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Frontiers in Psychology 01 frontiersin.org
A practical introduction to using
the drift diusion model of
decision-making in cognitive
psychology, neuroscience, and
health sciences
Catherine E. Myers 1,2*, Alejandro Interian
3,4 and
Ahmed A. Moustafa
5,6
1 Research and Development Service, VA New Jersey Health Care System, East Orange, NJ, United
States, 2 Department of Pharmacology, Physiology and Neuroscience, New Jersey Medical School,
Rutgers University, Newark, NJ, United States, 3 Mental Health and Behavioral Sciences, VA New
Jersey Health Care System, Lyons, NJ, United States, 4 Department of Psychiatry, Robert Wood
Johnson Medical School, Rutgers University, Piscataway, NJ, United States, 5 Department of Human
Anatomy and Physiology, The Faculty of Health Sciences, University of Johannesburg,
Johannesburg, South Africa, 6 School of Psychology, Faculty of Society and Design, Bond University,
Robina, QLD, Australia
Recent years have seen a rapid increase in the number of studies using
evidence-accumulation models (such as the drift diusion model, DDM) in the
fields of psychology and neuroscience. These models go beyond observed
behavior to extract descriptions of latent cognitive processes that have been
linked to dierent brain substrates. Accordingly, it is important for psychology
and neuroscience researchers to be able to understand published findings
based on these models. However, many articles using (and explaining) these
models assume that the reader already has a fairly deep understanding of
(and interest in) the computational and mathematical underpinnings, which
may limit many readers’ ability to understand the results and appreciate
the implications. The goal of this article is therefore to provide a practical
introduction to the DDM and its application to behavioral data – without
requiring a deep background in mathematics or computational modeling.
The article discusses the basic ideas underpinning the DDM, and explains
the way that DDM results are normally presented and evaluated. It also
provides a step-by-step example of how the DDM is implemented and used
on an example dataset, and discusses methods for model validation and for
presenting (and evaluating) model results. Supplementary material provides R
code for all examples, along with the sample dataset described in the text, to
allow interested readers to replicate the examples themselves. The article is
primarily targeted at psychologists, neuroscientists, and health professionals
with a background in experimental cognitive psychology and/or cognitive
neuroscience, who are interested in understanding how DDMs are used in
the literature, as well as some who may to go on to apply these approaches
in their own work.
TYPE Review
PUBLISHED 09 November 2022
DOI 10.3389/fpsyg.2022.1039172
OPEN ACCESS
EDITED BY
Prakash Padakannaya,
Christ University,
India
REVIEWED BY
Lesley K. Fellows,
McGill University,
Canada
John Almarode,
James Madison University,
UnitedStates
*CORRESPONDENCE
Catherine E. Myers
Catherine.Myers2@va.gov
SPECIALTY SECTION
This article was submitted to
Cognitive Science,
a section of the journal
Frontiers in Psychology
RECEIVED 07 September 2022
ACCEPTED 27 October 2022
PUBLISHED 09 November 2022
CITATION
Myers CE, Interian A and
Moustafa AA (2022) A practical introduction
to using the drift diusion model of
decision-making in cognitive psychology,
neuroscience, and health sciences.
Front. Psychol. 13:1039172.
doi: 10.3389/fpsyg.2022.1039172
COPYRIGHT
© 2022 Myers, Interian and Moustafa. This
is an open-access article distributed under
the terms of the Creative Commons
Attribution License (CC BY). The use,
distribution or reproduction in other
forums is permitted, provided the original
author(s) and the copyright owner(s) are
credited and that the original publication in
this journal is cited, in accordance with
accepted academic practice. No use,
distribution or reproduction is permitted
which does not comply with these terms.
Myers et al. 10.3389/fpsyg.2022.1039172
Frontiers in Psychology 02 frontiersin.org
KEYWORDS
drift diusion model, speed-accuracy tradeo, computational model, decision
making, evidence accumulation model, reaction time
Introduction
An important domain in cognitive psychology and cognitive
neuroscience is decision-making: the process of recognizing
features of the situation in which wend ourselves, considering
numerous possible alternative responses, selecting and executing
one response, observing the outcomes, and adjusting our behavior
accordingly. Disruption to any of these processes can aect
decision-making, with real-world consequences; examples include
addiction, where individuals make decisions to sacrice long-term
health for short term benets of the addictive substance or
behavior (Bechara, 2005; Balodis and Potenza, 2020), but
abnormal decision-making has also been implicated in disorders
ranging from depressive and anxiety disorders (Chen, 2022) to
borderline personality disorders (Hallquist et al., 2018) to
Parkinson’s disease (Frank etal., 2007; Moustafa etal., 2008) to
suicidality (Jollant etal., 2011; Brenner etal., 2015; Dombrovski
and Hallquist, 2017). Better understanding of the cognitive and
brain substrates of abnormal decision-making in these populations
is key to improving both psychological and pharmacological
treatments as well as treatment adherence.
One approach to understanding decision-making is through
computational models, such as the dri diusion model (DDM;
Ratcli, 1978; Ratcli and McKoon, 2008; Ratcli etal., 2016),
which was originally developed to describe how well-trained
participants make rapid decisions between two possible response
alternatives. Such computational models attempt to impute
information about latent cognitive processes based on observable
decision-making behavior. By providing a mathematical
framework to describe behavior, computational models can allow
researchers to make explicit the underlying mechanistic processes
that give rise to observable actions (Montague etal., 2012; Millner
etal., 2020).
Although rst described over 50 years ago, the DDM has
recently enjoyed widespread use, partly due to the development of
powerful and freely-available soware implementing
computationally-intensive model-tting algorithms, and partly
due to an accumulating literature documenting that the DDM can
indeed shed light on latent cognitive processes that are not
necessarily evident from traditional hypothesis-driven methods
of behavioral data analysis (Deghan etal., 2022), and that have
been linked to specic brain regions (Mulder etal., 2012; Mueller
etal., 2017; Weigard and Sripada, 2021; Gupta etal., 2022).
e DDM is thus of broad interest in cognitive psychology
and cognitive neuroscience (Evans and Wagenmakers, 2020),
which has led to a burgeoning literature including many primary
research reports that use the DDM to complement traditional
statistical analysis of the behavioral data. Unfortunately, most if
not all such articles assume readers’ familiarity with modeling
jargon and graphical conventions (such as “parameter recovery
studies,” “non-decision times,” and “hairy caterpillars”), hindering
the ability of many readers to fully understand these results and
their implications. Our own experience in dierent research
institutions has suggested that many in the elds of psychology
and neuroscience are somewhat intimidated by unfamiliar
computational models, or by the time and eort that seems to
berequired to understand these models and their use.
e current article thus aims to provide a reader-friendly
introduction to the DDM and its use. e article discusses the
basic ideas underpinning the DDM, provides a step-by-step
example of how the DDM is implemented and used on an example
dataset, and discusses methods for model validation and
conventions for presenting (and evaluating) model results. e
goal is to provide sucient background for a reader to critically
read and evaluate articles using the DDM – without necessarily
mastering the detailed mathematical underpinnings. However, for
those readers who wish to go a little deeper, the Supplemental
Material provides R script to allow readers to run the examples
discussed in the text, and to generate the data gures and tables
shown in the article (see Appendix).
Importantly, the current article is not meant to oer a
comprehensive review of the DDM literature (for good reviews,
see, e.g., Forstmann etal., 2016; Ratcli etal., 2016; Evans and
Wagenmakers, 2020; Gupta etal., 2022), nor a general tutorial on
good computational modeling practices (see, e.g., Daw, 2011;
Heathcote etal., 2015; Wilson and Collins, 2019); however, it may
provide a useful springboard for cognitive psychologists and
neuroscientists considering the use of the dri diusion model,
and related computational models, in their own research.
Overview of the drift diusion
model
Speeded decision-making tasks in cognitive psychology
require well-trained participants to rapidly choose between two or
more competing responses. Examples include lexical decision
tasks (press one key if the stimulus is a word or another if it is a
non-word), Stroop tasks (press a key corresponding to the color
in which a word is printed, ignoring the semantic content), and
saccadic anker tasks (move the eye in the direction indicated by
a central stimulus, ignoring the directionality of anker stimuli).
On such tasks, even well-trained participants show a speed-
accuracy tradeo: they may increase accuracy at the expense of
Myers et al. 10.3389/fpsyg.2022.1039172
Frontiers in Psychology 03 frontiersin.org
slower (more careful) responding, or make very quick decisions
that are more likely to beerroneous (Schouten and Bekker, 1967;
Wickelgren, 1977). is speed-accuracy tradeo appears to beat
least partially under conscious control, because a participant can
perform dierently when instructed to emphasize speed vs.
accuracy (Ratcli and Rouder, 1998; Voss etal., 2004; Milosavljevic
et al., 2010; Katsimpokis et al., 2020). is complicates the
interpretation of behavioral data. Additionally, dierences in
response time across groups might reect dierent underlying
mechanisms (Voss etal., 2013). For example, two patient groups
might both have slower mean reaction times (RTs) than a healthy
comparison group, but in one patient group this might reect
more cautious decision-making, and in the other it might reect
disease-related slowing of motor responses. Ideally, what is needed
is a way to evaluate data that considers not only accuracy and
speed, but the interaction between them.
To address these issues, a complementary approach to
analyzing the observed behavioral data is to use computational
models that attempt to extract latent cognitive parameters that,
together, could produce the observed distribution of RT and
accuracy data.
e dri diusion model (DDM), rst described by Ratcli
and colleagues (Ratcli, 1978; Ratcli and McKoon, 2008; Ratcli
etal., 2016), is one example of a broader class of models called
evidence accumulation models. ese models conceptualize
decision-making as a process in which, on each trial, individuals
accumulate evidence favoring one or another possible response,
until enough evidence accumulates to reach a criterion or
threshold, at which point a decision is made and the corresponding
response is initiated. Evidence accumulation models are
sometimes called sequential sampling models, reecting the idea
that the nervous system repeatedly (sequentially) obtains bits of
information (samples) from the environment, until a threshold of
evidence is reached. e speed-accuracy tradeo reects a balance
point determining when to stop sampling the environment and
make a decision based on the data at hand.
Like all computational models, the DDM is dened by a series
of mathematical equations, containing a number of parameters,
that can beassigned dierent values. An easy way to think of
parameters is as dials (or control bars) on a stereo system that each
control one aspect of the sound (e.g., treble, bass, volume), and
can beadjusted individually so that together they result in the
desired eect. Similarly, parameters in an evidence accumulation
model may control aspects such as how fast evidence is
accumulated, a built-in bias for one response alternative over
another, and a tendency to emphasize speed or accuracy. Each of
these parameters can beadjusted in the model, aecting how the
model behaves.
In the sections below, wewalk through major steps in the
modeling process, following the ow-chart in Figure1. First,
the remainder of this section provides a high-level description
of the DDM, its parameters, and its use in cognitive psychology.
en weconsider a concrete example of how the DDM might
beapplied to a simple two-choice decision-making task, and
step through the process of model-tting to estimate parameter
values for each participant in our dataset, followed by model
validation and model selection approaches. Finally, wediscuss
how the model results can be reported and subjected to
statistical analysis. We conclude with some thoughts about
evaluating published research using the DDM and other
computational models.
