Estimating ratio scale values when units are unspecified

Multiplication of a ratio scale by a positive constant (a similarity transformation) causes the values to change to a different unit of measure without upsetting relative ratios between objects. Because the relative ratios of any two objects are uniquely represented, a column vector estimating the same objects can be expressed as a comparison matrix which has no effect of any similarity transformation. Using these principles for several vector estimates, this paper evaluates column averaging approaches for aggregating estimates with unknown units into overall values. The geometric mean is shown to be the only method that is truly independent of the arbitrary unit of measure. Measures of clarity are proposed for the derived scale from the multiple columns. These new modeling ideas are compared with the common techniques used in the Analytic Hierarchy Process.