Non-Additive Quantity Measurement Model

Abstract

This work considers a model for measuring non-additive quantities, in particular a model for subjective measurement. The purpose of this work was to develop the measurement theory and form of a measurement model that uses the corrected S. Stevens measurement model.A generalized structure was considered that included an empirical system, a mathematical system, and a homomorphism of the empirical system into a numerical system. The main shortcomings of classical measurement theories seem to be: 1) homomorphism does not display operations (in this case, one cannot speak of the meaningfulness of the model); and 2) there is no empirical measurement model that could confirm the existence of a homomorphism. To overcome the shortcomings of existing theories a definition of the measurement equation is given. As a result a measurement model is obtained that is free from the shortcomings of classical measurement theories. The model uses the corrected model of S. Stevens and the reflection principle of J. Barzilai.The measurement model was tested using laws that were obtained empirically. Using the model it is shown that Fechnerʼs empirical law is equivalent to Stevensʼs empirical law. This means that the problem which has attracted attention of many researchers for almost a century, has been solved.A numerical example demonstrates the possibilities of the proposed measurement model. It is shown that the model can be used for extended analysis of expert assessments.

Keywords

• measurement theory
• fechnerʼs law
• stevensʼ law
• rasch model
• concept of meaningfulness