Abstract

In a paired comparisons experiment a subject has to indicate which of two ‘treatments’ Ti, Tj is preferred. We observe Oij, the frequency with which Ti is preferred to Tj.in nij comparisons. Under a class of models for such data, which include the Bradley Terry and Thurstone models, P(Ti is preferred to Tj) = F( i – j), where F(.) is a symmetric distribution function and ( i) is a treatment index. For identifiability purposes constraints must be imposed on parameters. One is to assume that ipi = 1, where pi = ln( i); an alternative is ipi = 1. Thus theorems identifying optimal design weights and algorithms for determining them carry over to the maximum likelihood estimation of these parameters. Of course these tools can also be used to determine locally optimal designs for such models. We will explore this fusion of topics, taking the opportunity to expand on the class of models, both for simple paired comparisons data and also for data consisting of orderings or rankings. In particular we will exploit multiplicative algorithms for maximum likelihood estimation.