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Ex-Gaussian Model (Classical Parameterization)

Source: R/model_exgaussian.R
rcogmod_exgaussian.Rd

Density, random generation, and brms custom family for the Ex-Gaussian distribution, using the "classical" parameterization familiar to experimental psychologists, in which mu and sigma are the mean and SD of the Gaussian component alone, and tau is the mean of the exponential component (the tail). This is unlike brms's built-in exgaussian() family, in which mu indexes the mean of the entire distribution (Gaussian + exponential components combined).

Functions:

  • rcogmod_exgaussian(): Simulates random draws from the Ex-Gaussian distribution.

  • dcogmod_exgaussian(): Computes the density (likelihood) of the Ex-Gaussian distribution.

  • pcogmod_exgaussian(): Computes the cumulative distribution function (CDF) or survival.

  • cogmod_exgaussian(): Creates a brms::custom_family() for use in brms models.

Usage

rcogmod_exgaussian(n, mu = 0.5, sigma = 0.1, tau = 0.2)

dcogmod_exgaussian(x, mu = 0.5, sigma = 0.1, tau = 0.2, log = FALSE)

pcogmod_exgaussian(
  q,
  mu = 0.5,
  sigma = 0.1,
  tau = 0.2,
  lower.tail = TRUE,
  log.p = FALSE
)

cogmod_exgaussian(
  link_mu = "identity",
  link_sigma = "softplus",
  link_tau = "softplus"
)

cogmod_exgaussian_lpdf_expose()

cogmod_exgaussian_stanvars()

log_lik_cogmod_exgaussian(i, prep)

posterior_predict_cogmod_exgaussian(i, prep, ...)

posterior_epred_cogmod_exgaussian(prep)

Arguments

n

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

mu

Mean of the Gaussian component. Unbounded - it is a location, not a scale - though for RT data it is normally positive. Range: (-Inf, Inf).

sigma

SD of the Gaussian component. Must be positive. Range: (0, Inf).

tau

Mean of the exponential component (the tail). Must be positive. 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 (reaction times, in seconds).

lower.tail

Logical; if TRUE (default), probabilities are P[X <= q], otherwise P[X > q] - the survival, which is what a right-censored response contributes to the likelihood under brms::bf(rt | cens(x) ~ ...). See the Censoring section of rcogmod_invgaussian().

log.p

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

Character of the type of link used to model the ex-Gaussian parameters. Defaults to "identity" for mu and "softplus" for sigma and tau (see Details).

i, prep

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

...

Additional arguments.

Value

rcogmod_exgaussian() returns a numeric vector of n simulated reaction times, in seconds. dcogmod_exgaussian() returns the density at each element of x - the log density if log = TRUE - recycled to the length of the longest argument. pcogmod_exgaussian() returns the cumulative probability at each element of q, honouring lower.tail and log.p. cogmod_exgaussian() returns a brms::custom_family object, to put on a brms::bf() formula. cogmod_exgaussian_stanvars() returns a brms::stanvars object holding the family's Stan functions block, to pass to brms::brm(), and cogmod_exgaussian_lpdf_expose() compiles that Stan code and returns it as an R function, for checking the density outside of a model. The remaining functions are brms post-processing methods, called by brms rather than directly: log_lik_cogmod_exgaussian() returns a numeric vector holding one log-likelihood value per posterior draw for observation i, posterior_predict_cogmod_exgaussian() a draws x 1 matrix of reaction times simulated for observation i, and posterior_epred_cogmod_exgaussian() a draws x observations matrix of expected reaction times.

Details

The Ex-Gaussian distribution is the sum of an independent Normal (Gaussian) random variable with mean mu and SD sigma, and an Exponential random variable with mean tau (rate 1 / tau). Unlike brms's built-in exgaussian() family - in which mu indexes the mean of the entire distribution (Gaussian + exponential components combined) - here mu is the mean of the Gaussian component alone, so that the mean of the full distribution is mu + tau.

This distinction matters because changes in the Gaussian location (mu) and changes in the exponential tail (tau) can offset one another at the level of the overall mean, so effects estimated on brms's default mu can lead to different (and potentially incorrect) inferences than effects estimated on this classical mu.

In the brms custom family (cogmod_exgaussian()), sigma and tau use a "softplus" link (log(1 + exp(x))) rather than "log". Both are scales and must be strictly positive for the density to exist at all. A "log" link would enforce that too, but its curvature explodes as the linear predictor departs from zero, producing extreme gradients and making priors and sampling harder to calibrate - and tau is on the RT scale (seconds), where it can take comparatively large values. "softplus" is positive-constrained like "log" but behaves almost linearly (softplus(x) ~ x) away from zero, so weakly-informative priors can be stated directly on the RT scale.

mu is different, and uses "identity". It is the location of the Gaussian component, not a scale: the convolution is well defined for any real value, and the density integrates to one at mu = 0 or below just as it does above (the Stan lpdf has always accepted a non-positive mu - it checks only sigma and tau). Nothing is gained by constraining it, and two things are lost. First, interpretability, which is most of the reason to prefer the ex-Gaussian in the first place: behind a softplus link a coefficient is not in seconds, and the conversion factor moves with the intercept - the local slope is 0.33 at mu = 0.4 s, 0.39 at 0.5 s and 0.63 at 1 s, so the same effect reads as a different number depending on where the intercept sits. On "identity" a coefficient is seconds, full stop. Second, fidelity: for fast, heavily-tailed data the Gaussian component genuinely belongs near or below zero with tau carrying the mass, and forcing mu > 0 distorts the mu/tau split in exactly the cases where that decomposition is the thing being estimated. This also matches every other implementation - brms's own exgaussian(), retimes, and the estimates reported in the literature - so fitted values are directly comparable.

The cost is that brms's default intercept prior is no longer sensible for mu (student_t(3, 0, 2.5) centred at zero seconds), which is why cogmod_priors() now supplies one.

References

  • 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

Examples

# Simulate 1000 RTs
rts <- rcogmod_exgaussian(1000, mu = 0.5, sigma = 0.1, tau = 0.2)
hist(rts, breaks = 50, main = "Simulated Ex-Gaussian RTs", xlab = "Reaction Time")