The simplest complete model of choice response time: Linear ballistic accumulation
Introduction
The psychological processes that govern decision making have been the focus of detailed study for over half a century. The choices under study are usually quite simple: for example, a perceptual decision about brightness; a choice about one’s memory for a probe stimulus; or even a simple judgment of the legality of word vs. non-word letter strings. Occasionally, choices between more than two alternatives are considered (e.g., Busemeyer and Townsend, 1992, Busemeyer and Townsend, 1993, Lacouture and Marley, 1991, Lacouture and Marley, 1995, Lacouture and Marley, 2004, Usher et al., 2002). Theories of choice are tested on their ability to accommodate empirical patterns of choice probability and latency (response time, RT, although some models have also considered confidence: Vickers and Lee, 1998, Vickers and Lee, 2000). Even when we restrict our focus to binary choices between simple alternatives in a well-controlled environment, the richness of empirical observations is overwhelming. Complicated effects are observed on the shape of RT distributions, the relative speed of correct and incorrect responses, and the interaction of all these with error rates. This richness has resulted in models of choice RT becoming increasingly complicated over the past 50 years (amongst others, see: Laming, 1968, Link and Heath, 1975, McClelland, 1979, McMillen and Holmes, 2006, Ratcliff, 1978, Ratcliff, 1988, Stone, 1960, Usher and McClelland, 2001, Usher et al., 2002, Van Zandt et al., 2000; for reviews see Bogacz et al., 2006, Luce, 1986, Ratcliff and Smith, 2004).
We hope to advance this research effort with a new theory of choice RT—the linear ballistic accumulator (LBA). The LBA is simpler than the leading models in the field, and yet it accommodates all the important empirical phenomena. The LBA also comes with detailed but simple analytic solutions, making it easy to apply. The model’s success in accounting for empirical phenomena is surprising, given it shares essential properties with older models that have proven inadequate. The LBA uses linear, independent response accumulators, as in (Vickers, 1970, Vickers, 1978, Vickers, 1979, Smith and Vickers, 1988, Smith and Vickers, 1989) and some others (e.g., Audley and Pike, 1965, Laberge, 1962, Townsend and Ashby, 1983, Van Zandt et al., 2000). This arrangement has proven inadequate for modeling very fast errors that are sometimes observed (e.g., Ratcliff & Smith, 2004), as well as the non-normality of RT distributions in long-RT tasks. The LBA also uses linear deterministic accumulation, similar to the models of Reddi and Carpenter, 2000, Grice, 1972, Reeves et al., 2005; and yet each of these models also provides an inadequate account of incorrect responses.
The LBA is a greatly simplified instance of the dominant theoretical framework for models of choice RT for the past 50 years: sequential sampling. Beginning with Stone (1960), and continuing through to Ratcliff and Smith (2004), theoretical accounts for the variability in responses and RTs have assumed that a decision is made by the accumulation of “evidence” that varies randomly from moment to moment. A canonical example is illustrated in the left panel of Fig. 1. Suppose a choice is to be made between two competing alternatives—perhaps to classify the letter string “SIRF” as either a valid word, or as a non-word. The two possible responses (either “word” or “non-word”) are identified with separate evidence accumulators. These accumulators gather evidence for each response, increasing their amount of evidence with time. When the amount of evidence in either accumulator reaches a response threshold, the decision corresponding to that accumulator is produced, and the decision time is the amount of time taken to reach the response threshold.1 Most importantly, the evidence accumulation process is stochastic—there are random moment-to-moment fluctuations in the amount of evidence supporting each response alternative. This randomness explains the variability in RTs (e.g., different response times are observed even if an identical stimuli is repeated) and variability in the responses made (e.g., incorrect responses are also observed).
The model we have described is Usher and McClelland’s (2001) “leaky competing accumulator” (also known as the “mutual inhibition” model). As with all successful theories of choice RT, the model includes several extra components in addition to the evidence accumulation process. There have been four particularly important additions: two extra sources of random variability, and two nonlinear processes. The sources of extra random variability allow that the initial amounts of evidence in favor of each response (“start points”) and the average speed of evidence accumulation for each response (“drift rates”) to fluctuate from trial to trial. The nonlinear processes used by Usher and McClelland were passive decay of accumulated evidence, and response competition—evidence accumulating in favor of one response decreases the evidence for the other response.
