# Cognitive Models

This lesson demonstrates how to control the “golem” (McElreath, 2016), the canonical linear ballistic accumulation (LBA) model (Brown & Heathcote, 2008). Please refer to the LBA paper for more details. Here we focus only on how to use this model in the context of Bayesian MCMC.

The LBA model posits a latent matching (M) factor and a response factor (R) on top of regular experimental factors. For most people who are not familiar with the LBA model, the two factors are unfortunately confusing.

Also for the modelling technicalities, the LBA model must fix one of the parameters in the mean_v or sd_v in at least one design cell. This is to serve as scaling purpose, similar to the moment-to-moment variability in the decision diffusion model. For example, the following code fixes sd_v = 1.

require(ggdmc)
model <- BuildModel(
p.map     = list(A = "1", B = "1", t0 = "1", mean_v = "M", sd_v = "1",
st0 = "1"),
match.map = list(M = list(s1 = 1, s2 = 2)),
factors   = list(S = c("s1", "s2")),
constants = c(st0 = 0, sd_v = 1),
responses = c("r1", "r2"),
type      = "norm")


In the above model, I define only one experimental factor, S, for stimulus, which has two levels, s1 and s2. The accuracy, reflected by the M factor, is mapped by s1 = 1 and s2 = 2, meaning that a correct response for s1 (or s2) stimulus is response r1 (or r2) and an error response for s1 (or s2) stimulus is r2 (or r1). Below I use simulate to generate an example data set.

p.vector <- c(A = .75, B = 1.25, t0 = .15, mean_v.true = 2.5, mean_v.false = 1.5)
dat <- simulate(model, nsim = 30, ps = p.vector)
dmi <- BuildDMI(dat, model)  ## DMI stands for data model instance.

dplyr::tbl_df(dmi)
# A tibble: 60 x 3
##    S     R        RT
##    <fct> <fct> <dbl>
##  1 s1    r1    0.608
##  2 s1    r1    0.972
##  3 s1    r2    0.817
##  4 s1    r1    0.718
##  5 s1    r1    0.618
##  6 s1    r1    1.17
##  7 s1    r1    0.730
##  8 s1    r2    0.727
##  9 s1    r1    0.711
## 10 s1    r1    0.688


I use an imaginary experiment with a design of one binary stimulus factor (S), such as left vs. right motion random dots.

match.map = list(M = list(left = “LEFT”, right = “RIGHT”)), responses = c(“LEFT”, “RIGHT”),

In another tutorial, I will fit the model to an empirical data set (Cox & Criss, 2017) to demonstrate fitting HLBA model.

The above match.map code shows the usage of strings, instead of numbers. The “left” and “LFET” could mean the random dots moving left and a left response. From an experimenter’s perspective, this imaginary experiment only has one stimulus (S) factor, which has two levels, random dots moving towards right and moving towards left as defined below.

factors = list(S = c(“left”, “right”)),

Below is the complete model definition.

model <- BuildModel(p.map = list(A = "1", B = "1", t0 = "1", mean_v = "M",
sd_v = "M", st0 = "1"),
constants = c(st0 = 0, sd_v.false = 1, mean_v.false = 0),
match.map = list(M = list(left = "LEFT", right = "RIGHT")),
factors   = list(S = c("left", "right")),
responses = c("LEFT", "RIGHT"),
type      = "norm")


The first option in the BuildModel function, p.map indicates the experimental design. In this example, I assumed the S factor does not affect any LBA latent variables operations. Therefore, I entered p.map as:

p.map = list(A = “1”, B = “1”, t0 = “1”, mean_v = “M”, sd_v = “M”, st0 = “1”),

The notations of the parameters in the LBA model refer to:

1. A, the variability of the starting point,
2. B, the travelling distance of accumulators,
3. b, (not shown in the p.map) the decision threshold,
4. t0, the non-decision time
5. mean_v, the means of the drift rates,
6. sd_v, the standard deviations of the drift rates,
7. st0, the variability of the non-decision time component.

The A = “1”, for instance, indicates that the variability of the starting point is fit by the intercept, 1. The M factor, because it is defined by the LBA model as a latent factor, you still see it in the p.map. It indicates there are two drift rate means in mean_v, one for each accumulator: the accumulator for the correct / matched responses and the accumulator for the error / mismatched responses. Similarly, this is also applied to the standard deviation of the drift rates, sd_v.

The only effect in the model defined in the p.map is that the drift rate for a correct response is larger than that for an error response. This is an assumption based on, in general, psychological literature. This is artificially set at

constants = c(st0 = 0, sd_v.false = 1, mean_v.false = 0),

which enforces mean_v.false = 0. This is to presume (also frequent observed phenomenon) that manifested accuracy rate should usually be greater than chance (50%).

mean_v.false stands for the mean of the drift rate of the error (false) accumulator. Because it is always zero, the correct drift rate, mean_v.true, if drawn from a truncated normal distribution bounded by 0 and Inf, will always be larger than the error drift rate.

### Demo 1

1. Fast and error prone performance This demonstration shows how I control the LBA golem to simulate fast and error prone RT distributions. I defined a true parameter vector, defining sd_v.true = (0.66), which is smaller than sd_v.false = 1. This seems often seen in empirical data.
pvec1 <- c(A = 1, B = 0, t0 = .2, mean_v.true = 1, sd_v.true = 0.66)
dat1  <- simulate(model, ps = pvec1, nsim = 1e4)
dmi1  <- BuildDMI(dat1, model)


In the following, I used functions in dplyr to print out the mean response times and accuracy for each stimulus types. The results showed:

1. Error and correct responses have similar average RTs.
2. Stimulus type 1 and stimulus type 2 have similar rates of correctness.
library(dplyr)

## dplyr
library(dplyr)
dat1$C <- dat1$S == tolower(dat1\$R)
d <- dplyr::tbl_df(dat1)

## Print average RTs and accuracy rates for each condition
group_by(d, S, C) %>% summarize(m = mean(RT))
## A tibble: 4 x 3
## Groups:   S [?]
##   S     C         m
##   <fct> <lgl> <dbl>
## 1 left  FALSE 0.624
## 2 left  TRUE  0.645
## 3 right FALSE 0.639
## 4 right TRUE  0.634

group_by(d, S, C) %>% summarize(m = length(RT) / 1e4)
## A tibble: 4 x 3
## Groups:   S [?]
##   S     C         m
##   <fct> <lgl> <dbl>
## 1 left  FALSE 0.391
## 2 left  TRUE  0.609
## 3 right FALSE 0.392
## 4 right TRUE  0.608

## data.table
library(data.table)
DT <- data.table(dat1)

## Print average RTs and accuracy for each condition
DT[, .(MRT = round(mean(RT), 3)), .(S, C)]
##        S     C   MRT
## 1:  left  TRUE 0.645
## 2:  left FALSE 0.624
## 3: right  TRUE 0.634
## 4: right FALSE 0.639

prop <- DT[, .N, .(S, C)]
prop[, NN := sum(N), .(S)]
prop[, acc := round(N/NN, 2)]
## Print accuracy rates for each condition
prop

##        S     C    N    NN  acc
## 1:  left  TRUE 6092 10000 0.61
## 2:  left FALSE 3908 10000 0.39
## 3: right  TRUE 6079 10000 0.61
## 4: right FALSE 3921 10000 0.39