# Generalized Bradley-Terry model

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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