The Overconstraint of Response Time Models: Rethinking the Scaling Problem

Article excerpt

Theories of choice response time (RT) provide insight into the psychological underpinnings of simple decisions. Evidence accumulation (or sequential sampling) models are the most successful theories of choice RT. These models all have the same "scaling" property-that a subset of their parameters can be multiplied by the same amount without changing their predictions. This property means that a single parameter must be fixed to allow the estimation of the remaining parameters. In the present article, we show that the traditional solution to this problem has overconstrained these models, unnecessarily restricting their ability to account for data and making implicit-and therefore unexamined-psychological assumptions. We show that versions of these models that address the scaling problem in a minimal way can provide a better description of data than can their over-constrained counterparts, even when increased model complexity is taken into account.

Many psychological experiments involve a choice between two alternatives. Despite their apparent simplicity, there are many complicated empirical regularities associated with the speed and accuracy of such choices. Response time (RT) distributions take on characteristic shapes that differ systematically, depending on whether the associated response is correct or incorrect, and depending on any number of experimental manipulations of stimulus properties or of instructions to the participants. A range of theories have been proposed to account for both choice probability and RT when making simple decisions (for reviews, see Luce, 1986; Ratcliff & Smith, 2004). Over the past 40 years, evidence accumulation (or "sequential sampling") models have dominated the debate about the cognitive processes underlying simple decisions (see, e.g., Busemeyer & Townsend, 1993; Ratcliff, 1978, Ratcliff & Smith, 2004; Smith, 1995; Stone, 1960; Usher & McClelland, 2001; Van Zandt, Colonius, & Proctor, 2000).

More recently, evidence accumulation models have been applied more widely, for example, as general tools to measure cognition in the manner of psychometrics (Schmiedek, Oberauer, Wilhelm, Süß, & Wittmann, 2007; Vandekerckhove, Tuerlinckx, & Lee, 2009; Wagenmakers, van der Maas, & Grasman, 2007), and as models for the neurophysiology of simple decisions (see, e.g., Forstmann et al., 2008; Ho, Brown, & Serences, 2009; Smith & Ratcliff, 2004). In light of this growing influence, it is especially important that users of these models are not misled by implicit-and hence unexamined-assumptions.

Evidence accumulation models all share a basic framework wherein, when making a decision, people repeatedly sample evidence from the stimulus. This evidence is accumulated until a threshold amount is reached, which triggers a decision response. These models naturally predict the response made (depending on which response has accumulated the most evidence) and the latency of the response (depending on how long the evidence took to accumulate). We illustrate these models using the example of a lexical decision task, in which a participant must decide whether a string of letters is a valid word (e.g., dog) or not (e.g., dxg). The participant samples information from the stimulus repeatedly and finds some evidence that suggests that the stimulus is a word, and other evidence to suggest that the stimulus is not a word. The participant accrues this information, waiting until there is enough evidence for one of the two options before responding. His or her choice corresponds to the response with the most evidence, and the time taken for this evidence to be accumulated is the response latency.

Over the past four or five decades, dozens of evidence accumulation models have been proposed, and all of them share a mathematical "scaling property": One can multiply a subset of their parameters by an arbitrary amount, without changing any of the model's predictions. …


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