RTDISTS: DISTRIBUTION FUNCTIONS
FOR ACCUMULATOR MODELS IN R
Henrik Singmann
Matthew Gretton
Scott Brown
Andrew Heathcote
SPEED VERSUS ACCURACY IN OBSERVED BEHAVIOR
Lexical Decision Task
Evidence Accumulation Models
Latent Parameters:
 quality/rate of evidence accumulation
 response caution (position of threshold)
 (non-decision time, position of start point, …)
Exp. 1; Wagenmakers, Ratcliff, Gomez, & McKoon, 2008
RT DISTRIBUTIONS
Types of Errors
Ratcliff/Drift Difussion Model
v:
t0
Slow Errors
Fast Errors
for identifiability: swithin-trials = 1 [or 0.1]
THE MATH: UGLY AND SLOW(ISH)
Density of simple diffusion (an infinite sum, converges slowly for large t)
f
Another approximation (better for large t, Navarro & Fuss, 2009)
f
Between-trial variability's require numerical integration (very slow!), the best code for this has been made available by
the Voss brothers.
Initial fast-dm, Voss & Voss (2007), fast method for CDF, Voss & Voss (2008), solve PDE:
initial conditions
and boundary conditions
Voss, Voss & Lerche (2015) released fast-dm-30 likelihoods, which does likelihoods.
LBA
Linear Ballistic Accumulator
The “Race Equation”
"simplest complete model of choice response time"
A
+ nondecision time t0 and variability st0
Brown & Heathcote (2008)
The density function
for choice i reaching
threshold at time t.
The probability that
choice j has not
reached threshold
by time t.
RTDISTS
R package providing distribution functions for Diffusion model and LBA:
 density functions (PDF): d… [n1PDF for LBA]
 cumulative distribution function (CDF): p… [n1CDF for LBA]
 random derivatives (RNG): r…
already available from CRAN: install.packages("rtdists")
 Diffusion model uses C code by Voss brothers.
 LBA model for four different drift rate distributions (Normal, Gamma, Frechet, log-normal)
Contains no fitting interface, but allows any type of fitting: maximum likelihood,
Bayesian, Chi-Square
 random derivatives allow assessment of model fit via simulation
> require(rtdists)
> s1 <- rlba_norm(1000, A = 0.5, b = 1, t0 = 0.4,
mean_v = c(2, 1), sd_v = 1)
> prop.table(table(s1$response))
#
1
2
# 0.714 0.286
>
#
#
#
#
#
#
#
head(s1)
rt response
1 1.1367528
2
2 0.8015599
2
3 0.8084981
2
4 0.7680607
2
5 0.6380967
1
6 0.6702449
1
> s1 <- rlba_norm(1000, A = 0.5, b = 1, t0 = 0.4, mean_v = c(2, 1), sd_v = 1)
> prop.table(table(s1$response))
#
1
2
# 0.714 0.286
ll_lba <- function(pars, rt, response) {
ds <- split(rt, f = response)
d1 <- n1PDF(ds[[1]], A = pars["A"], b = pars["A"]+pars["b"], t0 = pars["t0"], mean_v = pars[c("v1", "v2")], sd_v = 1, silent=TRUE)
d2 <- n1PDF(ds[[2]], A = pars["A"], b = pars["A"]+pars["b"], t0 = pars["t0"], mean_v = pars[c("v2", "v1")], sd_v = 1, silent=TRUE)
d <- c(d1, d2)
if (any(d == 0)) return(1e6)
else return(-sum(log(d)))
}
start <- c(runif(3, 0.3, 3), runif(2, 0, 0.2))
names(start) <- c("A", "v1", "v2", "b", "t0")
nlminb(start, ll_lba, lower = 0, rt=s1$rt, response=s1$response)
# $par
#
A
v1
v2
b
t0
# 0.4336521 1.9278475 0.9856471 0.5177903 0.3962212
#
# $objective
# [1] 137.3075
#
# $convergence
# [1] 0
#
# $iterations
# [1] 41
#
# $evaluations
# function gradient
# 57
242
#
# $message
# [1] "relative convergence (4)"
> require(rtdists)
> s1 <- rlba_norm(1000, A = 0.5, b = 1, t0 = 0.4,
mean_v = c(2, 1), sd_v = 1)
> s2 <- rlba_norm(1000, A = 0.5, b = 0.6, t0 = 0.4,
mean_v = c(2, 1), sd_v = c(0.5, 1))
> prop.table(table(s2$response))
#
1
2
