Out of Sight Out of Mind?
The Effects of Prior Study and Visual Attention on Word Identification
THESIS
Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in
the Graduate School of The Ohio State University
By
Charlette Lin, B.S.
Graduate Program in Psychology
The Ohio State University
2015
Master's Examination Committee:
Roger Ratcliff, Advisor
Ian Krajbich
Alex Petrov
Copyrighted by
Charlette Lin
2015
Abstract
Presentation of stimuli has been shown not to facilitate later forced-choice perceptual
identification, in which two response alternatives were presented on the screen at test.
Several models were developed to explain performance in these paradigms, but none
addressed the time course of processing. In this study, we examined the effect of prior
study on performance by modeling accuracy and response times as well as eye fixation
data. The model assumes two racing diffusing processes in which evidence accumulation
rate (drift rate) differs as a function of the response alternative being currently viewed.
Change in performance between the different study conditions (studied target, studied
foil, or studied neither) was accounted for by a change in drift rate. Even without the eye
tracking data, the model successfully fit choice behavior, response time distributions, and
many of the eye fixation results.
ii
Dedicated to all those who seek more knowledge
and push back the boundaries of the unknown.
iii
Acknowledgments
I am grateful to my advisor, Roger Ratcliff for giving me the resources, help, and
patience to complete this project. I am also sincerely thankful to Alex Petrov for inviting
me to study at The Ohio State University, and for all of his kind and constructive
criticisms of my work. Additionally, I greatly appreciate all the opinions and ideas of Ian
Krajbich, from which this study benefited immensely. Lastly, I would not have finished
this without the support of my friends and family, so this work is also a product of their
kindness.
iv
Vita
July 2005 ........................................................Torrey Pines High School
2011................................................................B.S. Psychology, University of California,
San Diego
2012 to present ..............................................Graduate Associate, Department of
Psychology, The Ohio State University
Publications
McKenzie, C. R. M., Sher, S., Müller-Trede, J., Lin, C., Liersch, M. J., & Rawstron, A.
G. (2015). Are Longshots Only for Losers? A New Look at the Last Race
Effect. Journal of Behavioral Decision Making.
Fields of Study
Major Field: Psychology
v
Table of Contents
Abstract .......................................................................................................................... ii
Dedication ...................................................................................................................... iii
Acknowledgments.......................................................................................................... iv
Vita..................................................................................................................................v
List of Tables ................................................................................................................ vii
List of Figures ............................................................................................................... viii
Introduction .....................................................................................................................1
Methods..........................................................................................................................20
Results ............................................................................................................................27
Discussion ......................................................................................................................31
Appendix: Figures and Tables .......................................................................................39
References ......................................................................................................................70
Footnotes ........................................................................................................................75
vi
List of Tables
Table 1. Example words for each study condition .........................................................66
Table 2. Reaction time and accuracy results from data .................................................67
Table 3. Estimated drift rates and chi-square values achieved ......................................68
Table 4. Estimated parameters and chi-squared values by subject ................................69
vii
List of Figures
Figure 1. Examples of visually similar pictures.............................................................40
Figure 2. Illustration of the Counter Model ...................................................................41
Figure 3. Graphical summary of the diffusion model ....................................................42
Figure 4. Diffusion model depiction of speed-accuracy trade off .................................44
Figure 5. Krajbich et al. (2010)'s experiment and model ...............................................46
Figure 6. Instruction slides for experiment ....................................................................48
Figure 7. Sequence of displays for study list and a word identification trial.................50
Figure 8. Estimated drift rates for each condition..........................................................52
Figure 9. Quantile probability function for data and model predictions........................53
Figure 10. Model assumptions and fixation characteristics from data ..........................55
Figure 11. More model assumptions and fixation characteristics..................................56
Figure 12. Model predictions and fixation characteristics from data ............................58
Figure 13. Probability of picking left and time advantage.............................................60
Figure 14. Probability of picking first fixated and first fixation duration .....................62
Figure 15. Time advantage of the not-last item .............................................................65
viii
Introduction
It is plausible that seeing something before will help one identify it later,
especially if one is familiar with repetition priming. There may be evolutionary
advantages to having such a mechanism, since encountering a specific predator yields a
higher likelihood of encountering that same threat later. Thus, being able to identify a
repeated stimulus faster could be beneficial. Priming, defined as improved processing of
a repeated stimulus, is a ubiquitous finding, so many researchers have investigated this
phenomenon and several theories have been built off of it (e.g. Roediger & McDermott,
1993; Tulving & Schacter, 1990). However, Ratcliff, McKoon, and colleagues (Ratcliff
& McKoon, 1996, 1997; Ratcliff, Allbritton, & McKoon, 1997; Ratcliff, McKoon, &
Verwoerd, 1989) showed that prior presentation does not improve subjects' abilities to
perceive the stimulus; subjects are actually biased to choose what they perceived before.
In order to demonstrate this phenomenon, Ratcliff and colleagues utilized forced
choice perceptual identification (FCPI) tasks. The objective of the FCPI task is to
identify a stimulus presented in a way that makes identification difficult. For example, an
auditory stimulus could be played with white noise. A visual stimulus could be flashed
very briefly (e.g. 16 milliseconds) before being masked. Immediately after the target is
given, two alternatives are displayed, and the participant is instructed to select the
alternative that matches the target.
1
Prior presentation can be in the form of memorization lists or various judgment
tasks, like lexical decision. In many of Ratcliff and colleagues' studies, the study
conditions of interest were where the correct alternative was studied (target-studied), the
wrong alternative was studied (foil-studied), or neither alternative was studied (neitherstudied). 1 Accuracy in the neither-studied condition acted as the baseline. As expected,
when the target was studied, participants were more accurate, meaning they chose the
target more often than baseline. However, when the foil was studied, participants were
less accurate by choosing the foil more often than baseline. Thus, prior presentation of
stimuli can hurt performance as well as improve it. Furthermore, the increase in accuracy
in the studied-target condition is the same size as the decrease in accuracy in the studiedfoil condition. These are consistent with a bias interpretation of the phenomenon, as
opposed to a priming interpretation. Priming would predict that prior presentation of a
stimulus would only cause an improvement in identification of it, by facilitating
processing.
Another important manipulation in the FCPI task is the similarity of the
alternatives. Similar auditory stimuli may be words like "disc" and "bisque." Similar
visual stimuli may be words like "spared" and "spaced" or similar-looking pictures as in
Figure 1. Performance in FCPI tasks are not affected by prior presentation if the
alternatives are dissimilar. For example, if the target was the word "spared," and the foil
was a dissimilar word like "buffet," then prior presentation of "spared" would not
2
increase the probability of participants identifying the target as "spared", and prior
presentation of "buffet" would not increase the probability of participants misidentifying
the target as "buffet" (Masson & Macleod, 1996; McKoon & Ratcliff, 1996; Ratcliff,
Allbritton, & McKoon, 1997; Ratcliff & McKoon, 1996; 1997; Ratcliff, McKoon, &
Verwoerd, 1989; Rouder, Ratcliff, & McKoon, 2000). In other words, there was no
advantage of studying the target, and no disadvantage of studying the foil if the
alternatives were not similar.
Since the original theories of priming could not account for these findings,
Ratcliff and McKoon (1997) developed the Counter Model in hopes of spurring
development of competing models. In this model, each alternative is represented by a
counter, which accumulates counts as evidence for that alternative. During each unit of
time in the decision process, exactly one of the counters receives a single count. An
alternative is chosen when its counter collects k more counts than its opposition. To
illustrate, suppose that a visual target is the word "spared," and the similar foil is
"spaced." If the counter for "spared" collects k more counts than the counter for
"spaced," then the decision-making system identifies the target word as "spared" (Figure
2A).
