Dixon-Coles Match Predictor
The Poisson model treats the two teams' goals as independent. In 1997 Mark Dixon and Stuart Coles found that, in English football, this got low-scoring draws wrong, and added one number, ρ, to correct it.
Expected goals from 0.1 to 5.
Try a value of ρ
Dixon-Coles score grid. Each cell is the chance of that exact score, in percent. The outlined cell is the most likely.
| Outcome | Poisson | Dixon-Coles | Change, points |
|---|
Only 0–0, 1–0, 0–1 and 1–1 change. Every other scoreline is exactly as Poisson has it.
How it works
Dixon-Coles takes the Poisson grid and multiplies just the four lowest-scoring cells by a correction factor τ. With λ and μ the home and away expected goals:
$$\begin{aligned} \tau(0,0) &= 1 - \lambda\mu\rho \\ \tau(0,1) &= 1 + \lambda\rho \\ \tau(1,0) &= 1 + \mu\rho \\ \tau(1,1) &= 1 - \rho \end{aligned}$$In plain football
- \(\rho\) is how strongly the two teams' low scores are linked. At \(\rho = 0\) there is no link and you get plain Poisson.
- A negative \(\rho\) makes 0–0 and 1–1 more likely and 1–0 and 0–1 less likely: tight games stay tight.
- The four changes cancel out exactly. Probability moves between those four scores; none is added or lost.
Because no cell can have a negative chance, ρ has limits that depend on the expected goals. The model stops at them and tells you.
In practice: does it help?
On the English data Dixon and Coles studied, it did. On Scottish Premiership data since 2000/01 it hardly does, and the reason is instructive. Here are the four low scores, as counted and as predicted, from the SPFL Prediction Board's evaluation:
| Score | Happened | Poisson expected | Dixon-Coles expected |
|---|---|---|---|
| 0–0 | 401 | 439.4 | 445.3 |
| 1–0 | 566 | 608.4 | 602.4 |
| 0–1 | 491 | 511.6 | 505.6 |
| 1–1 | 659 | 657.6 | 663.6 |
Here Poisson over-predicts 0–0, 1–0 and 0–1, and gets 1–1 almost exactly right: taken together, the low-scoring block is too big. Dixon-Coles can only move probability between those four scores, so no value of ρ can shrink the block as a whole. Fitted season by season, ρ came out small and negative (median −0.011, negative in 22 of 26 seasons): it improved 1–0 and 0–1 by making 0–0 and 1–1 worse. The overall error on the four cells barely moved, from 102.8 to 100.0, and the model's log loss got 0.0001 worse.
The lesson is not that Dixon-Coles is wrong. It is that a correction built on one league's data has to be tested on yours. If this league's low scores ever need fixing, it will take a different model, one that can move probability out of the low-scoring block altogether.
Where it goes wrong
- It shares Poisson's other assumptions: a steady scoring rate, and expected goals you supply.
- ρ is one number for a whole league. Real matches differ; a cup final is not a mid-table league game.
These probabilities come from a statistical model and are for analysis and education. Football remains uncertain and model predictions will frequently be wrong.
Start with the basics: How many goals will we score? The Poisson distribution