Brown and Heathcote (2005a) proposed a simplification of Usher and McClelland’s (2001) model. In a break from 50 years of stochastic sequential sampling models, our “ballistic accumulator” omitted the within-trial randomness from the evidence accumulation process, as shown in the center panel of Fig. 1. We demonstrated that the ballistic accumulator accommodated all of the important empirical phenomena, using only the four processes described above (two trial-to-trial variabilities, and two nonlinearities) without any variability in the evidence accumulation itself. Here, we propose a further simplification: omitting the nonlinearities from the ballistic accumulator. This new model, which we call the linear ballistic accumulator (LBA), is illustrated in the right panel of Fig. 1. Evidence accumulates linearly for both responses, without moment-to-moment variability, continuing until the response threshold is reached for one response. Evidence accumulation for each response is also independent of evidence accumulating for other responses.
Linear and independent evidence accumulation is a rare assumption amongst models of choice RT, which usually include response competition explicitly (as in Usher & McClelland, 2001) or implicitly (as in single accumulator models, such as Ratcliff’s, 1978, diffusion model), or assume passive decay of accumulated evidence (Smith & Ratcliff, 2004). Even with its very basic architecture, the LBA model accounts for all the most important empirical phenomena, including RT distribution shape, speed–accuracy tradeoffs, and the relative speed of correct vs. incorrect responses. The simplicity of the LBA provides an important advantage relative to all other models of choice RT: analytic solutions for the predicted distributions and probabilities. Similar solutions are available for some other models of choice RT, although these are not always easy to use. The unique advantage of the LBA model is that, unlike other choice RT models, the analytic solutions extend to choices between any number of response alternatives. In the following sections, we describe how the LBA model relates to previous attempts at theoretical simplification, and then describe its most important mathematical properties (with details in the Appendix A). Finally, we show that the LBA provides a good description of data from five previously published experiments.
Section snippets
Simpler models of RT
The increasing complexity of theories for choice RT has inspired several previous attempts at simplification, with reduced assumptions about variability and nonlinearity in evidence accumulation. We briefly describe five attempts, ending with the ballistic accumulator of Brown and Heathcote (2005a). We highlight the strengths and weakness of each model, and compare them to our proposed LBA model.
The most recent attempt at theoretical simplification is the EZ-diffusion model of Wagenmakers, van
The linear ballistic accumulator
The LBA model represents a choice between N alternatives (N = 2, 3, …) using N different evidence accumulators, one for each response. This is illustrated for the binary (N = 2) case in Fig. 3, which shows two evidence accumulators, one for “Response A” and one for “Response B”. Each evidence accumulator begins the decision trial with a starting amount of evidence (k) that increases at a speed given by the “drift rate” (d). Accumulation continues until a response threshold (b) is reached. The first
Predicting fast and slow errors
The relative speeds of correct and incorrect response times have become important for theories of choice RT (see, e.g., Luce, 1986, Ratcliff and Rouder, 1998, Ratcliff and Smith, 2004). When choices are easy, and participants are told to respond quickly, incorrect responses are faster than correct responses, but when choices are difficult, and accuracy is emphasized, incorrect responses are slower than correct responses. As in other complete models of RT, the LBA accommodates this pattern via
Benchmark tests
We test the LBA model against several sets of benchmarks, using data from five previously published experiments. We first analyze three lexical decision experiments, which illustrate the model’s basic predictions and show that the LBA accounts for response accuracy and for the shape and speed of RT distributions for both correct and incorrect choices. We then extend the model analyses to a brightness discrimination experiment, using unaveraged (individual participant) data. Those data exhibit a
Neurophysiological considerations
The LBA is a simplification of Usher and McClelland’s (2001) leaky competitive accumulator. Usher and McClelland’s model, along with other sequential sampling models, has received support from arguments of neural plausibility. The noisy accumulation of evidence in these models can be likened to changes in the firing rates or membrane potentials of neurons. Indeed, direct recordings from neurons involved in perceptual decisions have been modeled using noisy sequential sampling (see, e.g., Gold
Conclusions
We have developed the linear ballistic accumulator (LBA) model as a complete model of choice RT data, for decisions with any number of alternatives, N = 1, 2, 3, … The model is the only one of its kind to have simple analytic solutions for RT distributions in the N > 2 cases. We have demonstrated that the model accounts for the distribution of RTs for correct and incorrect responses using data from five previously published experiments. The LBA model accommodates complex patterns in the relative speed
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