The modulating influence of category size on the classification of exception patterns

Abstract

Generalization gradients to exception patterns and the category prototype were investigated in two experiments. In Experiment 1, participants first learned categories of large size that contained a single exception pattern, followed by a transfer test containing new instances that had a manipulated similarity relationship to the exception or a nonexception training pattern as well as distortions of the prototype. The results demonstrated transfer gradients tracked the prototype category rather than the feedback category of the exception category. In Experiment 2, transfer performance was investigated for categories varying in size (5, 10, 20), partially crossed with the number of exception patterns (1, 2, 4). Here, the generalization gradients tracked the feedback category of the training instance when category size was small but tracked the prototype category when category size was large. The benefits of increased category size still emerged, even with proportionality of exception patterns held constant. These, and other outcomes, were consistent with a mixed model of classification, in which exemplar influences were dominant with small-sized categories and/or high error rates, and prototype influences were dominant with larger sized categories.

We would like to thank the Department of Chemistry at Indiana University for providing us with the source deck and write-up of STEPIT 7.4 (copyright 1975 by J. P. Chandler), and T. P. Roo for his comments on an earlier version of this manuscript.

Notes

¹ The stimuli used in each experiment are distorted polygon-like patterns (Homa, 1978). Distances for these stimuli are better described by a Euclidean (r = 2) than city-block (r = 1) metric. This is consistent with previous modelling and multidimensional scaling of these stimuli (e.g., Homa et al., 1979). Also, participants invariably describe the distorted forms used here in terms of higher order perceptual properties rather than in terms of individual dimensions.

² Predicted gradients were generated using the distances obtained from a multidimensional scaling of selected patterns used in learning and transfer and by selecting values of c and g that closely matched those found in the quantitative fits reported in this manuscript (c = g = 2.50). Selecting different values of c and g does not substantially change the shapes of these functions and does not measurably alter the predictions contained in the Introduction.

³ Because we were ultimately interested in deriving similarity values for quantitative fits, a second group of 24 participants provided pairwise similarity ratings to selected patterns from the same prototypes as those used in Experiment 1. These ratings were then subjected to a nonmetric multidimensional scaling analysis (Kruskal, 1964; Shepard, 1962), whose distances were then used to fit the quantitative predictions. The stimulus pool consisted of 30 different patterns, 10 each from each of three prototypes. The full matrix of 435 interpattern ratings was then multidimensional scaled (Kruskal, 1964; Shepard, 1962) in Dimensions 1–6, using a program called KYST. The stress function declined from about .35 in 1 dimension to less than .10 in 6 dimensions (stress formula 1). The 3-dimensional solution was deemed reasonable (stress = .148) and was selected for all subsequent analyses.

⁴ The quantitative model fitted to the data included classification into the none category, mirroring the form of Busemeyer et al. (1984). Therefore, predicted classification into the prototype, feedback, other, and none category summed to 1.00. However, the use of a “none” category in Experiment 1 proved to be uninformative, with less than 5% of all category patterns classified into this category. The “none” option was dropped in Experiment 2.

⁵ Parameter values were obtained by a program called STEPIT, which finds local minima of a smooth function in several parameters.

⁶ This analysis was also performed for the results from Experiment 1. Although category size was 45, an optimal fit was obtained with a function category size of 7. The prototype contribution, β = .78, was higher than any obtained in Experiment 2.

⁷ Our modelling did not include either differential weighting of the dimensions (obtained from the multidimensional scaling task), as is sometimes done with exemplar modelling, or implementation of the λ response-scaling parameter recently added to the generalized context model. The use of λ, introduced to explain deterministic responding by individual participants, does not always appear in exemplar models (e.g., Zaki & Nosofsky, 2004). Nonetheless, our conclusions are restricted to the models explored and not to other variations that could be considered.

⁸ Zaki and Nosofsky (2004) recently argued that gradient effects obtained with dot pattern stimuli (related to our 9-point forms) may be artificially produced by procedures used to generate distortions. This assertion is false, since their demonstration applies to a 1-category paradigm and cannot be extended to a paradigm in which multiple categories are learned. When participants learn multiple categories, rather than inspecting a single category (as done in their paradigm), correct assignments cannot be based on potential elongation (or other) properties that might produce the gradient (according to their argument) since, presumably, all categories would have these properties. If accuracy of classification cannot be based on this property, then obviously chance performance cannot produce gradients to the prototype. Furthermore, routine inspection of all prototypes and distortions results in the elimination of the occasional peculiar pattern. Finally, we have argued here and elsewhere that prototype gradient effects are manifested most broadly when category size is increased, not generally and less likely when category size is small. In a related vein, the claim of learning during transfer by Zaki and Nosofsky (2004) may be true under some conditions (as when prototypes are repeated in the transfer test, as in their experiment), but their proof is based on an analysis that does not discriminate a response bias interpretation from a learning interpretation. To demonstrate learning within the transfer set, it is necessary to demonstrate performance differences among progressive segments of the transfer test, as in a simple split-half analysis.