subliminal visual oddball stimuli evoke a p300 component
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Subliminal visual oddball stimuli evoke a P300 component
Edward Bernat*, Howard Shevrin, Michael Snodgrass
Psychology Department, University of Minnesota, Elliot Hall, 75 East River Road, Minneapolis, MN 55455, USA
Accepted 18 August 2000
Abstract
Objective: To provide evidence that a P300 component can be elicited by subliminal stimuli in an oddball paradigm.
Methods: The words LEFT and RIGHT were presented in a frequent-rare ratio (80±20%), counterbalanced between subjects. Stimuli were
presented at the objective detection threshold (d 0 � 0, via unmasked 1 ms presentations), a stringent measure for detecting any conscious
perception.
Results: A signi®cantly larger amplitude component was found for rare vs. frequent stimulus presentations across electrodes Fz, Cz, and
Pz using both a broad 200±900 ms window (F�1; 27� � 5:75, P , 0:012, h2 � 0:18; one-tailed), and a more narrow 400±760 ms window
de®ned using principal component analysis (F�1; 27� � 10:10, P , 0:002, h2 � 0:27; one-tailed). No signi®cant component latency effects
were found. An analysis of the conscious perception index (d 0) and the oddball effect (rare-frequent amplitude difference) revealed a negative
relationship, further supporting the contention that conscious perception does not account for the ®nding, and suggesting that any conscious
stimulus detection may inhibit this subliminal effect.
Conclusions: Results provide evidence that an endogenous component can be elicited by undetectable subliminal stimuli in an oddball
paradigm. Implications are discussed for comparing conscious and unconscious information processing, unconscious learning, and the
measurement of ERPs to subliminal stimuli. q 2001 Elsevier Science Ireland Ltd. All rights reserved.
Keywords: Event-related potential; Oddball; P300; Subthreshold; Subliminal; Unconscious; Nonconscious
1. Introduction
A substantial body of work is now available supporting
the notion that considerable mental activity can happen
without conscious awareness (Bunce et al., 1999; Esteves
et al., 1994; Fox, 1991; Izard, 1993; LeDoux, 1995; Murphy
et al., 1995; Ohman et al., 1995; Shevrin and Fritzler, 1968;
Shevrin et al., 1996; Wong et al., 1994, 1997). This is
consistent with a broader understanding developing in
psychology that a great many mental processes must
occur outside conscious awareness (Kihlstrom, 1987;
Miller, 1996; LeDoux, 1995; Shevrin and Dickman,
1980). The current study was designed to determine whether
a subliminal event-related potential (ERP) could be
obtained in one of the most commonly investigated and
replicated ERP procedures, the oddball paradigm. Speci®-
cally, would an enhanced P300 component at an appropriate
location be found to rare stimuli when both frequent and
rare stimuli are presented entirely outside of consciousness?
The importance of demonstrating a subliminal oddball
P300 has not escaped the interest of other investigators.
Brazdil et al. (1998) presented stimuli in a supra- and
subliminal oddball design and reported ®nding an enhanced
oddball P300 in response to rare relative to frequent sublim-
inal stimuli. On methodological grounds (which will be
discussed in more detail below) their result remains ques-
tionable because conscious perception of the stimuli can not
be ruled out. Devrim et al. (1997) compared P300 responses
to supraliminal rare (target) stimuli in conditions using
either subliminal frequent stimuli or no frequent stimuli
(single-stimulus oddball design, e.g. Polich, 1996), and
reported an enhanced P300 in the subliminal frequent stimu-
lus condition relative to no frequent stimuli, suggesting that
the processing of subliminal frequent stimuli enhanced rare
P300 responses. However, like Brazdil et al., problems with
their subliminal methodology complicate and weaken their
®ndings (which will also be discussed in more detail below).
In the experiment to be reported, the method for establishing
an absence of awareness is based on a standard signal detec-
tion theory (SDT; Green and Swets, 1966) forced-choice
discrimination task, and stimuli meet the most stringent
criterion for subliminality, the objective detection threshold.
