single-LBA: Single accumulator of linear ballistic accumulator (LBA)
In rtdists: Response Time Distributions

Description Usage Arguments Details Value Note References Examples

Density, distribution function, and random generation for a single accumulator of the LBA model with the following parameters: A (upper value of starting point), b (response threshold), t0 (non-decision time), and driftrate (v). All functions are available with different distributions underlying the drift rate: Normal (norm), Gamma (gamma), Frechet (frechet), and log normal (lnorm).

dlba_norm(rt, A, b, t0, mean_v, sd_v, posdrift = TRUE, robust = FALSE)

plba_norm(rt, A, b, t0, mean_v, sd_v, posdrift = TRUE, robust = FALSE)

rlba_norm(n, A, b, t0, mean_v, sd_v, st0 = 0, posdrift = TRUE)

dlba_gamma(rt, A, b, t0, shape_v, rate_v, scale_v)

plba_gamma(rt, A, b, t0, shape_v, rate_v, scale_v)

rlba_gamma(n, A, b, t0, shape_v, rate_v, scale_v, st0 = 0)

dlba_frechet(rt, A, b, t0, shape_v, scale_v)

plba_frechet(rt, A, b, t0, shape_v, scale_v)

rlba_frechet(n, A, b, t0, shape_v, scale_v, st0 = 0)

dlba_lnorm(rt, A, b, t0, meanlog_v, sdlog_v, robust = FALSE)

plba_lnorm(rt, A, b, t0, meanlog_v, sdlog_v, robust = FALSE)

rlba_lnorm(n, A, b, t0, meanlog_v, sdlog_v, st0 = 0)

`rt`	a vector of RTs.
`A`	start point interval or evidence in accumulator before beginning of decision process. Start point varies from trial to trial in the interval [0, `A`] (uniform distribution). Average amount of evidence before evidence accumulation across trials is `A`/2.
`b`	response threshold. (`b` - `A`/2) is a measure of "response caution".
`t0`	non-decision time or response time constant (in seconds). Lower bound for the duration of all non-decisional processes (encoding and response execution).
`mean_v, sd_v`	mean and standard deviation of normal distribution for drift rate (`norm`). See `Normal`
`posdrift`	logical. Should driftrates be forced to be positive? Default is `TRUE`. (Uses truncated normal for random generation).
`robust`	logical. Should robust normal distributions be used for `norm` and `lnorm`? Can be helpful in rare cases but is approximately three times slower than the non-robust versions. Default is `FALSE`.
`n`	desired number of observations (scalar integer).
`st0`	variability of non-decision time, such that `t0` is uniformly distributed between `t0` and `t0` + `st0`. Only available in random number generation functions `rlba_`.
`shape_v, rate_v, scale_v`	shape, rate, and scale of gamma (`gamma`) and scale and shape of Frechet (`frechet`) distributions for drift rate. See `GammaDist` or `frechet`. For Gamma, scale = 1/shape and shape = 1/scale.
`meanlog_v, sdlog_v`	mean and standard deviation of lognormal distribution on the log scale for drift rate (`lnorm`). See `Lognormal`.

These functions are mainly for internal purposes. We do not recommend to use them. Use the high-level functions described in /link{LBA} instead.

All functions starting with a d return the density (PDF), all functions starting with p return the distribution function (CDF), and all functions starting with r return random response times and responses (in a matrix).

Density (i.e., dlba_), distribution (i.e., plba_), and random derivative (i.e., rlba_) functions are vectorized for all parameters (i.e., in case parameters are not of the same length as rt, parameters are recycled). Furthermore, the random derivative functions also accept a matrix of length n in which each column corresponds to a accumulator specific value (see rLBA for a more user-friendly way).

Brown, S. D., & Heathcote, A. (2008). The simplest complete model of choice response time: Linear ballistic accumulation. Cognitive Psychology, 57(3), 153-178. doi:10.1016/j.cogpsych.2007.12.002

Donkin, C., Averell, L., Brown, S., & Heathcote, A. (2009). Getting more from accuracy and response time data: Methods for fitting the linear ballistic accumulator. Behavior Research Methods, 41(4), 1095-1110. doi:10.3758/BRM.41.4.1095

Heathcote, A., & Love, J. (2012). Linear deterministic accumulator models of simple choice. Frontiers in Psychology, 3, 292. doi:10.3389/fpsyg.2012.00292

## random number generation using different distributions for v:
rlba_norm(10, A=0.5, b=1, t0 = 0.5, mean_v=c(1.2, 1), sd_v=c(0.2,0.3))
rlba_gamma(10, A=0.5, b=1, t0 = 0.5, shape_v=c(1.2, 1), scale_v=c(0.2,0.3))
rlba_frechet(10, A=0.5, b=1, t0 = 0.5, shape_v=c(1.2, 1), scale_v=c(0.2,0.3))
rlba_lnorm(10, A=0.5, b=1, t0 = 0.5, meanlog_v=c(1.2, 1), sdlog_v=c(0.2, 0.3))

# use somewhat plausible values for plotting:
A <- 0.2
b <- 0.5
t0 <- 0.3

# plot density:
curve(dlba_norm(x, A=A, b=b, t0=t0, mean_v = 1.0, sd_v = 0.5), ylim = c(0, 4),
      xlim=c(0,3), main="Density/PDF of LBA versions", ylab="density", xlab="response time")
curve(dlba_gamma(x, A=A, b=b, t0=t0, shape_v=1, scale_v=1), add=TRUE, lty = 2)
curve(dlba_frechet(x, A=A, b=b, t0=t0, shape_v=1,scale_v=1.0), add=TRUE, lty = 3)
curve(dlba_lnorm(x, A=A, b=b, t0=t0, meanlog_v = 0.5, sdlog_v = 0.5), add=TRUE, lty = 4)
legend("topright", legend=c("Normal", "Gamma", "Frechet", "Log-Normal"), 
      title = expression("Distribution of"~~italic(v)), lty = 1:4)


# plot cdf:
curve(plba_norm(x, A=A, b=b, t0=t0, mean_v=1.0, sd_v=1.0), 
      xlim = c(0, 3),ylim = c(0,1), 
      ylab = "cumulative probability", xlab = "response time",
      main = "Distribution/CDF of LBA versions")
curve(plba_gamma(x, A=A, b=b, t0=t0, shape_v=1,scale_v=1), add=TRUE, lty = 2)
curve(plba_frechet(x, A=A, b=b, t0=t0, shape=1, scale=1), add=TRUE, lty = 3)
curve(plba_lnorm(x, A=A, b=b, t0=t0, meanlog_v=0.5, sdlog_v = 0.5), add=TRUE, lty = 4)
legend("bottomright", legend=c("Normal", "Gamma", "Frechet", "Log-Normal"), 
       title = expression("Distribution of"~~italic(v)), lty = 1:4)