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Inverse Gaussian Model (Shifted Wald)

Source: R/rt_invgaussian.R, R/rt_invgaussian_brms.R
rrt_invgaussian.Rd

Density, distribution function, and random generation for the Shifted Wald distribution, also known as the Shifted Inverse Gaussian distribution. This distribution is commonly used in modeling reaction times in cognitive tasks. It is characterized by a drift rate (drift), a decision threshold (bs), and a non-decision time (ndt).

Functions:

  • rrt_invgaussian(): Simulates random draws from the Shifted Wald model.

  • drt_invgaussian(): Computes the density (likelihood) of the Shifted Wald distribution.

  • prt_invgaussian(): Computes the cumulative distribution function (CDF).

Usage

rrt_invgaussian(n, drift = 3, bs = 0.5, ndt = 0.2)

drt_invgaussian(x, drift = 3, bs = 0.5, ndt = 0.2, log = FALSE)

prt_invgaussian(
  q,
  drift = 3,
  bs = 0.5,
  ndt = 0.2,
  lower.tail = TRUE,
  log.p = FALSE
)

rt_invgaussian_lpdf_expose()

rt_invgaussian_stanvars()

rt_invgaussian(
  link_mu = "softplus",
  link_bs = "softplus",
  link_tau = "logit",
  link_minrt = "identity"
)

log_lik_rt_invgaussian(i, prep, ...)

posterior_predict_rt_invgaussian(i, prep, ...)

posterior_epred_rt_invgaussian(prep)

Arguments

n

Number of observations. If length(n) > 1, the length is taken to be the number required.

drift

Drift rate. Must be positive. Represents the average speed of evidence accumulation. Range: (0, Inf).

bs

Decision threshold (boundary separation). Must be positive. Represents the amount of evidence needed to make a decision. Range: (0, Inf).

ndt

Non-decision time (shift parameter). Must be non-negative. Represents time for processes like stimulus encoding and response execution. Range: [0, Inf).

x

Vector of quantiles (observed reaction times).

log

Logical; if TRUE, probabilities p are given as log(p).

q

Vector of quantiles (observed reaction times).

lower.tail

Logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

log.p

Logical; if TRUE, probabilities p are given as log(p). Defaults to FALSE.

Link functions for the parameters.

i, prep

For brms' functions to run: index of the observation and a brms preparation object.

...

Additional arguments.

Details

The Shifted Wald distribution describes the time it takes for a Wiener diffusion process starting at 0 to reach a threshold bs > 0, given a positive drift rate drift > 0. The resulting time is then shifted by a non-decision time ndt >= 0.

The distribution is mathematically equivalent to shifting an Inverse Gaussian distribution with mean mu = bs / drift and shape parameter lambda = bs^2. That is, ShiftedWald(drift, bs, ndt) = InverseGaussian(mean = bs/drift, shape = bs^2) + ndt.

The random generation algorithm implemented here is based on the method described by Michael, Schucany, and Haas (1976), as used in the statmod package.

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

  • Anders, R., Alario, F., & Van Maanen, L. (2016). The shifted Wald distribution for response time data analysis. Psychological Methods, 21(3), 309–327. doi:10.1037/met0000063

  • Matzke, D., & Wagenmakers, E. J. (2009). Psychological interpretation of the ex-Gaussian and shifted Wald parameters: A diffusion model analysis. Psychonomic Bulletin & Review, 16(5), 798–817. doi:10.3758/PBR.16.5.798

  • 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

# Simulate 1000 RTs
rts <- rrt_invgaussian(1000, drift = 3, bs = 0.5, ndt = 0.2)
# hist(rts, breaks = 50, main = "Simulated Shifted Wald RTs", xlab = "Reaction Time")