Parameters in the drift diusion model
e DDM starts with the assumption that the RT on each trial,
dened as the time from stimulus onset until execution of the
motor response, can bedecomposed into three parts (Figure2A):
the time required for the nervous system to detect or encode the
stimulus (oen denoted Te), the time to reach a decision about
how to respond to that stimulus (Td), and the time required for
the nervous system to execute the chosen motor response (Tr).
us, on a given trial, the observed reaction time is the sum of
these three components: RT = Te + Td + Tr.
Although it might in principle bepossible to measure Te and
Tr separately, normally the encoding and response time are
lumped together into a single parameter representing non-decision
time (Ter): that portion of the RT that occurs independently of the
decision-making process Td. Given this simplication,
RT = Ter + Td. Typical values of Te r lie in the range 0.1–0.5 s, partly
depending on the complexity of stimuli and the specic motor
responses involved (e.g., people can generally execute saccades
faster than keypresses). It’s usually assumed that Ter may dier
across individuals, but is relatively constant across trials for one
individual performing one task.
e other component of RT is decision time (Td ), which is the
time to make a decision (aer the stimulus is encoded, but before
the chosen response is executed). On a single trial, noisy
information is accumulated across time while the process travels
along a corridor bounded by the two possible responses
(schematized by red line Figure 2B). As progressively more
information is accumulated, evidence in favor of one response will
“push” the decision process closer to the corresponding boundary.
When one of the boundaries is reached, the corresponding
response is selected, and time to reach that boundary denes the
decision time Td on that trial.
By convention, the lower boundary is assigned a value of 0 on
the y-axis and distance to the upper boundary is dened by a
parameter representing boundary separation (a). Larger values of
a mean that the decision-making process must travel further (up
or down) to reach a boundary. e eect of larger a is thus that
decision-making will beslower and more cautious: slower because
more evidence is required before a distant boundary is reached
and a response is triggered, and higher accuracy because it will
berare for the decision process to “mistakenly” cross the wrong
boundary (Lerche etal., 2020). Boundary separation is in arbitrary
units, but is oen assumed to range from about 0.5–2. It is oen
assumed that the degree of boundary separation is at least partly
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under conscious control, depending on whether there is an
emphasis on speed (low a) or accuracy (high a).
On each trial, the decision process starts from a location on
the y-axis dened by a parameter denoting a relative starting point
(z) that ranges from 0 (lower axis) to 1 (upper axis). If z = 0.5, the
starting point is equidistant from the two boundaries. However, if
z approaches 1 (or 0), the decision process starts o close to the
upper (or lower) boundary on every trial, meaning that less
information is required in order to reach that boundary and
initiate the corresponding response. e starting point z therefore
reects a response bias in favor of one or the other response.
e decision-making process in the DDM is assumed to
be noisy (schematized by the jagged red line in Figure 2B),
reecting noisy sensory inputs, stochastic variation in the ring
rate of neurons in the decision-making centers of the brain, and
even momentary uctuations in attention. is noise means that
the same stimulus may not generate the same decision time, or
even the same response, every time it occurs – leading to
variations in RT and response accuracy across trials; multiple
trials with dierent decision times Td are schematized as multiple
red lines in Figure2C. Across many such trials, the average rate at
which evidence accumulates toward the correct boundary is
dened by a parameter denoting dri rate (d), schematized as the
slope of the heavy black line in Figure2C. Dri rate is a measure
of speed of information processing, which may vary depending on
task diculty. For easy tasks with highly discriminable stimuli,
there should bea high dri rate (steep slope up or down), and the
evidence should accumulate quickly and reliably toward the
correct boundary, resulting in fast RTs and high accuracy. For
more dicult tasks or more ambiguous stimuli, the dri rate may
be lower (less steep), meaning that evidence accumulation is
slower and noisier, resulting in slower and more variable RTs.
As summarized in Table1, then, the parameters of the DDM
map onto dierent cognitive processes: speed-accuracy settings
(boundary separation a), response bias (starting point z),
information processing speed (dri rate d), and non-decision time
(Ter). ese parameters are sometimes called “free parameters,” in
the sense that they can take on dierent values (“freely”) – and just
like the knobs on a stereo, changing each parameter aects
DDM behavior.
For example, let us consider a task, schematized in Figure3A,
in which the subject is instructed to execute one response r1 as
quickly as possible whenever stimulus s1 is shown, but a dierent
response r2 whenever stimulus s2 is shown.
FIGURE1
Overview of key steps in the modeling process. Typically, choices relating to model design and implementation should bemade at task design
stage, before collection of empirical data. Then, after data collection and data cleansing, model-fitting is conducted to estimate “best-fitting”
parameters for each participant that allow the model to most closely replicate that participant’s accuracy and reaction time (RT) distributions,
followed by model validation and model selection procedures, before the modeling results are reported and subjected to conventional statistical
analysis.
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As shown in Figure3B, increasing the starting point (z) will
move the starting point closer to the upper boundary, meaning
that the evidence accumulation process has farther to travel to
reach the r1 boundary than to the r2 boundary, making it easier
(and faster) to decide in favor of r2 on any trial. Such a prepotent
response bias for r2 might becreated if, say, r2 responses are much
more frequent or highly-rewarded in the task.
As shown in Figure3C, decreasing the boundary separation
(a) would make both responses faster, without necessarily
favoring one over the other. It also increases the error rate,
because it’s easy to for noise to push the decision-making process
across either boundary. A reduced boundary separation might
happen if, say, the subject had been instructed to respond quickly,
even at the expense of reduced accuracy. Increasing a would have
the opposite eect of increasing response caution and producing
slower RTs.
As shown in Figure3D, increasing the dri rate for one type
of stimulus (here, d.s2) would result in faster evidence
accumulation on s2 trials, while decreasing the other (here, d.s1)
would result in slower evidence accumulation on s1 trials. Dri
rates are typically slower (less steep) under more dicult task
conditions, with decreased stimulus discriminability, or in the
presence of distracting stimuli.
Finally, increasing or decreasing non-decision time Ter would
aect overall RT, without otherwise aecting the decision-making
process. For example, patients with motor dysfunction might have
increased Te r (and overall RT), independent of decision-
making considerations.
Together, the values of the DDM parameters Ter, a, z, and
d interact to affect overall task performance, including both
accuracy and RT. To use the stereo example again, the auditory
effect of changing one parameter (bass) may bevery different
A
B
C
FIGURE2
Schematic of the drift diusion model (DDM). (A) Total reaction time (RT) on each trial is assumed to reflect the time required for the nervous
system to encode the stimulus (Te), the time to make a decision (Td ), and the time to execute the selected motor response (Tr). The encoding and
response time are typically combined into a single parameter, Ter , representing non-decision time on each trial, so that RT = Ter + Td. (B) On each
trial, the DDM assumes a process of noisy evidence accumulation, represented here as a red line, traveling from a starting point (z) toward
boundaries representing two possible responses. When the decision-making process encounters one of the boundaries, the corresponding
response is triggered, and the time to reach that boundary is the decision time Td for that trial. In this schematic, the upper boundary is crossed
and Response 2 is chosen. The separation between the two boundaries (a) is one factor determining Td: the greater the boundary separation, the
further the evidence accumulation process has to travel to reach a boundary, leading to longer decision times on average. The starting point z also
influences Td: if z is placed closer to one boundary, it is easier to reach that boundary than the opposite boundary, leading to a response bias
favoring the nearer boundary. (C) Schematic of the DDM decision-making process for several trials on which the correct response is Response 2
(upper boundary). Noise in the evidence accumulation process means that trials with the same stimulus may have dierent Td (and hence dierent
RT), represented here as various red lines, and may on occasion even reach the opposite boundary, triggering an incorrect response, represented
here as a blue line. The drift rate (d) is the average slope of the evidence accumulation process across a large number of trials.
Myers et al. 10.3389/fpsyg.2022.1039172
Frontiers in Psychology 06 frontiersin.org
depending on whether the value of another (volume) is low or
high. To understand these complex interactions, rather than
using schematics such as Figure3, the DDM can beinstantiated
as a computer program, which includes equations describing
the drift diffusion process, and specific values of each
parameter Ter, a, z, and d. The model is then applied to the
behavioral task: On each trial, the model is presented with a
stimulus (e.g., s1 or s2), and the diffusion process starts from
z and is allowed to run (in effect, creating a trace such as the
ones shown in Figure2C) until it crosses one or the other
boundary at time Td, and triggers the corresponding motor
response (r1 or r2), resulting in reaction time RT = Ter + Td .
Over a series of such trials, usually the same stimuli in the
same order as experienced by a human participant, the model’s
responses and RTs are recorded, producing “simulated data.”
The accuracy and RT distributions in the simulated data can
then becompared against the accuracy and RT distributions
in the empirical data, to see how well the model replicates or
“fits” the empirical data. Typically, the goal is to fine-tune the
parameter values until the model’s accuracy and RT
distributions are as close as possible to the empirical data – a
process called model-fitting or parameter estimation.
Elaborations of the drift diusion model
Before going on, it’s worth noting that the above description
considers a “standard” DDM with four free parameters Ter, a, z,
and d. More elaborate versions can beconsidered. For example, in
many cases, it makes sense to consider dierent dri rates for
dierent trial types or conditions. For example, a lexical decision
task might have two types of stimuli (s1 = nonwords and
s2 = words) but also have easy and hard trials, depending on
whether the trigrams are visually degraded or not, or whether the
non-words are pronounceable or not. In such cases, it may make
sense to allow dierent dri rates for each combination of stimulus
conditions, with the expectation that (for most participants), there
will bea steeper dri rate for trials under the easy condition than
the harder condition. In the schematic of Figure3A, then, instead
of having one dri rate for each stimulus (d.s1 and d.s2), wemight
have one for each conguration of stimulus and condition (d.
s1.hard, d.s1.easy, d.s2.hard, d.s2.easy).
Versions of the DDM have also been considered that include
additional free parameters specifying the amount of trial-by-trial
variation in the DDM parameters (Forstmann et al., 2016);
however, simpler models ignoring this variability can oen
account for observed behavior as well as (and more parsimoniously
than) more complex models (e.g., Dutilh etal., 2019).
Drift diusion model parameters
correspond to latent cognitive processes
e purpose of performing model-tting to estimate
parameter values is to provide some insight into the underlying
cognitive processes. ese processes are latent in the sense that
wecannot observe them directly, only impute them based on the
participant’s pattern of behavior. ese latent processes may
bevery important for understanding the cognitive neuroscience
of decision-making. For example, they may map onto dierent
brain systems, and may vary in principled ways in patient groups
with dierent neuropsychological disorders.
In a computational model, wehave the advantage that those
cognitive processes are made explicit, in the form of the parameter
values governing the model’s behavior. is can provide a way to
identify specic cognitive mechanisms that underlie
group dierences.
A main reason for the recent popularity of the DDM is that
the linkage between these DDM parameters and cognitive
processes has been validated in a number of cognitive
psychology studies, showing that changes in task conditions
can alter DDM parameters in a principled way (e.g.,
Milosavljevic etal., 2010; Mulder etal., 2012). For example,
participants instructed to “work especially carefully and avoid
mistakes,” that is, to emphasize accuracy over speed, show
larger boundary separation a, corresponding to greater
response caution (Voss et al., 2004), while participants
working under time pressure, i.e., emphasizing speed over
TABLE1 Free parameters in a “standard” drift diusion model (DDM),
and associated latent cognitive processes.
