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 than or equal to 2 object or attribute weights from a set of n(n - 1)/2 pairwise preferences expressed on a ratio scale. Saaty's classical solution to this problem (Saaty, 1977, Journal of Mathematical Psychology, 15, 234-281) is to use as an estimate the normalized positive eigenvector corresponding to the Perron eigenvalue of the binary comparison matrix. An alternative procedure, rooted in the statistical theory of linear models, involves minimizing the total sum of square deviations measured on the logarithmic scale (de Jong, 1984, Journal of Mathematical Psychology, 28, 467-478; Crawford and Williams, 1985, Journal of Mathematical Psychology, 29, 387-405). In recent years, extensive simulation studies have revealed that the two methods have a tendency to produce similar estimates. This observation is theoretically strengthened here. In addition, a statistically valid interpretation is provided for Saaty's eigenvalue-based index of cardinal consistency in a set of responses. This makes it possible to derive an approximate distribution 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 matrices that exhibit too large a degree of cardinal inconsistency. (C) 1994 Academic Press, Inc.