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

Writing ξ_{s}(x) for the stimulus intensity judged greater (louder, heavier, brighter) than stimulus intensity x with criterion s, Iverson (2006) proposed a law of similarity ξ_{s}(λx) = γ(λ,s)ξ_{η(λ,s)}(x) to model the dependence of ξ_{s}(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 (2006) analyzed this model assuming the representation s = u(ξ_{s}(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(ξ_{s}(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 non-constancy or of dependence on only one input variable. We find that for some solutions, w has the same form as u, reflective of the context for which u = w, while for other solutions, w takes a different form than u. Comparisons are made to Iverson (2006) and to other work.