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Shifted Inverse Weibull Model

Source: R/model_invweibull.R
rcogmod_invweibull.Rd

Density, random generation, and brms custom family for the shifted Inverse Weibull distribution. A Inverse Weibull-distributed decision time is shifted by a non-decision time ndt, and a fixed proportion poutlier of responses is generated by an outlier process instead of by the decision process.

Functions:

  • rcogmod_invweibull(): Simulates random draws.

  • dcogmod_invweibull(): Computes the density (likelihood).

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

  • cogmod_invweibull(): Creates a brms::custom_family().

  • cogmod_invweibull_stanvars(): Generates the stanvars to pass to brm().

Usage

rcogmod_invweibull(n, mu = 3, sigma = 0.4, ndt = 0.2, poutlier = 0)

dcogmod_invweibull(
  x,
  mu = 3,
  sigma = 0.4,
  ndt = 0.2,
  poutlier = 0,
  log = FALSE
)

pcogmod_invweibull(
  q,
  mu = 3,
  sigma = 0.4,
  ndt = 0.2,
  poutlier = 0,
  lower.tail = TRUE,
  log.p = FALSE
)

cogmod_invweibull(
  link_mu = "softplus",
  link_sigma = "softplus",
  link_ndt = "log",
  link_poutlier = "logit",
  predict_outliers = FALSE
)

cogmod_invweibull_lpdf_expose()

cogmod_invweibull_stanvars()

log_lik_cogmod_invweibull(i, prep)

posterior_predict_cogmod_invweibull(i, prep, predict_outliers = NULL, ...)

posterior_epred_cogmod_invweibull(prep, predict_outliers = NULL)

Arguments

n

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

mu

Shape of the inverse Weibull (Frechet) decision time. Must be positive.

sigma

Scale of the inverse Weibull (Frechet) decision time. Must be positive.

ndt

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

poutlier

Proportion of responses generated by the outlier process rather than by the decision process. Range: [0, 1]. At poutlier = 0 the distribution reduces to the plain shifted LogNormal.

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 (see the Censoring section).

log.p

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

Link functions for the parameters.

predict_outliers

Logical; whether posterior_predict() and posterior_epred() should include the outlier component. FALSE (the default) fixes poutlier to zero for prediction, so predictions describe the decision process alone; the likelihood is always the full mixture either way. See with_outliers().

i, prep

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

...

Additional arguments.

Value

rcogmod_invweibull() returns a numeric vector of n simulated reaction times, in seconds. dcogmod_invweibull() returns the density at each element of x - the log density if log = TRUE - recycled to the length of the longest argument. cogmod_invweibull() returns a brms::custom_family object, to put on a brms::bf() formula. cogmod_invweibull_stanvars() returns a brms::stanvars object holding the family's Stan functions block, to pass to brms::brm(), and cogmod_invweibull_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_invweibull() returns a numeric vector holding one log-likelihood value per posterior draw for observation i, and posterior_predict_cogmod_invweibull() a draws x 1 matrix of reaction times simulated for observation i. posterior_epred_cogmod_invweibull() returns a draws x observations matrix of expected reaction times, with Inf wherever the mean does not exist.

Details

mu is the shape and sigma the scale of the inverse Weibull (Frechet) decision time, whose mean is sigma * gamma(1 - 1 / mu) and exists only for mu > 1.

ndt and poutlier mean exactly what they do in cogmod_lognormal(), and with_outliers(), without_outliers(), p_outlier() and cogmod_priors() all work here too. See ?rcogmod_lognormal for why ndt is expressed directly in seconds rather than as a fraction of the fastest observed response, what the half Normal outlier component is for, and why the outlier component's scale is a constant rather than a dpar, and why reaction times have to be in seconds.

posterior_epred() returns Inf where mu <= 1, because the Frechet has no mean there. cogmod_loggamma() nests this family at shape = -1.

Examples

rts <- rcogmod_invweibull(1000, mu = 3, sigma = 0.4, ndt = 0.3, poutlier = 0.02)
hist(rts, breaks = 100, xlab = "RT (s)")


# Responses faster than ndt keep positive density, unlike the unmixed model
dcogmod_invweibull(0.1, ndt = 0.3, poutlier = 0.02)
#> [1] 0.07041307
dcogmod_invweibull(0.1, ndt = 0.3, poutlier = 0)
#> [1] 0