Extensions of Narens' theory of ratio magnitude estimation

Abstract

Stevens postulated that the responses of a participant in a ratio scaling experiment can be used directly to construct a psychophysical function. Today, it is generally accepted that the axioms of commutativity and multiplicativity are crucial for the interpretation of the subjects' ratio scaling behaviour. Empirical findings provide evidence that commutativity holds, whereas multiplicativity fails to hold across different sensory modalities. This shows that, in principle, Stevens' direct scaling methods yield measurements on a ratio scale level, but that the numerals occurring in a ratio scaling experiment cannot be taken at face value. Thus, Narens and others introduced a transformation function f, which converts the numerals used in an experiment into the latent mathematical numbers. The aim of the present paper is to specify the (unknown) shape of the transformation function f, by analysing different extensions of the multiplicative property. The results provide evidence that f is either a power function or a logarithmic function.

What are we estimating when we fit Stevens’ power law?

“…Therefore, as multiplicativity has been rejected in every domain in which it has been evaluated (loudness, brightness, contrast, perception of time, size of circles, and more), also W (1) = 1 has been rejected (as a general rule) empirically. Augustin (2006Augustin ( , 2010 presented theoretical extensions of Narens (2006) giving support to the fact that W (p) is either a power or a logarithmic function.…”

“…An important more recent achievement developed in this stream of literature includes the axiomatization of various novel theories which comprehend Stevens' model as a special case (Augustin, 2006(Augustin, , 2010Luce, 2002Luce, , 2008Narens, 1996Narens, , 2002Narens, , 2006. Following a terminology introduced by Luce (2002), we say formally that a psychological scale of subjective intensities can be represented in a separable form if there exist a psychophysical function ψ and a subjective weighting function W such that p is in the following relation with d 1 and d 2 :…”