DDM
parameter
Parameter
name
Typical range
of values
Cognitive
processes
aBoundary
separation
0.5–2 (in arbitrary
units)
Response caution:
higher a emphasizes
accuracy over speed,
lower a emphasizes
speed over accuracy.
zStarting point 0…1 (as
proportion of a)
Response bias: starting
point nearer to one
boundary leads to
faster and more
common decisions
favoring that response.
dDri rate −5…+5 (values
<0 slope down to
lower boundary)
Speed of evidence
accumulation
processing: can
beaected by task
diculty, stimulus
discriminability,
attention.
Ter Non-decision
time
0.1–0.5 s (cannot
exceed total RT)
Neurological processes
for registering
(encoding) sensory
stimuli and for
executing motor
responses.
Naming conventions for these parameters sometimes vary across soware packages; for
consistency, the above parameter names are used throughout this article.
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accuracy, show reduced boundary separation a (Milosavljevic
etal., 2010). When trials are manipulated so that one response
is more frequent or more highly rewarded, the starting point
z shifts to favor that response (Ratcliff and McKoon, 2008;
Mulder et al., 2012; Arnold et al., 2015). When stimulus
discriminability is varied, making the task harder or easier,
this is reflected in changes to the drift rate d (Ratcliff and
McKoon, 2008), while participants deprived of sleep for 24 h
also show decreased drift rate (Johnson et al., 2021).
Introducing a motor response handicap, such as requiring a
single finger beused for all keyboard responses (Voss etal.,
2004) or requiring multiple keypresses for each response
(Lerche and Voss, 2019), increases Te r; similarly, varying the
response modality (so that participants respond by eye
movements, key pressing, or pointing on a touchscreen),
affects Ter but not the other parameters (Gomez etal., 2015).
Together, all these studies suggest that the DDM parameters
do capture recognizable – and at least partly separable –
cognitive processes.
e DDM has also been used to explore cognitive processes
even when dierences in observable behavior alone (e.g.,
participants’ response accuracy and RT) do not discriminate
groups (Zhang etal., 2016). e DDM can also help disentangle
dierent processes of information processing; for example, it
has been repeatedly documented that older adults have longer
RT than younger adults and that this is associated not only with
higher non-decision times (Ter ) but also with increased
response caution (larger a; see iesen etal., 2021, for meta-
analysis). Although originally developed to address data
representing asymptotic performance on speeded response
tasks where RT is fast (e.g., <1 or 1.5 s) and within-session
learning is negligible (i.e., the decision rule is already known
and practice eects are minimal), the DDM is increasingly also
being applied to more complex tasks with longer (e.g., 1–3 s)
RTs (Palada etal., 2016; Lerche and Voss, 2019; Lerche etal.,
2020), to tasks that involve explicit learning across trials
(Millner etal., 2018; Miletić etal., 2021), and even to data
obtained from non-human animals (Brunton et al., 2013;
Schriver etal., 2020).
In sum, there is a considerable and growing body of literature
using the DDM to elucidate cognitive processes that aect
decision-making.
AB
CD
FIGURE3
Graphical illustration of the eects of changing DDM parameters. (A) A “standard” DDM model, defined by boundary separation a, relative starting
point z, and two drift rates d.s1 and d.s2 for trials associated with stimulus s1 or s2, respectively. The remaining parameter, non-decision time (Ter),
is not shown but is assumed to represent a fairly constant contribution to overall RT, independent of the decision-making processes. Increasing or
decreasing Ter will aect average RT regardless of other task considerations. (B) Changing the starting point z, by moving it closer to one
boundary, introduces a response bias. Here, z moves closer to the upper boundary; on each trial, the evidence accumulation process has farther
to go to reach r1 than r2, creating a response bias favoring r2 responses. (C) Reducing boundary separation a reduces the distance that the
evidence accumulation process has to travel to cross a boundary. This will tend to result in faster responses and potentially more errors, since it is
easier for noise to push the evidence accumulation process across the wrong boundary. Reduced a is therefore interpreted as reduced response
caution: less evidence required before selecting either response. Increasing a has the opposite eect of increasing response caution: more
evidence will berequired before the evidence accumulation process crosses either boundary. (D) Increasing drift rate (here, d.s2) means that the
evidence accumulation process will travel more quickly (steeply) toward the corresponding boundary: evidence accumulation is more ecient
and, in eect, the task condition is easier. As a result, RTs on s2 trials will generally befaster. Decreasing a drift rate (here, d.s1) has the opposite
eect: evidence accumulation on s1 trials is less ecient and the process proceeds more slowly toward the r1 boundary. As a result, RTs on s1
trials will generally beslower. Manipulating any model parameter individually can thus aect both accuracy and speed; model-fitting procedures
find the configuration of parameter values (a, z, d.s1, d.s2, and Ter ) that together provide the most accurate description of the observed accuracy
and RT distributions.
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Getting started with the drift
diusion model: A concrete
example
Suppose that weare going to collect data from human
participants on a simple task with two stimuli (or classes of
stimuli) s1 and s2 that map to two responses r1 and r2,
respectively. The task has 500 trials, including 150 s1 and 350
s2 trials, so that s2 (and r2) occur more frequently than s1
(and r1). Wewill assume that participants could realistically
achieve 100% accuracy, but the requirement to respond as
quickly as possible introduces errors due to the speed-
accuracy tradeoff. Wemight beinterested in comparing two
groups of participants (say, patients vs. healthy controls), and
we might plan to analyze the behavioral data using one
ANOVA (or non-parametric equivalent) to compare accuracy
across groups, and another to compare RTs across groups.
However, being aware of the speed-accuracy tradeoff, and also
because weare interested in the underlying cognitive processes
that produce any observed group differences, wealso plan to
apply a computational model, the DDM.
Model definition
Having decided to use the DDM, the first question is:
which free parameters will weconsider? As noted in Table1,
DDMs usually include at least four free parameters:
non-decision time Ter , boundary separation a, response bias
z, and (at least one) drift rate d. Given that our task involves
two different stimuli (or classes of stimuli) s1 and s2, welikely
want to allow separate drift rates (d.s1 and d.s2) for trials with
our two types of stimuli.
Both simpler and more complicated models are possible.
For example, we already noted above an example where
wemight want multiple dri rates corresponding to multiple
task conditions. On the other hand, if wethink s1 and s2 should
beequally discriminable, wemight assume only a single dri
rate (equivalent to assuming d.s1 = −d.s2: i.e., the two dri rates
are of equal steepness, but slope in opposite directions, one
down to r1 and the other up to r2). Our resulting model would
therefore have only three free parameters that could vary across
subjects: Te r, a, and a single dri rate d. Similarly, if wethink
there is no reason to assume a response bias, wemight consider
consider xing z at 0.5. In this case, z would no longer be“free”
to vary across participants, but would have the same value
for everyone.
For now, though, let us focus on a “default” DDM with ve
free parameters: Ter, a, z, d.s1, and d.s2, while noting that other
versions are possible. In fact, our DDM will look very much like
that in Figure 3A, although dierent participants may have
dierent values of the free parameters – such as the examples
schematized in Figures3B–D – that in turn produce individual
dierences in behavior.
At this point in the process, wewould also typically consider
what approach weplan to use for model-tting, and whether
soware is available; for the DDM, several options exist that will
beconsidered in more detail in the section on “Model-tting in
the DDM,” below.
Empirical data
Continuing our example, let us assume we have a test
dataset obtained from one participant on our task. (e test
datale is provided in the Supplemental material: see Appendix).
From these behavioral data, wecould plot this participant’s
overall accuracy. As shown in Figure4A, this participant made
about 76% correct responses to s1 but about 83% correct responses
to the more frequent stimulus, s2. We could also plot the
distribution of RTs for correct and incorrect responses to each
stimulus. Figure4B shows unimodal RT distributions with right
skew: most responses occur within 250–750 msec, but a small
proportion take 1 s or longer (and none occur faster than
220 msec). ere also tends to beslightly faster mean RT on error
responses, which is oen the case when participants sacrice
accuracy for speed.
Sometimes, the RT histograms of Figure 4B are instead
presented as probability density functions (PDFs), as shown in
Figure4C, in which the height at any given point on the x-axis
corresponds to the likelihood of a response occurring at that
RT. PDFs are oen plotted with correct and error responses on the
same graph, scaled so that the area under the curves sums to 1.
is makes it easy to see not only the relative rates, but also the
relative timing of correct and incorrect responses.
Figure4D shows a slightly dierent way of visualizing the
same data: cumulative distribution functions (CDFs) which plot
at each RT the probability that a response has occurred at or
before that RT. No RTs occur faster than about 0.2 s, and all RTs
– for both stimuli, correct and incorrect responses – have occurred
by about 1.5–2 s. (In Figure4D, the curves are scaled to asymptote
at 1, indicating that 100% of responses have occurred; sometimes,
CDFs are instead scaled so that the height at asymptote represents
the likelihood of each response; in this case, the curve is called a
“degraded CDF.”)
While the plots in Figure4 do convey useful information
about the empirical data, they do not take into account the
speed-accuracy tradeoff. In this simple task, it seems likely
that our participant could have achieved 100% accuracy if
instructed to work slowly and carefully – and conversely,
might have made even more errors if instructed to respond
even more quickly. This makes it problematic if wewished to
compare a group of subjects (e.g., healthy controls) against
another (e.g., neurological or psychiatric patients): perhaps
one group has higher accuracy, but is this because they were
more willing to sacrifice speed? And can weinfer anything
useful about the underlying cognitive processes in each group?
For this, weturn to the DDM.
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Data cleansing
Before applying the DDM to our empirical data, wehave one
more step to consider: data cleansing. RT distributions are
typically unimodal but right-skewed, as illustrated in Figure4B,
with a few very long RTs but no very short RTs.
It is widely assumed that genuine RTs have a theoretical
minimum of about 100–150 msec, representing the time needed
for the physiological processes of stimulus perception and motor
response execution, and bounded by the speed of neuronal
transmission in the nervous system (Luce, 1986; Whelan, 2008;
Woods etal., 2015). Decision-making time (Td) would add (at
least) tens of msec to this lower bound. However, empirical data
les oen contain a few “very fast” RTs of <100 msec, which could
bethe result of anticipatory responding initiated before stimulus
onset, or even a very delayed response from a prior trial. ere
may also be“extremely slow” RTs, which could occur because the
participant is briey inattentive (e.g., a distracting noise in the
background or a sneeze). Unfortunately, such outlier RTs can
strongly inuence the outcome of hypothesis tests on RT data
(Ratcli, 1993) as well as biasing any attempts at model-tting
(Ratcli and Tuerlinckx, 2002). erefore, it is common to
perform data cleansing to attempt to reduce the eect of outlier
RTs (Ratcli, 1993; Whelan, 2008).
AB
CD
FIGURE4
Test dataset. Here, the participant completed a task that intermixes 150 trials with stimulus s1 (correct response = r1) and 350 trials with
stimulus s2 (correct response = r2). (A) In this example, the participant achieved 76% accuracy on s1 and 83% accuracy on s2. (B) Frequency
histograms showing right-skewed RT distributions for each stimulus and response. (C) The RT data plotted as probability density functions
(PDFs), in which height of the curve at any point on the x-axis reflects the probability of a response occurring at that RT. Here, PDFs are
scaled so that the total area under the curves sums to one, resulting in “taller” PDFs for the more frequent correct than incorrect response.
