Fits the Bradley-Terry model for paired-comparison preference tests (R.W. Payne).

### Options

`PRINT` = string tokens |
What to print (`model` , `deviance` , `summary` , `estimates` , `correlations` , `fittedvalues` , `accumulated` , `monitoring` , `confidence` , `preferenceprobabilities` ); default `mode` , `summ` , `esti` |
---|---|

`GROUPS` = factor |
Factor representing different test circumstances |

`COVARIATE` = variates |
Other covariates to include in the model |

`NOMESSAGE` = string tokens |
Which warning messages to suppress (`dispersion` , `leverage` , `residual` , `aliasing` , `marginality` , `vertical` , `df` , `inflation` ); default `*` |

`FPROBABILITY` = string token |
Printing of probabilities for variance and deviance ratios (`yes` , `no` ); default `no` |

`TPROBABILITY` = string token |
Printing of probabilities for t-statistics (`yes` , `no` ); default `no` |

`SELECTION` = string tokens |
Statistics to be displayed in the summary of analysis produced by `PRINT=summary` (`%variance` , `%ss` , `adjustedr2` , `r2` , `dispersion` , `%meandeviance` , `%deviance` , `aic` , `bic` , `sic` ); default `disp` |

`DISPERSION` = scalar |
Dispersion parameter to be used as estimate for variability in s.e.s etc; default 1 |

`PROBABILITY` = scalar |
Probability level for confidence intervals for parameter estimates; default 0.95 |

### Parameters

`WINNERS` = factors |
Specifies the winners in the tests |
---|---|

`LOSERS` = factors |
Specifies the loser in the tests |

`NWINS` = variates or scalars |
Number of wins; default 1 |

`NBINOMIAL` = variates or scalars |
Number of trials; default 1 |

`PREFERENCEPROBABILITIES` = matrices or pointers |
Saves the estimated probability that each object is preferred to other objects |

`LOWERPREFERENCEPROBABILITIES` = matrices or pointers |
Saves the lower values of the confidence intervals for the preference probabilities |

`UPPERPREFERENCEPROBABILITIES` = matrices or pointers |
Saves the upper values of the confidence intervals for the preference probabilities |

`SAVE` = identifiers |
To save the regression save structure |

### Description

In a paired-comparison trial, assessers are given pairs of objects to assess and asked to indicate which of the two they prefer. They occur, for example, in sensory testing of food items, where the aim may be to establish preferred recipes or methods or cooking. Many other activities, including sports matches (where the items are teams that complete in pairs), can be analysed in the same way.

The results of the trial are specified by the `WINNERS`

, `LOSERS`

, `NWINS`

and `NBINOMIAL`

parameters. You can specify the comparisons individually, by setting the `WINNERS`

and `LOSERS`

parameters to a pair of factors, with a unit for every competition. `WINNERS`

specifies the object that was preferred, and `LOSERS`

specifies the one with which it was compared.

Alternatively, it is more efficient to group the comparisons between each pair of objects together. You nominate one as winner and the other as loser, and record them in the corresponding element of the `WINNERS`

and `LOSERS`

factors. You define the number of times that they were compared in a variate to be specified by the `NBINOMIAL`

parameter, and the number of wins in a variate to be specified by the `NWINS`

parameter.

The data are analysed using the Bradley-Terry model (Bradley & Terry 1952), which is fitted as a generalized linear model with binomial distribution and logit link. The underlying assumption is that each item has an underlying “ability” score, which is estimated by the analysis on the log scale. The logit of the probability that one item is preferred to another is estimated by the difference in their estimated scores. For further details, see the *Methods* Section.