There are three different types of counts. A diagnostic count corresponds to a
feature of the target that is informative in distinguishing between the two alternatives,
because only one of the alternatives has that feature. For instance, if "spared" was the
3
flashed target, and "spaced" was the foil, then a diagnostic count could be given by the
letter "r" to the "spared" counter. A cohort count corresponds to a feature of the target
that is uninformative between the two alternatives, because both alternatives share that
feature. In our example, the letter "s" would yield a cohort count, and could go to either
counter. A null count does not depend on the target stimulus presentation, and could also
go to either counter. Null counts introduce noise to the system, and allow a decision to
be reached even when little or no information was extracted from the stimulus.
Prior presentation of a word allows that word's counter to steal nondiagnostic
counts (cohort or null counts) from the counters of similar words. Continuing with our
example, if "spaced" was studied before, then it could steal counts from "spared" (Figure
2B). This mechanism of theft allows the model to explain the bias to pick alternatives
that were previously studied. Importantly, stimuli can only steal counts from similar
neighbors. If "spaced" was studied, but the foil was dissimilar like "buffet," then
"spared" cannot reach the counts of "buffet." This is how the model explains why there is
bias when alternatives are similar, but not when they are dissimilar.2 The counters of
high-frequency words are also assumed to have a higher resting level than counters of
low-frequency words, an assumption made by other models of word identification too
(e.g. McClelland & Rumelhart, 1981; Morton, 1969). This means that counters for highfrequency words start with some number of counts, which results in bias to identify
targets as those higher frequency stimuli, relative to lower frequency stimuli.
4
Since Ratcliff and McKoon's challenge in 1997, more studies were conducted and
more models were developed (e.g. the ROUSE model by Huber, Shiffrin, Lyle, & Ruys,
2001; and the criterion-shift model by Wagenmakers, Zeelenberg, Schooler, &
Raaijmakers, 2000). It was found that when both alternatives were low-frequency words,
prior study leads to an overall increase in accuracy in the FCPI task with similar
alternatives instead of just a bias (Bowers, 1999; McKoon & Ratcliff, 2001;
Wagenmakers, Zeelenberg, & Raaijmakers, 2000). To accommodate this, the Counter
Model's assumption about high-frequency words was simply changed to have increased
probability of diagnostic counts instead of a higher resting level. There are many
possible explanations for why this could be the case theoretically (Wagenmakers et al.,
2000), but this does not affect the model, because the implementation remains the same.
Crucially, these older models only considered accuracy, and ignored reaction time
(RT), even though both are important for evaluating performance due to the speedaccuracy trade off. Decision-makers often trade accuracy for speed or vice versa
depending on the situation. If Participant A was 90% accurate, and Participant B was
87% accurate on the same task, then we may be tempted to conclude that Participant A
performed better. However, if Participant A's average RT was 2400 ms while Participant
B's average RT was 550 ms, then our assessment may change dramatically. Put another
way, an individual's speed does not correlate with accuracy (Ratcliff, Thompson, &
McKoon, 2015). If Participant C is faster than Participant D, Participant C does not
5
necessarily have to be more accurate too. For this reason, even though it's plausible that
accurate participants are also fast participants, this is probably not the case. Accuracy
and RT must be jointly considered when studying decision-making, but researchers
continue to ignore one or the other in their studies. Using a sequential sampling model
like Ratcliff's diffusion model (1978; Ratcliff & McKoon, 2008), we can concurrently
account for both of these factors and more appropriately analyze performance.
Ratcliff's diffusion model could be said to be a unifying framework on simple
decision-making, which allows us translate behavioral data to components of processing.
It has been successfully applied to many different areas within psychology including
psychopathy (e.g. White, Ratcliff, Vasey & McKoon, 2010a, 2010b), sleep deprivation
(e.g. Ratcliff & Van Dongen, 2009; 2011), aging (e.g. Ratcliff, Thapar, & McKoon,
2010, 2011; Starns & Ratcliff, 2012), and development (e.g. Cohen-Gilbert et al., 2014;
Ratcliff, Love, Thompson, & Opfer, 2012). It has also reached other fields like
neuroscience (e.g. Ratcliff, Philiastides, & Sajda, 2009; Resulaj, Kiani, Wolpert, &
Shadlen, 2009) and economics (e.g. Clithero & Rangel, 2014). Moreover, the model
helped develop understandings of various decision processes like multi-choice decisions
(e.g. Krajbich & Rangel, 2011; Leite & Ratcliff, 2010), confidence judgments (e.g.
Ratcliff & Starns, 2009; 2013), single-choice decisions (e.g. Gomez, Ratcliff, & Perea,
2007; Ratcliff & Strayer, 2014), and response signal/deadline tasks (e.g. Jahfari,
Ridderinkhof, & Scholte, 2013; Ratcliff, 2006). Other successful decision-making
6
models, like the linear ballistic accumulator (Donkin, Brown & Heathcote, 2011) and the
leaky competing accumulator (Usher & McClelland, 2001) share the same general
structure as the diffusion model. We build off of the diffusion model in this study to
examine the effects of prior study on performance in a FCPI task.
The standard diffusion model was designed for simple two-alternative tasks, with
reaction times of about 1000 ms. This time consists of non-decision time and decision
time. Non-decision time includes all processes that are not part of the decision process,
like encoding the stimulus, memory or lexicon access, and motor response (Figure 3A).
Decision time is the amount of time it takes for one to accumulate enough evidence to
make a choice. The process starts at a point 𝑧 between the two alternatives' boundaries.
One choice is represented by the upper boundary 𝑎 and the other choice is represented by
the lower boundary 0 (Figure 3B). The system nosily accumulates evidence towards one
or the other boundary, and a response is initiated once the process touches a boundary.
These decision boundaries represent how much evidence a decision-maker demands
before making a choice. If someone is being careful, then more evidence is desired to
decide on one or the other alternative, which corresponds to wider boundaries (Figure
4A). Conversely, if the decision-maker is in a rush, and cares more about making the
decision quickly, then less evidence is required before reaching a decision, which
translates to narrower boundaries (Figure 4B).
7
The rate of evidence accumulation is drift rate 𝑣, so higher drift rates lead to
faster reaction times. Drift rate is higher for an alternative when there is bias in favor of
that alternative and/or when information is more readily available for it (e.g. White,
Kapucu, Bruno, Rotello, & Ratcliff, 2014). Bias for choices due to imbalanced stimulus
frequencies or different payoffs can also be represented by a shift in the starting point
(Leite & Ratcliff, 2011). The decision process starts closer to whichever choice it is
biased for, so there is a shorter distance to travel before reaching this boundary,
corresponding to faster response times. This means that the decision process has a longer
distance to travel before reaching the boundary of the less frequent direction, which
makes responses for that direction slower. Additionally, the decision process is much
more likely to accidentally hit the boundary of the more frequent direction because its
boundary is closer than the boundary of the less frequent direction. Shifting the starting
point of the decision process can produce bias behavior from the model, though it is
mathematically equivalent to move one boundary further and the other boundary closer.
Krajbich, Rangel, and colleagues found evidence that drift rate is linked with
visual attention (Krajbich & Rangel, 2011; Krajbich, Armel, & Rangel, 2010; Krajbich,
Lu, Camerer, & Rangel, 2012). Their model, the attentional drift-diffusion model
(aDDM), tracks where the decision-maker is attending and penalizes the drift rates of unattended stimuli. In other words, alternatives have higher drift rates when they are being
looked at, than when they are not being looked at. In Krajbich et al.'s 2010 study, hungry
8
participants were presented two high-resolution images of food items side-by-side and
were instructed to choose which one they would prefer to eat at the end of the experiment
(Figure 5A). At the end of the experiment, one trial was randomly selected, and the
participant's choice on that trial was given to him/her. In a diffusion model framework,
they represented the left food item with the top boundary and the right food item with the
bottom boundary (Figure 5B). The relative decision value 𝑉 of the two options is the
evidence in the decision system and changes over time. At time 𝑡 of a decision process,
if the left item is fixated, then the relative decision value is
𝑉𝑡 = 𝑉𝑡−1 + 𝑑(𝑟left − 𝜃𝑟right ) + ℰ𝑡
Where ℰ𝑡 is a term for normally distributed noise, while 𝑟left and 𝑟right are the
values of the left and right items respectively. Taking the difference between 𝑟left and
𝑟right yields a measure of difficulty for that trial; the smaller it is, the harder the decision.