(D) The same data plotted as cumulative distribution functions (CDFs), showing the probability that a response has occurred at or before
each RT; here, curves asymptote at about 2 s, showing that all RTs have occurred by that point. (C) Plotted using functions in the DMC
package for R.
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Various techniques for reducing the eect of outlier data have
been proposed, but a common solution is to dene absolute
cut-points for very-short and very-long RTs, and drop from the
dataset any trials with RTs outside those limits. Ideally, cutos
should bechosen that eliminate obvious outliers while retaining
as many data points as possible as possible (ideally, no more than
1%–2% of trials should bedropped). e choice of appropriate
cuto criteria for a specic study will of course vary, but common
cutos are oen RT < 100 or 200 msec and RT > 1 or 1.5 s; studies
have suggested that ndings may berelatively robust to minor
dierences in the exact cuto values (e.g., Ratcli etal., 2018).
Alternate methods for reducing the eect of outliers, such as
transforming the data to normalize the data, or using cutos based
on standard deviation or interquartile range, are possible, and can
seem less ad hoc, but may greatly reduce power and can introduce
biases of their own (Ratcli, 1993; Ulrich and Miller, 1994).
In our test dataset, weinspect our data le (e.g., histograms of
Figure4B), and nd no obvious outlier RTs, and wecan move on.
Model-fitting in the drift diusion
model
At this point, weare ready to use the DDM to estimate the
parameter values that best describe our empirical data and –
wehope – the participant’s underlying cognitive processes. e
task at hand can bethought of as nding a conguration of
parameter values in the DDM that, together, cause it to generate
simulated data that are as close as possible to the empirical data
shown in Figure4, accounting for both accuracy rates and for the
distributions of correct and incorrect RTs.
Overview of the process
We start by proposing some “reasonable” values for the DDM
parameters. For example, wemight set boundary separation at an
arbitrary value of a = 1 and starting point at z = 0.5 (no a priori
bias for either response); given that RTs in the data seem to range
from about 0.25–1.0 s, wemight estimate non-decision time at
Ter = 200 msec (assuming the decision time Td always takes at
least an additional few dozen msec, so our fastest RTs would
be about 220–250 msec); for dri rate, wemight set d.s1 < 0,
reecting that the evidence accumulation on s1 trials should
proceed downward (toward r1), and d.s2 > 0, so that evidence
accumulation on s2 trials should proceed upward (toward r2).
Here, for illustration, we’ll set d.s1 = −1 and d.s2 = +1.25.
To simulate a single trial with stimulus s1, we plug these
parameter values into the DDM equations, start the evidence process
at z with a boundary separation of a and dri rate of d.s1, allow the
diusion process to operate until a boundary is crossed at time Td,
and then record both accuracy (was the correct response “r1”
generated?) and overall RT (Ter + Td ) for this trial. In this example,
wend that the response r1 is indeed chosen, with an RT of 0.382 ms.
Since the evidence accumulation process in the DDM is
noisy, wewould repeat a second time, and get a slightly dierent
RT (and potentially even a dierent response). In all, werepeat
500 times (150 with s1 and 350 with s2, just as in the empirical
data), resulting in distributions of predicted RTs for correct and
incorrect responses, and a predicted accuracy rate (how oen the
correct response was selected). ese predictions (from the
DDM) are then compared to the accuracy and RT distributions
in the empirical data (from the participant) to determine how
well the model ts the empirical data.
For example, Figure5A plots accuracy (how oen the DDM
chose the correct response to each stimulus) and Figure5B shows
the distribution of RT obtained for each stimulus, for correct and
incorrect responses. ese predicted data can be visually
compared with the empirical data (Figures4A,B).
In this example, on s1 trials, the DDM chose the correct
response (r1) 77% of the time – very close to the 76% accuracy in
the empirical data. On s2 trials, the DDM chose the correct
response (r2) 75% of the time – a little lower than the 83%
accuracy in the empirical data. e histograms for the predicted
data also look similar to those from the empirical data, although
not perfect: for example, the mean for correct responses to s1 is
about 0.44 s, which is a little faster than the mean 0.54 s in the
empirical data.
We could also plot PDFs and CDFs of the predicted data, to
see how closely these overlap the PDFs and CDFs from the
empirical data. For example, the empirical data could bedivided
into quantile “bins,” for example, based on the RT at which the
fastest 10%, 30%, 50%, 70%, 90%, and 100% of RTs have occurred.
For our test dataset, this would result in an approximation to a
degraded CDF shown by the black lines in Figure 5C. e
simulated data generated by the DDM can beplotted in the same
way (red lines in Figure5C). In this example, the correspondence
between empirical and predicted data is not great. For example,
according to Figure 5C-le, 90% of incorrect responses to
stimulus s1 have occurred by about 800 msec; the model,
however, predicts that 90% of error responses to s1 will have
occurred by about 600 msec. In other words, the particular set of
DDM parameter values used to generate these predicted data do
not reproduce the empirical RT data very accurately.
All of the above has allowed us to assess how well the DDM
reproduces a single participant’s data, given a single set of
parameter values in the model. Wecould then slightly perturb one
or more of our parameter values – say, changing a from 1 to 1.01
– and run the DDM again to see if the new combination of
parameter values provided a better approximation to the
participant’s data. Aer iterating over a large number of possible
parameter values, wewould obtain the best possible t: a set of
values for a, z, Ter and dri rates d.s1 and d.s2 that together allow
the DDM to most closely replicate the empirical data.
Needless to say, estimating the optimal values for multiple free
parameters is a formidable computational challenge, one that is
complicated by the fact that changes to one parameter may aect
the optimal value of another parameter. Fortunately, several
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methods have been devised, many of which are currently available
as open-access soware packages.
Specific methods and computational
packages for parameter estimation using
drift diusion model
This section reviews several methods of parameter
estimation for DDM that have been widely used in the
literature, and that have been implemented as freely-
available software.
χ2 method
An early, and mathematically tractable, approach to estimating
DDM parameters is the χ2 method (e.g., Ratcli and Tuerlinckx,
2002), which compares a histogram of RT distributions in the
empirical data to those predicted from the model under a given
set of parameter values (as in Figure5C). For those familiar with
the χ2-test in inferential statistics, that method distributes the
A
C
B
FIGURE5
Results from a single run of the DDM, given a particular set of parameter values. On each trial, evidence accumulation occurs in the DDM until a
boundary is crossed, triggering a response; the response and RT = Td + Te r are recorded for each trial. Just as in the behavioral task completed by
our participant, there are 150 s1 trials and 350 s2 trials. (A) The predicted data has accuracy rate similar to, but not identical to, the actual accuracy
in the empirical data (red dots). (B) The predicted data also has RT distributions similar to, but not identical to, those in the empirical data (compare
Figure4B). (C) One way of quantifying model fit is by dividing the empirical RT distribution into quantile “bins,” e.g., the RT at which the fastest 10,
30, 50, 70, 90, and 100% of RTs have occurred, plotted here as black triangles (for correct and incorrect responses to each stimulus). The
simulated (predicted) data obtained from the DDM are divided into the same bins, plotted here as red circles, and connected by red lines to
simulate a CDF. If the predicted data fit the empirical data very well, the red circles would overlie the black triangles. Here, the fit between
empirical and predicted data is not great. For example, the model predicts too many correct responses and also too few slow RTs, compared to
the empirical data. Predicted data obtained using the RWiener and rtdists package for R.
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empirical data into a number of bins or categories, and then
compares that distribution against the distribution predicted by
the null hypothesis; the χ2 statistic is a measure quantifying the
dierence in distributions, and if χ2 is large enough, the null
hypothesis of no dierence between distributions can berejected.
Conversely, if the distributions are similar (i.e., small χ2), then the
model is said to provide a good t to the empirical data.
e goal is to identify a set of DDM parameter values that,
together, minimize χ2 – minimizing the dierence between
empirical data and DDM predictions. Without delving too
deeply into the mathematical methods, suce to say that this
can bedone via an optimization routine. Many optimization
routines involve iterative search: start with an initial rough
estimate of the parameter values, calculate how well the model
using those values ts the data, and then iteratively perturb one
or more of the values, to see if this provides a better t: if yes,
then the new values are adopted for the next pass; if no, then the
old values are retained and a dierent perturbation is tried.
Early in the search, when the t is likely to bepoor, minor
perturbations of parameter values may produce large
improvements in t; but the search is said to converge when
values for all parameters have stabilized and no further
perturbations can beidentied that improve t by more than a
very small value (e.g., 10−7). At this point, the value of χ2 is
returned as a metric of goodness-of-t, and the corresponding
parameter values are taken as the optimal or “best-tting”
parameter estimates for the empirical data.
e χ
2
method is one of the estimation methods instantiated
in the freely available fast-dm package (Voss and Voss, 2007; Vo ss
etal., 2015) and it can also beused with the rtdists package in R
(Singmann etal., 2016).
e main advantages of the χ
2
approach are computational speed
and relative robustness to outlier RTs. e robustness to outliers
reects the fact that the rst and last bins are “open” in the sense that
there is no lower bound for RT in the quantile bin that contains the
fastest responses, and no upper bound for RT in the bin that contains
the slowest responses; therefore even a very extreme outlier (say,
RT = 0 s or RT = 10,000 s) would not dramatically aect the results.
However, the χ2 approach requires a large number of trials to produce
a reliable estimate (e.g., at least 500 trials) and may beespecially
problematic if there are relatively few error responses (e.g., <12 trials
in any quantile bin; Voss etal., 2015).
For these reasons, the χ2 approach to parameter tting has
become less widely used in recent years, as other methods have
become available, and as computing power has increased.
However, it’s worth understanding this method because RT
quantiles (such as those in Figure 5C) are oen plotted in
publications that have used other model-tting methods.
Maximum likelihood estimation
A popular method for estimating DDM parameters uses
maximum likelihood estimation (MLE) to generate estimates for
each parameter. MLE may bemost familiar to readers as a means
to identify parameter values (beta weights) in regression models,
to minimize dierence (error) between the predicted and
empirical outcomes. e principle is the same here.
Formally, MLE tries to nd a set of parameter values that,
together, maximize the probability that the outcome of the model
matches the empirical data on all trials. is probability is referred
to as the likelihood estimate, L; more commonly, researchers
report the log of that value, which is called the log-likelihood
estimate LLE.
e goal then becomes to nd the set of parameter values that,
together, maximize LLE for the model applied to a given dataset.
Again, this is normally done by optimization routines that work
by iterative search: constructing a (possibly random) set of
“starting values” for the parameters and evaluating LLE, then
perturbing one or more parameters by a small amount and
re-evaluating LLE, until no further improvements in LLE can
be obtained by perturbing any of the parameters. (Some
researchers prefer to speak in terms of minimizing negative LLE,
rather than maximizing positive LLE, but the resulting parameter
estimates will bethe same.)
For example, Figure6 shows accuracy and RT distributions
obtained after using MLE to optimize the model parameters
against our test dataset. The figures show that the predicted
data matches the overall accuracy of the empirical data pretty
well, although predicting slightly too few correct responses to
s1 and slightly too many correct responses to s2, and also
captures the modes and general shape of the RT distributions
reasonably well.
