When Parameters Collide: A Warning about Categorization Models
Smith, J. David, Psychonomic Bulletin & Review
Similarity-choice (S-C) models of categorization contain two principal mathematical transformations: an exponential-decay similarity function and a choice rule. However, there is a tension between the psychological processes that models emulate and the mathematics they use to do so. To illustrate this, I show that in these models an unappreciated interaction occurs between the mathematical transformations so that the stages of the model essentially cancel each other out. The result is that the model's output reflects its input linearly. This cancellation phenomenon has potentially serious implications regarding the interpretation and use of S-C models. The phenomenon also raises questions about the simplification and psychological grounding of categorization models. Modelers broadly might benefit from an internal analysis of their models, such as that described here.
Mathematical models are part of theory building in cognitive science. At their best, models help us to instantiate, in parameter values, states and processes of mind and to evaluate psychological assumptions by testing their fit to data. However, there is a tension between the psychological processes that models emulate and the mathematics that they use to do so. Data transformations are crucial to modeling. We transform physical measures of stimulus relations into psychical measures, such as similarity. We transform psychical measures into predicted-performance measures. There is always the question of how faithful these transformations are to psychological representation and process. This question is sharpened because we ignore the psychological aspect at the critical time of evaluating a model's fit. This evaluation is purely mathematical, with fit measures comparing predicted performance and observed performance.
One goal of the present article is to illustrate this tension through the use of the influential class of similarity-choice (S-C) categorization models. I will show that an unappreciated interaction often occurs between the mathematical transformations in such a model so that the model's stages essentially cancel each other out. The result is that the model's output reflects its input linearly, despite the presumption of nonlinear psychological transformations. This has potentially serious implications regarding the interpretation and use of these models. This goal resonates with the concern expressed by others (e.g., Maddox & Ashby, 1998) that the psychological grounding of S-C models is tenuous. However, the present article also gives reason for optimism that we might simplify these models and make them more expressive of psychological process and representation.
The second goal of this article is to raise a more general issue about evaluating cognitive models. There is a tendency to evaluate models using an external criterion-that is, their fit to data-to index their success and strengthen their attendant theory (also see Roberts & Pashler, 2000). This tendency has become stronger as models have become more complex, with internal structures that are difficult to query and understand. In contrast, in this article I emphasize that evaluating a model's internal structure and behavior is as important as evaluating its external fit, because a model's internal misbehavior can undermine its interpretative value as surely as poor fit can. Therefore, beyond fit, it is also important to ask whether or not (1) a model's components preserve their psychological meaning, (2) the processes presumably instantiated in the model are expressed by the model, and (3) the components of a model interact in a way that changes the model's character.
The ideas presented in this article apply equally to prototype and exemplar S-C models. They neither promote nor are selectively critical of any representational theory of categorization (e.g., exemplar theory, prototype theory) or any body of categorization research. In this balanced spirit, the article's simulations are based on both kinds of category representation. …