Multidimensional Models of Perception and Cognition

By F. Gregory Ashby | Go to book overview
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.
Summary
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 A
Proof 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 the inequality:
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 when
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)

-145-

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