model has five parameters per stimulus and five for the ideal. For one distribution, both means can be set arbitrarily to 0 and both variances to 1. Therefore,
with n stimuli there are 5n + 1 free parameters. There are also n(n -1)/2 df in a
paired comparison experiment. Thus, the model is testable if n(n - 1)/2 > 5n +
1, which holds whenever n > 11.
GRT is capable of predicting identification judgments as well as or better than the
biased-choice model ( Luce, 1963a). However, GRT need not assume that similarity is symmetric. In terms of similarity, GRT contains the traditional Euclidean scaling models as special cases, but is not constrained by any of the distance
axioms. It can also predict violations of the dominance and consistency axioms
that constrain many versions of the feature-contrast model ( Tversky, 1977). As a
theory of preference, it contains the traditional Euclidean unfolding models as
special cases, yet is more general than these models in several respects. For
example, the GRT preference model is not restricted to predicting single-peaked
preference functions and is not constrained by any distance axioms.The value of GRT is that the theory unifies several separate research areas in
perceptual and cognitive psychology. The theory allows for exploration of the
relationships between identification, similarity, preference, and categorization
(see chap. 16). GRT offers a deeper interpretation of the psychological processes
underlying these judgments while avoiding some of the problems associated with
other models of identification, similarity, and preference.
APPENDIX AProof of Proposition 6.2:
|1. ||The dominance axiom states that|
δ(ap,bq) > max[δ(ap,aq), δ(aq,bq)].
Rewriting the dissimilarities in terms of the feature-contrast model, we have for
αg(ap) + βg(bq) -- θg(0 + ̿)
> max[αg(p) + βg(q) - θg(a) , αg(a) + βg(b) - θg(q)],
which will hold when the saliency function is nondecreasing and nonnegative or
g(ap) > g(p), g(ap) > g(a), g(bq) > g(q), and g(bq) > g(b).
|2. ||The consistency axiom states that|
δ(ap,bp) > δ(cp,dp) iff δ(aq,bq) > δ(cq,dq)