Freely-available soware implementing MLE for DDM
includes the fast-dm package (Voss and Voss, 2007; Voss etal.,
2015) as well as the RWiener (Wabersich and Vandekerckhove,
2014) and rtdists (Singmann etal., 2016) packages for R.
MLE approaches have been successfully used in a large
number of DDM studies, and can beused even when there are
relatively few (e.g., <50) trials available from each participant
(Lerche etal., 2017); however, MLE can bevery sensitive to
outlier RTs (especially very fast RTs), and so careful thought
must begiven to data cleansing. MLE algorithms are also
vulnerable to local minima, meaning that they can converge
to a set of parameter values where no small perturbations can
further improve LLE, even though this may not bethe optimal
solution. For this reason, it’s often a good idea to run the MLE
procedure multiple times, with different starting values, to
make sure the same solution is found each time.
Bayesian approaches
Recently, the advent of open-source platforms for Bayesian
statistics have given rise to a number of Bayesian approaches for
estimating DDM parameter values (for readable introductions to
Bayesian methods, see Kruschke and Liddell, 2018; Wagenmakers
etal., 2018b). In brief, these methods follow a Bayesian approach
starting with initial estimates (i.e., “prior distributions” or simply
“priors”) about reasonable values for each parameter that are
iteratively updated to produce “posterior distributions” (or simply
“posteriors”) for those parameters.
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Oen, very vague and uninformative priors are used, so that
minimal pre-existing knowledge is assumed and even a small
amount of data will “overwhelm” the priors, meaning that the
posteriors depend much more on the data than on the researcher’s
choice of priors; additionally, several dierent priors may betried,
to show that the prior assumptions do not greatly inuence the
posteriors/conclusions. For example, the prior for z might simply
specify that it is a value somewhere in the range from 0.0 to 1; the
priors for dri rate might specify broad normal distributions with
mean 0; etc. (see Figure7A).
Whereas the χ
2
and MLE methods attempt to estimate a single
“best-tting” value for each parameter, Bayesian approaches
generate posterior distributions for each parameter, including
both a point estimate (e.g., mean or median) and a measure of
condence in that estimate (e.g., the standard deviation SD or
condence interval CI). For example, Figure7B shows posterior
distributions (or simply “posteriors”) for the ve DDM parameters;
for visual comparison, the priors are plotted as red lines, and at
this scale they now appear as nearly at lines near y = 0. For
example, whereas the prior estimate for z treated all values in the
range from 0 to 1 as equally plausible, the posterior estimate for z
has mean 0.60, indicating a mild but denite response bias (toward
the upper boundary and r2); the narrow width (SD = 0.2) indicates
high condence that the true value lies in the body of the posterior.
In this case, we can say that information from the data has
“overwhelmed” the priors, resulting in much more closely
specied posteriors.
Calculating posterior distributions is extremely
computationally intensive, and direct solution is generally
intractable: i.e., there is no known mathematical way to directly
calculate the posteriors from the priors and the data. Instead,
approaches such as Markov Chain Monte Carlo (MCMC)
methods leverage computer power to generate approximate
solutions (for a readable introduction to MCMC methods, see van
Ravenzwaaij etal., 2018).
In brief, MCMC methods estimate a distribution by repeatedly
drawing “sample values” to form a “chain” of values for each
parameter. A simple version of MCMC might run as follows: At
the rst step or iteration of the chain, a value is selected at random
for each parameter from the prior distributions, and the resulting
model is evaluated (here, the DDM would be run and LLE
computed). At the next iteration, the distribution of one of the
parameters is perturbed slightly, perhaps by slightly altering its
mean or SD, and new parameter values are drawn at random from
the distributions, and the model is run again. If the result is an
improvement (e.g., improved LLE), then the updated parameter
values are used for the next step in the chain; otherwise, the old
parameter values are retained. e process is repeated hundreds
or thousands of times, until no further improvements are
discovered, at which point the process is said to have converged
on a solution: a set of distributions (posteriors) for each parameter.
If weare examining a DDM with ve free parameters (a, z,
Ter, d.s1, d.s2), then typically all the parameter values are updated
at each iteration (oen, by holding all the other parameters
constant at their current values while weperturb and evaluate
each one in turn). erefore, each sample in the chain contains
updated values for all the parameters.
Typically, multiple chains are run, oen using the rule of
thumb to run three times as many chains as there are free
parameters; thus, for a DDM with ve free parameters wemay run
15 chains. Results from 15 such chains are shown in Figure7B,
one colored line per chain. e gure shows that, at the rst
iteration, there is a wide variety parameter values drawn from the
priors; but within a few dozen iterations, the parameter values in
all chains begin to converge to common values, and LLE rapidly
increases for all the chains.
For example, the prior for Ter is a uniform distribution in the
range from 0.1–1.0 s (refer Figure7A). At the start of each chain,
a value for Ter is chosen at random from that distribution;
because the prior is vague, starting points can vary widely, as
shown at the extreme le of the Ter traceplot in Figure7B. Across
200 iterations, though, all the chains gradually converge to new
estimates of Ter with means close to about 0.2 s. is rapid ne-
tuning of Ter and the other parameters produces corresponding
improvement in LLE, reecting progressively better model t.
Here, by about the 200th iteration, all the chains are meandering
around the same point, without any large deviations or upward
or downward trends.
Because of the wide variability in possible starting points, the
beginning of each chain (e.g., the rst 200 iterations shown in
Figure7B) is oen discarded (as “burn-in”), and then the chains
run for a few more hundreds (or thousands) of iterations, so that
the posterior predictions are based on the (hopefully stable) ends
of the chains, as shown in Figure7C. Note the change in scale on
the y-axes from Figures 7B,C as the chains “zero in on” a very
narrow distribution of values for each parameter, resulting in only
minor uctuations in LLE from one iteration to the next.
Formally, the state when the distribution does not change
(much) across iterations is known as convergence. Convergence
can bevisually assessed by inspecting the traceplots of the chains,
which should look like the “fat, at, hairy caterpillars” of
Figure 7C: minor uctuations around a mean value with no
systematic upward or downward tendencies (compare the
unconverged chains in the early iterations of Figure7B).
Convergence can also beassessed quantitatively. One widely-
used measure of convergence is the Gelman-Rubin Rˆ (“R-hat”)
statistic (Gelman and Rubin, 1992) which assesses the similarity
of within-chain vs. between-chain variability. Values of
Rˆ approaching 1 indicate convergence; a common criterion for
good convergence is Rˆ < 1.1. (In the example from Figure7, aer
1,000 iterations, Rˆ = 1.03.)
Given successful convergence, the posterior distributions can
bereported (e.g., Figure7D), and/or the mean or median of the
posteriors can beused as point estimates of the parameters.
Freely-available soware implementing Bayesian approaches
to DDM includes the python-based HDDM (Wiecki etal., 2013)
and the Dynamic Models of Choice (DMC) package for R
(Heathcote etal., 2019).
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Bayesian approaches to DDM may be more robust in
recovering model parameters than other methods, such as MLE
and χ2 methods, when a limited numbers of trials are available
(Wiecki etal., 2013). Bayesian approaches also provide not only
parameter estimates (mean or median of the posterior
distributions), but also quantify the uncertainty in those estimates
(standard deviation or 95% condence interval of the posterior
distributions). Like MLE algorithms, Bayesian methods based on
iterative sampling may bevulnerable to getting stuck inlocal
minima, although this risk is ameliorated by use of multiple chains
to ensure convergence (e.g., the “hairy caterpillars” of Figure7C
indicate that all the chains are converging around the same
stable estimates).
Hierarchical methods
e above methods for estimating DDM parameters all
assume that parameters are t to each participant’s data
independently. An alternate approach is hierarchical modeling,
which addresses individual dierences while also pooling
information across individuals to generate group-level parameter
estimates (Vandekerckhove et al., 2011; Wiecki et al., 2013;
Johnson etal., 2017). Hierarchical approaches may beparticularly
useful where within-group variability is much lower than
between-group variability, or where only a small number of trials
are available for each participant; however, hierarchical models
may not bevalid if there are only a few participants in each group.
e Bayesian model approaches in some soware packages,
including HDDM and DMC, provide for hierarchical
model tting.
So, which method should beused?
Each of these methods for estimating parameters has strength
and weaknesses. Table 2 summarizes a few of the key
considerations when determining whether to use χ2, MLE, or
Bayesian methods. Choice of an appropriate method for a given
dataset typically represents a compromise among these
considerations, as well as the researchers’ familiarity with a
particular approach and access to soware. Fortunately, the
availability of well-documented soware packages, and
widespread availability of powerful computers, means that all
these methods are within the reach of an investigator willing to
invest the time required to learn their use.
In most cases, an investigator with access to reasonable
computer power is likely to choose between MLE and Bayesian
approaches. ese approaches do not require the large number of
trials required by the χ2 method, although they are more
vulnerable to outlier RTs. For this reason, data cleansing is
important to mitigate the eects of outliers.
In general, MLE and Bayesian approaches should return
comparable results, although parameter estimates may differ
both due to randomness (noise) in the estimation routines and
also due to scaling factors adopted by different software
packages.
For example, both MLE (via the RWiener package in R) and
Bayesian MCMC (via the DMC package in R) were used to
estimate the ve DDM parameters for our test dataset. As shown
in Table3, both methods return nearly identical estimated values
parameters for a, z and Ter ; estimated values of dri rate dier
AB
FIGURE6
Results of model-fitting using maximum likelihood estimation (MLE). The set of DDM parameter values is identified that, together, maximizes the
likelihood of the model producing same response as the participant on each trial. Given these “best-fitting” parameters, (A) accuracy levels for s1
and s2in the predicted data were quite similar to those of the empirical data (red dots), and (B) RT distributions for each combination of stimulus
and response were also quite similar in predicted and empirical data. DDM results obtained using RWiener package in R.
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AB
CD
FIGURE7
Bayesian approaches to estimating DDM model parameters. (A) First, prior distributions or “priors” (initial estimates) are generated for each model
parameter. Here, the prior for non-decision time Ter simply specifies that it is a value in the range from 0.1 to 1.0 s, with all values in that range
equally likely (uniform distribution); similarly, the prior for starting point z is a value in the range from 0 to 1.0 (as a proportion of boundary
separation, a). The priors for drift rates are specified as broad normal distributions with means of 0; the prior for a is a normal distribution with
mean of 1, truncated at 0 and 2 (because boundary separation is in arbitrary units but can never be<0). These priors are all intentionally vague, so
that they have minimal influence on the posteriors. Note that, in these graphs, drift rates d.s1 and d.s2 are plotted as >0 if they slope in the
direction of the correct response. (B) MCMC is applied to the dataset from Figure4, using priors as defined in (A): 15 “chains” are formed starting
from the prior distributions and progressively updating, with model fit evaluated at each step. Traceplots of values for the 5 DDM parameters over
the first 200 iterations are shown as one colored line per chain. Initially, there is wide variation across chains in the initial values drawn for each
parameter from the priors; however, over the first 200 iterations, parameter values start to converge to similar values in all chains. For example, Ter
(Continued)
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slightly, but the relative relationships are preserved: d.s1 is in the
opposite direction to, and steeper than, d.s2 in both methods.
e bottom line is that, in many cases, the general conclusions
from the DDM should beroughly the same, regardless of which
specic approach (or soware package) is used to generate parameter
estimates. For many purposes, the choice will simply reect the
approach and/or soware package with which the investigator is
most comfortable.
