cannot be negative, and win/lose outcomes are yes or no. A GLM handles all of these with two choices.

First, pick a distribution for the outcome: normal for times, binomial for yes/no, Poisson or negative binomial for counts, gamma for skewed positive amounts. Second, pick a link, a function that connects the weighted sum to the average outcome; for counts this is usually the log, which keeps predictions positive.

Logistic regression is a GLM. So is the classic Maher-style football model, where each team's goals are Poisson with a log rate built from attack strength, the opponent's defence and home advantage.

The maths

g\big(E[Y]\big) = \beta_0 + \beta_1 x_1 + \dots + \beta_k x_k

\text{Football Poisson GLM:} \quad \ln \lambda_{\text{home}} = \mu + h + a_{\text{home}} - d_{\text{away}}, \qquad \ln \lambda_{\text{away}} = \mu + a_{\text{away}} - d_{\text{home}}

E[Y]: the average outcome, such as expected goals.
g: the link function (log for counts, logit for probabilities).
λ: expected goals for a side.
μ: league baseline; h: home advantage.
a: attack strength; d: defence strength (higher means better defence).

In words: build a score from the inputs, then use the link to turn it into a sensible average for that kind of outcome.

Worked betting example

A fitted Poisson GLM gives μ = 0.10 and h = 0.25. Home side: attack 0.30, defence 0.15. Away side: attack 0.05, defence −0.10.

Home log rate: 0.10 + 0.25 + 0.30 − (−0.10) = 0.75, so λ home = e^0.75 ≈ 2.12 goals.
Away log rate: 0.10 + 0.05 − 0.15 = 0.00, so λ away = e^0 = 1.00 goal.
P(home scores) = 1 − e^(−2.12) ≈ 88.0%. P(away scores) = 1 − e^(−1.00) ≈ 63.2%.
Treating the two as independent, P(both teams to score) ≈ 0.880 × 0.632 ≈ 55.6%. Fair price ≈ 1.80.
Betfair offers 1.85 on BTTS yes. £10 stake at 2% commission: a win pays £8.50 × 0.98 = £8.33. EV ≈ 0.556 × £8.33 − 0.444 × £10 ≈ +£0.19.
At 1.92 EV rises to about +£0.57. At 1.85 the edge is too small to trust once model error is considered.

Where it's good

Goals, corners, cards and shots markets, using Poisson or negative binomial GLMs.
Team attack and defence ratings with home advantage in one fitted model.
Yes/no markets via the binomial (logistic) GLM.
Adding context such as weather, referee or rest days as extra terms.
A transparent baseline that is hard to beat for many football markets.

Limitations and pitfalls

Poisson assumes the variance equals the mean; cards and corners are usually more spread out, so use negative binomial.
Independent home and away Poisson goals slightly misprice low scores and draws; Dixon-Coles corrects this.
Straight-line effects on the link scale can miss curves; GAMs relax this.
Team strengths change through a season; without time weighting the model lags.
Many team parameters with few matches overfit early in the season.
BTTS and goal markets in top leagues are efficient; a GLM rarely finds big edges there.

How to build it

Python: statsmodels GLM (family=Poisson, NegativeBinomial, Binomial), penaltyblog for ready-made football models; R: glm.
Data: one row per team per match, with goals, team, opponent and a home flag.
Add exponential time decay weights, as in Dixon and Coles (1997), so recent matches count more.
Tip: check for overdispersion by comparing the residual deviance to the degrees of freedom before trusting Poisson.

Linear regression: the special case with a normal outcome and no link.
Logistic regression: the binomial GLM.
Poisson distribution: the standard outcome for goal counts.
Negative binomial: for counts more spread out than Poisson allows.
Dixon-Coles model: a corrected Poisson GLM for football scorelines.