1
𝑑 is a positive constant in units of 𝑚𝑠 which converts the value difference into the speed
of the decision process—drift rate. Thus, more difficult decisions have smaller drift
rates.
𝜃 is a parameter between 0 and 1 which discounts the value of the non-fixated
item. If the right item is fixated, then 𝑟left is instead penalized with the discount
parameter 𝜃.
𝑉𝑡 = 𝑉𝑡−1 + 𝑑(𝜃𝑟left − 𝑟right ) + ℰ𝑡
9
If 𝜃 is set to 1, then the model is a regular diffusion model
𝑉𝑡 = 𝑉𝑡−1 + 𝑑(𝑟left − 𝑟right) + ℰ𝑡
With this model, it does not matter which item is fixated, and the drift rate is just
a function of the difference between the two item values.
For Krajbich and co's studies and the present study, a "fixation" on an item is
defined as the total duration of the gaze within the item's specified region. Multiple
fixations within an item's region are still counted as one fixation to that item. The
fixation does not end until the gaze leaves the item's region and enters another item's
specified region.
When the aDDM was fit to data in previous studies (Krajbich & Rangel, 2011;
Krajbich et al., 2010; Krajbich et al., 2012), a value of 𝜃 between 0 and 1 fit data better
than 𝜃 values of 0 or 1. Therefore, some discounting of the non-fixated item was more
plausible than complete discounting of the non-fixated item (𝜃 = 0) and no discounting
of the non-fixated item (𝜃 = 1). This model makes many predictions. First, the last item
fixated is more likely to be chosen than the item that was not fixated last. This is because
the decision process during the last fixation will on average drift towards the fixated
item's boundary, unless the fixated item's value is so much worse relative to the nonfixated item's that its drift rate becomes negative. Second, last fixations tend to be shorter
than middle fixations, which are the fixations that are not the first or last of a trial. This
10
is because last fixations are cut short when a decision boundary is reached. Third, an
alternative's accumulated evidence is generally positively correlated with the amount of
time it was looked at, which predicts many other trends. For example, the more time a
given item was looked at during a trial, the more time the alternative item should be
looked at, in order for the alternative to be chosen over the given item. Similarly, longer
fixations on an item correlates positively with the probability of that item being chosen.
Another prediction is that the more time spent looking at an item above the amount of
time spent looking at its opposition, the more likely the item with the time advantage will
be chosen. Additionally, the duration of the last fixation should be positively correlated
with the time advantage that the other item has before the last fixation. This is due to the
idea that the last fixated item tends to be chosen, and it would have had to overcome the
excess evidence that the other item accumulated—which is usually more if that other
item had more time advantage. Krajbich et al.'s data exhibited these predicted patterns,
so their model seems to provide a good foundation for understanding the relationship
between visual attention and decision-making.
Schotter, Berry, McKenzie, and Rayner (2010) observed that the effect of
fixations may depend on the task. In their study, they used a two-alternative forced
choice task in which the stimuli were photographs. In one condition, participants were
instructed to pick the picture they liked more, and in another condition, they were
instructed to pick the picture they liked less. In the control conditions, the participants
11
were instructed to pick the picture that looked older or more recent. Schotter et al. found
that participants indeed tended to look longer at the photographs that were chosen
compared to photographs that were not chosen, which they called the gaze bias effect.
Importantly, this effect was attenuated when the task was to choose the disliked
photograph. Results showed that in general, participants also tended to look longer at
photographs that they liked.
Armel, Beaumel, and Rangel (2008) conducted a similar study, but controlled the
amount of time participants were able to look at the alternatives. Two items were
displayed one at a time, for six times each. One item was displayed for 300 ms each time
and then the other item was displayed for 900 ms time, so in total, the former would be
on screen for 1800 ms while the later would be on screen for 5400 ms. They conducted
three experiments, each of which had different types of choices. Experiment 1 used
appetitive snacks (junk food such as popular chocolate bars), Experiment 2 used aversive
foods (e.g. Spam and baby food), and Experiment 3 used art posters. The participants'
tasks were to choose the food that they would rather eat for Experiments 1 and 2, and the
poster they would rather have for Experiment 3. The experimenters found that appetitive
foods and posters were more likely to be chosen when they were displayed longer.
However, aversive foods were less likely to be chosen when displayed longer. This is
consistent with the previous study, where the gaze bias effect was lessened during the
task to choose the disliked photo.
12
These findings show that the nature of the task and the type of stimuli change how
the non-fixated alternative is affected. There may be an interaction between the task and
the valence of the stimuli. If the task is to choose a preferred alternative out of two
desirable items, or to choose the disliked alternative out of two aversive items, then the
gaze bias effect would manifest strongly. Conversely, if the task is to choose the less
preferred alternative of two desirable items, or to choose the preferred item from two
aversive items, then the gaze bias effect may disappear or reverse. In the model, this
means that a fixated alternative does not always have a higher drift rate relative to when it
is not fixated (disregarding the effects of noise). This means that an alternative is not
always more likely to be chosen when it is fixated, and it may even have a smaller
probability of being chosen when fixated.
However, a lot of decision-making is not based on preference and does not have
highly desirable or aversive choices. In the FCPI task, one alternative is correct and the
other is incorrect, so preference should be irrelevant. If the stimuli are neutral words,
then there should also be little effect of preference relative to when the stimuli are photos,
foods, or consumer products. Evidence accumulation here might not be as fungible since
the task is less dependent on subjective value, and the stimuli do not have clear utilities.
The FCPI task is also different in that gathering information about one of the
alternatives gives corresponding information about the other alternative. Evidence for
alternative A being correct is the same as evidence for alternative B being incorrect.
13
During a given trial, a participant might not look at both alternatives before making a
decision in the FCPI task, because if the first alternative seen is correct, then the other
alternative must be incorrect. Likewise, if the first one is incorrect, then the other one
must be correct, and a second fixation is not necessary. This is unlike preference based
tasks in which the participant must look at both alternatives in order to make an informed
decision about which one is preferred.
In the aDDM, the novel discount parameter 𝜃 represents how much less the
decision-maker is accumulating evidence for the non-fixated alternative. However, in the
FCPI task, evidence against the fixated alternative is perfect evidence for the non-fixated
alternative. Looking at one alternative theoretically should not harm the evidence
accumulation of the other alternative, since they are perfect complements of each other
for determining which one should be chosen. Therefore, visual attention might not
matter, and the discount parameter 𝜃 close to a value of 1 may fit this task better. If this
is the case, then data from a FCPI task should not exhibit a positive correlation between
visual attention on a given item and probability of choosing it.
Conversely, an accurate decision can be made even if only one of the alternatives
was attended, and the non-fixated alternative was completely ignored. If this is what
participants do, then there should be a strong positive correlation between time spent
looking at an item and probability of picking it. Given that accuracy is better than
chance, then participants should also spend more time looking at correct alternatives,
14
unless it is possible to look at an option, and decrease evidence for it as in the preference
based studies by Schotter et al. (2010) and Armel et al. (2008). However, the FCPI task
is not preference-based, so perhaps parameter estimation of 𝜃 would then be closer to 0 if
participants are spending more time looking at target words relative to foil words. It is
unclear what the aDDM should predict about evidence accumulation of the non-fixated
items here. In this study, we investigated how visual attention affects decision-making in
an FCPI task and estimated discount parameter 𝜃.
