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Journal of Experimental Psychology:
Learning, Memory, and Cognition
2012, Vol. ●●, No. ●, 000 – 000
© 2012 American Psychological Association
0278-7393/12/$12.00 DOI: 10.1037/a0026948
OBSERVATION
How to Say “No” to a Nonword:
A Leaky Competing Accumulator Model of Lexical Decision
Stéphane Dufau, Jonathan Grainger, and Johannes C. Ziegler
Centre National de la Recherche Scientifique, Marseille, France, and Aix-Marseille University
We describe a leaky competing accumulator (LCA) model of the lexical decision task that can be used
as a response/decision module for any computational model of word recognition. The LCA model uses
evidence for a word, operationalized as some measure of lexical activity, as input to the YES decision
node. Input to the NO decision node is simply a constant value minus evidence for a word. In this way,
evidence for a nonword is a function of time from stimulus onset (as in standard deadline models)
modulated by lexical activity via the competitive dynamics of the LCA. We propose a simple mechanism
for determining the value of this constant online during the first trials of a lexical decision experiment,
such that the model can rapidly optimize speed and accuracy in discriminating words from nonwords.
Further optimization is achieved via trial-by-trial adjustments in response criteria as a function of task
demands and list context. We show that the LCA model can simulate mean response times and response
distributions for correct and incorrect YES and NO decisions for a number of benchmark experiments that
have been shown to be fatal for deadline models of lexical decision. Finally, using lexical activity
calculated by a computational model of word recognition as input to the LCA decision module, we
provide the first item-level simulation of both word and nonword responses in a large-scale database.
Keywords: lexical decision, leaky competing accumulator, response criteria, item-level simulations,
nonword responses
of data obtained with this task (see Balota & Chumbley, 1984).
One solution to this general problem, proposed by Grainger and
Jacobs (1996), is to develop computational models that incorporate
a key distinction between task-specific and task-independent core
processes. In doing so, Grainger and Jacobs applied the notion of
functional overlap to illustrate how different tasks used to investigate a given phenomenon can each contribute to uncovering the
basic mechanisms underlying the phenomenon of interest, as long
as the contribution of task-specific mechanisms is well understood.
Continuing this tradition, we argue that lexical decision involves
some very important differences with respect to normal, everyday,
word recognition and that understanding these differences is absolutely critical for making progress in this field. However, we
also believe that when performing the lexical decision task, participants apply very general principles of human information processing expressed in terms of optimization of performance (Anderson, 1990; Oaksford & Chater, 1998). In our opinion, these
general principles do not dispense with a more fine-grained analysis of the specific mechanisms involved in performing a particular task, and it is such fine-grained analyses that advance understanding, once the more general principles have been accepted
(e.g., Perry, Ziegler, & Zorzi, 2010).
The lexical decision task, first used by Rubenstein, Garfield, and
Millikan (1970), remains one of the most widely used behavioral
measures of silent word reading. A recent smartphone application
even enables lexical decision data to be collected from thousands
of participants all over the world (Dufau et al., 2011). Participants
in this task are requested to classify as rapidly and accurately as
possible a given string of letters as being a word or not. Variation
in response times and in accuracy to different word stimuli is taken
to reflect variation in the ease with which these words are read.
Performance in the lexical decision task has therefore played a
primary role in developing and testing theories of visual word
recognition. However, as with that for any other measure of word
recognition, lexical decision performance is prone to measurement
biases and response strategies, which complicate the interpretation
Stéphane Dufau, Jonathan Grainger, and Johannes C. Ziegler, Centre
National de la Recherche Scientifique and Laboratoire de Psychologie
Cognitive, Aix-Marseille University, Marseille, France.
This research was supported by European Research Council Advanced
Grant 230313, awarded to Jonathan Grainger. We thank Max Coltheart,
Pablo Gomez, and Eric-Jan Wagenmakers for constructive comments on an
earlier version of this work. Matlab code for the LCA decision module is
available from Stéphane Dufau ([email protected]).
Correspondence concerning this article should be addressed to Jonathan
Grainger, Laboratoire de Psychologie Cognitive, Aix-Marseille Université, 3,
Place Victor Hugo, 13331 Marseille, France. E-mail: [email protected]
Lexical Decision as Evidence Accumulation
The present work follows the tradition of recent accounts of
performance in the lexical decision task (Norris, 2006; Ratcliff,
Gomez, & McKoon, 2004; Wagenmakers, Ratcliff, Gomez, &
1
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DUFAU, GRAINGER, AND ZIEGLER
2
Fn1
The Present Approach
a straightforward implementation of what could be conceptualized
as a dynamic deadline account of lexical decision and, therefore,
an alternative to standard deadline models, such as the multiple
read-out model (MROM; Grainger & Jacobs, 1996). One key
aspect of the MROM is that the temporal deadline is adjusted as a
function of lexical activity generated by the stimulus, such that the
deadline is adjusted upward with greater values of lexical activity
and downward with lower values of lexical activity (see also
Coltheart, Davelaar, Jonasson, & Besner, 1977). In this way, the
temporal deadline in the MROM is essentially a mechanism for
combining evidence for a word with elapsed time (time from
stimulus onset) in order to generate a NO response. A NO response
is generated if not enough evidence for a YES response has
accumulated before the deadline has been reached. However, Ratcliff et al. (2004) and, more recently, Wagenmakers et al. (2008)
have outlined a number of deficiencies with this particular mechanism. The deficiencies that were pinpointed in these studies are
summarized in Table 1. All concern either correct response times
(RTs) to nonword stimuli or error RTs to word stimuli. Thus, these
deficiencies are all related to performance driven by the temporal
deadline mechanism. Here we describe a solution for generating
NO responses in the LCA model that is conceptually similar to the
temporal deadline mechanism. We examine whether this dynamic
deadline model overcomes the deficiencies listed by Ratcliff et al.
(2004) and Wagenmakers et al. (2008) with respect to standard
deadline models such as the MROM (Grainger & Jacobs, 1996).
