Journal of Experimental Psychology: General 1992, Vol. 121, No. 3, 364-381 Copyright 1992 by the American Psychological Association, Inc. 0096-3445/92/S3.00 Selectivity, Scope, and Simplicity of Models: A Lesson From Fitting Judgments of Perceived Depth James E. Cutting Cornell University Nuala P. Brady Cornell University Nicola Bruno University of Trieste Trieste, Italy Cassandra Moore Columbia University When comparing psychological models a researcher should assess their relative selectivity, scope, and simplicity. The third of these considerations can be measured by the models' parameter counts or equation length, the second by their ability to fit random data, and the first by their differential ability to fit patterned data over random data. These conclusions are based on exploration of integration models reflecting depth judgments. Replication of Massaro's (1988a) results revealed an additive model (Bruno & Cutting, 1988), and Massaro's fuzzy-logical model of perception (FLMP) fit data equally well, but further exploration showed that the FLMP fit random data better. The FLMP's successes may reflect not its sensitivity in capturing psycholog- ical process but its scope in fitting any data and its complexity as measured by equation length. Good scientific theories are usually thought to have several properties: They are accurate, simple, broad in scope, inter- nally consistent, and have the ability to generate new research (Kuhn, 1977). When models can be used to instantiate theo- ries, they might reflect these same properties. For our purposes the two key concepts in this set are simplicity and scope. Simplicity can be measured in several ways. We measure it in two: by the number of parameters in a model and, in a way not customary to experimental psychology, by the length of the equation that instantiates a model. Scope can also be measured in various ways, but here we consider how theory or model accounts for all possible data functions, where those functions are generated by a reasonably large sample of ran- dom data sets. Under this construal, broad scope is a mixed blessing. A model with greater scope than another may fit more data functions of interest to the researcher, but simul- taneously it may also fit more functions of no interest. Thus, we propose a new criterion for testing and comparing models: selectivity. We define selectivity as the relative ability of a This research was supported by National Science Foundation Grant BNS-8818971 to James E. Cutting. Results without modeling or simulations were reported briefly at the 28th annual meeting of the Psychonomic Society, Seattle, Washington, November 1987. We thank Dominic W. Massaro for helping us understand and implement the fuzzy-logical model of perception; Michael S. Landy and Mark J. Young for insights into implementing other models; James L. McClelland for a general discussion about modeling; Wil- liam Epstein, James A. Ferwerda, and Mary M. Hayhoe for random discussions related to the topics presented here; Carol L. Krumhansl, Michael S. Landy, Geoffrey R. Loftus, Dominic W. Massaro, and an anonymous reviewer for comments on previous versions of this article; and Nan E. Karwan for sustained interest in and discussions about the project. Correspondence concerning this article should be addressed to James E. Cutting, Department of Psychology, Uris Hall, Cornell University, Ithaca, New York 14853-7601. model to fit data functions of interest with its ability to fit random data factored out. This perspective on modeling arises out of our struggles with three different sources of evidence: first, our continuing empirical study of how individuals use multiple sources of information about the perception of objects laid out in depth (see also Bruno & Cutting, 1988); second, our study of the properties of the models used to fit those data; and third, our investigation of why those models have the data-fitting prop- erties they do. Thus, our presentation is divided into these three parts, followed by a set of suggestions about how future research with psychological models might be conducted. Models of Information Integration and Their Fits to Human Judgments of Depth How do we perceive the layout of objects in depth? This is among the oldest questions in psychology, and answers to it have been myriad. One reason for persistent interest in, and debate over, this query is the existence of multiple sources of information in any scene, which can contribute to perceived depth. One list, reworded and reorganized from Gibson (1950, pp. 71-73), includes binocular disparity, convergence, accom- modation, linear perspective, apparent size, relative motion, occlusion, aerial perspective, height in plane, shading, and texture gradients. To be sure, some theorists deny the exis- tence of multiple sources of information (e.g., Burton & Turvey, 1990), based largely on Gibson's later thoughts about invariance and one-to-one mappings between information and objects or events (see Cutting, 1986, 199la, 1991b). However, for those who accept their existence for the percep- tion of objects in depth (and for the perception of many other properties) a major question arises: How is all this information used? Two general possibilities about information use emerge: Either perceptual information is selected, one source from 364