A continuum of paired comparisons models

SUMMARY An approach to building models for paired comparisons experiments based on com- parisons 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. The most common analysis of paired comparisons experiments associates a parameter with each object in the experiment such that the probability that one object is preferred to a second object is a function of the difference between the associated parameters. This linear model approach is described in detail by David (1988). Some nonparametric techniques are discussed by Kendall & Babington-Smith (1939), Remage & Thompson (1966) and Gokhale, Beaver and Sirotnick (1983). A bibliography of the paired com- parisons literature is given by Davidson & Farquhar (1976). The present paper considers a subset of the linear models which are obtained from a physical model of the comparison procedure. We suppose that the outcome of a paired comparison is determined by comparing the waiting time for r events to occur in each of the two processes being compared. Gamma random variables are used to describe these waiting times. The resulting models include the Bradley-Terry (Bradley & Terry, 1952) and Thurstone-Mosteller (Thurstone, 1927; Mosteller 1951) linear models. The following section provides a motivation for models based on gamma random variables. An independent increments gamma process extends the range of applicable models by permitting noninteger as well as integer values of r. The models which are developed for particular values of r are described in ? 3. Relationships with currently used methodologies are described wherever possible. Section 4 describes inference under the gamma model including estimation procedures and goodness-of-fit measures. The gamma approach is applied to several paired comparisons data sets in ? 5.