On the functional forms in a psychophysical law of similarity under a subtractive representation

Abstract

Writing ξs(x) for the stimulus intensity judged greater (louder, heavier, brighter) than stimulus intensity x with criterion s, Iverson (2006b) proposed a law of similarity ξs(λx)=γ(λ,s)ξη(λ,s)(x) to model the dependence of ξs(x) on x. This model, which has η(λ,s) and γ(λ,s) as parameters, is quite general and may be applied in a number of situations in psychophysics. Iverson (2006b) analyzed this model assuming the representation s=u(ξs(x))−u(x) and derived the possible functional forms for the scale u. In the present work, we extend the analysis to the more general s=u(ξs(x))−w(x) and derive the forms for the scales u and w. We avoid the assumption of differentiability and replace it with an assumption either of nonconstancy or of dependence on only one input variable. We find that for some solutions, w has the same form as u, possibly with different parameters, while for other solutions, w takes a different form than u. Comparisons are made to Iverson (2006b) and to other work.