Mathematical Psychology
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Linear Ballistic Accumulator

The Linear Ballistic Accumulator model describes choice as a race between deterministic, linearly increasing accumulators with between-trial variability in drift rate and starting point.

xᵢ(t) = kᵢ + dᵢ·t (deterministic accumulation)

The Linear Ballistic Accumulator (LBA), introduced by Brown and Heathcote in 2008, is an elegantly simple model of multi-alternative choice. Unlike the DDM or LCA, the LBA has no within-trial noise — each accumulator rises linearly and deterministically from its starting point to a common threshold. All variability in RT and choice comes from between-trial variation in drift rates and starting points.

Model Architecture

LBA Model Accumulation: xᵢ(t) = kᵢ + dᵢ · t
Starting point: kᵢ ~ Uniform(0, A)
Drift rate: dᵢ ~ Normal(vᵢ, s)
Decision time: Tᵢ = (b − kᵢ) / dᵢ
Response: choose i with smallest Tᵢ

Advantages

The LBA's deterministic accumulation makes it mathematically tractable — the RT distribution for each accumulator has a closed-form expression (a shifted inverse Gaussian), and the likelihood function can be computed analytically. This makes parameter estimation fast and reliable, enabling application to large datasets and complex experimental designs. The model naturally extends to any number of alternatives without additional assumptions.

Despite its simplicity, the LBA produces the full range of benchmark phenomena: right-skewed RT distributions, the speed-accuracy tradeoff, correct responses being faster than errors (on average), and realistic patterns of individual differences. The LBA has become one of the two dominant evidence accumulation models (alongside the DDM) in cognitive psychology.

Related Topics

References

  1. 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
  2. Donkin, C., Brown, S., Heathcote, A., & Wagenmakers, E.-J. (2011). Diffusion versus linear ballistic accumulation: Different models but the same conclusions about psychological processes? Psychonomic Bulletin & Review, 18(1), 61–69. https://doi.org/10.3758/s13423-010-0022-4
  3. Heathcote, A., & Love, J. (2012). Linear deterministic accumulator models of simple choice. Frontiers in Psychology, 3, 292. https://doi.org/10.3389/fpsyg.2012.00292

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