A primary goal in the present work is to understand the mechanism used by participants in a lexical decision task to discriminate
words from nonwords. After all, that is the definition of the lexical
decision task. As described above, deadline models provide a
straightforward solution here. For models like the diffusion model
and the Bayesian Reader, the solution is not so straightforward
(e.g., the virtual nonword mechanism of the Bayesian Reader).
Here we offer an easily implementable solution within the framework of LCA models.
Another important goal in the present study is to investigate the
role of adjustments in response criteria within the LCA framework
as a means of capturing effects of list context in the lexical
decision task. This second goal is a straightforward extension of
the criterion adjustment approach developed by Grainger and
Jacobs (1996) in their account of list-context effects on lexical
decision (see also Perea, Carreiras, & Grainger, 2004). The approach adopted here is not to optimize data fitting via parameter
tuning routines but to understand changes in model performance
(mimicking changes in human performance) by controlled adjustments of a handful of parameters. Put differently, the philosophy
of our approach is not to seek which parameters need to be
adjusted in order to accurately fit data but rather to test whether
theoretically motivated adjustments of parameters allow the model
to account for the data.2
Finally, it is important to note that our aim is not to compare an
LCA model of lexical decision with other eminent approaches,
In the present work we apply an alternative framework for
understanding speeded binary decision making, the leaky competing accumulator (LCA) model of Usher and McClelland (2001).
The specific architecture of the LCA model provides an ideal
framework for conceptualizing the basic mechanisms involved in
making a lexical decision. In particular, the LCA framework offers
1
The lexical decision task is not the only task in this category of binary
decisions, one other prominent example being the standard old–new decision task used to study recognition memory.
2
We are not claiming that our approach is superior in any way, just that
the focus is different.
McKoon, 2008) in applying the basic notion of noisy accumulation
of evidence over time. Input to the decision process is therefore a
measure of evidence in favor of a word response and evidence in
favor of a nonword response. However, although it is fairly
straightforward to specify what might constitute evidence for a
word, defining what might be evidence for a nonword is not so
straightforward. This is one specificity of the lexical decision task,
as opposed to other binary decision tasks, such as deciding whether
a stimulus is moving to the left or to the right, is a square or a
circle, or is red or blue. In a sense, the problem here is that
evidence for a nonword is just less of the same measure that is used
to provide evidence for a word. This is unlike binary decision
tasks, where qualitatively different kinds of evidence are associated with the two responses, and it is this particular characteristic
of the lexical decision task that was the major motivation for
deadline accounts of nonword decision making (Grainger & Jacobs, 1996).1
Ratcliff et al. (2004) described a diffusion model account of the
lexical decision task. They assigned a positive value to word
stimuli, whose strength depended on word frequency, and a negative value to nonword stimuli, whose strength was determined by
how much the nonword resembled real words. Of course, this
approach begs the question of how a lexical processor could
actually transform the evidence extracted from a string of letters
into positive and negative values. One possibility would be that the
system learns, via exposure to words and nonwords during practice, the amounts of activation generated by word and nonword
stimuli and, as a function of that experience, sets a zero point on
the activation (wordness) dimension separating positive values
from negative values. This corresponds to the drift rate criterion in
the diffusion model.
Norris’ (2006) model, the Bayesian Reader, performs lexical
decision by calculating the posterior probability that the stimulus is
a word and comparing this with the probability that it is a nonword.
The posterior probability of a word response for a particular
stimulus is calculated using the probabilities of each individual
word, given the stimulus (something akin to a global lexical
activation value). The posterior probability of a nonword response
is calculated by assuming the existence of a single virtual nonword
that has a frequency approximately equal to the average word
frequency (of words that are encountered in an experiment) and
whose distance from the input is greater than the distance of the
nearest word by a value that varies as a function of the similarity
of words and nonwords in an experiment. This implies that participants in a lexical decision experiment must evaluate the average frequency of the nonword stimuli in the experiment. They
could do so by estimating the amount of lexical activation that
nonwords compared with the word stimuli generate, in a way
similar to the setting of the drift rate criterion in the diffusion
model.
T1
Fn2
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LCA MODEL OF LEXICAL DECISION
3
Table 1
The Four Deficiencies of the Multiple Read-Out Model (MROM) of Lexical Decision
1. In MROM with word targets, error RTs cannot be faster than correct RTs without having RTs in the leading edge of the distribution (.1 quantile)
that are too fast.
2. In MROM with word targets, as word frequency increases, correct RTs get faster and accuracy increases but error RTs do not get faster.
3. In MROM with nonword targets, the RT distributions for correct RTs are not skewed like the distributions for correct RTs to words.
4. In MROM with nonword targets, correct RTs cannot be faster than error RTs, and in general MROM cannot generate fast correct responses to
nonwords while keeping response accuracy at an acceptable level.
Note. The first three points are from Ratcliff et al. (2004, p. 178), and the fourth point is from Wagenmakers et al. (2008). Italics highlight where the
problems arise. RTs ⫽ response times.
such as the diffusion model (Ratcliff et al., 2004). This particular
comparison has already been made in the work of Ratcliff and
Smith (2004). We also acknowledge the obvious similarities across
these different approaches and the fact that they can be shown to
be mathematically equivalent under certain ranges of parameter
settings (e.g., Bogacz, Brown, Moehlis, Holmes, & Cohen, 2006).
However, we believe that important differences emerge at a conceptual level and that in this respect the LCA framework offers a
more straightforward implementation of the concepts we wish to
express. Most important, the LCA model can be seen as a module
for lexical decision making that can be appended to any computational model of visual word recognition that outputs a measure of
lexical activation for a given string of letters as input (see Ziegler,
Grainger, & Brysbaert, 2010, for a review of such models). In this
way, lexical input values can be fed into the LCA response/
decision module on a trial-by-trial basis, thus providing the opportunity for performing item-level analyses of simulation results.
