The bpcs package performs Bayesian estimation of Paired Comparison models utilizing Stan, such as variations of the Bradley-Terry (Bradley and Terry 1952) and the Davidson models (Davidson 1970).
Package documentation and vignette articles can be found at: https://davidissamattos.github.io/bpcs/
For the bpcs package to work, we rely upon the Stan software and the rstan package (Stan Development Team 2020).
To install the development version of the bpcs package, install directly from the Github repository.
After installing, we load the package with:
The main function of the package is the bpc function. For the simple Bradley-Terry model, this function requires a specific type of data frame that contains:
We will utilize the tennis dataset available (Agresti 2003). The dataset can be seen below and is available as data(tennis_agresti):
|
player0 |
player1 |
y |
id |
|---|---|---|---|
|
Seles |
Sanchez |
0 |
14 |
|
Sabatini |
Sanchez |
1 |
42 |
|
Seles |
Graf |
0 |
1 |
|
Graf |
Sabatini |
1 |
22 |
|
Sabatini |
Sanchez |
1 |
41 |
|
Seles |
Navratilova |
1 |
12 |
|
Graf |
Sanchez |
0 |
32 |
|
Sabatini |
Navratilova |
0 |
35 |
|
Sabatini |
Sanchez |
0 |
39 |
|
Seles |
Graf |
1 |
4 |
Based on the scores of each contestant, the bpc function computes automatically who won the contest. Alternatively, you can provide a vector of who won if that is already available (for more information see ?bpc.
For the simple Bradley Terry Model we specify the model type as 'bt'. Here we hide the MCMC sampler chain messages for simplicity in the output.
m<-bpc(data = tennis_agresti, #datafrane
player0 = 'player0', #name of the column for player 0
player1 = 'player1', #name of the column for player 1
result_column = 'y', #name of the column for the result of the match
model_type = 'bt', #bt = Simple Bradley Terry model
solve_ties = 'none' #there are no ties in the dataset so we can choose none here
)
#>
#> SAMPLING FOR MODEL 'bt' NOW (CHAIN 1).
#> Chain 1:
#> Chain 1: Gradient evaluation took 8.7e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.87 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1:
#> Chain 1:
#> Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
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#> Chain 1:
#> Chain 1: Elapsed Time: 0.299618 seconds (Warm-up)
#> Chain 1: 0.312838 seconds (Sampling)
#> Chain 1: 0.612456 seconds (Total)
#> Chain 1:
#>
#> SAMPLING FOR MODEL 'bt' NOW (CHAIN 2).
#> Chain 2:
#> Chain 2: Gradient evaluation took 3.9e-05 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.39 seconds.
#> Chain 2: Adjust your expectations accordingly!
#> Chain 2:
#> Chain 2:
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#> Chain 2:
#> Chain 2: Elapsed Time: 0.284584 seconds (Warm-up)
#> Chain 2: 0.344625 seconds (Sampling)
#> Chain 2: 0.629209 seconds (Total)
#> Chain 2:
#>
#> SAMPLING FOR MODEL 'bt' NOW (CHAIN 3).
#> Chain 3:
#> Chain 3: Gradient evaluation took 4.1e-05 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.41 seconds.
#> Chain 3: Adjust your expectations accordingly!
#> Chain 3:
#> Chain 3:
#> Chain 3: Iteration: 1 / 2000 [ 0%] (Warmup)
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#> Chain 3:
#> Chain 3: Elapsed Time: 0.305793 seconds (Warm-up)
#> Chain 3: 0.292243 seconds (Sampling)
#> Chain 3: 0.598036 seconds (Total)
#> Chain 3:
#>
#> SAMPLING FOR MODEL 'bt' NOW (CHAIN 4).
#> Chain 4:
#> Chain 4: Gradient evaluation took 4.2e-05 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.42 seconds.
#> Chain 4: Adjust your expectations accordingly!
#> Chain 4:
#> Chain 4:
#> Chain 4: Iteration: 1 / 2000 [ 0%] (Warmup)
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#> Chain 4:
#> Chain 4: Elapsed Time: 0.30643 seconds (Warm-up)
#> Chain 4: 0.300302 seconds (Sampling)
#> Chain 4: 0.606732 seconds (Total)
#> Chain 4:If rstan is available and correctly working this function should sample the posterior distribution and create a bpc object.
