The Analysis of Subjective Judgment Matrices

A popular method for quantifying subjective judgment utilizes the dominant eigenvector of a matrix of paired comparisons. The eigenvector yields a scale of the importance of each element of a collection relative to the others. The scale is based on a matrix of subjective paired comparisons. Thomas Saaty has shown this to be a useful tool for analyzing hierarchical structures in many military and industrial applications: by estimating the scale at each level of a structured problem, this procedure yields the relative importance of the elements at the bottom level of the hierarchy to the goals or output at the top level. The geometric mean vector is computationally easier than and statistically preferable to the eigenvector. Further, the geometric mean vector is applicable to a wider class of problems and has the advantage of arising from common statistical and mathematical models. The statistical advantages are theoretically and empirically demonstrated.