Running a Bradley-Terry model for pairwise comparisons with XLSTAT-MX

An Excel sheet with both the data and the results can be downloaded by clicking here.

The data correspond to those used in the book Agresti [Agresti, A. (1990). Categorical Data Analysis. Wiley]. These are the results of baseball games of the American League of the year 1987.

The goal is to fit a Bradley-Terry model by taking home-field advantage into account.

Data format for a Bradley-Terry model

The classical data format for a Bradley-Terry model is the following:

data Bradley-Terry

Each pair of teams is compared and the number of wins for each team are counted in the two columns following. In ties are taken into account, a third column has to be added. If homefield advantage are used, the team 1 is considered to be at home.

Goal of a Bradley-Terry model for pairwise comparisons

In this tutorial, we apply a Bradley-Terry model in order to model the probabily of winning or loosing of baseball teams in the baseball league. The results of the plays are entered in the analysis and a model is obtained with probability of winning or loosing for each team. It enables to take into account the home field advantage and the ties.

Setting up a Bradley-Terry model with home-field advantage

After opening XLSTAT, select the XLSTAT / XLSTAT-MX/ Generalized Bradley-Terry model command, or click on the corresponding button of the XLSTAT- MX toolbar.

menu Bradley-Terry

Once you've clicked on the button, the dialog box appears.

The format of the dataset is such that you have Pairs/Variable table. The dataset is composed by two tables. The first one is the meeting table and the second table corresponds to the results: the first column is the number of wins and the other one the number of losses. The option Two-way table can be used when data are presented in a contingency table.

dialog box Bradley-Terry general

In the Options tab, three approaches are proposed to infer model parameters with some option (Home, Ties, stop conditions and Gamma hyperparameters). The confidence interval level can also be modified. Here, we choose the Bayesian EM algorithm with Home option and we have left all the other options to their default values.

dialog box Bradley-Terry options

The computations begin once you have clicked on OK. The results will then be displayed in a new sheet.

Interpreting the results of a Bradley-Terry analysis

The first results displayed are the statistics for the various teams.

Bradley-Terry summary statistics

Hereafter are the results for the available dataset. We can notice that the value of Home parameter is 1.76. It means that being at home improve the probability of winning.

Bradley-Terry parameters

For instance, given that Detroit team is at home, the probability that this team beats the Cleveland team is 0.776.

Bradley-Terry probability

The generalized Bradley Terry model can be applied in other cases. For example, to compare products in a sensory test.

About KCS

Kovach Computing Services (KCS) was founded in 1993 by Dr. Warren Kovach. The company specializes in the development and marketing of inexpensive and easy-to-use statistical software for scientists, as well as in data analysis consulting.

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