# Measurement by Subjective Estimation: Testing for Separable Representations

### Abstract

Studying how individuals compare two given quantitative stimuli, say $d_1$ and $d_2$, is a fundamental problem. One very common way to address it is through ratio estimation, that is to ask individuals not to give values to $d_1$ and $d_2$, but rather to give their estimates of the ratio $p = d_1 / d_2$. Several psychophysical theories (the best known being Stevens' power-law) claim that this ratio cannot be known directly and that there are cognitive distortions on the apprehension of the different quantities. These theories result in the so-called separable representations [Luce, R. D. (2002). A psychophysical theory of intensity proportions, joint presentations, and matches. Psychological Review, 109, 520-532; Narens, L. (1996). A theory of ratio magnitude estimation. Journal of Mathematical Psychology, 40, 109-788], which include Stevens' model as a special case. In this paper we propose a general statistical framework that allows for testing in a rigorous way whether the separable representation theory is grounded or not. We conclude in favor of it, but reject Stevens' model. As a byproduct, we provide estimates of the psychophysical functions of interest.

Publication
Journal of Mathematical Psychology, 52(3), 184-201

##### Raffaello Seri
###### Professor of Econometrics

My research interests include statistics, numerical analysis, operations research, psychology, economics and management.