Model file to estimate the Linear Ballistic Accumulator (LBA) in EMC2.
Details
Model files are almost exclusively used in design().
Default values are used for all parameters that are not explicitly listed in the formula
argument of design().They can also be accessed with LBA()$p_types.
| Parameter | Transform | Natural scale | Default | Mapping | Interpretation |
| v | - | [-Inf, Inf] | 1 | Mean evidence-accumulation rate | |
| A | log | [0, Inf] | log(0) | Between-trial variation (range) in start point | |
| B | log | [0, Inf] | log(1) | b = B+A | Distance from A to b (response threshold) |
| t0 | log | [0, Inf] | log(0) | Non-decision time | |
| sv | log | [0, Inf] | log(1) | Between-trial variation in evidence-accumulation rate |
All parameters are estimated on the log scale, except for the drift rate which is estimated on the real line.
Conventionally, sv is fixed to 1 to satisfy scaling constraints.
The b = B + A parameterization ensures that the response threshold is always higher than the between trial variation in start point of the drift rate.
Because the LBA is a race model, it has one accumulator per response option.
EMC2 automatically constructs a factor representing the accumulators lR (i.e., the
latent response) with level names taken from the R column in the data.
The lR factor is mainly used to allow for response bias, analogous to Z in the
DDM. For example, in the LBA, response thresholds are determined by the B
parameters, so B~lR allows for different thresholds for the accumulator
corresponding to left and right stimuli (e.g., a bias to respond left occurs
if the left threshold is less than the right threshold).
For race models, the design() argument matchfun can be provided, a
function that takes the lR factor (defined in the augmented data (d)
in the following function) and returns a logical defining the correct response.
In the example below, the match is simply such that the S factor equals the
latent response factor: matchfun=function(d)d$S==d$lR. Then matchfun is
used to automatically create a latent match (lM) factor with
levels FALSE (i.e., the stimulus does not match the accumulator) and TRUE
(i.e., the stimulus does match the accumulator). This is added internally
and can also be used in model formula, typically for parameters related to
the rate of accumulation.
Brown, S. D., & Heathcote, A. (2008). The simplest complete model of choice response time: Linear ballistic accumulation. Cognitive Psychology, 57(3), 153-178. https://doi.org/10.1016/j.cogpsych.2007.12.002
Examples
# When working with lM it is useful to design an "average and difference"
# contrast matrix, which for binary responses has a simple canonical from:
ADmat <- matrix(c(-1/2,1/2),ncol=1,dimnames=list(NULL,"d"))
# We also define a match function for lM
matchfun=function(d)d$S==d$lR
# We now construct our design, with v ~ lM and the contrast for lM the ADmat.
design_LBABE <- design(data = forstmann,model=LBA,matchfun=matchfun,
formula=list(v~lM,sv~lM,B~E+lR,A~1,t0~1),
contrasts=list(v=list(lM=ADmat)),constants=c(sv=log(1)))
#>
#> Sampled Parameters:
#> [1] "v" "v_lMd" "sv_lMTRUE" "B" "B_Eneutral"
#> [6] "B_Eaccuracy" "B_lRright" "A" "t0"
#>
#> Design Matrices:
#> $v
#> lM v v_lMd
#> TRUE 1 0.5
#> FALSE 1 -0.5
#>
#> $sv
#> lM sv sv_lMTRUE
#> TRUE 1 1
#> FALSE 1 0
#>
#> $B
#> E lR B B_Eneutral B_Eaccuracy B_lRright
#> speed left 1 0 0 0
#> speed right 1 0 0 1
#> neutral left 1 1 0 0
#> neutral right 1 1 0 1
#> accuracy left 1 0 1 0
#> accuracy right 1 0 1 1
#>
#> $A
#> A
#> 1
#>
#> $t0
#> t0
#> 1
#>
# For all parameters that are not defined in the formula, default values are assumed
# (see Table above).