Some aspects of rank ordering and scaling with paired comparisons data

Abstract

Some aspects of rank ordering and scaling with paired comparisons data are examined. In terms of the mathematical concept “relation” the characterizing features of a “simple ordering” are set out to make explicit those properties which must be exhibited by paired comparisons data if the objects compared are to be rank orderable. Rank ordering is a necessary but not a sufficient condition for scaling. Cases arise where ordered elements are not scalable by Thurstone's procedures. It is argued that the Law of Comparative Judgement is only one of a number of possible scaling models for paired comparisons data; and an alternative is proposed which relates scale distance to the relative incidence of “error” in comparative judgement.