The model we used has roots in other models. It was inspired by the Counter
Model (Ratcliff & McKoon, 1997) and adopts the discount parameter 𝜃 from the aDDM
(Krajbich & Rangel, 2011; Krajbich et al., 2010; Krajbich et al., 2012), but it is most
closely related to Ratcliff and Starns' (2013) RTCON2 model (Response-Time
Confidence 2). RTCON2 is able to accommodate more than two alternatives, and though
we only use two alternatives in our current study, the structure of RTCON2 will allow us
to easily accommodate more alternatives as well.
In this model, each alternative is represented by an evidence accumulator 𝑥𝑖
which is incremented by this equation:
Δ𝑥𝑖 = 𝑎𝑣𝑖 Δ𝑡 + 𝜎𝜂𝑖 √Δ𝑡
15
Where 𝑎 is a scalar, 𝑣𝑖 is the drift rate of the 𝑖th accumulator, Δ𝑡 is the time step,
and 𝜎 is the standard deviation of the within-trial variability in drift rate. On each time
step, one accumulator is randomly chosen to be incremented.
Importantly, when one accumulator is incremented, all of the other accumulators
are decremented by a total amount equal to the increment. As a result, the total amount
of evidence in the system does not change. All accumulators other than the incremented
𝑥𝑖 are decremented by this equation:
Δ𝑥𝑗 = − (
= −(
1
) (𝑎𝑣𝑖 Δ𝑡 + 𝜎𝜂𝑖 √Δ𝑡)
𝑁−1
1
) Δ𝑥𝑖
𝑁−1
for 𝑗 ≠ 𝑖
𝑁 is the number of accumulators (i.e. alternatives or choices).
There are five parameters plus 𝑁 drift rates (𝑣𝑖 ), 𝑁 between-trial variability in
drift rates (𝜂𝑖 ), and 𝑁 boundaries (𝑏𝑖 ). 𝑎 is the scaling parameter that converts 𝑣𝑖 into
drift rates, and the drift rates vary normally within-trial with standard deviation 𝜂𝑖 . Each
boundary vary uniformly across trials with range 𝑠𝑏 . Non-decision time is captured by
parameter 𝑇𝑒𝑟 , which is uniformly distributed across trials with range 𝑠𝑡 .
In Ratcliff and Starns' (2013) study, they had reason to posit 𝑁 different
boundaries, because they assumed that their alternatives required different amounts of
evidence to be chosen. In our FCPI task, we assumed that participants have equal
16
tendency to choose the left and right alternatives, so all accumulators had the same
boundary (𝑏). Our other parameters were: scale on drift rate (𝑎), drift rates for each
condition (𝑣𝑖 ), across-trial variability in drift rate (𝜂𝑖 ), within-trial variability of drift rates
(𝜎), across-trial variability of boundaries (𝑠𝑏 ), non-decision time (𝑇𝑒𝑟 ), and variability in
non-decision time (𝑠𝑡 ). Our incrementing equation was the exact same as RTCON2's:
Δ𝑥𝑖 = 𝑎𝑣𝑖 Δ𝑡 + 𝜎𝜂𝑖 √Δ𝑡
Since there were two alternatives in the FCPI task, 𝑁 was 2, and the complement
decrementing equation was simply:
Δ𝑥𝑗 = −Δ𝑥𝑖
for 𝑗 ≠ 𝑖
This means that when one accumulator is incremented, the other accumulator is
decremented by the same amount.
To incorporate the discount parameter 𝜃, which represents the bias against the
non-fixated alternative, there was a single change if the incremented accumulator's word
was not fixated:
Δ𝑥𝑖 = 𝜃𝑎𝑣𝑖 Δ𝑡 + 𝜎𝜂𝑖 √Δ𝑡
The decrementing equation remained unchanged. 𝜃 is a parameter which takes
values between 0 and 1, so when the incremented accumulator's word is not fixated, the
change in evidence is smaller. In the FCPI task, evidence accumulation for the nonfixated alternative could be near 0, because most of the evidence accumulation could be
17
construed in terms of the correctness of the current fixated word. Or, the evidence
accumulation of the non-fixated item could be completely unrestrained by the discount
parameter (𝜃 = 1), because evidence against the fixated alternative is perfect evidence
for the non-fixated alternative and vice versa.
In the past, non-decision time (𝑇𝑒𝑟 ) has been assumed to occur partially before the
decision process (e.g. for stimulus encoding or memory access) and partially after the
decision process (e.g. for response execution), but such assumptions were not explicit in
implementations of the Diffusion Model. 𝑇𝑒𝑟 was normally just added to the decision
time, without specifying that some portion of it happened before the decision process and
some happened after, because it was mathematically equivalent. Krajbich et al. (2010)
assumed that non-decision time was the time spent not looking at an item. In other
words, their 𝑇𝑒𝑟 for a specific trial was the sum of all of the time before, after, and in
between fixations on items. We preferred not to make any assumptions about the
cognitive processes during those times, so we specified that 0 to 100 ms of the first
fixation is non-decision time, and 0 to 100 ms of the last fixation is non-decision time.
The range 0 to 100 ms is not integral to our model, as changing it did not affect the model
fits.
The goal of this study is to investigate the cognitive processes underlying the
influences of visual attention and prior presentation in a FCPI task. Prior research on the
bias effect of prior presentation has been incomplete by analyzing only accuracy and not
18
reaction times. Research on the effect of visual attention on decision-making is still new,
and has not been studied with a FCPI task yet. Thus, we tracked the eye movements of
participants while they performed a FCPI task and fit our model to the data.
19
Methods
Participants
Twenty-two undergraduate students from Ohio State University participated and
earned $12 or course credit for the session.
Apparatus
Participants' fixation patterns were recorded at sampling rate of 1000 Hz using an
EyeLink 1000 desktop-mounted eye tracker. Viewing was binocular but only movements
of the right eye were recorded. Following calibration, eye position errors were less than
1°.
The stimuli were displayed on a Compaq P110 Color Monitor (19.8 × 20.6 × 20.1
in) with a pixel resolution of 640 × 480 and an update rate of 60 Hz. Participants
responded using a Microsoft SideWinder® Plug & Play Game Pad.
Materials
The stimuli were drawn from pools of orthographically similar word pairs (e.g.
"cart" and "card"). We kept account of word frequency, though word frequency was not
a key manipulation in our experiment. Four word pools were made: 61 pairs of two highfrequency words, 54 pairs of low-frequency words, 51 pairs of one high-frequency word
and one low-frequency word, and 428 pairs of medium-frequency words. Highfrequency words ranged from 78 to 10,595 occurrences per million (M = 323.25; Kučera
20
& Francis, 1967). Medium-frequency words ranged from 6 to 77 occurrences per million
(M = 150). Low-frequency words ranged from 4 to 5 occurrences per million (M = 4.41).
18 study lists were constructed by taking one word of 16 random word pairs.
Then, 18 word identification blocks were constructed with 16 word pairs of a
corresponding study list plus 16 random pairs, which made 32 trials per word
identification block. Finally, three memory-test lists were constructed with 8 studied
words from six corresponding study lists (as expounded on later) and 8 new words drawn
from the medium-frequency words, which made 16 trials for each memory test.
Stimuli were displayed in a monospaced font, such that each character was 0.8
degrees across at the experiment’s viewing distance of 27.56 inches (70 cm). For the two
word alternatives during word identification trials, the left and right words were separated
by about 28.75 degrees, which was as far apart as they could be separated on the display.
Procedure
At the beginning of a session, the participant was presented with instructions
verbally along with the PowerPoint slides in Figure 6, followed by a practice block.