An LCA Account of Lexical Decision
F1
Our LCA model of lexical decision is shown in Figure 1.
Applied to the specific case of lexical decision, bottom-up input to
the YES response node is evidence for a word that is derived from
the stimulus. In this respect we follow standard practice. What is
original in our approach is that the input to the NO response node
is simply a constant value minus the evidence for a word. Our
model can therefore be thought of as implementing a dynamic
deadline mechanism for nonword decision making, in that, like the
MROM deadline mechanism (Grainger & Jacobs, 1996), the time
it takes to make a nonword response depends only on lexical
activity (i.e., evidence for a word) and elapsed time.
In our LCA model there are two sources of lexical influences on
activity in the NO response node (see Figure 1). The first influence
comes from the standard practice of having total input to the two
response nodes equal to a constant (Usher & McClelland, 2001).3
This implies that in the absence of any positive evidence for a
nonword (the solution adopted here), the bottom-up input to the
NO response node is equal to the constant total input value minus
the evidence for a word extracted from the stimulus. The use of a
total constant input also differentiates the current LCA model from
previous proposals (Davis, 2010; Grainger, Dufau, & Ziegler,
2009), in which input to the NO decision node was only a function
of elapsed time. The second source of lexical influences in our
LCA model arises via the mutually inhibitory connections that
exist between the two response nodes, such that a rise in activity
in one automatically causes a reduction in activity in the other and
vice versa. Both of these mechanisms are fundamental to the LCA
approach in general. Finally, the other ingredient of LCA models
is the “leak” or decay of information that is associated with the
noisy accumulation of information over time.
Implementation
We implemented a linear version of LCA in which accumulators
are updated according to the following equations:
yYES 共i ⫹ 1兲 ⫽ yYES 共i兲 ⫹ ⌬YES 共i兲
(1)
yNO 共i ⫹ 1兲 ⫽ yNO 共i兲 ⫹ ⌬NO 共i兲
(2)
where yYES and yNO are variables representing the activities of the
YES and NO accumulators (here, the calculation of the ith ⫹ 1
cycle), and deltas are variation terms of the corresponding cycle.
Deltas are calculated with the following equations:
⌬ YES 共i兲 ⫽ cYES 共i兲 䡠 关⫺w 䡠 yNO 共i兲 ⫺ k 䡠 yYES 共i兲 ⫹ IYES ⫹ GYES 共i兲兴
(3)
⌬ NO 共i兲 ⫽ cNO 共i兲 䡠 关⫺w 䡠 yYES 共i兲 ⫺ k 䡠 yNO 共i兲 ⫹ INO ⫹ GNO 共i兲兴
(4)
Figure 1. General architecture of the LCA model of lexical decision.
YES and NO decision nodes receive bottom-up information from their
associated inputs: lexical input for the YES node and constrained input
(constant minus YES input) for the NO node. The decision nodes mutually
inhibit each other. LCA ⫽ leaky competing accumulator.
3
Note that the constant total input is not applied here in order to solve
the scaling problem, as is typically done when applying parameter-fitting
routines to LCA models (Donkin, Brown, & Heathcote, 2009). As shown
in Table 2, a majority of the parameters are fixed in our approach.
Furthermore, in the General Discussion we discuss how this constant input
value might change as a function of experimental conditions, hence avoiding the problems noted by Donkin et al.
Fn3
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DUFAU, GRAINGER, AND ZIEGLER
4
Figure 2. An illustration of processing in the LCA model. Panel A: YES and NO activities with three parameter
sets illustrating the noisy accumulation of information and the between-node inhibition (10 trials per simulation;
from left to right: YES input ⫽ .12, .11, .08; NO input ⫽ .08, .09, .12). The first parameter set gives a greater
advantage to YES activity than the second one and therefore inhibits NO activity more quickly. The third
parameter set gives an advantage to NO activity. Panel B: Blowup of the second parameter set simulation (left,
YES activity; right, NO activity). Each activity is associated with a decision criterion (black solid lines), whose
value is drawn from a normal distribution (starting mean ⫽ .5; standard deviation ⫽ .05) and is subject to a
trial-by-trial adjustment. A response is made when one of the two activities crosses its threshold, generating a
distribution of simulated response times expressed in cycles for both the YES and NO activities. In this
simulation (Panel B), the parameter set represents weak evidence for a word, and the two distributions
correspond to correct (YES) and error (NO) responses to word targets. LCA ⫽ leaky competing accumulator.
Fn4
where k and w are the leakage and inhibition factors, IYES and INO
are the input to the YES and NO accumulators, GYES and GNO are
Gaussian noise terms associated with activity in the YES and NO
accumulators, and cYES and cNO are bounding factors ensuring that
yYES and yNO values stay within [0;1].4
IYES ⫹ INO is a constant.
F2
(5)
The model therefore makes decisions on the basis of the accumulation of noisy, leaking, and competing information over time (see
Figure 2, Panel A). Whenever the accumulation of activation in
either the YES or the NO decision node reaches the criterion value
specified for that trial by the setting of response thresholds, a
lexical decision response is made and processing is terminated.
The response corresponds to the decision node whose threshold
was reached first, and the response time corresponds to the time
from stimulus onset to the time at which the threshold was reached.
In our simulations, we also maintained the principle of adding
noise to the decision criteria that was an essential ingredient of
MROM (Figure 2, Panel B; Grainger & Jacobs, 1996; Jacobs &
Grainger, 1992). One key aspect of the model is the implementation of trial-by-trial adjustments of response criteria that occur
within a simulation. For example, in simulating a lexical decision
experiment where the focus is on response speed rather than
4
Following the equations of interactive activation (McClelland & Rumelhart, 1981), activation is bounded between 0 and 1, but removing this
constraint does not affect the performance of our LCA model.