Clinical Neurophysiology 112 (2001) 159±171
1388-2457/01/$ - see front matter q 2001 Elsevier Science Ireland Ltd. All rights reserved.
PII: S1388-2457(00)00445-4
www.elsevier.com/locate/clinph
CLINPH 99183
* Corresponding author. Tel.: 11-612-624-5063; fax: 11-612-626-2079.
E-mail address: ebernat@umn.edu (E. Bernat).
1.1. Establishing subliminality
Establishing subliminality (absence of stimulus aware-
ness) must be based on sound methodological grounds.
Thus, it is important to explain the methodology of the
current study in the broader context of subliminal research,
and to de®ne some terms and issues. We will ®rst summar-
ize the approach to measuring subliminal thresholds
employed in this study, then review some important issues
about these methods, and ®nally discuss the recent attempts
to investigate subliminal oddball P300 responses in this
context.
In this study, an inference of unconscious perception is
made by assessing activity on a direct measure of conscious
perception (forced-choice stimulus discrimination,
expressed as d 0) and an indirect measure of unconscious
perception (P300 amplitude from subliminal rare vs.
frequent stimuli) using a form of the classic dissociation
paradigm (CDP; e.g. Dagenbach et al., 1989; Greenwald
et al., 1995; Holender, 1986; Snodgrass, 2000). The premise
of the CDP is that if null sensitivity can be demonstrated on
a direct measure (i.e. d 0 � 0), while signi®cant effects are
present on an indirect measure (e.g. greater P300 to rare vs.
frequent amplitude), this would be evidence for activation
without phenomenal awareness, or unconscious perception.
Relevant to understanding CDP approaches, SDT has
offered an important distinction between a participant's
subjective criterion to report a stimulus (b; response bias)
and their sensitivity to discriminate stimuli (d 0). While d 0
re¯ects a participant's more fundamental sensitivity to
discriminate two stimulus states, the criterion re¯ects a
non-sensory decision process that participants apply to
their sensory experience in order to select a response
(Macmillan and Creelman, 1991; Snodgrass, 2000). These
SDT parameters correspond closely to the subjective thresh-
old and objective threshold approaches (Cheesman and
Merikle, 1986) commonly used in subliminal research. In
subjective threshold approaches, participants are asked to
indicate when they have a phenomenal awareness of stimu-
lus discrimination, and stimuli are arranged such that parti-
cipants achieve null sensitivity with regard to their
subjective criterion for phenomenal awareness. For objec-
tive threshold approaches, participants complete a forced-
choice discrimination task, and stimuli are arranged such
that participants achieve null sensitivity on the d 0 measure.
Discrimination in either approach can be either an identi®-
cation-based task (discriminations of one stimulus from
another) or a detection-based task (discriminations of the
presence or absence of a stimulus). Subjective thresholds
and identi®cation tasks generally correspond to higher d 0
values than objective thresholds and detection tasks respec-
tively; that is, they involve more conscious perception. For
example, it is easier to discriminate whether or not any
stimulus was presented (detection task) than to discriminate
whether stimulus A or B was presented (identi®cation task).
Similarly for thresholds, in varying the intensity of a stimu-
lus presentation from maximum to zero (e.g. no stimulus
presented), participants will report being `unable to see' a
stimulus (subjective threshold) with higher intensities than
those required for participants to perform at chance on a
forced-choice discrimination task (objective threshold). In
this hierarchy, the objective detection threshold is the most
stringent criterion for subliminality. An important dif®culty
of the subjective threshold approach is that the subjective
criterion (b) is confounded with the participant's sensitivity
(d 0). This confound, often called the `criterion artifact,'
served as the basis for fundamental criticisms against
early subliminal work using subjective threshold
approaches in which null sensitivity was incorrectly inferred
from subjective threshold methods (Cheesman and Merikle,
1986; Eriksen, 1960; Goldiamond, 1958; Macmillan, 1986;
Ohman, 1999; Snodgrass, 2000). Objective threshold
approaches, on the other hand, make it possible to determine
a true null discrimination sensitivity for the direct measure
of conscious perception.