There were two tasks in the experiment: a memory task and a word identification
task. Participants were instructed to remember the words presented in the study lists for
later memory tests. During a study list, words were shown one at a time for 1000 ms
each with a 300 ms interstimulus interval (Figure 7A). After each study list, there was a
word identification block. A word identification trial was initiated with a "+" symbol in
21
the center of the screen, which was displayed until it was fixated. Then the target word
was flashed for 20 ms, before being masked by a row of "@" symbols (Figure 7B). The
word alternatives were displayed below the mask until a response was made. Participants
were instructed to pick the word that was just flashed.
After every six study lists and six word identification blocks, there was a memory
test which covered material from those six study lists. At the beginning of each memory
test, instructions were displayed on the screen: "Right-click for words you've seen in this
test; Left-click for new words you've not seen." Test words were displayed one at a time
until a response was made. “ERROR” messages were displayed for 500 ms after
incorrect responses.
There was a practice study list, a practice word identification block, and a practice
memory test before the experimental blocks. The practice study list was exactly like an
experimental study list. Unlike the later memory tests that covered three study lists each,
the practice memory test only covered the one practice study list. The practice word
identification block provided error feedback, but the experimental blocks did not.
Each study list, word identification block, and memory test was initiated by the
participant, and they were allowed to rest as desired in between. Participants were
instructed to "go as quickly and as accurately as possible." For all trials, if participants
responded faster than 250 ms from the stimulus onset, then “TOO FAST” was displayed
on the screen for 1500 ms. If they responded slower than 1500 ms, then “TOO SLOW”
22
was displayed on the screen for 500 ms. An experimental session lasted approximately
45 minutes.
Design
There were three experimental conditions: target studied, foil studied, and neither
studied. In the target studied condition, the target word (the flashed word to be
identified) was on the latest study list. In the foil studied condition, the foil word (the
similar-looking, but incorrect distractor) was on the most recent study list. In the neither
studied condition, neither word was on the latest study list nor any of the previous study
lists (See Table 1). This last condition served as the baseline condition. Every wordidentification block comprised of eight target-studied trials, eight foil-studied trials, and
16 neither-studied trials, in a random order determined by the computer. Each participant
partook in all of the conditions.
Data analysis
We enforced stringent criteria for a participant's data to be included in our
analyses; performance had to be significantly above chance on both the word
identification task and the memory task. Two participants were excluded for performing
at or below chance on the memory task, and four participants were excluded for
performing at or below chance on the word identification task.
Fixations were only recorded during word identification blocks. A fixation on a
word was defined as the total gaze duration within that item's specified region, which was
23
a box around the area in which the word would appear. Fixations to different parts of the
same word still only counted as one fixation. A fixation did not qualify as terminated
until the gaze left the word's specified region and into the other word's region, or until the
trial ended. For example, fixations to the left item, then to nothing (i.e. not fixated on the
left item nor the right item), and then to the left item again would be counted as a single
left fixation. This is based on the assumption that it is unlikely for a participant to look
at a word, look away, and then look back at the same item, and that such gaze patterns are
due the machine losing track of the gaze location. After this consolidation process, if a
fixation was less than 40 ms, then it was excluded from the data.
Fixation pools were constructed for each participant by taking all fixations from
that participant, excluding last fixations. The model sampled from these fixation pools to
simulate fixating on an item before switching to the other item.
Model fitting
The data was split by accuracy (correct and error responses) and by study
condition (studied target, studied foil, and studied neither). Then, the number of
observations and the 0.1, 0.3, 0.5, 0.7, and 0.9 reaction time quantiles were computed for
each bin, yielding 36 data points. For each participant, the model was simultaneously fit
to all 36 data points, but since there are no exact solutions for this model, we used Monte
Carlo methods to simulate 20,000 trials per study condition per iteration.
24
We estimated best-fitting parameters with a simplex algorithm which minimized
the chi-square statistic through parameter adjustments (Ratcliff & Tuerlinckx, 2002).
The process used initial parameters approximated from similar tasks for the first iteration,
and the chi-square statistic was calculated with observed data points against expected
model predictions.
Hypothesis
The main parameters of interest were the drift rates of the target in each condition.
The drift rates of the foils were simply the converse of the target drift rates (𝑣lure =
1.0 − 𝑣target), so they were not free parameters. Our hypothesis was that the bias for
studied words would be exhibited as an increase in its drift rate. Thus, drift rates for the
target should be highest in the studied-target condition, lowest when its opposition was
studied (studied-foil condition), and in-between in the studied-neither condition. The rest
of the parameters remained constant across conditions.
Of note, bias could not be exhibited in the form of closer boundaries for the
studied item, because participants had no way of knowing whether the studied item
would be on the right or the left. (This depiction of bias is equivalent to having the
starting point be closer to the biased item's boundary.) If the boundary for the studied
item was closer than that of the unstudied items, then the entire RT distributions of
studied items should shift down and be faster than the RT distributions of unstudied
items. Changes in boundaries or starting points affect reaction time more than accuracy,
25
whereas changes in drift rates affect accuracy more than reaction time (Ratcliff &
McKoon, 2008). If the bias is due to a change in drift rate, then there should be a change
in accuracy between the conditions.
We implemented the non-fixated discount parameter (𝜃) in two ways. First, we
fit the model five times, fixing 𝜃 at 0.00, 0.25, 0.50, 0.75, and 1.00. Then, we fit the
model again, letting 𝜃 vary and using the aforementioned routine find the best fitting
values.
26
Results
The three conditions differed on which word was previously studied: targetstudied, foil-studied, and neither-studied. Table 2 shows the reaction time and accuracy
data from each condition. There was no significant difference between conditions for
correct RTs, F(2,45) = 0.066, p > .05, nor for error RTs, F(2,45) = 0.987, p > .05. If we
had only considered reaction times, we may have concluded that there was no effect of
study condition. For accuracy, there was significant difference among the conditions,
F(2,45) = 3.283, p < .05. Only the increase in accuracy from studying the foil to studying
the target was significant; t(30) = 2.61, p = 0.01, but the accuracy of the baseline studiedneither condition was in between the other two conditions as expected, so we replicated
the bias effect of prior study. As the Counter Model predicted, the difference in accuracy
between studied-target and studied-neither (0.03) was also about the same size as the
difference between studied-neither and studied-foil (0.04).
Table 3 shows the estimated drift rates for each condition with the non-fixated
discount parameter (𝜃) at 0.00, 0.25, 0.50, 0.75, and 1.00. Drift rates were highest in the
target-studied condition and lowest in the foil-studied condition. Also, drift rate appeared
to be negatively correlated with 𝜃 (Figure 8); as the value of 𝜃 goes down, drift rates
generally increases, possibly to compensate for the greater suppression of drift rates when
𝜃 is small. When we let our simplex routine attempt to find values of 𝜃 that would best
27
fit the reaction time and choice probability data, the average value returned was 0.548
(SE = 0.033, Table 4).
A concise way of viewing the reaction time and choice probability data and model
predictions are quantile probability functions (QPFs) as described by Ratcliff and
McKoon (2008). Probability of response is on the x-axis. Since correct responses are
more likely than error responses, all points to the right of the figure are correct responses,
and all points to the left of the figure are error responses. For example, the right-most
group of vertical numbers/dots represents the correct responses in the studied-target
condition. The horizontal position of this group is the response probability (i.e. 0.82 was
the probability of responding correctly in the studied-target condition). Reaction time
(ms) is on the y-axis. The quantiles 0.1, 0.3, 0.5, 0.7, and 0.9 are plotted for the correct
and error responses of each condition, so there are five points for each group. QPFs of
the data and model predictions show that our model captures the reaction time quantiles
and choice probabilities fairly well for all values of 𝜃 (Figure 9).