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LCA MODEL OF LEXICAL DECISION
T2, AQ:1
accuracy, the YES and NO threshold values decrease by a constant
value on each trial as long as a correct response is given by the
model. When an error is made, thresholds are reset to their initial
values. In experiments where task instructions stress accuracy
rather than speed, a smaller constant value for threshold adjustments is used (see Table 2).
Parameter Settings
T3
Following the parameter setting approach adopted by Grainger
and Jacobs (1996), the core parameters of the LCA model were
hand-tuned to a target set of data (Experiments 1 and 2 from
Ratcliff et al., 2004). These parameter values are given in Table 3.
Once the core parameters of our LCA model were fixed, the only
parameters that were allowed to vary in order to accommodate
specific patterns of results were (a) the values of bottom-up input
to the model (to capture changes in stimulus-driven factors) and
(b) the values of response criteria, which were modified on a
trial-by-trial basis (to capture influences of task and list context).
Bottom-Up Input
The different word frequency conditions were simulated by a
gradient of lexical input values ranging from .11 (very lowfrequency words) to .12 (high-frequency words; see Table 2).
Lexical input differed for the two classes of nonword stimuli tested
in the simulations. Orthographically regular, pronounceable pseudowords were assigned a lexical input value of .0875, whereas
nonwords formed of random strings of letters were assigned a
lexical input value of .0825. Input to the NO decision node was
simply a constant value (.2) minus the lexical input. For example,
a lexical input of .11 would generate a NO input of .09 (word
stimulus), and one of .09 would generate a NO input of .11
(nonword stimulus).
Response Criteria Adjustments
As noted above, one key feature of the present theoretical work
is the implementation of trial-by-trial adjustments in response
criteria. The initial settings for YES and NO response criteria were
drawn independently from a normal distribution of mean .5 and
standard deviation .01. The initial value of the mean and the value
5
Table 3
Values of the Core (Nonmodifiable) Parameters of the LCA
Model of Lexical Decision
Parameter
Value
Activity curves
w (inhibition)
k (leakage)
IYES ⫹ INO (total input)
GYES and GNO mean (Gaussian noise)
GYES and GNO SD (Gaussian noise)
yYES and yNO intervals
Measurements
Response criteria SD
No. trials per run
No. runs per simulation
Note.
.15
.05
.2
0
.05
[0;1]
.01
1,500
50
LCA ⫽ leaky competing accumulator; SD ⫽ standard deviation.
of the standard deviation did not change across simulations. At the
end of every trial, the mean values of both response criteria were
adjusted as a function of response accuracy, such that if a correct
response was made these values were decreased by a small constant. Whenever an error occurred, the mean values of response
thresholds were reset to their initial values. The threshold adjustment constant varied as a function of task instructions, type of
nonwords, and percentage of word versus nonword stimuli (see
Table 2).
Simulation Studies
Our LCA model was first evaluated against the results of Experiments 1 and 2 of Ratcliff et al. (2004) in Simulation Study 1.
This seemed the most appropriate set of benchmark phenomena for
an initial evaluation, given that (a) these were the data used to
falsify the MROM and (b) these data were used as the unique
benchmarks in a recent evaluation of the Bayesian Reader model
(Norris, 2009). The model was also evaluated against the experiments of Wagenmakers et al. (2008) in Simulation Study 2 and the
English Lexicon Project (ELP) lexical decision data (Balota et al.,
2007) in Simulation Study 3. The results of Wagenmakers et al.
provide another important benchmark for models of the lexical
Table 2
Bottom-Up Lexical Input Values (Evidence for a Word), Initial Threshold Values, and (in Parentheses) the Corresponding
Trial-by-Trial Adjustment Values Used in the Simulations
Lexical input values
Simulation
1
2
3
Stimuli/conditions
Words
Words
Words
Words
Words
Words
Words
and
and
and
and
and
and
and
pseudowords
random strings
pseudowords (accuracy stressed)
pseudowords (speed stressed)
pseudowords (75% words)
pseudowords (75% pseudowords)
pseudowords
HF
LF
VLF
NW
Threshold values
YES / NO (adjustment)
.12
.12
.12
.12
.12
.12
BIAM
.115
.115
.115
.115
.115
.115
BIAM
.11
.11
.11
.11
.11
.11
BIAM
.0875
.0825
.0875
.0875
.0875
.0875
BIAM
.5 (.003)/.5 (.003)
.5 (.02)/.5 (.03)
.5 (.003)/.5 (.003)
.5 (.05)/.5 (.05)
.5 (.02)/.5 (.003)
.5 (.003)/.5 (.02)
.5 (.001)/.5 (.001)
Note. HF ⫽ high-frequency; LF ⫽ low-frequency; VLF ⫽ very low-frequency; NW ⫽ nonwords; BIAM ⫽ bimodal interactive activation model
(Diependale et al., 2010).
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6
DUFAU, GRAINGER, AND ZIEGLER
decision task, and the ELP simulation provided an initial test of the
possible integration of the LCA decision mechanism within the
larger framework of models of word recognition and lexical decision (Coltheart, Rastle, Perry, Langdon, & Ziegler, 2001; Grainger
& Jacobs, 1996).
In the experiments of Ratcliff et al. (2004), high-frequency
(HF), low-frequency (LF), and very low-frequency (VLF) words
were intermixed with regular pronounceable nonwords (pseudowords) in Experiment 1 and random letter strings in Experiment 2.
Wagenmakers et al. (2008) tested HF, LF, and VLF words intermixed with pseudowords while manipulating speed versus accuracy instructions in their Experiment 1. In these experiments,
correct responses to words were faster and accuracy was higher
when nonwords were random strings than when they were pseudowords, and responses were faster but accuracy was lower when
speed was stressed in the instructions to participants. Wagenmakers et al. (2008) reported the results of a second experiment in
which they manipulated the proportion of word versus nonword
stimuli, with one condition having 75% words and another condition having 75% nonwords. Overall, their results showed that RTs
were faster and accuracy was higher for the stimulus category
(word vs. nonword) that occurred the most frequently in a given
condition.
