Equivalence: A Novel Basis for Model Analysis

Equivalence: A Novel Basis for Model Analysis Terrence C. Stewart (terry@ccmlab.ca) Robert L. West (robert_west@carleton.ca) Institute of Cognitive Science, Carleton University 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6 Canada Abstract Repeated Binary Choice One of the simplest tasks in experimental psychology is the Repeated Binary Choice (RBC) task. Here, participants make a series of either/or decisions, usually represented by pressing one of two buttons. After each decision, reinforcement feedback of some kind is provided to the participant, indicating how good that choice was. This allows them to refine their choices. For example, choice A may be correct 70% of the time, while choice B is correct the remaining 30% of the time. By giving feedback as to whether the correct choice was made, the subject will eventually choose A more often than B. Of particular interest is how much more often A is chosen over B, and how this changes over time. In much of the early work on this task (e.g., Myers et al., 1963), the feedback was limited to a simple correct/incorrect indication. In such situations, the overall result in human participants is generally characterized as “probability matching”. That is, if A is correct 70% of the time, then the participant will choose A around 70% of the time. This is somewhat surprising, given that the optimal behaviour in this condition is to choose A 100% of the time. It is also possible to include numerical rewards within the task feedback. This can either involve just a single number indicating the reward given for the button that was pressed (the “minimal information” paradigm), or two values can be provided, indicating both the reward the participant has received as well as the reward they would have received if they had have pressed the other button (the “complete feedback” paradigm). These rewards can be probabilistic; for example, pressing button A may give 1 point half of the time and 10 points the other half of the time, while button B may always give 7 points. This work has resulted in a large collection of experimental results about the frequency of choices in various conditions. However, the precise pattern of these choices and the mechanism(s) underlying this performance are unclear. Although the idea that participants tend to match probabilities is prevalent, Friedman and Massaro (1998) note that “probability matching in binary choice . is less robust than most psychologists seem to believe. ” The actual observed behaviour is much more complex, and a variety of models have been proposed to account for it. As cognitive models are developed that are meant to apply to a broad range of phenomena, it is necessary to evaluate how successfully they do so. This is commonly done by measures such as the Mean Squared Error. We propose and demonstrate an alternate approach based on a measure of statistical equivalence. Instead of using sample means, this method uses confidence intervals, and places an upper bound on how wrong the model may be, given the uncertainties in the data. We apply this to the RELACS model in various different repeated binary choice tasks. We show that the equivalence measure identifies ranges of canonical parameter settings that are equally equivalent. It also identifies experimental conditions that are not yet modelled well. Keywords: modeling; statistics; equivalence; repeated binary choice; RELACS; parameter spaces; philosophy of modeling Introduction When evaluating a cognitive model of a phenomenon, it is common practice to determine how closely the model's behaviour matches that of the real system being modelled. This is typically done through finding the mean squared error, and there is often a determination of a “best fit” parameter setting for the model. Other approaches (e.g., Bayesian methods) requiring assumptions about statistical distributions and prior probabilities are not considered here. In (Stewart, 2006) and (Stewart, in press), we have argued for supplementing this approach by also determining the set of equivalent models. This set consists of all models that cannot be statistically distinguished from the original data at a given probability (p<0.05). This set of models (one for each possible combination of parameter settings) determines the space of potential explanations for the phenomenon at hand. 1 In contrast, the best fitting model (however determined) can be seen as the best predictor of the particular data set that was used. In this paper, we apply this approach to a well-established model of repeated binary choice behaviour in humans. Specifically, we examine the Reinforcement Learning Among Cognitive Strategies (RELACS) model (Erev & Barron, 2005), and determine what potential explanations arise from it. This analysis is contrasted with the more typical fitting analysis performed by the original developers of the model. This work is a part of a project examining various cognitive models of this phenomenon to determine key experiments that may distinguish between them. Existing Models Given the simplicity of this task, a wide variety of existing cognitive models could be applied. Indeed, any reinforcement-based model would be a natural approach, including all Reinforcement Learning (RL) systems. There are, of course, further criteria that would be needed before a model can be safely considered to be an explanation. Statistical equivalence (to a specified degree) on the set of relevant measures is necessary for an explanation, but is not sufficient.