Relative measurement and the power function
Relative measurement is considered in the context of Nomology, a theory that defines and relates categories of activity that arise in qualitative decision making. A nomological variable is defined to correspond to the amount, weight or importance of any naturally forming cluster or category of qualitative information. This article shows that, in general, the mind measures nomological variables relatively, and changes in a nomological variable relative to its current value. The way the mind assesses the relationship between two related nomological variables generally should be modelled as a power function, and it is appropriate to use a geometric mean to average multiple observations. Empirical evidence from psychophysics is shown to support these results. When synthesising direct and indirect relative comparisons of nomological variables it is proven that the indirect comparisons should be given a weight that is inversely proportional to the number of relative measurements made in the indirect comparison. This result is shown to be consistent with, but not identical with, the use of the row geometric means approach for synthesising pairwise relative comparison matrices such as in the Analytical Hierarchy Process. In the case of more than three variables other terms should be included in the synthesis. The article reports on empirical research that endorses the use of nomological structuring and relative weighting of criteria trees. It shows that decision-makers generally prefer scoring within intervals when comparing alternatives, but can also use relative measurement where there are difficulties with identifying intervals. The former leads to modelling by an additive value function, the latter to a power function.
relative measurement power function nomology psychophysics analytic hierarchy process decision theory ratio scale