The simulations reported here were structured like a behavioral
experiment. Each simulation study consisted of 50 consecutive
runs, equivalent to 50 virtual participants. Each run was composed
of 750 words and 750 nonwords randomly ordered, except for the
simulations in which word/nonword proportion was manipulated,
where there were 1,125 words and 375 nonwords in the 75% word
simulation and 375 words and 1,125 nonwords in the 75% non-
word simulation. Number of cycles to reach either the YES or the
NO decision threshold was recorded on each trial, as well as the
winning response (YES or NO).
Simulation Study 1
Simulation Study 1 used the conditions tested in Experiments 1
and 2 of Ratcliff et al. (2004), in which HF, LF, and VLF words
were either intermixed with orthographically regular and pronounceable nonwords (pseudowords, Experiment 1) or with random letter strings (random strings, Experiment 2). The effects of
this manipulation of type of nonword were simulated by a change
in the rate of the trial-by-trial adjustments of response criteria (see
Table 2). The rationale behind this particular simulation is discussed below. Figure 3 shows the scatterplots of simulated and
experimental mean RTs for the different conditions tested in these
experiments. The 16 mean RTs are derived from the combination
of stimulus type (HF, LF, and VLF words, and nonwords), correctness of response (correct and error), and type of nonword
(pseudowords and random strings). The results of these simulations clearly show that our LCA model provides a good fit to the
experimental data. It should be noted that the LCA model does not
exhibit the same problem as does the Bayesian Reader model,
which systematically underestimated error RTs in these experiments (see Norris, 2009, Tables 1 and 2).
Most important, as can be seen in Figure 3, is that the LCA
model accurately captures the pattern of RTs generated by the NO
decision node (correct responses to nonwords and errors to words).
This is exactly where the MROM (Grainger & Jacobs, 1996)
failed, which demonstrates that a dynamic deadline account of
Figure 3. Performance of the LCA model in simulating the response time (RT) condition means from Ratcliff
et al. (2004, Experiments 1 and 2). R2 is the percentage of explained variance. Correct responses are in dark gray,
errors are in light gray. PW ⫽ pseudowords; HF ⫽ high frequency; LF ⫽ low frequency; VLF ⫽ very low
frequency; RS ⫽ random strings; LCA ⫽ leaky competing accumulator.
F3
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LCA MODEL OF LEXICAL DECISION
lexical decision, as implemented in our LCA model, can overcome
the fundamental limitations of standard deadline models. We return to this point in greater detail in the General Discussion.
Simulation Study 2
F4
Simulation Study 2 tested the LCA against the data from
Wagenmakers et al. (2008) in which HF, LF, and VLF words were
intermixed with orthographically regular, pronounceable pseudowords. The same parameter set as in Simulation Study 1 was
maintained in these simulations. In Experiment 1 of Wagenmakers
et al. (2008), participants were given either instructions that
stressed accuracy of responding or instructions that stressed speed
of responding. The effects of this manipulation were simulated by
changes in the rate of trial-by-trial adjustments in response criteria.
In the speed condition, adjustment of both response criteria was
greater than in the accuracy condition (see Table 2). In Experiment
2, Wagenmakers et al. manipulated the percentage of words versus
pseudowords in the experiment (75% words in one condition and
25% in the other condition). Once again, these manipulations were
simulated in LCA by trial-by-trial adjustments of response criteria.
In the condition with a high proportion of words, adjustment of the
YES decision node was greater than adjustment of the NO decision
node, and in the condition with a high proportion of nonwords, the
opposite adjustment was implemented (see Table 2).
Figure 4 presents the results of simulations run on the LCA
model. The core parameters of the LCA model were the same as
those used in Simulation Study 1. Quite remarkably, by simply
modifying the rate of adjustment of response criteria, the LCA
7
model captured practically the same amount of variance in mean
RT (82%) as in Simulation Study 1 that involved not only a
different experimental manipulation (type of nonword), but also
different stimuli and participants. Once again, the LCA model
captures key patterns in the data that the MROM could not capture.
First, in the speed instructions condition tested by Wagenmakers et
al. (2008), the LCA model correctly simulates the faster RTs
obtained to error responses than to correct responses (diamond
shapes in Figure 4), and one can also see a clear frequency effect
in both the correct and the error RTs that is well captured by the
LCA model. Figure 5 (left panel) shows the distribution of simulated correct and error RTs to word targets in the speed instructions
condition, and it therefore demonstrates that RTs can be faster on
errors than on correct responses without RTs in the leading edge of
the distribution being too fast. Second, correct RTs to nonword
stimuli can be faster than error RTs to nonword stimuli, as was the
case in the accuracy instructions condition of Wagenmakers et al.
and as captured by LCA (circle shapes in Figure 4). This is further
demonstrated in Figure 5 (right panel), showing the distribution of
simulated RTs for correct and error responses to nonwords in the
accuracy instructions condition of Wagenmakers et al. Here we
clearly see a shift in the complete RT distribution, with correct RTs
being faster than error RTs for nonword stimuli. Furthermore,
Wagenmakers et al. demonstrated that this pattern of correct and
error RTs to nonword stimuli is exaggerated in the 75% nonword
condition they tested, but the opposite pattern is seen in the 75%
word condition. Figure 4 shows that LCA perfectly captures this
pattern. Finally, Figure 6 shows the mean accuracy per condition
Figure 4. Performance of the LCA model in capturing the pattern of mean response times (RTs) per condition
in Wagenmakers et al. (2008). R2 is the percentage of explained variance. Correct responses are in dark gray,
errors are in light gray. PW ⫽ pseudowords; HF ⫽ high frequency; LF ⫽ low frequency; VLF ⫽ very low
frequency; acc ⫽ accuracy instructions; speed ⫽ speed instructions; 75W ⫽ 75% word stimuli; 75NW ⫽ 75%
nonword stimuli; LCA ⫽ leaky competing accumulator.
