Multidimensional Models of Perception and Cognition

By F. Gregory Ashby | Go to book overview

appear the weakest of the four operational tasks discussed. The assumptions of the Restricted Classification Task are not easily tested, and, even if satisfied, the task is too susceptible to extraneous effects (e.g., stimulus, task, and observer). The Redundancy Task is also susceptible to extraneous variables, such as the subject's level of experience with the task. In addition, the ideal observer frequently violates decisional separability in the redundancy condition, thereby invalidating the Redundancy Task as a test of perceptual separability. The assumptions of Direct Dissimilarity Scaling are more easily tested, but have weak empirical validity (e.g., Tversky, 1977). Despite this fact, consistency, a relation that can be tested on dissimilarity data, is related to perceptual separability and decisional separability (see Proposition 7.2). Of the four operational tests reviewed in this chapter, the Filtering Task provides the best test of perceptual separability, because the ideal observer frequently utilizes decisional separability in the filtering condition.

Data from an identification task provide the most rigorous tests of perceptual separability because converging evidence from several quantitative tests (e.g., marginal response invariance or transmitted information) can be obtained. In addition, the parameters of each perceptual distribution can be estimated, and fits between general and more constrained models can be compared.


APPENDIX A

Proof of Proposition 7.1: The first several steps of the proof are identical to those of Ashby and Perrin ( 1988, Theorem 1) and will not be repeated. Begin with their equation stating

s(A, B) =Φ{-1/2[μA - μB)′∑-1(μA - μB)]1/2}, (A1)

where Φ represents the standard normal cumulative distribution function. Now, since = σr2I, s(A, B) = Φ(-dAB/2), where dAB is the Euclidean distance between means.

Denote the monotonically decreasing function that relates dissimilarity and similarity by g. Then

(A2)

As dAB increases, Φ(-dAB/2) decreases, so g[Φ(-dAB/2)] is increasing, which proves the theorem.

Proof of Proposition 7.2: For simplicity, I derive only Equation 8. The proof for Equation 9 follows analogously. If components A and B are perceptually separable, then

-179-

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