Now it’s time for a big reveal: the original “empirical” test
dataset (shown in Figure4) was itself generated from a DDM
with predefined parameters: a = 1.2, d.s1 = −1.25 (sloping
down toward the lower boundary), d.s2 = +1.1 (sloping
toward the upper boundary), z = 0.6, and Te r = 0.2. That
means that wehave the luxury of knowing the correct answer:
if the DDM is functioning as advertised – if it is truly able to
“infer” parameter values by inspecting the data generated –
then the parameter estimates should bevery close to the true
(or “generating”) parameter values. In fact, Table3 shows that
both instantiations of the DDM do, indeed produce
parameter estimates that are very close to the
generating parameters.
Of course, if this dataset had been generated by a human
being, wewould not have the luxury of knowing the generating
parameters – the entire point of using the DDM would beto
attempt to infer these latent parameter values from the empirical
data. But the ability of the DDM to accurately recover parameters
from a test dataset greatly increases our condence that that the
methods can beapplied to this type of data.
Running the drift diusion model on a
group of empirical data files
All the work wehave done so far has rested on attempting
to fit the DDM to a single data file, and a “simulated” data file
at that. Now let us assume wewere going to fit the model to a
group of data files obtained from multiple participants an
experiment. For the purposes of example, let us consider
n = 10 empirical data files obtained from our simple task
(included in the Supplemental material; see Appendix).
Wewill also assume that wehave already performed a data
cleansing step, to identify any trials with very-short or very-
long RT (none were identified). Wethen run our DDM (with
five free parameters a, z, d.s1, d.s2, Ter) using MLE (via the
RWiener package in R) to find best-fitting parameters for each
data file. The results are shown in Table4, along with the
Figure 7 (Continued)
(which the priors merely specify as ranging between 0.1–1.0 s) quickly converges to a mean value near 0.2 s. As the parameters begin to converge,
log-likelihood estimates (LLE) for each chain also improve rapidly, indicating improving model fit. (C) Following the initial 200 trials (which are
discarded as “burn-in”), an additional 1,000 iterations are run; the traceplots for each parameter now resemble “fat, flat, hairy caterpillars”: relatively
horizontal with small, random scatter around a common mean value, which is a visual indicator that estimates for the parameters have converged.
Meanwhile, LLE for the chains is also relatively stable within and across chains: compare scale of y-axes in (C) vs. (B). (D) The resulting posterior
distributions or “posteriors” for each parameter. The mean/median of posteriors can betaken as a point estimate of the parameter value, and the
width is a measure of uncertainty in that estimate. Compared to the priors, posteriors should beunimodal and fairly narrow: Note the dierence in
x-axis and y-axis scales from (A) to (D). For visual comparison, the priors are plotted in (D) as red lines, and at this scale they appear as nearly flat
lines near y = 0. For example, whereas the prior estimate for z treated all values in the range from 0 to 1 as equally plausible, the posterior estimate
for z has narrowed to a distribution with mean 0.60 and SD 0.02, indicating a mild but definite response bias (toward the upper boundary and r2).
DDM results and figures from DMC package for R; all plots are arbitrary units except Ter , which is in sec.
TABLE2 Comparison of some key features among three approaches
to estimate DDM parameters.
χ2 method Maximum
likelihood
estimation
(MLE)
Bayesian
approaches
(based on
MCMC
sampling)
Computational
tractability
Relatively fast Moderate Can bevery slow
Sample size
required
Requires large
number of trials
per subject (e.g.,
at least 500 trials
with at least 12+
per quantile bin)
Can beused with
as few as about
40–50 trials per
subject, at least 10
trials per condition
May bemore
robust than other
methods when
limited number of
trials available
Outlier RTs Relatively robust
to outlier RTs
Very sensitive to
outlier RTs
(especially fast
RTs )
Moderately
sensitive to outlier
RTs
Other (+ and −)
considerations
− Loss of
information due
to use of
quantile bins,
rather full RT
distribution
+ General
principles of MLE
are likely familiar
to a wide swath of
researchers
+ Allows to
quantify not only
parameter
estimates, but
uncertainty
(variability) in
those estimates
+ Available
methods for
hierarchical model-
tting
− Steep learning
curve for
researchers not
familiar with
Bayesian methods
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maximum LLE obtained for each data file, given those best-
fitting parameters.
Validating the model
e estimated parameters in Table 4 represent the
conguration of parameter values (a, z, d.s1, d.s2 and Ter ) that,
together, allow the DDM to most closely approximate each
individual participant’s behavior on the task. At this point, there
are a few ways in which weshould validate our model, bolstering
condence that the DDM is actually discovering parameter values
that describe the underlying processes that generated the data.
Sanity check
Before going any further, the importance of a simple sanity
check cannot beoverstated. Do the model results even make sense?
One type of sanity check has already been mentioned: in the
context of Bayesian MCMC methods, researchers oen report Rˆ
and/or show “hairy caterpillars” to document convergence (e.g.,
Figure7C), and may plot posterior distributions to show that the
parameter estimates are unimodal and (ideally) low in variance
(e.g., Figure7D). For MLE methods, optimization routines also
usually report whether a predened criterion for convergence has
been met for each le.
If the parameter estimation process did not converge, then
obviously the parameter estimates cannot betrusted. Sometimes,
failure to converge just indicates an unlucky choice of starting
point, and if the optimization routine is re-run with a dierent
randomly selected starting point, convergence may beachieved.
In this case, it’s customary for authors simply to note how many
attempts (re-starts) were required before convergence
was achieved.
Assuming convergence, the next step should always bea
sanity check of the parameter estimates obtained. For example,
non-decision time Ter cannot be lower than the empirically-
observed behavioral RT (since RT = Ter + Td and Td cannot beless
than zero), and boundary separation a is in arbitrary units but
cannot be<0 (since it represents a distance or separation). In
Table4, the values of a, z and Ter all meet these minimal criteria.
If there is more than one dri rate, sign (direction) and
steepness (slope) should beconsistent with observed accuracy and
relative response speeds. In Table4, for all subjects, d.s1 < 0 and
d.s2 > 0, meaning that in each case the dri rate slopes toward the
correct boundary. e magnitude (absolute value) of the dri rates
suggests that the evidence accumulation process on s1 trials is
somewhat steeper than on s2 trials. is might reect something
about the underlying nature of the task (e.g., perhaps it is harder
to distinguish and decide to respond to s2 stimuli than s1 stimuli).
In any case, the dri rates in Table4 look reasonable too.
In sum, then, our parameter estimates all appear plausible. In
general, any parameter values that violate common sense likely
indicate that the model has failed, regardless of what t metrics
may bereported.
Predictive check
A next important step to establish model validity is a
predictive check, in which the parameter estimates obtained from
the empirical data are used to generate simulated datasets.
Specically, a DDM with the estimated parameter values is run on
the same task (same number and type of trials as in the original
task) and the model’s predicted response and RT recorded for each
trial. is could bedone one or more times for each data le, or
for a few representative data les selected at random, or even using
the group means for the parameter estimates.
e simulated data should mimic key features of the
behavioral data, such as accuracy and mean/SD of RTs for correct
and incorrect responses to each stimulus. PDFs (or RT histograms)
and CDFs for simulated data can also bevisually compared against
the empirical data (similar to comparison of predicted vs.
empirical data in Figure6).
TABLE3 Parameter estimates for DDM with five free parameters,
applied to the dataset of Figure1, using MLE and Bayesian MCMC
(means of posterior estimates are shown), and “true” generating
parameters.
Median estimated parameter values
a d.s1 d.s2 z Ter
MLE 1.17 −1.35 0.94 0.6 0.201
Bayesian MCMC 1.23 −1.49 1.16 0.6 0.202
True (generating)
values
1.2 −1.25 1.1 0.6 0.2
MLE, maximum likelihood estimate (calculated via RWiener package in R); MCMC,
Markov Chain Monte Carlo (calculated via DMC package in R); Ter given in seconds;
other DDM parameters (a, d.s1, d.s2, z) given in arbitrary values.
TABLE4 Parameter estimates obtained from the DDM, using MLE to
maximize LLE for each data file separately.
Estimated parameters LLE
a z d.s1 d.s2 Te r
1 1.09 0.46 −3.93 1.27 0.21 185.2
2 1.91 0.48 −3.66 1.97 0.25 70
3 1.85 0.51 −5.3 1.8 0.16 128.5
4 1.19 0.55 −3.62 2.09 0.17 287.1
5 1.1 0.46 −4.01 1.27 0.21 189.7
6 1.65 0.49 −2.41 2.43 0.25 140.5
7 1.2 0.44 −2.01 2.02 0.21 129.5
8 0.78 0.4 −1.01 1 0.23 220.4
9 0.83 0.53 −3.2 1.53 0.18 331.9
10 0.85 0.43 −1.2 0.11 0.18 135.6
Mean 1.25 0.48 −3.04 1.55 0.2 181.8
SD 0.42 0.05 1.36 0.67 0.03 79.6
Parameter estimates obtained using RWiener package in R.
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If the model predicts the empirical data well, the plots for
simulated and empirical data for each participant will behighly
overlapping. If wehave a large dataset, rather than inspect each
simulated data le individually, it may beenough to “spot-check”
a few representative cases, and then look at some summary
statistics, such as mean percent accuracy and median RT in the
simulated vs. empirical data. Yet another possibility is shown in
Figure 8: For each data le, there should be very close
correspondence between the percent accuracy to s1 and to s2in
the empirical vs. simulated data, and similarly the median RT for
correct and incorrect responses in the empirical data should
beclosely matched in the simulated data. In the current example,
Figure8 shows extremely high correlation between empirical and
simulated data on percent accuracy and median RT on correct
responses (all r > 0.97); for RT on incorrect responses, the
correlation is lower, particularly for s1, reecting the relatively
low number of error responses on which these calculations
are based.
e fact that the simulated data share many of the features
with the empirical data cannot, of course, prove that they were
generated in the same way – but failure would almost certainly
argue that the model is not valid. So, predictive checks are a
conventional rst step in model validation.
Parameter recovery study
Aer generating the simulated data, a parameter recovery
study can beconducted, in which the DDM is applied to the
simulated data, to see whether the parameter values which
generated those simulated data can becorrectly recovered by the
DDM (e.g., Ratcli and Childers, 2015; Lerche etal., 2017; White
etal., 2018b).
Table5 shows the results of just such a parameter recovery
study: using the estimated parameters from each participant
(Table4) to generate 10 simulated datasets, and then running the
DDM on those simulated datasets to infer or “recover” those
parameter values. In a perfect world, the parameter values
estimated from the simulated data will match the generating
parameters quite closely: high correlation (Pearson’s r) between
generating and recovered parameters is considered “good” if
r > 0.75 or “excellent” if r > 0.90 (White etal., 2018b). For the
current example, as shown in Figure9, the correlations between
generating and recovered parameters would all beconsidered
excellent (all Pearson’s r > 0.9).