# 0.723 0.277
> s2 <- rlba_norm(1000, A = 0.5, b = 0.6, t0 = 0.4,
mean_v = c(2, 1), sd_v = c(0.5, 1))
> prop.table(table(s2$response))
#
1
2
# 0.723 0.277
ll_lba <- function(pars, rt, response) {
ds <- split(rt, f = response)
d1 <- n1PDF(ds[[1]], A = pars["A"], b = pars["A"]+pars["b"], t0 = pars["t0"], mean_v = pars[c("v1", "v2")],
sd_v = c(pars["sv"], 1), silent=TRUE)
d2 <- n1PDF(ds[[2]], A = pars["A"], b = pars["A"]+pars["b"], t0 = pars["t0"], mean_v = pars[c("v2", "v1")],
sd_v = c(1, pars["sv"]), silent=TRUE)
d <- c(d1, d2)
if (any(d == 0)) return(1e6)
else return(-sum(log(d)))
}
start <- c(runif(3, 0.3, 3), runif(3, 0, 0.2))
names(start) <- c("A", "v1", "v2", "b", "t0", "sv")
nlminb(start, ll_lba, lower = 0, rt=s2$rt, response=s2$response)
# $par
#
A
v1
v2
b
t0
sv
# 0.5832459 2.2496939 1.2850416 0.1522494 0.3838269 0.4766951
#
# $objective
# [1] -601.8689
#
# $convergence
# [1] 0
#
# $iterations
# [1] 51
#
# $evaluations
# function gradient
# 87
341
#
# $message
# [1] "relative convergence (4)"
data(speed_acc)
# Exp. 1; Wagenmakers, Ratcliff, Gomez, & McKoon, 2008
# remove excluded trials:
speed_acc <- speed_acc[!speed_acc$censor,]
speed_acc$corr <- with(speed_acc, 1-as.numeric(stim_cat == response))
p11 <- speed_acc[speed_acc$id == 11 & speed_acc$condition == "accuracy" & speed_acc$stim_cat == "nonword",]
prop.table(table(p11$corr))
#
0
1
# 0.95833333 0.04166667
ll_lba <- function(pars, rt, response) {
ds <- split(rt, f = response)
d1 <- n1PDF(ds[[1]], A = pars["A"], b = pars["A"]+pars["b"], t0 = pars["t0"], mean_v = pars[c("v1", "v2")], sd_v = c(pars["sv"], 1),
silent=TRUE)
d2 <- n1PDF(ds[[2]], A = pars["A"], b = pars["A"]+pars["b"], t0 = pars["t0"], mean_v = pars[c("v2", "v1")], sd_v = c(1, pars["sv"]),
silent=TRUE)
d <- c(d1, d2)
if (any(d == 0)) return(1e6)
else return(-sum(log(d)))
}
start <- c(runif(3, 0.5, 3), runif(2, 0, 0.2), rnorm(1))
names(start) <- c("A", "v1", "v2", "b", "t0", "sv")
p11_norm <- nlminb(start, ll_lba, lower = c(0, 0, -Inf, 0, 0, 0), rt=p11$rt, response=p11$corr)
p11_norm
# $par
#
A
v1
v2
b
t0
sv
# 0.1182946 1.0449834 -2.7409705 0.4513529 0.1243451 0.2609940
#
# $objective
# [1] -211.4202
#
# $convergence
# [1] 0
#
# $iterations
# [1] 65
#
# $evaluations
# function gradient
# 109
474
#
# $message
# [1] "relative convergence (4)"
q <- c(0.1, 0.3, 0.5, 0.7, 0.9)
p11_q_c <- quantile(p11[p11$corr == 0, "rt"], probs = q)
#
10%
30%
50%
70%
90%
# 0.4900 0.5557 0.6060 0.6773 0.8231
p11_q_e <- quantile(p11[p11$corr == 1, "rt"], probs = q)
#
10%
30%
50%
70%
90%
# 0.4908 0.5391 0.5905 0.6413 1.0653
max_pred <- n1CDF(Inf, A = p11_norm$par["A"], b = p11_norm$par["A"]+p11_norm$par["b"],
t0 = p11_norm$par["t0"], mean_v = c(p11_norm$par["v1"], p11_norm$par["v2"]),
sd_v = c(p11_norm$par["sv"], 1))
# [1] 0.9581341
inv_cdf <- function(x, A, b, t0, mean_v, sd_v, value) {
abs(value - n1CDF(x, A=A, b = b, t0 = t0, mean_v=mean_v, sd_v=sd_v, silent=TRUE))
}
pred_correct <- sapply(q*max_pred, function(y) optimize(inv_cdf, c(0, 3), A = p11_norm$par["A"],
b = p11_norm$par["A"]+p11_norm$par["b"], t0 = p11_norm$par["t0"],
mean_v = c(p11_norm$par["v1"], p11_norm$par["v2"]), sd_v = c(p11_norm$par["sv"], 1),
value = y)[[1]])
# [1] 0.4871693 0.5510531 0.6081732 0.6809982 0.8301352
pred_error <- sapply(q*(1-max_pred), function(y) optimize(inv_cdf, c(0, 3), A = p11_norm$par["A"],
b = p11_norm$par["A"]+p11_norm$par["b"], t0 = p11_norm$par["t0"],
mean_v = c(p11_norm$par["v2"], p11_norm$par["v1"]), sd_v = c(1, p11_norm$par["sv"]),
