Multidimensional Models of Perception and Cognition

decision bound that best separates the subject's A and B responses? Second, which model best predicts the data?

Across a variety of experiments, Ashby and Maddox ( 1990, 1992) tested whether each subject's A and B responses were best separated by the optimal bound, the most accurate linear bound, the minimum distance bound, or the independent decisions bounds. Although subjects responded suboptimally in some experiments, the best predictor of the categorization performance of experienced subjects, across all the experiments, was the optimal bound. This was true regardless of whether the optimal rule involved minimum distance, general linear, or nonlinear classification. Similar results were found with the rectangular and circular stimuli, and thus it made little difference whether the components of the stimuli were integral or separable.

Ashby and Maddox ( 1991a) examined the ability of a number of categorization models to account for the data collected in these experiments. When the optimal bound was linear, the best fits were obtained with the general linear classifier or, equivalently, with the probabilistic weighted prototype model. The deterministic exemplar model performed almost as well, but the GCM fit worse than the other models. The poor performance of the GCM appeared to result because responding was less variable than predicted by the GCM.

When the optimal bound was quadratic, the general quadratic classifier provided fits that were substantially better than any other model. The GCM and the deterministic exemplar model each fit the data from one experiment about equally well, but in a second experiment the deterministic exemplar model performed better than the GCM. The prototype model and the general linear classifier provided the poorest fits. The poor fits of the exemplar models in the quadratic conditions apparently occurred because subjects responded suboptimally in these experiments, but in a fashion that required the full flexibility of the general quadratic classifier to mimic. Specifically, a manipulation of attention weights or response biases proved inadequate.

When models of the categorization process are constructed, it is vital to carefully distinguish between the construct of a category and the act of categorization. The Classical Theory does not explicitly separate these two components. Prototype and Exemplar Theories have traditionally represented a category as a set of points in a multidimensional perceptual space. Exemplar Theory has made no explicit assumptions about the distribution of these points but Prototype Theory assumes that the category representation is dominated by the prototype. Both theories have been more explicit about the act of categorization, specifying it as a process in which the presented stimulus is globally matched to the memory representation of one or more category exemplars, and then a response is selected on the basis of these matching operations. General Recognition Theory also

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