Use this tool to fit a Bradley-Terry model to data obtained from pairwise comparisons. Available in Excel using the XLSTAT software.

## Description of the generalized Bradley-Terry model

The generalized Bradley-Terry model is used to describe possible outcomes when elements of a set are repeatedly compared with one another in pairs.

For example, in a marketing study, k products are evaluated by consumers. The products are submitted in pairs and the consumers are asked to indicate which product they prefer or if they cannot decide.

## Options of the generalized Bradley-Terry Model in XLSTAT

It is possible to choose different inference methods:

Numerical: The model is rewritten as a logistic regression. Ties are not allowed.

Bayesian EM: The parameters are supposed to be distributed as a Gamma distribution. The inference is done via an EM algorithm which aims at updating the prior distributions. The parameters of the complete model (with home-field advantage and ties) cannot be inferred with this algorithm.

Sampling: The parameters are supposed to be distributed as a Gamma distribution. The posterior distribution is obtained by a Gibbs sampler.

Two other model options can also be selected:

Home: Select this option to take home-field advantage into account. In this case, the order of the elements in the pairs table is of importance. The first element is supposed to be at home.

Ties: Select this option if ties are allowed. If the option is enabled, the variables table must have 3 columns.

## Results of the generalized Bradley-Terry model in XLSTAT

Summary statistics: This table displays the descriptive statistics for each element

Estimated parameters: the estimates of the model parameters are given in this table. The standard error and the confidence interval are also provided for each parameter.

Likelihood-based criterion: In this table, several likelihood-based criteria are given (-2*log(Likelihood), BIC, AIC).

Probabilities of winning: This table provides the probability that element ii (in row) beats element jj (in column), given the model parameters.

Convergence graph: This chart displays for each parameter the evolution of the parameter and the corresponding confidence interval.

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