Although many researchers subscribe to the logic of the
CDP (e.g. Greenwald et al., 1989; Dagenbach et al., 1989;
Groeger, 1988; Snodgrass, 2000), Reingold and Merikle
(Reingold and Merikle, 1988, 1990; Merikle and Reingold,
1998) have suggested that even objective detection tasks
may not `exhaustively' measure all task-relevant conscious
perception. The essence of Reingold and Merikle's concern
(see also Jacoby, 1991) is that whenever different tasks are
used to index conscious (direct measure, e.g. d 0) and uncon-
scious perception (indirect measure, e.g. subliminal priming
in their work), task differences and process differences are
confounded. Thus, even if null discrimination sensitivity
were satisfactorily attained, effects on the indirect measure
might merely re¯ect aspects of conscious perception not
indexed by the direct measure, rather than truly unconscious
effects.
However, there are substantial reasons to think that detec-
tion is exhaustively sensitive to any conscious perception
that might account for the target indirect effects (see Snod-
grass, 2000 for a full treatment of these issues, which are
summarized here). First, the subliminal effects of interest
usually require that the stimuli be semantically analyzed
(e.g. semantic priming). For such semantic analysis to
occur, in turn, requires at least partial stimulus identi®cation
(differentiating one stimulus from another). Second, and
crucially, SDT models identi®cation as multidimensional
detection (see, e.g. Green and Birdsall, 1978; Macmillan
and Creelman, 1991). Speci®cally, identi®cation is
conceived of as the multidimensional distance between
the individual detection sensitivities. That is, SDT regards
each stimulus as having its own detection dimension, and
identi®cation between two stimuli is represented as the third
leg of the triangle connecting the two individual detection
dimensions (see Fig. 1). In this way, identi®cation arises
from, depends on, and cannot occur without non-zero detec-
tion. In short, given the SDT framework, detection is
exhaustively sensitive to all identi®cation-relevant informa-
E. Bernat et al. / Clinical Neurophysiology 112 (2001) 159±171160
tion. In particular, when the objective detection threshold is
attained, identi®cation should be impossible (presuming
that there is only one, presumably conscious, perceptual
process at work). Thus, when effects requiring some degree
of stimulus identi®cation are nonetheless obtained on the
indirect measure, this is strong evidence for unconscious
perception. As Macmillan (1986), p. 39) concluded,
ªAbove-chance recognition (i.e. identi®cation) performance
(or other evidence of activation) when detection d 0 � 0
would be, for almost everyone, persuasive evidence for
unconscious perception.º
Next we will review recent advances in subliminal
research methods which advocate modeling the relationship
between d 0 and the indirect measure, providing additional
testable hypotheses for putatively unconscious effects. First,
a direct implication of the SDT model is that it predicts a
positive relationship between detection and identi®cation
(Snodgrass, 2000). Using the geometric model described
above (Fig. 1), as the detection dimensions get larger, the
identi®cation dimension becomes larger as well. With this
in mind, a single-process (e.g. conscious perception only)
SDT model predicts that the (identi®cation-dependent)
indirect effects should correlate positively with the direct
detection measure, and that if null detection sensitivity is
attained, the indirect effects should be greatly reduced or
completely eliminated (Fig. 2, top). Notably, Reingold and
Merikle (Reingold and Merikle, 1988, 1990; Merikle and
Reingold, 1998; Cheesman and Merikle, 1984) have repeat-
edly stressed just this idea ± that indirect effects should
decline to zero as d 0 approaches zero. Pritchard (1981) has
similarly suggested, in a review of signal detection related
P300 ®ndings, that P300 is absent until d 0 is greater than
zero.