A successful parameter recovery study conrms that the
model-tting procedure is able to reliably estimate (“recover”) the
parameter values that generated the simulated data. is cannot,
of course, guarantee that the model has accurately estimated
parameter values from the empirical data, but it does increase our
condence that the model is (at least) capable of correctly
recovering parameters given data such as these. If the model
cannot even replicate the generating parameters for simulated
data, where weknow what the true values for each parameter are,
then wecertainly cannot trust that it is accurately estimating the
parameters for human participants, where the generating
parameters are not known!
Model selection
e previous section focused on how wend a set of estimated
parameter values that provide the “best possible” t for each le in
our empirical data. It’s also important to ask just how good that
“best possible” t actually is. Model selection typically refers to a
process of systematically comparing dierent instantiations (or
versions) of that model, with dierent free parameters, to
determine which provides the best way of understanding the data.
For example, in the preceding sections, weused a DDM with
ve free parameters, including a, z, Ter and separate dri rates
d.s1 and d.s2 for s1 and s2 trials, and weobtained pretty good
results, validated both by predictive check and parameter
recovery study; but could wehave obtained (almost) as good
results with a simpler model, say, assuming only a single dri rate
d, regardless of stimulus?
Other things being equal, wewould typically favor the simpler
model with fewer free parameters, as a simpler way to describe the
data, and also because models with more free parameters have a
higher risk of overtting the data. Overtting refers to situations
where a model can describe an existing dataset with high accuracy,
but does not generalize well to other datasets. is is a concern not
only in the DDM, but in all kinds of model tting, such as linear
regression: adding a large number of predictor variables to make
a more complex model may result in overtting the sample data,
such that the regression equation obtained makes very accurate
predictions on the sample data, but not for new datasets. A simpler
regression model, with fewer predictor variables may sacrice
some accuracy but provide better generalization.
So, do wereally need two separate dri rates in our model?
Are the benets (better model t) worth the costs (complexity and
potential overtting)? To answer this question, weneed to evaluate
two things: rst, exactly how “good” a t does each version of the
model provide to the data? Second, is one t “meaningfully” better
than the other?
Assessing model goodness-of-fit
In order to quantify model t, several goodness-of-t metrics
are available. We have already discussed one: the maximal
LLE. e maximal LLE is simply the value of LLE that was
obtained using the best-tting parameter values (and it was
reported for our example study in Table4).
e trouble with this metric is that the value of LLE depends
not only on the model goodness-of-t, but also on the number of
trials, so it’s not clear how to interpret an arbitrary value of LLE,
nor what magnitude of dierence in LLE values constitutes a
“meaningful” dierence.
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Varying the number of free parameters
Although wecannot necessarily interpret an arbitrary value of
LLE, wedo know that larger values are better, signifying closer t
of model to data. One thing wecan do is ask whether the model,
as currently described, is the simplest possible description of the
data: Can wedo even better with more free parameters, or could
we do nearly as well with fewer? What is the right level of
complexity in our model?
For purposes of discussion, let us use the nickname DDM−5
to refer to our DDM with ve free parameters: a, z, d.s1, d.s2, and
Ter. e “best-tting” parameters for DDM-5 (using MLE) were
presented in Table 4, which also showed the maximal LLE
obtained for each data le, using those “best-tting” parameters.
We might then consider a DDM with only four free
parameters: a, z, and Ter but only a single dri rate d to beused
on all trials regardless of the stimulus. (In this case, wewould
likely assume a sign change: dri rate −d on s1 trials so that the
evidence accumulation process tends downward to r1, and dri
rate +d on s2 trials so that the evidence accumulation process
tends upward to r2, but the magnitude of d does not dier for s1
and s2 trials, so it can bedescribed by a single free parameter.) For
purposes of discussion, let us call this version DDM-4 (DDM with
four free parameters).
We could then conduct model-tting on our dataset with
DDM-4, just as wedid with DDM-5. Assuming both the model-
tting process converges, and that the parameter estimates survive
an initial sanity check, wecould then compare the maximal LLE
obtained under DDM-4 with that obtained DDM-5 (Table6).
A rst, important point is that the maximal LLE obtained
under DDM-4 will (by denition) beless than or equal to that
obtained by DDM-5. is is because any solution explored by
DDM-4 (which constrains d.s1 = −d.s2) should also beexplored
by DDM-5 (which allows the two dri rates to vary independently
– not excepting those cases where they happen to have the same
magnitude but dierent sign).
So, the question here is not whether DDM-4 can provide a
better t: weknow that it cannot. e question is: can DDM-5
provide a suciently better t than DDM-4, enough to justify its
added complexity?
For example, Table4 showed that the estimated parameters for
participants #6, #7, and #8 under DDM-5 had dri rates d.s1 and
d.s2 that were nearly equal in magnitude, though oppositely
signed, and so DDM-4 (where the two dri rates are forced to
have the same magnitude) provides just as large LLE as does
DDM-5. And so, at least for these three participants, there does
not appear to be much “advantage” to using the more
complex model.
FIGURE8
Validating the model: Predictive Check. Using the estimated parameters for each data file, shown in Table4, the DDM is used to generate
“simulated data.” For each empirical file, the simulated data should closely reproduce (left) accuracy as well as features of the RT distribution, such
as median RT on (center) correct and (right) incorrect responses.
TABLE5 Parameter recovery test: estimated parameters “recovered”
from simulated data.
Recovered parameters LLE
a z d.s1 d.s2 Ter
1 1.06 0.48 −4.34 1.2 0.21 198.8
2 1.89 0.5 −3.91 1.8 0.25 53.2
3 1.74 0.52 −5.14 1.73 0.17 129
4 1.23 0.55 −3.69 1.97 0.16 264.5
5 1.09 0.47 −3.69 1.27 0.21 156.8
6 1.62 0.49 −2.31 2.5 0.25 144.8
7 1.22 0.43 −1.82 1.78 0.21 57.6
8 0.8 0.41 −0.83 0.64 0.23 180.3
9 0.82 0.53 −2.87 1.51 0.18 334.1
10 0.85 0.41 −1.07 0.27 0.18 127
Mean 1.23 0.48 −2.97 1.47 0.2 164.6
SD 0.39 0.05 1.43 0.65 0.03 86.3
Parameter estimates obtained using RWiener package in R.
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FIGURE9
Validating the model: results from parameter recovery study. Scatterplots showing correspondence between estimated parameters (Table4) and
recovered parameters (Table5), for the 10 simulated data files. Correlations of estimated and recovered values are excellent (Pearson’s r > 0.9 for all
parameters). This increases our confidence that the DDM can accurately recover parameter values for this type of data.
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On the other hand, the larger DDM-5 provides a much better
t (larger LLE) for participants #1, #2 and #3. Averaged across all
10 participants, DDM-5 does provide numerically better mean
LLE than DDM-4: 181.8 vs. 163.5. What weneed is a way to
quantify whether this 20-unit improvement in LLE is “signicant”
or “meaningful” – enough to justify our use of the more
complex model.
Is the more complex model “worth it”?
ere are a number of metrics that can beused to address this
question. One of the most commonly used is Akaike’s Information
Criterion (AIC), which is an attempt to compare LLE between
models while penalizing more complex models (Akaike, 1974):
specically, AIC = 2 k − 2*LLE, where k is the number of free
parameters (5 for DDM-5 and 4 for DDM-4). e smaller AIC,
the better; therefore, the addition of 2k to the LLE results in a
larger “penalty” (increasing AIC) for models with more free
parameters. Using this formula, the mean AIC for DDM-5 is
−354, and that for DDM-4 is −319, so wewould conclude that
the larger model, despite its added complexity, is a better
description of the dataset.
A related metric, the Bayesian Information Criterion (BIC),
considers number of parameters k as well as the number of trials
n in the dataset (Schwartz, 1978): BIC = k*ln(n) − 2*LLE; again,
lower (more negative) is better. BIC is only valid if n>> k (i.e.,
number of trials much larger than number of free parameters). A
nice feature of BIC is that there are conventions for interpreting
BIC values (Kass and Raery, 1995): as a rule of thumb, if the
dierence in BIC between two models is <2, then the more
complex model is “not worth it” (more formally, there is no
positive evidence in favor of the more complex model, and so the
simpler model should bepreferred); a dierence in BIC of >2
indicates positive evidence in favor of the more complex model,
while BIC dierence of >6 is considered strong evidence and >10
indicates very strong evidence in favor of the complex model.
In our example, DDM-5 has mean BIC of −333 and DDM-4
has mean BIC of −302. e dierence is >30, so weconclude that
there is very strong evidence favoring the more complex model
with separate dri rates.
e above results assume that weused MLE as our model-
tting procedure. When Bayesian methods are used, AIC and BIC
can be reported, but some articles instead report Deviance
Information Criterion (DIC), which is a generalization of AIC for
use when posterior distributions have been obtained via MCMC
methods (Spiegelhalter et al., 2002), or the Watanabe-Akaike
Information Criterion (WAIC; Watanabe, 2010), which is a
generalized version of the AIC that can beused when the posterior
distributions are not normal. In all cases, lower values indicate
better t aer penalizing for model complexity.
If Bayesian methods have been used, it is also possible to
report a Bayes Factor (BF), which is a ratio of the marginal
likelihood of the two models, interpretable as the relative strength
of the evidence that each model is correct. Values of BF = 1 mean
the two models are equally likely, while larger values of BF make
us increasingly condent in supporting the rst hypothesis (or
model). As a rule of thumb, BF > 3 is considered weak evidence
and BF > 10 is considered strong evidence (e.g., Wagenmakers
etal., 2018a).
All these metrics – AIC, BIC, DIC, WAIC, BF – are used
to compare how well two models describe the same data
file(s). As discussed earlier, there may besome participants for
whom one model has a much lower metric than the other
model, but some participants where the reverse is true. Often,
a decision favoring one model or the other is based on a simple
majority vote: which model results in best metrics for a
majority of participants. Always, the burden is on the more
complex model to justify its use, so if the complex model does
not clearly provide a better fit, then the simpler model would
likely bepreferred.
And remember: model selection methods only tell us which
of the models under consideration ts the data best. It does not
guarantee that any of them are correct (nor that they are better
than any of the innitely many other models that might have
been evaluated).
Now (and only now), present the
model results
So far, wehave estimated best-tting parameters for our data;
wehave conducted predictive checks and parameter recovery
studies to assure ourselves that DDM-5 can accurately recover
parameters for this type of data; and wehave compared DDM-5
vs. DDM-4 (and possibly some other variations also), and
concluded that DDM-5 is the best description of our data,
TABLE6 Model comparison: Results of model-fitting with DDM-4; for
ease of comparison, maximal LLE for each file under DDM-5 is also
shown (reprinted from Table4).