The aDDM made some assumptions regarding the visual attention data. The
number of fixations should not differ between the study conditions, and they did not;
F(2,45) = 0.06, p > 0.05 (Figure 10A). In all the conditions, the probability of looking at
the target word first should not be significantly different from chance, and it was not;
studied target: t(15) = 0.64, p > 0.05; studied foil: t(15) = 1.42, p > 0.05; studied neither:
t(15) = 1.21, p > 0.05 (Figure 10B). The duration of middle fixations (fixations that were
28
not the first or last fixations) on the target word and the foil words should not differ, and
they did not; t(30) = -0.19, p > 0.05 (Figure 11A). Middle fixation durations also should
not change with condition, and again, they did not (F(2,45) = 0.06, p > 0.52) (Figure
11B). The model is under predicting the average number of fixations per trial, and is
under prediction the duration of fixations. This may be due to the fact that our model
only simulates fixations during decision processes, while participants may be making
fixations during non-decision time.
Regarding model predictions of the visual data, last fixations (M = 291 ms, SE =
12 ms) were shorter than middle fixations (M = 398 ms, SE = 14 ms); t(25) = 6.8, p <
0.001 (Figure 12A), as predicted by all of the models. First fixations (M = 285 ms; SE =
9 ms) were also shorter than middle fixations (t(25) = 6.81, p < 0.001), and our models
also predicted this, but we had not hypothesized this beforehand. Excluding singlefixation trials from this analysis eliminated the effect in the model predictions, but it did
not change the trend in the data (labeled as "First*" in Figure 12A).
The probability of picking the last fixated item was greater than chance (M = 0.58,
SE = 0.02); t(15) = 4.21, p < 0.05 (Figure 12B). All models predicted this, except for the
standard diffusion model with 𝜃 = 0. The model with 𝜃 = 0.75 generated predictions
that best fit the data in this case. Along the same lines, accuracy was higher when the last
fixation was on the target word (M = 0.82, SE = 0.02) than when the last fixation was on
the foil (M = 0.74, SE = 0.03); t(29) = 2.37, p < 0.05. The left item also had a greater
29
probability of being picked when it was fixated last regardless of whether was the target
(t(26) = 3.25, p < 0.05) or the foil (t(24) = 3.28, p < 0.05).
The probability of picking the left item increased as the time advantage for the left
increased (time advantage for the left is total time looking left minus total time looking
right for a given trial) (Figure 13A). This trend was still there after correcting for the
probability of picking left when the target was on the left (Figure 13B). The range of 𝜃
values produced different predictions, and values of 0.25 and 0.50 seemed to match the
data the best for this analysis.
The probability of picking the first fixated word generally increased as the
duration of the first fixation became longer (Figure 14A). This trend still seemed to exist
when corrected for the probability of picking the first fixated item when the target was
fixated first (Figure 14B). Unexpectedly, none of the models matched the data well here.
Last fixations were longer when the not-last-fixated item had more time
advantage (see Figure 15A). In other words, the last fixation to an item A was longer
when there was more time fixating item B in excess of time fixating item A during that
trial. Again, none of the models seemed to be able to generate this trend (Figure 15B).
30
Discussion
It was originally thought that prior presentation of stimuli would facilitate later
identification of those stimuli, a phenomenon that was labeled a sort of "priming." Later,
researchers found that prior presentation did not simply provide advantages for
perceptual identification; it can also be disadvantageous when similar alternatives exist.
Forced-Choice Perceptual Identification (FCPI) tasks have been used to demonstrate this
outcome, in which participants identify a target stimulus by choosing from two displayed
alternatives. Relative to baseline, performance improves when the target was previously
encountered, but it worsens when the foil was previously encountered. However, this
effect of prior presentation requires that the alternatives are similar.
Many models have devised explanations for this, but they had only accounted for
accuracy. Performance in decision-making cannot be evaluated by accuracy alone or
reaction time alone, because of the speed-accuracy trade-off. Sequential sampling
models are able to model both factors in conjunction, and translate them into underlying
cognitive processes, like rate of evidence accumulation. The fact that reaction times were
not significantly different is consistent with our hypothesis that bias for the studied item
is not due to a shift in starting point or differing boundaries. If prior presentation biased
the decision process to start closer to the studied item's boundary, then reaction times
should be faster for studied items than non-studied items. However, this was not the
case. Reaction time did not significantly differ between the study conditions, but
31
accuracy did. This is consistent with our hypothesis that prior presentation biases the
decision process by speeding up evidence accumulation for the studied items relative to
unstudied items.
Additionally, visual attention has been empirically implicated in decision-making,
but to our knowledge, this has only been examined in preference-based value judgments.
Much of decision-making are not based on preference, and choices are explicitly correct
or incorrect. Analysis of eye-tracking data from a FCPI task expands the visual attention
field to include this prevalent category of decision-making.
In our study, we conducted a diffusion model analysis on performance in a FCPI
task. There were three conditions in our experiment: 1) studied-target, in which the target
stimuli were studied, 2) studied-foil, in which the foil stimuli were studied, and 3)
studied-neither, the baseline condition in which neither was studied.
There was no significant difference in average reaction times between the
conditions (Table 2). If we only compared the average reaction times, like many
researchers do in decision-making studies, then we may have concluded that there was no
effect of prior study. However, this lack of a difference is informative when considered
along with accuracy in a diffusion model analysis. As mentioned earlier, if the bias for
studied items is due to starting the decision process closer to the studied item instead of
the unstudied item, then the RTs of studied items should be faster than unstudied items.
We hypothesized that the bias is due to a change in drift rate, not starting point, so the
32
lack of significant difference in RT between study conditions supports our hypothesis.
This is especially supportive in conjunction with the difference in accuracies across study
conditions; accuracy was highest in the studied-target condition, and accuracy was lowest
in the studied-foil condition, which is consistent with our drift rate account of bias.
In fitting our model to the accuracy and reaction time data, the behavioral changes
between the conditions were successfully captured as a change in the drift rates. Drift
rates for the target (𝑣target ) were highest in the studied-target condition and lowest in the
studied-foil condition (Figure 8). Since drift rate for the foil (𝑣foil) is 1 − 𝑣target, a
lower target drift rate indicates a higher foil drift rate. In other words, studying an item
accelerates evidence accumulation for that item. Thus, not only does this methodology
include both aspects of the speed-accuracy trade-off, but it also provides a cognitive
explanation of bias for studied words.
Krajbich and colleagues' Attentional Drift Diffusion Model (aDDM) introduced a
non-fixated discount parameter 𝜃, which penalizes the drift rate of the alternative(s) that
is not fixated during the given fixation. A value of 1.00 for 𝜃 denotes that the full drift
rate of the non-fixated item is used, so there is no drawback for being un-fixated. This
would be the same as the standard diffusion model without 𝜃. A value of 0.00 for 𝜃
denotes that the drift rate of the non-fixated item is reduced to just noise, which would be
the maximum penalty for being un-fixated. If 𝜃 has a value between 0.00 and 1.00, then
33
the non-fixated alternative would only be partially penalized. Previous research supports
a value between 0.00 and 1.00 for 𝜃.
Incorporating 𝜃 into our model at 0.00, 0.25, 0.50, 0.75, and 1.00, the average
drift rates for each condition still consistently exhibited the predicted pattern: highest drift
rates in the studied-target condition and lowest drift rates in the studied lure conditions
(Table 3). This was also true for when 𝜃 was allowed to vary as a free parameter (Table
4). The model fit the accuracy and reaction time data fairly well regardless of the 𝜃
value, so the drift rate account of choice bias is fairly robust to different amounts of
visual attention bias (Figure 9). Therefore, the standard diffusion model is sufficient to
explain accuracy and reaction time.
The standard diffusion model can also explain why last fixations are shorter than
middle fixations on average. Last fixations are interrupted by a decision being made, so
any sequential sampling model should make this prediction.
Interestingly, the data and all the model predictions also have first fixations
shorter than middle fixations on average (Figure 12A). Krajbich et al. (2010) also found
this result in their data, but had no explanation for it. They incorporated this in their
model by having their model sample from the empirical pool of first fixations separately
from middle fixations. However, our models sampled from a fixation pool composed of
both first and middle fixations, and yet all of them still produced this phenomenon.