Several researchers have begun to examine this idea by
investigating the relationship between d 0 for various direct
measures and indirect effects using subliminal priming para-
digms. Greenwald et al. (1995) found using a regression
analysis that under some conditions when the direct measure
(d 0) was zero the intercept on the indirect measure was
signi®cantly above zero. They suggest that this is strong
evidence that indirect effects do not approach zero as the
detection d 0 approaches zero, thus supporting the inference
that unconscious processing is present. Dagenbach et al.
(1989) found that the relationship between the direct and
indirect measures can be non-monotonic, and suggest that
such irregularities indicate a single conscious process can
not explain the ®ndings. Greenwald et al. (1995) similarly
suggested such a non-monotonic relationship, proposing
that any degree of conscious perception may actually reduce
subliminal effects on the indirect measure above the objec-
tive detection threshold (d 0 slightly above zero), which will
then recover when the objective detection threshold is
achieved (d 0 � 0).
Drawing on the ideas suggested by Greenwald et al.
(1995) and Dagenbach et al. (1989), Snodgrass (2000) has
proposed a non-monotonic model of the relationship
between direct (conscious) and indirect (unconscious)
effects, conceptualized in the hierarchical terms of SDT
discrimination tasks (see Fig. 2, bottom). The relationship
is hypothesized to be strongly positive from full conscious
awareness to somewhere below the subjective identi®cation
threshold. Then the indirect effect is hypothesized to bottom
out and become non-signi®cant somewhere above the
objective detection threshold (due to interference of slight
conscious perception on unconscious processing), but that
as the stimuli meet more stringent threshold criteria
E. Bernat et al. / Clinical Neurophysiology 112 (2001) 159±171 161
Fig. 1. Signal detection theory (SDT) model of identi®cation between two
stimuli as a function of ability to detect each of the stimuli independently.
Fig. 2. Comparison of a single-process model (e.g. conscious perception
only) (top), in which the direct and indirect measures are simply positively
correlated, to the proposed non-monotonic relationship (bottom) which
makes the opposite negative correlation prediction in the range of data
obtained from data in this study (gray part, top and bottom).
(towards the objective detection threshold of d 0 � 0), the
indirect effect will begin to recover (due to the disappear-
ance of even slight bits of interfering conscious perception).
Data in this study occupy this lowest end of possible of d 0
values (not different from objective detection d 0 of zero).
Thus, for data in this study, the single-process and non-
monotonic models of this relationship make exactly oppo-
site predictions (Fig. 2, gray part top and bottom); the
single-process model predicts a positive relationship while
Snodgrass's non-monotonic model predicts a negative rela-
tionship. We thus hypothesized that (1) a negative relation-
ship would be found between d 0 on an objective detection
task and the indirect subliminal oddball P300 effects, and (2)
at d 0 � 0 these indirect subliminal effects will be present as
indicated by a regression y-intercept term signi®cantly
above zero. In this study we will test both parts of this
hypothesis in seeking to demonstrate that an oddball P300
effect will be found at the objective detection threshold.
1.2. Studies of P300 and subliminal stimuli
Finally, here we will assess the P300 studies discussed
earlier in the context of more general subliminal research
methods. Brazdil et al. (1998) reported an increased P300
response to rare versus frequent subliminal stimuli using a
CDP design. However, stimulus conditions were manipu-
lated to meet only the subjective identi®cation threshold, the
weakest of subliminal criteria. As discussed earlier, the
subjective method for establishing subliminality confounds
sensitivity (d 0) with a subjective criterion for consciousness,
the `criterion artifact' problem. For example, the partici-
pants in the Brazdil et al. (1998) study may simply have
had a high subjective criterion for reporting awareness even
though some awareness was present. There is thus no way to
assure that while participants may have reported no aware-
ness of the stimulus, that they were additionally unable to
discriminate the stimuli above chance.