The `COVARIATE`

option allows you to specify additional covariates to include in the model. The `GROUPS`

option can specify a factor to define different trials; different ability scores are then estimated for each group. The other options `(PRINT`

, `NOMESSAGE`

, `FPROBABILITY`

, `TPROBABILITY`

, `SELECTION`

, `DISPERSION`

and `PROBABILITY`

) all operate as in the standard regression directives like `FIT`

etc, except that the `PRINT`

option has an additional setting `preferenceprobabilities`

to print a matrix showing the probability that each object is preferred to every other one. These can also be saved using the `PREFERENCEPROBABILITIES`

parameter, and lower and upper values of their confidence intervals can be saved using the `LOWERPREFERENCEPROBABILITIES`

and `UPPERPREFERENCEPROBABILITIES`

parameters. If there are no groups, each of these saves a matrix, with losers on the rows and winners on the columns. If there are groups, they save pointers containing a matrix for each group.

After `RBRADLEYTERRY`

you can use the standard regression output commands, `RDISPLAY`

, `RKEEP`

and so on, in the usual way. The `SAVE`

parameter allows you to save the regression save structure.

Options: `PRINT`

, `GROUPS`

, `COVARIATE`

, `NOMESSAGE`

, `FPROBABILITY`

, `TPROBABILITY`

, `SELECTION`

, `DISPERSION`

, `PROBABILITY`

.

Parameters: `WINNERS`

, `LOSERS`

, `NWINS`

, `NBINOMIAL`

, `PREFERENCEPROBABILITIES`

, `LOWERPREFERENCEPROBABILITIES`

, `UPPERPREFERENCEPROBABILITIES`

, `SAVE`

.

### Method

The model assumes that each object *i* has an underlying “ability” score, τ_{i} say, and that the probability that object i is preferred to object j is given by

p_{ij} = τ_{i} / (τ_{i} + τ_{j} )

= (τ_{i} / τ_{j}) / (1 + (τ_{i} / τ_{j}))

= exp(λ_{i} – λ_{j}) / (1 + exp(λ_{i} – λ_{j}))

where λ_{i} = log(τ_{j}). So

p_{ij} / (1 – p_{ij}) = exp(λ_{i} – λ_{j})

and therefore

logit(p_{ij}) = λ_{i} – λ_{j}.

### Action with `RESTRICT`

You can analyse a subset of the data by restricting any of the factors or variates in the data set.

### Reference

Bradley, R.A., Terry, M.E. (1952). Rank analysis of incomplete block designs I: The method of paired comparisons. *Biometrika*, 39, 324-45.

### See also

Procedures: `GENPROCRUSTES`

, `SAGRAPES`

.

Commands for: Regression analysis.

### Example

CAPTION 'RBRADLEY example',!t('Baseball results from 1987.',\ 'See Agresti, A. (2002). Categorical Data Analysis',\ '(Second edition), Wiley, p.438.'); STYLE=meta,plain FACTOR [LABELS=!t(Milwaukee,Detroit,Toronto,NewYork,Boston,\ Cleveland,Baltimore); NVALUES=42; REFERENCE=7] Winner,Loser READ Winner 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 4 4 4 5 5 6 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 4 4 4 5 5 6 : READ Loser 2 3 4 5 6 7 3 4 5 6 7 4 5 6 7 5 6 7 6 7 7 2 3 4 5 6 7 3 4 5 6 7 4 5 6 7 5 6 7 6 7 7 : VARIATE [VALUES=21(1,-1)] Home VARIATE [NVALUES=42] Wins,Total READ Wins 4 4 4 6 4 6 4 4 6 6 4 2 4 4 6 4 4 6 5 6 2 3 5 3 1 5 5 3 1 5 3 5 5 3 4 6 2 3 4 2 6 4 : READ Total 7 6 7 7 6 6 6 7 6 7 7 6 7 6 6 7 6 7 7 6 6 6 7 6 6 7 7 7 6 7 6 6 7 6 7 7 6 7 6 6 7 7 : RBRADLEY [PRINT=summary,estimates,preference; COVARIATE=Home]\ Winner; LOSER=Loser; NWINS=Wins; NBINOMIAL=Total