Measurement by Subjective Estimation: Testing for Separable Representations

Studying how individuals compare two given quantitative stimuli, say d1 and d2, 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 d1 and d2, but rather to give their estimates of the ratio p = d1/d2. 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 (Narens 1996, Luce 2002), 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 strongly reject Stevens’ model. As a byproduct, we provide estimates of the psychophysical functions of interest.