Estimated parameters, using
DDM-4
LLE
(from
DDM-
4)
LLE
(from
DDM-
5)
a z d.s1 d.s2 Te r
1 1.08 0.36 −1.87 1.87 0.22 150.6 185.2
2 1.88 0.22 −2.35 2.35 0.24 49.9 70
3 1.74 0.22 −2.36 2.36 0.15 69.2 128.5
4 1.17 0.43 −2.51 2.51 0.17 273.6 287.1
5 1.08 0.36 −1.88 1.88 0.22 153 189.7
6 1.65 0.3 −2.42 2.42 0.25 140.5 140.5
7 1.2 0.36 −2.02 2.02 0.21 129.5 129.5
8 0.78 0.51 −1 1 0.23 220.4 220.4
9 0.83 0.59 −2.01 2.01 0.18 319.4 331.9
10 0.85 0.47 −0.45 0.45 0.18 128.9 135.6
Mean 1.23 0.38 −1.89 1.89 0.2 163.5 181.8
SD 0.4 0.12 0.67 0.67 0.03 84.7 79.6
Myers et al. 10.3389/fpsyg.2022.1039172
Frontiers in Psychology 22 frontiersin.org
providing excellent ts to the data with no unnecessary complexity
(no more free parameters than are actually needed).
At this point, the parameter estimates from our DDM-5 can
(nally!) bereported for individual data les and/or summarized
for each group (e.g., patients vs. controls). ese could be the
value estimates for each parameter returned by MLE; or, if
Bayesian methods were used, the results could bepresented as
medians or means of the posterior distributions for each
parameter, or even plots of the posterior distributions for each
parameter. For example, perhaps subjects #1–5 constitute the
control group and subjects #6–10 constitute the experimental
group in our study; Figure 10 plots the median (and IQR)
parameter estimates for each group.
e point estimates for each parameter can bealso subjected to
statistical analysis, using analogous methods to those that were used
to analyze the behavioral data (accuracy and RT). To illustrate, in our
example study, it appears that a, z, and Ter are generally consistent
across groups, but that one group has much higher dri rate for s1
than s2, whereas the other group has equivalent dri rates for the two
stimulus types. is apparent interaction could beconrmed by
ANOVA or other inferential statistics.
In the real world, we might be interested in comparing
parameter estimates between patients vs. controls (e.g., higher
response caution favoring accuracy over speed in schizophrenia;
Moustafa et al., 2015) or across task conditions (e.g., mood
induction shis response bias in favor of mood-congruent
responses; White etal., 2018a).
Additionally, just as with the original behavioral data,
parameter estimates can beexamined for relationships with other
variables of interest (such as demographic, clinical, or
neurocognitive variables). Some researchers have suggested that
model parameters can beused as classiers to distinguish patients
from controls, and that addition of parameter estimates can
improve classication beyond standard demographic and clinical
variables and/or behavioral variables alone (e.g., Zhang etal.,
2016; Myers etal., 2022 July 28).
Reporting – And critically
evaluating – The model
e nal step in our study would bepublishing the results
and, as with any scientic method, it is important that the report
becomplete enough to allow the reader to critically evaluate the
methods, the results, and the authors’ interpretation, including
several key questions.
What modeling approach was used, and
was it appropriate for the job at hand?
Reporting requirements for the DDM should, at a minimum,
state what free parameters were considered, and what method of
model-tting was used (e.g., χ2, MLE, Bayesian MCMC). Ideally, the
authors should provide code for validation/replication. e reader
should beable to evaluate whether the model design was appropriate
for the cognitive task (and research hypothesis) under study; for
example, is there an existing literature using these methods with the
study population? If the authors present a new or modied approach,
did they explain and justify this process?
For example, the “standard” DDM described here assumes
rapid responding by well-trained participants, with little
expectation that learning, practice, or fatigue will modify behavior
across the study. Is this consistent with the behavioral task
demands? Also, dierent models (and model-tting methods)
have dierent requirements, such as number of responses, number
of total trials, and minimum number of correct/incorrect
responses per trial type (see Table2); were these requirements
met? More generally, is the model likely to shed any light on the
cognitive processes being investigated by the study hypothesis?
Was the model validated?
Before presenting parameter estimates from the model, the
authors should validate their model. First, did the authors present
any theoretical justication for the free parameters being
considered? Did they conduct predictive tests to show that the
model can, in fact, generate simulated data that at least
supercially captures important aspects of the behavioral data?
Did they conduct a parameter recovery study to demonstrate that
their model, as described, can accurately recover generating
parameters? Did they conduct any model selection studies, to
examine the eect of adding/deleting free parameters and did
these results convincingly support the version of the model that
the authors eventually reported?
Sometimes, this information is relegated to an appendix or an
online supplement, but it is an important part of the modeling
process, and the reader should beassured that this was done.
Do the model results survive a sanity
check?
Turning now to the model results, which are oen parameter
estimates (such as Figure10 or Table4), usually compared across one
or more groups or task conditions: Do the reported results seem
reasonable for this task, given what weknow from prior studies in
the literature? If violin plots or strip plots (or, for Bayesian methods,
posterior distributions) are presented, are the results unimodal? Is
there any evidence of oor/ceiling eects that might suggest a
broader range of possible parameter values needs to beexamined?
Is there a thoughtful discussion of model
limitations?
A good modeling paper (like all good science) will behonest
about its limitations. At present, the DDM is almost always used
post-hoc, rarely to test an a priori hypothesis. As such, overtting
Myers et al. 10.3389/fpsyg.2022.1039172
Frontiers in Psychology 23 frontiersin.org
is always a concern. Ideally, modeling results obtained in one
sample should bevalidated in a new sample, but failing this (or
until follow-up studies are conducted), authors can try techniques
such as cross-validation (or “out-of-sample” testing), to t the
model to dierent subsets of the data and test how well it
generalizes. At a bare minimum, the possibility of overtting
should beaddressed when interpreting model results.
A second general limitation of DDM is that it can show that
estimated parameters are sucient to explain the empirical data;
but can never prove that this is the case. Latent cognitive processes
remain latent. Ideally, results from the model identify candidate
cognitive processes that could then form the basis of future
hypothesis-driven studies.
e DDM also makes many simplifying assumptions about
the process of evidence accumulation and decision-making.
Simplicity is a virtue, but of necessity leaves out complicating
factors that can include variations in attention, emotion, and other
processes that may inuence decision-making. Standard versions
of the models assume that the empirical RT distribution reects a
large number (dozens if not hundreds) of repeated measurements
of a well-learned response under constant conditions. If the
subject learns new response strategies, or loses attention, as the
session proceeds, this assumption may not bevalid. is issue is
oen partially remediated by having a long practice phase before
“real” data collection starts, so that the RT and accuracy
measurements reect performance of a well-learned response.
Perhaps most important: Does the model
tell us anything non-trivial?
e main point of using computational models is (we hope) to
uncover information about latent cognitive processes, and possibly
to link these latent cognitive processes to brain substrates. So, did
the current results actually provide any insights that would not
beobvious from the behavior alone? For example, given that one
group performed more slowly than another, can weunderstand
this in terms of specic mechanisms such as increased boundary
separation, reduced dri rate, and/or increased non-decision time
– and if so, does this tell us anything interesting about the group in
question? Even more interesting, can these parameters bemapped
onto brain substrates or physiological processes?
Importantly, model results can sometimes beinformative even
in the presence of non-signicant behavioral results. For example,
even if two groups performed similarly in terms of accuracy and
RT, perhaps the model can suggest qualitatively dierent ways in
which the groups solved the speed-accuracy tradeo (perhaps one
group, with slower Ter due to motor dysfunction, “compensated”
by reducing boundary separation).
Additionally, while group dierences in model parameters can
sometimes suggest important dierences in underlying cognitive
processes, absence of parameter dierences can potentially show
where a theory falls short, in failing to describe the phenomena of
interest (Millner et al., 2020). In this way, the “failures” of a
computational model can sometimes beas insightful as its successes.
Conclusion
e above limitations notwithstanding, the DDM has become a
dominant model of speeded decision-making, and some have argued
that the DDM should replace mean RT and accuracy as default
measurement tools for cognitive psychology (Evans and
Wagenmakers, 2020). e DDM and other computational models are
a useful complement to verbal theories in that they require explicit
specication of cognitive components, and how these components
interact (Millner etal., 2020). e idea of DDM parameters that
correspond to fairly general (if latent) cognitive processes also aligns
with RDoC (Research Domain Criteria), a research framework
proposed by the U.S. National Institute of Mental Health (NIMH) for
FIGURE10
Box-and-whiskers plots showing DDM estimated parameter results from 10 data files, using MLE. (Left) Boundary separation a and drift rates d for
s1 and s2, all in arbitrary units. For easy comparison, drift rates are plotted as absolute values; the actual value of d.s1 < 0 reflects slope downward to
r1 boundary while d.s2 > 0 reflects slope upward to r2 boundary. (Right) Relative starting point z (as a proportion of a) and non-decision time Ter (in
seconds). Heavy lines indicate median; boxes indicate interquartile range (IQR); whiskers indicate range; dots indicate scores lying >1.5 IQR beyond
the box.
Myers et al. 10.3389/fpsyg.2022.1039172
Frontiers in Psychology 24 frontiersin.org
investigating mental disorders in the context of basic biological and
cognitive processes that contribute to a range of neurobehavioral
functions (Insel etal., 2010; Cuthbert and Insel, 2013).
e reader who has made it thus far will appreciate that using
and understanding the DDM can take considerable investment
of time, to understand the basic concepts, to acquire and master
soware, to design experiments that are consistent with planned
modeling, and to interpret and report results. Yet, like other
analysis methods in cognitive psychology and neuroscience,
computational models can repay this investment by providing
insightful and replicable results that complement standard
behavioral measures. It is our hope that this article will help
provide our colleagues in cognitive psychology and neuroscience
with the background to appreciate and critically evaluate research
articles that report modeling results, and even to consider using
these computational models in their own research.
Author contributions
CEM, AI, and AM contributed to the conceptualization,
design, and writing of the manuscript. All authors contributed to
the article and approved the submitted version.
Funding
is work was funded by Merit Review Award #I01 CX001826
(CEM) from the U.S. Department of Veterans Aairs Clinical
Sciences Research and Development Service. e funder had no
role in study design, data collection and analysis, decision to
publish, or preparation of the manuscript.
Conflict of interest
e authors declare that the research was conducted in the
absence of any commercial or nancial relationships that could
beconstrued as a potential conict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors
and do not necessarily represent those of their aliated organizations,
or those of the publisher, the editors and the reviewers. Any product
that may be evaluated in this article, or claim that may be made by its
manufacturer, is not guaranteed or endorsed by the publisher.
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Appendix
Supporting material
An R script le, used to generate the data-related gures and tables in this article, is provided online at https://osf.io/cpfzj/, along with
the test data le (testdata.csv) and the 10 empirical data les (empdata1.csv…empdata10.csv) used to generate the results. All data les
were generated for the purpose of this tutorial and do not constitute human subjects research data.
e R script was veried in October 2022 to run on a Macintosh iMac (Mac OS Catalina 10.15.7) running R version 4.1.2 (R Core
Team, 2021), and a Dell PC (Windows Enterprise 10) running R version 4.1.1. e script should also run under other releases of R and on
Linux machines; however, wedo not maintain (and cannot guarantee) the R packages used in the script.
is code may befreely used and/or modied for users to run with their own datasets; however, westrongly suggest that users making
their rst foray into computational modeling carefully consult the extensive tutorials and documentation provided with the soware
packages (especially DMC, rtdists, and RWiener, used in the examples here), as well as familiarizing themselves with best practices for
computational modeling (e.g., Daw, 2011; Heathcote etal., 2015; Wilson and Collins, 2019) and the many excellent DDM review articles
available (some of which are cited in this article).
Available via license: CC BY
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