F5
F6
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8
DUFAU, GRAINGER, AND ZIEGLER
Figure 5. Left panel: Distributions (quantile means) of simulated correct and error response times (RTs) in the
LCA model for high-frequency (HF), low-frequency (LF), and very low-frequency (VLF) words under the speed
instruction conditions of Wagenmakers et al. (2008). Right panel: Distributions (probability density) of
simulated correct and error RTs to nonwords under the accuracy instruction conditions of Wagenmakers et al.
(2008). LCA ⫽ leaky competing accumulator.
in Simulation Studies 1 and 2, in order to demonstrate that the
LCA model’s success in simulating mean RT is not achieved at the
cost of accuracy.
Summarizing the results of Simulations Studies 1 and 2, we
conclude that the LCA model provides an excellent fit to the major
data patterns presented by Ratcliff et al. (2004) and Wagenmakers
et al. (2008). Most important, with the same set of core (nonmodifiable) parameters, our model generated excellent fits to the data of
Wagenmakers et al., via theoretically motivated, trial-by-trial adjustments in the values of response criteria.
Simulation Study 3
Figure 6. Mean accuracy in the six experiments and simulations of the
present study. In black, pseudowords; in dark gray, very low-frequency
words; in light gray, low-frequency words; in white, high-frequency words.
PW ⫽ pseudowords (Ratcliff et al., 2004, Experiment 2); RS ⫽ random
strings (Ratcliff et al., 2004, Experiment 1); Accuracy ⫽ accuracy stressed
(Wagenmakers et al., 2008, Experiment 1); Speed ⫽ speed stressed
(Wagenmakers et al., 2008, Experiment 2); 75W ⫽ 75% words (Wagenmakers et al., 2008, Experiment 3); 75NW ⫽ 75% nonwords (Wagenmakers et al., 2008, Experiment 4).
Simulation Study 3 tested the LCA model against the data from
the English Lexicon Project (ELP; Balota et al., 2007) in which
words were intermixed with orthographically regular, pronounceable pseudowords. From the 40,000⫹ words in the ELP database,
we extracted the 5-letter and 6-letter words (842 words) that had
been tested in Ratcliff et al. (2004). We also randomly selected 842
5-letter and 6-letter nonwords from the ELP database. These words
and nonwords were then submitted as input to the bimodal version
of the interactive activation model (Diependaele, Ziegler, &
Grainger, 2010). For each stimulus, we took the activation level of
the most activated word unit at the 15th cycle (i.e., before asymptote). These activation values were then linearly transformed to fit
the range of input values used in Simulation Studies 1 and 2,
and these values were used as input to the YES response node in
LCA. In this simulation, the same core parameter values as in
Simulation Studies 1 and 2 were maintained (see Tables 2 and 3).
The value of the threshold adjustment parameter was set to a small
value for both thresholds, given the large number of trials in these
simulations.
Figure 7 shows the simulation run on LCA. Although the
percentage of explained variance (37%) is lower than in the
previous simulations, it must be remembered that variance is
explained here at the item level, as opposed to variance in mean
RT in Simulation Studies 1 and 2. One should also keep in mind
that the amount of reproducible variance in such large-scale data-
F7
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LCA MODEL OF LEXICAL DECISION
AQ: 2
9
Figure 7. LCA simulation of lexical decision response times (RTs) extracted from the English Lexicon Project
database (Balota et al., 2007) corresponding to the 842 words tested by Ratcliff et al. (2004) and 842 randomly
selected nonwords (left panel, behavioral RT distributions; bottom panel, simulated RT distributions; central
panel, scatterplot of mean RT per item). R2 is the percentage of explained variance. NW ⫽ nonwords; HF ⫽
high-frequency words; LF ⫽ low-frequency words; VLF: very low-frequency words. LCA ⫽ leaky competing
accumulator.
bases is probably no more than 40% (see Rey, Courrieu, SchmidtWeigand, & Jacobs, 2009). Furthermore, this result is quite encouraging, given that no parameter tuning was performed in order
to increase the amount of variance that could be explained. Figure 7 also shows that LCA generates RT distributions that change
as a function of word frequency and lexical status (word vs.
nonword), much in the same way as the empirical distributions. To
our knowledge, this is the first simulation of large-scale lexical
decision data that includes a simulation of nonword responses as
well as word responses. It is also the first simulation of large-scale
lexical decision data involving trial-by-trial adjustments in response criteria (see Perea et al., 2004, for similar simulations of a
standard lexical decision experiment). The relative success of this
simulation opens up a whole new perspective for analyzing sequential dependencies in such large-scale data and examining the
consequences of adjustments in response criteria on the evolution
of correct and error RTs as well as error rates in such data.
General Discussion
The lexical decision task is probably the most popular task used
to study silent single-word reading. Yet, we still have no clear
understanding of how participants in a lexical decision experiment
perform this task. In particular, it is not at all clear how participants
respond negatively when the stimulus is not a word. One simple
solution to this problem, the temporal deadline mechanism (implemented in the MROM; Grainger & Jacobs, 1996), was rejected
on the basis of recent analyses of the patterns of RTs obtained to
correct and error responses in the lexical decision task (Ratcliff et
al., 2004; Wagenmakers et al., 2008; see Table 1). In the present
work, we tested a dynamic deadline account of lexical decision
implemented within the framework of an LCA (Usher & McClelland, 2001). This model has two decision nodes, one for a YES
decision and one for a NO decision. Input to the YES decision node
is a function of lexical activity generated by the stimulus, whereas
input to the NO decision node is equal to a constant value minus
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10
DUFAU, GRAINGER, AND ZIEGLER
the input to the YES decision node. The activation dynamics of the
two decision nodes follow the standard equations of LCAs (Usher
& McClelland, 2001), with decay over time (leak) and lateral
inhibition between nodes (competition).
