A STATISTICAL LOOK AT SAATYS METHOD OF ESTIMATING PAIRWISE PREFERENCES EXPRESSED ON A RATIO SCALE

This article is concerned with the problem of estimating n greater tha n or equal to 2 object or attribute weights from a set of n(n - 1)/2 p airwise preferences expressed on a ratio scale. Saaty's classical solu tion to this problem (Saaty, 1977, Journal of Mathematical Psychology, 15, 234-281) is to use as an estimate the normalized positive eigenve ctor corresponding to the Perron eigenvalue of the binary comparison m atrix. An alternative procedure, rooted in the statistical theory of l inear models, involves minimizing the total sum of square deviations m easured on the logarithmic scale (de Jong, 1984, Journal of Mathematic al Psychology, 28, 467-478; Crawford and Williams, 1985, Journal of Ma thematical Psychology, 29, 387-405). In recent years, extensive simula tion studies have revealed that the two methods have a tendency to pro duce similar estimates. This observation is theoretically strengthened here. In addition, a statistically valid interpretation is provided f or Saaty's eigenvalue-based index of cardinal consistency in a set of responses. This makes it possible to derive an approximate distributio n for this quantity under reasonable assumptions and ultimately leads to a critical examination of the assumptions underlying the so-called 10% cut-off rule recommended by Saaty for identifying response matrice s that exhibit too large a degree of cardinal inconsistency. (C) 1994 Academic Press, Inc.