34
One possible explanation for first fixations being shorter than middle fixations, is
that sometimes, first fixations are also last fixations, so they can be cut short. However,
middle fixations are never last, so they must always take up the entire time of the
sampled fixation. Figure 12A includes first fixation durations after excluding trials that
only have a single fixation. The models indeed predicted first fixations to be as long as
middle fixations in these cases, but the corresponding first fixations from the data are still
short. As such, the relative brevity of the first fixations remains unexplained.
The standard diffusion model (𝜃 = 1.00) also cannot account for why the
probability of picking the last-fixated item is greater than chance. Since this model posits
no visual attention bias, knowing which item was last-fixated should reveal nothing about
what is chosen. Conversely, all the values of 𝜃 less than 1.00 predict greater probabilities
of picking the last-fixated alternative. With 𝜃 < 1.00, the fixated alternative would
generally have a greater drift rate than its opposition, so the decision process would more
often be moving towards the fixated alternative's boundary. As the value of 𝜃 decreases,
the non-fixated item penalty becomes greater, and the probability of picking the lastfixated item increases (Figure 12B).
Similarly, the standard diffusion model cannot predict the correlation between the
time advantage of an item and the probability of picking that item. Recall that time
advantage of the left item is the amount of time the left item is looked at longer than the
right item in a given trial. Figure 13A plots time advantage for the left against
35
probability of picking the left item. Figure 13B is the same as Figure 13A, except that it
is corrected for the probability of picking the left, conditional on whether the left was the
target or the foil. The data exhibit a positive slope, as do all of the model predictions
except for predictions from the standard diffusion model (blue line), which is flat. The
model with 𝜃 = 1.00 claims that time advantage of the left item has no predictive power
on the probability of choosing the left item. On the other hand, the aDDM posits that
alternatives tend to gain more evidence when they are being looked at than when they are
not. Consequently, time advantage loosely tracks an advantage in evidence, and ergo, an
increase in the probability of being chosen.
Figures 12A, 12B, and 13B are puzzling to us, because of the behavior of the
models with 𝜃 < 1.00. Figure 14 plots the probability of picking the first fixated item
depending on the duration of the first fixations. Figure 14B is the same as Figure 14A,
except that it is corrected for the probability of picking the first fixated item, given that it
was the target or the foil. The model with 𝜃 = 1.00 returned the expected prediction of
no relation between fixation duration and choice probability; the blue dashed line is flat.
However, we expected the other models to predict a positive correlation between fixation
duration and probability of picking the fixated item. When the non-fixated discount
parameter 𝜃 is less than 1.00, the fixated alternatives accrue advantage in evidence when
they looked at longer. Thus, longer fixations on a given item should be associated with
36
greater probabilities of choosing that item. Instead, the models with 𝜃 < 1 seem to have
a generally negative slope and a sigmoidal curve, which are contrary to the data.
Figure 15B plots the duration of the last fixation against the time advantage of the
item that was not fixated last (i.e. time spent looking at the not-last item minus time spent
looking at the last-fixated item, before the last fixation, as illustrated in Figure 15A). The
hypothesis is that since last fixated items tend to chosen, and since more time advantage
for the other item should indicate more evidence for that other item, then the last fixation
needs to be longer in order to overcome the disparity in evidence in order for the last
fixated item to be chosen. Of course, that is just for models with 𝜃 < 1.00. As expected,
the standard diffusion model predicted no relation between fixation duration and time
advantage, but the other models should have had a positive slope in Figure 15B. The data
does exhibit this generally positive slope, so theoretically, this supports the existence of
the non-fixated discount parameter 𝜃, even though our implementation of it defies
expectations.
An important difference between our model with 𝜃 and Krajbich et al.'s (2010;
2011; 2012), is how non-decision time was incorporated. Non-decision time in their
model was set to the time during a trial in which no item was fixated. We included it as a
parameter to be fit, along with variability in non-decision time. The benefit to fitting
non-decision time is that we make no assumptions about what cognitive process are
happening during saccades and eye-movements. However, Krajbich et al.'s method
37
would produce a stronger link between fixation time and decision processes. This should
help the model generate the expected effects of 𝜃, but it may also be perceived as
swaying model predictions in favor of the alternative hypothesis.
Regardless, the non-fixated discount parameter 𝜃 was not necessary to fit
accuracy and reaction time data from a FCPI task in which the target, the foil, or neither
alternative may have been studied. The bias to choose the studied alternative was
accounted for by a boost in drift rate for the studied item. Additionally, the fact that last
fixation durations were shorter than middle fixation durations did not need the extra
parameter. However, the addition of the parameter allowed the model to explain why the
last-fixated item is chosen more often than chance. With 𝜃 < 1 , the model could also
account for the positive correlation between time spent looking at an item and probability
of choosing that item. Theoretically, the models with 𝜃 < 1 should also have fit more of
the fixation trends. Therefore, we conclude that even for non-value-based judgments, the
non-fixated discount parameter is valuable for data that includes visual attention. "Out of
sight" does seem to be at least partially "out of mind."
38
APPENDIX: FIGURES AND TABLES
39
Figure 1. Examples of visually similar pictures from Ratcliff & McKoon (1996).
40
A
B
Figure 2. Illustration of the Counter Model. A) The "spared" counter wins here by
surpassing the "spaced" counter by k number of counts. B) The "spaced" counter wins
here after stealing nondiagnostic counters from "spared". Alternatives steal counts if
previously studied, but they can only steal from similar neighbors.
41
Figure 3. Graphical summary of the diffusion model. A) The reaction time of a trial is
the sum of the non-decision time and the decision time. Non-decision time refers to
processes like time to encode the stimulus at the beginning of a trial, and time to execute
a response after a decision is made. B) Decision processes start at starting point 𝑧 and
accumulate evidence toward boundary 𝑎 or 0 with drift rate 𝑣. Starting point 𝑧 varies
uniformly between trials with range 𝑠𝑧 . Drift rates vary normally across trials with
standard deviation 𝜂. The jagged profile of the lines depicts the within-trial variability of
the drift rate. A real decision process estimated from data would be much noisier, but for
the sake of clarity, the variability is reduced in the image. A decision is made once the
process hits a boundary.
42
Figure 3.
43
Figure 4. Diffusion model depiction of the speed-accuracy trade off. A) When accuracy
is emphasized, boundaries are farther apart, so more evidence is required before reaching
either one. Decisions take more time on average when the boundary separation is wide
relative to when the separation is narrow. However, with a wider boundary separation,
the processes are less likely to hit the incorrect boundary due to noise, so accuracy
increases as well. B) When speed is emphasized, boundaries are closer together, and less
evidence is needed to reach one. Here, the same decision process in panel C terminated
at the wrong boundary due to noise, showing how accuracy decreases, but reaction times
speed up.
44
Figure 4.
A
B
45
Figure 5. Figure 1 of Krajbich et al. (2010) illustrating their experimental design and
model. a) During a trial of the task, participants looked at a fixation point for 2000 ms,
and then had unlimited time to make a choice between two food items. After a choice
was made, the selected item was highlighted with a yellow box for 1000 ms. b) In the
model, the left food item is represented by the top boundary, and the right food item is
represented by the bottom boundary. Importantly, the item being fixated is also tracked,
as noted by the different colored regions. c) When the alternatives of a decision have
equal values, the average drift rate is zero. Then, the system has to reach a boundary by
noise, or more interestingly, by whichever item receives more visual attention. On
average, decision processes move towards the boundary of the item being fixated in these
cases. d) Here, the left item has such a low value that even when it is fixated, the process
does not move much closer to the left-item's boundary. When the high valued right item
is fixated, the process moves quickly towards that boundary, gaining much more ground
in less time, and ends up being chosen.
