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Biometrika 1990 77(2):265-273; doi:10.1093/biomet/77.2.265
© 1990 by Biometrika Trust
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A continuum of paired comparisons models

HAL STERN

Department of Statistics, Harvard University Cambridge, Massachusetts 02138, U.S.A.

An approach to building models for paired comparisons experiments based on comparisons of gamma random variables is considered. The probability that one object is preferred to a second object is taken to be the probability that one gamma random variable with shape parameter r is less than a second independent gamma random variable with the same shape parameter but a different scale parameter. This is motivated by considering a point scoring competition as the basis for each paired comparison. Different values of r provide different paired comparisons models including the Bradley-Terry model, having r equal to one, and the Thurstone-Mosteller model, achieved as r tends to infinity. Applications of the models to data sets are discussed.

Key Words: Bradley-Terry model • Gamma random variable • Paired comparison • Thurstone-Mosteller model


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