Devrim et al. (1997) reported an enhanced rare (target)
P300 in a subliminal frequent stimulus condition relative to
no frequent stimuli, suggesting that the processing of
subliminal frequent stimuli contributed to the rare P300
responses. However, their study also suffers from some
methodological limitations. First, the study does not address
(and it was not their aim) whether the entire process of
context updating, including the rare stimuli, can happen
without conscious awareness. Secondly, like Brazdil et al.
(1998), their method for establishing subliminality is
problematic. First, in the detection series, participants did
not discriminate between stimulus-plus-noise (SN) and
noise-only (N) stimuli, as would be the case for standard
SDT procedures. Rather, using an approach resembling the
classical method of limits, stimulus intensities were gradu-
ally reduced and participants were simply required to indi-
cate when they perceived that a stimulus had been
presented. This was done in two separate, blocked condi-
tions ± no subliminal stimuli (false alarm condition), or with
subliminal stimuli (hit condition). Accordingly, Devrim et
al.'s procedures did not allow computation of true d 0. Addi-
tionally, the hit and false alarm measures they did obtain
were substantially undersampled. Of the 30 presentations in
each condition, both mean hit and false alarm rates were 1.6,
suggesting a severe response bias (a more balanced response
selection would be 15). Even if their non-standard d 0 proce-
dures were reasonable, this response bias renders hit and
false alarm estimates unstable, making their claim of no
difference between them problematic. As in the Brazdil et
al. (1998) study, serious questions remain as to whether
conscious perception was in fact ruled out, and thus whether
the enhanced oddball P300 could have resulted from
conscious perception of the stimuli.
1.3. Hypotheses
1. The rare stimuli will evoke greater amplitude than
frequent stimuli in the de®ned component window
when all stimuli are presented at the objective detection
threshold
2. The relationship between d 0 and the oddball effect will be
negative
3. The y-intercept of a regression between d 0 and the
oddball effect will be signi®cantly above zero.
2. Method
2.1. Participants
Thirty-two right handed paid participants took part in the
study, although 4 were excluded due to equipment failure
during data collection. The remaining 28 participants had a
mean age of 21.7 years (SD 2.49), 18 were female, all had
vision correctable to 20/20 and all reported no history of
neurological or mental abnormalities. Participants were
recruited using a classi®ed advertisement in the student
newspaper at the University of Michigan.
2.2. Passive oddball paradigm
Because stimuli were presented below the objective
detection threshold, and participants were thus not
conscious of what had been presented, a passive oddball-
type paradigm was used in which participants are not asked
to respond in any way to the stimulus presentations. Polich
(1989) directly compared passive and active auditory
oddball paradigms, with no additional tasks. He reported
that compared to the active paradigm, the passive oddball
paradigm generated a P300 which was lower in amplitude
overall, and had a decreased, but present, parietal maxi-
mum. Polich cites a number of other studies which have
evidenced a frontally maximal P300 in a passive auditory
oddball paradigm, but contends that because such studies
generally use some other active task (e.g. the no-go part of a
E. Bernat et al. / Clinical Neurophysiology 112 (2001) 159±171162
go/no-go paradigm), the passive paradigm and the addi-
tional active task are confounded in any interpretation of
the source of the frontal maximum. Thus, the P300 topo-
graphic distribution in the passive paradigm employed here
is likely to be parietally maximal to a lesser degree than an
active paradigm. A frontally maximal effect, however,
cannot be entirely ruled out.
The words LEFT and RIGHT served as the frequent and
rare stimuli, counterbalanced between participants. Stimuli
were presented in 60 blocks containing 5 stimulus presenta-
tions (one frequent and 4 rare) in an 80±20 frequent-rare
ratio. The rare stimulus occurred randomly in position 3, 4,
or 5 in the block. Participants were told nothing about the
content of the stimulus presentations, including the block
structure of the presentations and the frequent-rare ratio of
the stimuli. During debrie®ng, participants uniformly
described being unable to see the stimuli and reported no
awareness of the content of the stimuli or the organization of
stimulus presentations.
