Simulate from the Two-Accumulator Racing Diffusion Model (RDM)

Simulates choice reaction times from a two-accumulator Racing Diffusion Model (RDM). This is a specialized version where exactly two accumulators race towards a common threshold. The model assumes variability in the starting point of the diffusion process, drawn from a uniform distribution. This version is optimized for performance using vectorized operations and allows one (but not both) drift rate to be zero.

Usage

rrdm(n, vzero, vone, bs, bias, ndt)

drdm(x, vzero, vone, bs, bias, ndt, log = FALSE)

Arguments

n: Number of trials to simulate. Must be a positive integer.
vzero: Drift rate for the first accumulator. Must be a single non-negative number.
vone: Drift rate for the second accumulator. Must be a single non-negative number. At least one of vzero or vone must be positive.
bs: Threshold parameter, defined as bs = b - bias, where b is the decision threshold and bias is the maximum starting point. Must be a single positive number.
bias: Maximum starting point parameter. The starting point for each accumulator on each trial is drawn from Uniform(0, bias). Must be a single positive number.
ndt: Non-decision time (encoding and motor time offset). Must be a single non-negative number.
x: Vector of quantiles (observed reaction times).
log: Logical; if TRUE, probabilities p are given as log(p).

Value

bias data frame with n rows and two columns:

rt: The simulated reaction time (minimum finishing time across the two accumulators).
choice: The index of the winning accumulator (1 for vzero, 2 for vone).

Details

The RDM implemented here follows the formulation where the two accumulators have drift rates vzero and vone. The diffusion process starts at a point z drawn from Uniform(0, bias). The process terminates when either accumulator reaches a threshold b. The parameter bs is defined as bs = b - bias, representing the distance from the maximum starting point bias to the threshold b. Therefore, the effective distance to threshold for a given trial is bs = b - z = bs + bias - z.

The finishing time for a single accumulator, given its drift rate v, bs, bias, and ndt, is simulated by:

Sampling a starting point z ~ Uniform(0, bias).
Calculating the distance bs = bs + bias - z.
If v > 0, simulating the time to reach bs from an Inverse Gaussian distribution with mean bs / v and shape bs^2. This simulation uses an internal implementation based on Michael et al. (1976).
If v = 0, the finishing time is considered infinite (Inf).
Adding the non-decision time ndt to finite finishing times.

The function simulates this process for both accumulators using vectorized operations. The accumulator that finishes first determines the choice (1 for vzero, 2 for vone) and the reaction time (RT) for that trial. If one drift rate is zero, the accumulator with the positive drift rate will always win.

This implementation is based on the description and parameters used in: 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, 911-936. doi:10.3758/s13423-020-01738-8 (specifically matching the WaldA component used within their RDM simulation).

References

Michael, J. R., Schucany, W. R., & Haas, R. W. (1976). Generating Random Variates Using Transformations with Multiple Roots. The American Statistician, 30(2), 88–90. doi:10.2307/2683801
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, 911-936. doi:10.3758/s13423-020-01738-8
Folks, J. L., & Chhikara, R. S. (1978). The inverse Gaussian distribution and its statistical application—a review. Journal of the Royal Statistical Society Series B: Statistical Methodology, 40(3), 263-275.

Examples

rdm_pos <- rrdm(n = 1000, vzero = 0.8, vone = 0.6, bs = 0.5, bias = 0.2, ndt = 0.15)