The Bradley-Terry Model

A Probabilistic Framework for Pairwise Comparisons

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The Bradley-Terry model is a statistical framework for modelling pairwise comparison data, first proposed by Bradley and Terry (1952). It converts a series of binary comparisons into a ranking. Pairwise comparisons reduce cognitive load compared with direct ranking of multiple items[1][4].

Model Definition

Given items \( \{1, \dots, n\} \) with positive parameters \( \pi_i \), the probability that item \( i \) is preferred over \( j \) is:

\[ P(i > j) = \frac{\pi_i}{\pi_i + \pi_j} \]

\( \pi_i \) represents the "strength" or "utility" of item \( i \).

Bradley & Terry (1952), Zermelo (1929)

Model Properties

Scale Invariance

\[ P(i > j) = \frac{c\pi_i}{c\pi_i + c\pi_j} = \frac{\pi_i}{\pi_i + \pi_j}, \quad c > 0 \]

Parameters are usually constrained (e.g., \( \sum \pi_i = 1 \)) for identifiability.

Bradley & Terry (1952)

Logit Formulation

\[ \log \frac{P(i > j)}{P(j > i)} = \log \pi_i - \log \pi_j \]

This shows the model is a special case of logistic regression.

Bradley & Terry (1952), Springall (1973)

Transitivity

If \( P(i > j) > 1/2 \) and \( P(j > k) > 1/2 \), then \( P(i > k) > 1/2 \).

Davidson (1970)

Parameter Estimation

Maximum likelihood estimates can be computed iteratively:

\[ \pi_i^{(k+1)} = \frac{w_i}{\sum_{j \neq i} \frac{n_{ij}}{\pi_i^{(k)} + \pi_j^{(k)}}} \]

\( w_i \) is the total wins for item \( i \); \( n_{ij} = y_{ij} + y_{ji} \).

Bradley & Terry (1952), Zermelo (1929)

Connection to Elo Rating

\[ P(i > j) = \frac{1}{1 + 10^{(R_j - R_i)/400}} \]

This shows equivalence to Elo ratings with scale factor 400.

Bradley & Terry (1952)

Extensions

ExtensionDescriptionReference
Plackett-LuceRank multiple itemsPlackett (1975), Luce (1959)
Davidson ModelAccommodates tiesDavidson (1970)
Covariate BTIncludes item featuresSpringall (1973)
BT Regression TrunkTree-based modeling of preferencesD'Ambrosio et al. (2023)

References

  1. Reips & Funke (2008). VAS Generator. Behavior Research Methods, 40(3), 699-704.
  2. Bradley & Terry (1952). Rank analysis of incomplete block designs. Biometrika, 39(3/4), 324-345. link
  3. Miller (1956). Magical number seven. Psychological Review, 63(2), 81-97.
  4. Thurstone (1927). Law of comparative judgment. Psychological Review, 34(4), 273-286.
  5. Zermelo (1929). Die Berechnung der Turnier-Ergebnisse. Mathematische Zeitschrift, 29(1), 436-460.
  6. Davidson (1970). Extending Bradley-Terry for ties. JASA, 65(329), 317-328.
  7. Plackett (1975). Analysis of permutations. JRSS C, 24(2), 193-202.
  8. Springall (1973). BT with covariates. Applied Statistics, 22(1), 59-68.
  9. D'Ambrosio et al. (2023). BT regression trunk. Psychometrika, 88(4), 1443-1465.