To see a summary of the results we can run the summary function. Here we get three tables:
summary(m)
#> Estimated baseline parameters with HPD intervals:
#>
#>
#> Parameter Mean HPD_lower HPD_higher n_eff Rhat
#> -------------------- ------ ---------- ----------- ------- -----
#> lambda[Seles] 0.51 -2.37 3.34 663.50 1.01
#> lambda[Graf] 0.94 -1.85 3.71 622.34 1.01
#> lambda[Sabatini] -0.33 -3.12 2.50 662.40 1.01
#> lambda[Navratilova] 0.04 -2.87 2.91 637.48 1.01
#> lambda[Sanchez] -1.12 -4.04 1.64 670.86 1.01
#> NOTES:
#> * A higher lambda indicates a higher team ability
#>
#>
#> Posterior probabilities:
#> These probabilities are calculated from the predictive posterior distribution
#> for all player combinations
#>
#>
#> i j i_beats_j
#> ------------ ------------ ----------
#> Graf Navratilova 0.70
#> Graf Sabatini 0.77
#> Graf Sanchez 0.86
#> Graf Seles 0.60
#> Navratilova Sabatini 0.58
#> Navratilova Sanchez 0.74
#> Navratilova Seles 0.41
#> Sabatini Sanchez 0.66
#> Sabatini Seles 0.32
#> Sanchez Seles 0.20
#>
#>
#> Rank of the players' abilities:
#> The rank is based on the posterior rank distribution of the lambda parameter
#>
#>
#> Parameter MedianRank MeanRank StdRank
#> -------------------- ----------- --------- --------
#> lambda[Graf] 1 1.42 0.64
#> lambda[Seles] 2 2.08 0.88
#> lambda[Navratilova] 3 3.04 0.91
#> lambda[Sabatini] 4 3.65 0.83
#> lambda[Sanchez] 5 4.81 0.48get_hpdi_parameters functionget_rank_of_players function.get_probabilities function.expand_aggregated_data.get_sample_posterior function.predict function.bt) (Bradley and Terry 1952)davidson) for handling ties (Davidson 1970)Options to add to the models:
-ordereffect). E.g. for home advantage (Davidson and Beaver 1977)-generalized). When we have contestant specific predictors (Springall 1973)-U). For example, to compensate clustering or repeated measures (Böckenholt 2001)E.g.:
btdavidson-Ubt-generalized-ordereffectNotes:
bt-U-ordereffect is equivalent to bt-ordereffect-U- is mandatoryexpand_aggregated_data.This package provides a series of small and self contained vignettes that exemplify the use of each model. In the vignettes, we also provide examples of code for data transformation, tables and plots.
Below we list all our vignettes with a short description:
Getting Started: This vignette shows a basic example on tennis competition data, covering how to run a Bradley-Terry model, MCMC diagnostics, posterior predictive values, ranking, predict new matches
Ties and home advantage: This vignette covers a soccer example from the Brazilian soccer league. Here, we first model the results using a Bradley-Terry model and the Davidson model to handle ties. Then, we extend both models to include for order effects, this allows us to investigate the home advantage in and without the presence of ties.
Bradley-Terry with random effects: This vignette covers the problem of ranking black-box optimization algorithms based on benchmarks. Since in benchmarking we often run the same optimization algorithm more than once with the same benchmark problem, we need to compensate for the repeated measures effect. We deal with this utilizing a simple Bradley-Terry model with random effects.
If you are interested you are welcome to contribute to the repository through pull requests.
We have a short contributing guide vignette.
If you find bugs, please report it in https://github.com/davidissamattos/bpcs/issues
Agresti, Alan. 2003. Categorical Data Analysis. Vol. 482. John Wiley & Sons.
Böckenholt, Ulf. 2001. “Hierarchical Modeling of Paired Comparison Data.” Psychological Methods 6 (1): 49.
Bradley, Ralph Allan, and Milton E Terry. 1952. “Rank Analysis of Incomplete Block Designs: I. The Method of Paired Comparisons.” Biometrika 39 (3/4): 324–45.
Davidson, Roger R. 1970. “On Extending the Bradley-Terry Model to Accommodate Ties in Paired Comparison Experiments.” Journal of the American Statistical Association 65 (329): 317–28.
Davidson, Roger R, and Robert J Beaver. 1977. “On Extending the Bradley-Terry Model to Incorporate Within-Pair Order Effects.” Biometrics, 693–702.
Springall, A. 1973. “Response Surface Fitting Using a Generalization of the Bradley-Terry Paired Comparison Model.” Journal of the Royal Statistical Society: Series C (Applied Statistics) 22 (1): 59–68.
Stan Development Team. 2020. “RStan: The R Interface to Stan.” https://mc-stan.org/.