Bradley–Terry model
Section of Paired comparisons, and how far a fit is from the data, part of Generalised linear models
Bradley–Terry model
| Type | Section |
| Taught in | Paired comparisons, and how far a fit is from the data |
| Part | Generalised linear models |
| Questions using it | 1 |
A head-to-head result is still a binary outcome. The Bradley–Terry model gives team j a skill \lambda_j and says \mathrm{odds}(j\text{ beats }k)=\exp(\lambda_j-\lambda_k). Swap the labels and the odds invert: \mathrm{odds}(k\text{ beats }j)=\exp(\lambda_k-\lambda_j)=\frac{1}{\exp(\lambda_j-\lambda_k)}. The model is forced by that consistency. Only differences of skills are identified, so one team can be set to zero and everyone else is measured from them. Software often sets the alphabetically first team to zero. We fit that, then shift the origin so the weakest team is zero. The ranking is unchanged.
Home advantage, or White moving first in chess, is one extra parameter \alpha on the log-odds of the home side: \mathrm{odds}(j\text{ beats }k\text{ at home})=\exp(\lambda_j-\lambda_k+\alpha), and the away fixture puts -\alpha on j. Each match contributes a Bernoulli likelihood with success probability given by that odds. Maximum likelihood is ordinary.
The 2024 NRL table, regular season through 11 September, with the Tigers shifted to zero, is
| team | \hat\lambda | team | \hat\lambda |
|---|---|---|---|
| Storm | 2.587 | Knights | 1.222 |
| Panthers | 2.192 | Dragons | 1.053 |
| Roosters | 1.962 | Dolphins | 0.985 |
| Cowboys | 1.738 | Broncos | 0.881 |
| Sharks | 1.669 | Warriors | 0.827 |
| Bulldogs | 1.594 | Rabbitohs | 0.510 |
| Eagles | 1.430 | Titans | 0.483 |
| Raiders | 1.253 | Eels | 0.310 |
| Tigers | 0 |
Home advantage is \hat\alpha=0.3957. The odds of winning at home, for any team against a given opponent, are \exp(0.3957)=1.49 times the odds of winning the same match away. That gap is large enough to overturn some neighbouring pairs. The Dolphins sit above the Broncos by 0.985-0.881=0.10, which is less than 0.40, so the Broncos at home are the favourites against the Dolphins. The same arithmetic applies to the Warriors at home against the Broncos.
The odds that the Panthers beat the Roosters at home are \exp(2.1915-1.9623+0.3957)=\exp(0.625)=1.87. Odds of 1.87 are a probability of 1.87/(1+1.87)\approx 0.65. The official ladder had the Sharks above the Cowboys on for-and-against. This fit has the Cowboys above the Sharks, because it credits who was beaten, not only how many. That is the point of a skill parameter.
A draw can be coded as half a win and half a loss, which says the two teams were equal that day, without adding a third multinomial outcome. If the competition itself scores a draw as one point and a win as three, as in the English Premier League, the honest code is one third of a win rather than one half. We will not build the three-outcome model unless draws are common and meaningful.
Used in
- Taught in Paired comparisons, and how far a fit is from the data
- Questions that use it (1) — C26
- All 4 questions on Paired comparisons, and how far a fit is from the data
See also
Paired comparison · Deviance · Three links, one cholesterol slope · Analysis of deviance · Residuals in a GLM · Saturated model · Residual deviance · Null deviance