Simulation Studies 1 and 2 provided a test of LCA on two
independent sets of experimental data. Simulation Study 1 was
used to hand-tune the core parameters of LCA. Having fixed these
core parameter values, LCA was able to capture the effects of type
of nonword (pseudowords vs. random strings) by adjusting the
lexical input generated by these two types of stimuli and by
adjusting the rate of adjustment of response criteria. LCA captured
83% of the variance across the 16 mean RTs reported by Ratcliff
et al. (2004, Experiments 1 and 2).
The same core parameters whose values were adjusted to fit the
empirical data in Simulation Study 1 were used in the second
simulation study. The only parameter that changed between these
two simulation studies was the value of the trial-by-trial adjustment of response criteria. Despite the absence of any systematic
parameter tuning, LCA accounted for 82% of the variance across
the 32 mean RTs reported in Wagenmakers et al. (2008). More
precisely, the model provided an accurate account of variations in
mean RT and error rate to word and nonword stimuli as a function
of changes in speed versus accuracy instructions and the percentage of words versus nonwords in the experiment. Overall, the
results of Simulation Studies 1 and 2 showed that the LCA model
easily passed the tests that had been fatal for the MROM (see
Table 1) and that proved problematic for Norris’ (2009) Bayesian
Reader model (i.e., incorrect predictions concerning error RTs).
We analyze this particular aspect of the simulation results in more
detail in the following section.
Simulation Study 3 tested LCA against a subset of words and
nonwords extracted from the English Lexicon Project database of
item-level lexical decision latencies. Input to the YES response was
word activation level as calculated by the bimodal interactive
activation model (Diependaele et al., 2010), which provides a
measure of word activity for word and nonword stimuli of varying
length. The core parameter values used in Simulation Studies 1 and
2 remained unchanged. The model provided a good fit to the
item-level data, suggesting that it can be usefully appended to
word recognition models in order to account for performance in
the lexical decision task.
In the following sections, we first analyze precisely how the
LCA model overcomes the deficiencies of its predecessor: the
multiple read-out model (MROM) of lexical decision (Grainger &
Jacobs, 1996). Then we discuss the main contributions of the
present approach in increasing our understanding of how participants actually perform the lexical decision task. In particular, we
describe the LCA solution to the hard question in modeling lexical
decision, that is, what constitutes evidence for a nonword? Finally,
we discuss a key novel aspect to the current approach, that is, the
implementation of trial-by-trial adjustments in response criteria.
The Four Deficiencies of MROM
In Table 1 we listed the four deficiencies of the MROM
(Grainger & Jacobs, 1996), as identified in the work of Ratcliff et
al. (2004) and Wagenmakers et al. (2008). Here, we analyze the
success of the LCA model in passing these tests.
Considering first of all Points 1 and 2, both of which concern
RTs in response to word stimuli. MROM was found to be incapable of producing fast error RTs to word targets without generating excessively fast RTs in the first decile of the RT distribution
(Point 1), and error RTs to word targets in MROM did not show
a word frequency effect that is seen in the empirical data (Point 2).
The LCA simulation of the speed instruction condition tested by
Wagenmakers et al. (2008) revealed faster error RTs than correct
RTs to word targets (see Figure 4) without distorting the RT
distribution (see Figure 5, left panel). Also, the results of Simulation Studies 1 and 2 both revealed a word frequency effect in the
error RTs to word stimuli (see Figures 3 and 4).
Next, let us consider Points 3 and 4, both of which concern RTs
in response to nonword stimuli. MROM was found to generate RT
distributions for correct responses to nonword stimuli that did not
have the same rightward skew as the distributions for correct
responses to words (Point 3), and MROM could not generate fast
correct RTs to nonwords (and never faster than error RTs to
nowords) without generating too many errors (Point 4). Figure 5
(right panel) reveals that the RT distributions for nonword stimuli
in the LCA model show the distinctive rightward skew. The
simulation results reported in Figure 3 clearly demonstrate that the
LCA model can generate RTs for correct responses to nonword
targets that are faster than the error RTs to the same targets.
Finally, accuracy in the LCA model compares favorably with
accuracy in the experiments, as shown in Figure 6.
Input to the NO Response
Having abandoned a deadline account of generating NO responses in lexical decision, as implemented in the MROM
(Grainger & Jacobs, 1996), we note that the hard question for
alternative accounts of lexical decision, such as the diffusion
model (Ratcliff et al., 2004), the Bayesian Reader (Norris, 2009),
or the present LCA model, is to define what constitutes evidence
for a nonword. Defining evidence for a word (i.e., a YES response)
is relatively straightforward. In the first two simulation studies of
the present work, we followed Ratcliff et al. in using word frequency to determine these input values. In the third simulation
study, we used a measure of lexical activity derived from a
computational model of visual word recognition (Diependaele et
al., 2010). The main contribution of the present work concerns our
proposal for how to calculate evidence for a nonword.
In the LCA model, input to the NO decision node (i.e., evidence
for a nonword) is equal to a constant value minus the input to the
YES decision node (i.e., a measure of lexical activity generated by
the stimulus). The model therefore has to know only two things in
order to calculate input to the NO decision node: the amount of
lexical activity generated by the stimulus and the total input value
that optimizes speed and accuracy of responses to words and
nonwords.
So, a key remaining question here concerns how the total input
value could be computed online during the first trials of a lexical
decision experiment? One simple solution is to monitor the lexical
activity generated by word and nonword stimuli on each trial
(these correspond to the lexical input values given in Table 3, and
to compute the sum of these two values. This sum can be used to
provide an initial estimate of the total input value after having seen
only one word and one nonword in a lexical decision experiment.
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This initial value can then be adjusted in order to optimize performance, much in the same manner as response criteria are
adjusted on a trial-by-trial basis in our model. Increasing the total
input value will generate faster RTs to nonwords but will also
generate more errors to word stimuli, as input to the NO response
will eventually be greater than input to the YES response. Errordriven adjustments of this value can therefore be used to rapidly
home in on an optimal setting. A similar mechanism could be
applied to the diffusion model (Ratcliff et al., 2004), where evidence for a word and evidence for a nonword would be averaged,
rather than summed, in order to provide an initial estimate of the
drift rate criterion.