value = y)[[1]])
# [1] 0.4684436 0.5529506 0.6273629 0.7233882 0.9314874
q <- c(0.1, 0.3, 0.5, 0.7, 0.9)
plot(pred_correct, q*max_pred, type = "b", ylim=c(0, 1), xlim = c(0.4, 1.3),
ylab = "Cumulative Probability", xlab = "Response Time (sec)")
points(p11_q_c, q*prop.table(table(p11$corr))[1], pch = 2)
lines(pred_error, q*(1-max_pred), type = "b")
points(p11_q_e, q*prop.table(table(p11$corr))[2], pch = 2)
legend("right", legend = c("data", "LBA predictions"), pch = c(2, 1), lty = c(0, 1))
speed_acc$boundary <- factor(speed_acc$corr, labels = c("upper", "lower"))
ll_diffusion <- function(pars, rt, boundary)
{
densities <- tryCatch(abs(
drd(rt, boundary=boundary, a=pars[1], v=pars[2], t0=pars[3], z=0.5,
sz=pars[4], st0=pars[5], sv=pars[6])
), error = function(e) 0)
}
if (any(densities == 0)) return(1e6)
return(-sum(log(densities)))
start <- c(runif(2, 0.5, 3), 0.1, runif(3, 0, 0.5))
names(start) <- c("a", "v", "t0", "sz", "st0", "sv")
p11_fit <- nlminb(start, ll_diffusion, lower = 0, rt=p11$rt, boundary=as.character(p11$boundary))
# $par
#
a
v
t0
sz
st0
sv
# 1.3203928 3.2746681 0.3385756 0.3504362 0.2019925 1.0567170
#
# $objective
# [1] -207.5513
#
# $convergence
# [1] 0
#
# $iterations
# [1] 38
#
# $evaluations
# function gradient
# 68
272
#
# $message
# [1] "relative convergence (4)"
max_pred <- do.call(prd, args = c(t = Inf, as.list(p11_fit$par), boundary =
"upper"))
# [1] 0.9389375
i_prd <- function(x, args, value, boundary) {
abs(value - do.call(prd, args = c(t = x, args, boundary = boundary)))
}
pred_correct <- sapply(q*max_pred, function(y) optimize(i_prd, c(0, 3), args =
as.list(p11_fit$par), value = y, boundary = "upper")[[1]])
# [1] 0.4963202 0.5736658 0.6361402 0.7147534 0.8815223
pred_error <- sapply(q*(1-max_pred), function(y) optimize(i_prd, c(0, 3), args
= as.list(p11_fit$par), value = y, boundary = "lower")[[1]])
# [1] 0.4480115 0.5222731 0.5824374 0.6665744 0.8829667
plot(pred_correct, q*max_pred, type = "b", ylim=c(0, 1),
xlim = c(0.4, 1.3), ylab = "Cumulative Probability",
xlab = "Response Time (sec)")
points(p11_q_c, q*prop.table(table(p11$correct))[1], pch = 2)
lines(pred_error, q*(1-max_pred), type = "b")
points(p11_q_e, q*prop.table(table(p11$correct))[2], pch = 2)
legend("right", legend = c("data", "Diffusion predictions"),
pch = c(2, 1), lty = c(0, 1))
nlminb(start, ll_lba, lower = c(0, 0, -Inf, 0, 0, 0), rt=p11$rt, response=p11$corr)
# $par
#
A
v1
v2
b
t0
sv
# 0.1182946 1.0449834 -2.7409705 0.4513529 0.1243451 0.2609940
# $objective
# [1] -211.4202
nlminb(start_frechet, ll_lba_frechet, lower = 0, rt=p11$rt, response=p11$corr)
# $par
#
A
v1
v2
b
t0
sv
# 1.781754 2.279085 1.959484 1.007131 0.353327 5.536361
# $objective
# [1] -174.0797
nlminb(start_gamma, ll_lba_gamma, lower = 0, rt=p11$rt, response=p11$corr)
# $par
# A
v1
v2
b
t0
sv
# 0.07309753 7.85791431 0.62676205 0.96593679 0.24999751 0.36519063
# $objective
# [1] -208.1622
nlminb(start, ll_lba_lnorm, lower = 0, rt=p11$rt, response=p11$corr)
# $par
#
A
v1
v2
b
t0
sv
# 0.001157895 2.056824618 0.140372535 2.360040196 0.315294542 0.453494110
# $objective
# [1] -204.272
Terry, Marley, Barnwal, Wagenmakers, Heathcote, & Brown (in press)
SUMMARY
rtdists intended to provide reference implementation of two most common
accumulator models in R: Diffusion model and LBA
Functions are fast and fully vectorized (allowing trialwise parameters)
Allows implementation of all major estimation methods.