46
Figure 5.
47
Figure 6. Instruction slides for experiment, presented along with verbal instructions. The
black box in Slide 2 was animated to display three words sequentially as a demonstration
of what the study list would be like. Slide 5 was also animated to show how a word
identification trial would proceed. The flashed word was displayed for longer than it was
be in a real trial to ensure that participants understood that there was a word being
flashed. Participants were informed that the flash would be faster during the task.
48
Figure 6.
1
2
3
4
5
6
49
Figure 7. Sequence of displays for the study list and a word identification trial. A)
During the study list, each word was presented for 1000 ms with a 300 ms gap in
between. There were 16 words per study list. B) For a word identification trial, a
fixation cross was displayed until looked at, and then the target word was flashed for 20
ms. The target word was masked by a row of "@" symbols, and the alternatives were
displayed below until a response was made. There were 32 trials per word identification
block.
50
Figure 7.
A
B
51
Figure 8. Estimated drift rates for each condition and value of 𝜃. Evidence accumulation
for the target was always fastest when the target was studied, and slowest when the
similar-looking foil was studied.
52
Figure 9. Quantile probability function for averaged data and model predictions.
Probability of response is on the x-axis. Correct responses are more likely than error
responses, so all points to the right of the figure are correct responses, and all points to
the left of the figure are error responses. Black points plot the data, and the colored
symbols with dashed lines plot the model predictions. The fits are generally good,
because the colored symbols are roughly on top of the black points. However, the 0.9
quantiles of the errors (top left group of points) are a bit off, and the model with 𝜃 = 0.75
appears to get the closest to the data, as reflected in the chi-square statistics in Table 3.
53
Figure 9.
54
A
B
Figure 10. Model assumptions and fixation characteristics from data. A) The number of
fixations do not differ between conditions. B) The probability of looking at the target
word first also do not differ between conditions.
55
Figure 11. More model assumptions and fixation characteristics from data. A) The
durations of middle fixations on target words are not different from the durations of
middle fixations on foil words. B) The duration of middle fixations are not different
between conditions.
56
Figure 11.
A
B
57
Figure 12. Model predictions and fixation characteristics from data. A) As predicted by
the models, last fixations were shorter than middle fixations. First fixations were also
shorter than middle fixations, but this was not something we had hypothesized
beforehand. The column labeled "First*" are for first fixations from all trials that had
more than one fixation. Removing single-fixation trials effectively made first fixations as
long as middle fixations in the model predictions, but not in the data. B) The probability
of picking the last fixated item was greater than chance. All models except for the
𝜃 = 1.00 model also predicted this, though 𝜃 = 0.75 gets the closest to the data.
58
Figure 12.
A
B
59
Figure 13. Black points with solid lines denote the data, and the colored points with
dashed lines denote model predictions. A) The probability of picking left increased with
time advantage for left, which is total time on left minus total time on right. B) The same
as A), but corrected for the probability of picking left when left was correct. The
standard diffusion model (𝜃 = 1.00, blue dashed line) predicted no effect of time
advantage, and so its graph is flat. 𝜃 = 0.50 or 𝜃 = 0.25 (yellow and orange dashed
lines respectively) matched the data the best here.
60
Figure 13.
A
B
61
Figure 14. Black points with solid lines denote the data, and the colored points with
dashed lines denote model predictions. A) The probability of picking the first fixated
item generally increased with first fixation's duration. B) Corrected for the probability of
picking the first item given that the first item was correct. The trend may still be there, if
we disregard the first two data points. We discarded fixations that were less than 40 ms
from our analysis of the data, so points at that the lower range of these analyses are not
reliable. It is unclear why the models with 𝜃 < 1.00 do not have generally positive
slopes here.
62
Figure 14.
A
B
63
Figure 15. Time advantage of the not-last item. A) Illustration of what the last fixation of
a trial and the not-last fixation of a trial is. The time advantage of the not-last item before
the last fixation is the time spent looking at the not-last item, minus the time spent
looking at the last-fixated item. This is prior to the last fixation, so the last fixation is not
included. B) Duration of last fixation plotted against time advantage of the not-last item.
Black points with solid lines denote the data, and the colored points with dashed lines
denote model predictions. In the data, the duration of the last fixation correlated
positively with the time advantage of the item that was not fixated last. The models had
trouble generating this pattern.
64
Figure 15.
A
B
65
Study condition
target studied
foil studied
neither studied
Target word
(flashed)
spared
spared
spared
Word alternatives
spared
spaced
spared
spaced
spared
spaced
Studied word
spared
spaced
buffet
Table 1. Example target word, choice words, and studied words for each study condition.
66
Condition
Target studied
Foil studied
Neither studied
ANOVA p-value
Accuracy
0.829
0.749
0.786
0.0467
Mean
correct
RT
851
856
868
0.937
Mean
error
RT
942
944
951
0.987
No. of
observ.
2301
2300
4595
Table 2. Reaction time and accuracy results from data. The accuracies reached
significant difference, but the reaction times did not.
67
𝜃
0.00
0.25
0.50
0.75
1.00
Average
Target
studied
0.947
0.798
0.761
0.767
0.701
0.795
Drift rates
Foil
studied
0.706
0.662
0.607
0.650
0.620
0.649
Neither
studied
0.827
0.757
0.702
0.708
0.685
0.736
Chi-square
113.9
79.5
104.2
67.1
77.7
Table 3. Estimated drift rates of each condition and the chi-square statistic achieved with
all parameters for the given 𝜃. Lower chi-square values indicate better fits, so the model
with 𝜃 = 0.75 wins here.
68
PN
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Average
discount
theta
0.533
0.550
0.413
0.687
0.544
0.584
0.522
0.519
0.498
0.586
0.290
0.926
0.572
0.500
0.525
0.512
0.548
drift
1
0.913
0.892
0.641
0.605
0.693
0.703
0.660
0.664
1.037
0.803
0.974
0.714
0.746
0.755
0.725
0.725
0.766
drift
2
0.600
0.710
0.614
0.570
0.631
0.618
0.642
0.603
0.493
0.741
0.269
0.705
0.646
0.649
0.589
0.632
0.607
drift
3
0.881
0.833
0.623
0.621
0.663
0.617
0.673
0.662
0.732
0.718
0.949
0.728
0.716
0.681
0.644
0.584
0.708
chisquare
283.4
50.1
47.3
75.6
30.5
61.0
59.9
37.3
302.6
62.5
162.4
30.6
34.4
70.7
46.3
42.1
87.3
Table 4. From left to right (excluding participant number), estimated values of 𝜃, drift
rates in target-studied condition, drift rates in foil-studied condition, drift rates in neitherstudied condition, and chi-square statistic achieved with all estimated parameters for the
participant.
69
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Footnotes
1
The prior presentation of the stimuli does not necessarily have to be the exact
same stimuli, which is the case for repetition priming. Many studies have also examined
effects of previously presented stimuli that were not exactly the same as the target or foil,
but related in other ways like semantically, orthographically, or associatively (e.g. Huber,
Shiffrin, Lyle, & Ruys, 2001). However, a review of the different effects by various
methods of priming is beyond the scope of the current article.
2
The Counter Model can also model a naming identification task and a "yes-no"
identification task. In a naming identification task, no alternatives are given to choose
from, so the decision-maker must name the flashed stimuli without the help of smaller
choice-sets. In this case, there would be multiple competing counters for each decision,
but everything else remains. Only one count is divvied out per time step, and a counter
needs k more counts than the rest of the competition to be selected. In a "yes-no"
identification task, the target stimulus is flashed, and then a single alternative is
presented. The decision-maker is instructed to respond "yes" if the presented alternative
is the flashed target, and "no" if it is not. The Counter Model represents this process with
a single counter for the single alternative. Evidence for the word gives counts, and
evidence against the word takes counts away, so null counts can be positive or negative.
75