It is important to note here that our proposed mechanism for
computing the optimal value of the total constant input in LCA is
analogous to the way in which response criteria are adjusted in our
model. This implies that the modifiable parameters in our LCA
model all involve either stimulus-driven changes in lexical input
values or trial-by-trial adjustments of the mean values of the YES
and NO decision thresholds and the total constant input in order to
optimize performance.
Online Adjustments of Response Criteria
One of the key features of the LCA model, in line with the
general approach to modeling adopted in the MROM (Grainger &
Jacobs, 1996), is that effects of list context and task instructions
are simulated via adjustments of response criteria. In the present
study, we demonstrated that such adjustments in the thresholds of
the YES and NO decision nodes provided an accurate account of
how different types of nonword affect responding to word stimuli
(an effect of list context), how responses to word and nonword
stimuli are affected by the proportion of words versus nonwords in
the experiment (another effect of list context), and how responses
to word and nonword stimuli are affected by stressing speed over
accuracy or vice versa (an effect of task instructions). The general
principle that was applied in this endeavor is that participants in a
lexical decision experiment are trying to optimize their performance in line with the instructions they receive and do so by
monitoring their performance on a trial-by-trial basis. This critical
information provided by such monitoring is whether or not a
mistake was made, and it is this information about accuracy of
responding that was used to adjust response criteria in the present
work.
In implementing such trial-by-trial adjustments of response criteria, we follow in the steps of a number of prior attempts to
describe such adjustments in the light of empirical data concerning
post-error responding (e.g., Laming, 1968; Rabbitt, 1966; Rabbitt
& Rodgers, 1977). In particular, conflict monitoring theory (Botvinick, Braver, Barch, Carter, & Cohen, 2001) accounts for errorrelated changes in performance by postulating the occurrence of
conflict in association with errors. Conflict is used to change the
activation level of output units, which is of course functionally
equivalent to modifying response thresholds, as in the present
approach. This allowed Botvinick et al. to account for the standard
pattern of slower RTs and higher levels of accuracy following an
error (but see Notebaert et al., 2009, for different patterns of
post-error performance).
We implemented a relatively simple version of this general
“monitor and adjust” principle, which despite its simplicity pro-
11
vided an accurate account of the behavioral data. The mean values
of the thresholds of YES and NO decision nodes were lowered on
every correct trial by a small constant value that was determined
by task instructions and list context, and the rate of change could
be different for the two decision nodes. Every time there was an
error, the values of both thresholds were reset to their initial
values.
To simulate the influence of type of nonword on responses to
both nonword and word targets, we increased the rate of adjustment of response thresholds in the easy nonword condition (i.e.,
random strings), and this increase was slightly greater for the NO
threshold than the YES threshold. The NO threshold can be lowered to a certain extent without an increase in false negative errors,
as long as the YES threshold is lowered at the same time. The YES
threshold can be lowered to a certain extent without an increase in
false positive errors because random string nonword stimuli have
high input activation to the NO response node. Similar adjustments
of the rate of change of response thresholds were used to simulate
the effects of the proportion of word versus nonword stimuli in an
experiment. With more words than nonwords, the YES response
threshold was lowered by a greater amount on each correct trial,
and with more nonwords than words it was the NO response
threshold that was adjusted by a greater amount. The lowering of
response thresholds as a function of the proportion of words and
nonwords enables optimization of responding for a majority of
trials in the experiment. Finally, with speed instructions, the trialby-trial threshold adjustment was increased by the same amount
for both decision nodes, thus simulating participants’ attempts to
respond faster to both word and nonword stimuli. This generated
more false negative and false positive errors, and the resulting
error RTs were faster due to the fact that they are mostly generated
with the lowest settings of response criteria. Because errors immediately generate a resetting of response criteria to their initial
values, correct responses in these conditions are therefore slower
on average than error responses.
Thus, overall our simple implementation of trial-by-trial adjustments in response criteria provided an accurate simulation of the
effects of list context and task instructions as tested in the studies
of Ratcliff et al. (2004) and Wagenmakers et al. (2008). Although
this study is clearly only a first approximation with respect to how
participants in a lexical decision experiment adjust to such contextual factors, the relative success of this approach encourages
further tests of more sophisticated adjustments in response criteria
with the aim to capture trial-by-trial changes in human performance.
Conclusions
In 1996, Grainger and Jacobs presented an influential account of
the lexical decision task (the multiple read-out model; MROM), in
which time from stimulus onset was used to trigger NO responses
via a temporal deadline. This account has been rejected on the
basis of results showing fast NO responses in certain conditions in
lexical decision experiments (Ratcliff et al., 2004; Wagenmakers
et al., 2008). Here we implemented and tested a dynamic deadline
model of the lexical decision task using the framework of leaky
competing accumulators (Usher & McClelland, 2001). The LCA
model was shown to overcome the deficiencies associated with the
MROM, and overall it provided an accurate account of variations
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DUFAU, GRAINGER, AND ZIEGLER
12
in RTs for correct and error responses to word and nonword
stimuli as a function of the type of nonword, the proportion of
words versus nonwords in the experiment, and whether or not task
instructions stressed speed or accuracy of responding. Finally, by
using lexical activity calculated by a computational model of
visual word recognition as lexical input to the LCA decision
module, we were able to provide the first item-level simulation of
both word and nonword responses in a large-scale database.
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Received February 18, 2011
Revision received December 1, 2011
Accepted December 5, 2011 䡲
JOBNAME: AUTHOR QUERIES PAGE: 1 SESS: 1 OUTPUT: Thu Dec 29 01:20:29 2011
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AUTHOR QUERIES
AUTHOR PLEASE ANSWER ALL QUERIES
AQ1: Author. Table 3 was cited before Table 2, so they were switched.
AQ2: Author. Figure 7. The caption could make a clearer distinction between the panels.
1