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

摘要

Writing xi(s)(x) for the stimulus intensity judged greater (louder, heavier, brighter) than stimulus intensity x with criterion s, Iverson (2006b) proposed a law of similarity xi(s)(lambda x) = gamma(lambda, s)xi(eta(lambda,s))(x) to model the dependence of xi(s)(x) on x. This model, which has eta(lambda, s) and gamma(lambda, 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(xi(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(xi(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 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. (c) 2020 Elsevier Inc. All rights reserved.