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An overview of different sequential sampling models

In my previous blogs about the diffusion decision model (DDM; Ratcliff, 1978), I already mentioned that this is a prominent model of a larger class called sequential sampling models (SSMs) or also commonly referred to as information accumulation models (IAMs). To describe the most important IAMs, it is helpful to categorize them into classes based on their underlying assumptions. The Figure below provides an overview of seven important models; it is a modified and extended figure from: Ratcliff & Smith, 2004. In this overview, I concentrate on models that simulate accumulation processes as diffusion processes, which include within-trial variability, thereby excluding linear ballistic accumulation models.

At a first level, IAMs can be distinguished based on whether they assume a relative or an absolute decision criterion. Absolute Evidence Accumulation Models introduce a separate decision process (i.e., accumulator) for each response alternative. Therefore, the accumulator that has first reached a decision criterion represents the chosen alternative. In contrast, Relative Evidence Accumulation Models introduce a single decision process for all response alternatives. In so doing, the accumulation of evidence over time represents the net evidence for one response alternative (relative to other response alternatives). A response is initiated once the accumulated net evidence reaches a decision criterion.

 

At a second level, IAMs can be distinguished based on whether they assume an evidence accumulation process in discrete time steps or in infinitesimal small time steps (i.e., continuous process).

 

At a third level, IAMs can be distinguished based on their specifications of the evidence accumulation process. Considering Relative Evidence Accumulation Models: Diffusion Decision Models (DDMs; Ratcliff, 1978) integrate evidence continuously over time. The path of evidence accumulation is modeled as a noisy wiener process with a mean (referred to as drift rate) and a variance (referred to as diffusion coefficient). In contrast, Ornstein-Uhlenbeck models (OUMs; Busemeyer & Townsend, 1992) integrate noisy evidence over time with a drift rate that either decreases or increases with time. Random Walk Models (RWMs; Ashby, 1983; Stone, 1960) are the discrete time counterparts of DDMs and OUMs. Considering Absolute Evidence Accumulation Models: Accumulator Models (AMs; Vickers et al., 1971) presume that for each accumulator, evidence is accumulated at discrete time steps (with evidence drawn from a normal distribution at each time step). There are different versions of AMs that vary in their underlying assumption about whether evidence accumulation processes are noisy and/or variable across trials (Linear Ballistic Accumulator Model; Brown & Heathcote, 2008; accumulator and counter models, Ratcliff & Smith, 2004; recruitment models, LaBerge, 1962). For models that assume a race of multiple continuous decision processes, there is an additional level that is important to distinguish. Namely, whether evidence accumulation for each response alternative evolves independently (models 5, 6) or dependently (model 7) from each other. Race Diffusion Models (RDM; Tillman et al., 2020) and Dual Diffusion Models (DuDM; Caplin & Martin, 2016) both assume that accumulators do not affect each other. RDMs are similar to DDMs in that both presume diffusion processes with constant drift rates. In contrast, DuDMs are similar to OUMs in that both presume diffusion processes with time-varying drift rates. Leaky Competing Accumulator Models (LCAM; Usher & McClelland, 2001) presume that accumulators affect each other. Specifically, the LCAM by Usher and McClelland presumes mutual inhibition between accumulators so that evidence for one accumulator is considered to represent evidence against the other accumulators. In that sense, this model is similar to DDMs but it provides an extension in that the LCAM framework can be extended to more than two response alternatives. The LCAM has been proposed for linking psychological and neural measures since it includes neural properties such as lateral inhibition and decay of incoming signals. Moreover, the LCAM incorporates a passive exponential decay of accumulated evidence (referred to as leakage).

 

The seven IAMs introduced above differ in their predictive ability to account for the speed of correct versus relative responses. For some models, this depends on the specification of parameters (e.g., OUMs). For other models, this depends on the type of variability parameters included (e.g., DDMs). Moreover, some of those models (at least in their classical form) are only applicable to tasks that involve binary choices (e.g., DDMs). Other models can be extended to account for tasks that involve multiple choice alternatives (e.g., AMs).

 

There are some fantastic papers that provide further fantastic overviews and comparisons: (Bogacz et al., 2006; Ratcliff & Smith, 2004; Voss et al., 2019).

 

References

Ashby, F. G. (1983). A biased random walk model for two choice reaction times. Journal of Mathematical Psychology, 27(3), 277–297. https://doi.org/10.1016/0022-2496(83)90011-1

Bogacz, R., Brown, E., Moehlis, J., Holmes, P., & Cohen, J. D. (2006). The physics of optimal decision making: A formal analysis of models of performance in two-alternative forced-choice tasks. Psychological Review, 113, 700–765. https://doi.org/10.1037/0033-295X.113.4.700

Brown, S. D., & Heathcote, A. (2008). The simplest complete model of choice response time: Linear ballistic accumulation. Cognitive Psychology, 57(3), 153–178. https://doi.org/10.1016/j.cogpsych.2007.12.002

Busemeyer, J. R., & Townsend, J. T. (1992). Fundamental derivations from decision field theory. Mathematical Social Sciences, 23(3), 255–282. https://doi.org/10.1016/0165-4896(92)90043-5

Caplin, A., & Martin, D. (2016). The Dual-Process Drift Diffusion Model: Evidence from Response Times. Economic Inquiry, 54(2), 1274–1282. https://doi.org/10.1111/ecin.12294

LaBerge, D. (1962). A recruitment theory of simple behavior. Psychometrika, 27(4), 375–396. https://doi.org/10.1007/BF02289645

Ratcliff, R. (1978). A theory of memory retrieval. Psychological Review, 85(2), 59–108. https://doi.org/10.1037/0033-295X.85.2.59

Ratcliff, R., & Smith, P. L. (2004). A Comparison of Sequential Sampling Models for Two-Choice Reaction Time. Psychological Review, 111(2), 333–367. https://doi.org/10.1037/0033-295X.111.2.333

Stone, M. (1960). Models for choice-reaction time. Psychometrika, 25(3), 251–260. https://doi.org/10.1007/BF02289729

Tillman, G., Van Zandt, T., & Logan, G. D. (2020). Sequential sampling models without random between-trial variability: The racing diffusion model of speeded decision making. Psychonomic Bulletin & Review, 27(5), 911–936. https://doi.org/10.3758/s13423-020-01719-6

Usher, M., & McClelland, J. L. (2001). The time course of perceptual choice: The leaky, competing accumulator model. Psychological Review, 108(3), 550–592. https://doi.org/10.1037/0033-295X.108.3.550

Vickers, D., Caudrey, D., & Willson, R. (1971). Discriminating between the frequency of occurrence of two alternative events. Acta Psychologica, 35(2), 151–172. https://doi.org/10.1016/0001-6918(71)90018-7

Voss, A., Lerche, V., Mertens, U., & Voss, J. (2019). Sequential sampling models with variable boundaries and non-normal noise: A comparison of six models. Psychonomic Bulletin & Review, 26(3), 813–832. https://doi.org/10.3758/s13423-018-1560-4

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