The interstimulus interval (ISI) in this study was variable
(routinely between 5 and 15 s, and infrequently as long as a
minute) and was not directly measured or analyzed. There
are two sources of the variable ISI: our use of a tachisto-
scope for stimulus presentations which required a person to
be in the recording booth with the participant changing the
stimulus card before each trial, and the logistical procedure
for stimulus delivery (a more detailed accounting of the
procedure for stimulus delivery is included below). Our
reason for using a tachistoscope was its ability to deliver
stimuli at a 1 ms duration, thus making it possible to reach
the objective detection threshold. It is important to note that
both the experimenter in the booth and the experimenter
monitoring the EEG machine (where individual trials were
triggered) were blind to the content of all trials. A longer ISI
has been shown to reduce P300 frequent-rare differences
(Donchin et al., 1986). Thus, to the degree our method
differs from many oddball studies in this respect, the differ-
ences should work against positive ®ndings.
2.3. Stimulus presentation
The stimuli were printed in Adobe 18 pt. Helvetica light
centered on 4 £ 6 inch white cards and presented in a 3-®eld
Gerbrands Model T3-8 tachistoscope. One ®eld of the
tachistoscope was used for all stimulus presentations. A
second ®eld was used for a ®xation point, visible at all
times except during the moments of stimulus presentations.
The distance from the stimulus ®eld to the eyepiece was 76
cm. The printed stimulus words LEFT and RIGHT were 14
and 19 mm wide, respectively, within the 58 visual angle for
foveal vision (Polyak, 1957). Field brightness was tested for
luminance level and pulse width and equated for both ®elds.
Luminance levels for the stimulus and ®xation ®elds, as well
as the ambient light levels in the booth, were set at 5
footlamberts. Stimuli were rendered subliminal by being
presented at 1 ms duration. Because the tachistoscopically
presented subliminal 1 ms stimuli (black print on a white
background) were preceded and followed by a ®xation ®eld
(black dot on a white background) of equal luminance, there
was very little disturbance in the visual ®eld.1 Presumably,
this contributed substantially to the small amplitude of the
observed ERPs.
2.4. Experimental procedure
After a brief introduction to the laboratory, participants
completed an informed consent statement. All electrodes
were attached prior to seating participants in a sound-
proof, electrically shielded, temperature-controlled booth.
Participants were instructed that during each stimulus
presentation they were to remain as still as possible, to
focus on the ®xation point, to pay attention, and to keep
eye blinks to a minimum. This instruction was repeated
periodically during the experiment. They were told that
the stimulus presentations were very brief so that they
may or may not be able to see them. After experiencing a
few practice trials, participants were reassured that some-
thing was being presented, and that we were interested in
their brain responses to the stimuli even though they may
not experience seeing something. Participants were
instructed that they were not required to respond in any
way to the stimuli, just to stay attentive, still, and keep
eye blinks to a minimum during the stimulus delivery and
ERP recording for each trial.
The stimulus delivery sequence was as follows. First
participants heard a single `ready' tone indicating that the
experimenters were ready to present a trial. Participants
would then respond by stating the number of the current
trial followed by the word ready (i.e. `one-ready,' `two-
ready,' and so on). After that verbal response, stimulus
delivery was triggered by an experimenter monitoring the
EEG record, who was blind to the stimulus content. After
the stimulus was presented and the 1 s of ERP activity was
recorded, a double tone signaled the end of that trial. Parti-
cipants were told they could move, blink, and relax between
that double tone and the next single ready tone. During the
time between the double tone and the next single ready tone,
the data from that trial was written to disk, and an experi-
menter in the booth, who was blind to the content, changed
the stimulus card.
Several time epochs in this sequence were variable,
constituting the variable ISI discussed earlier. The time
E. Bernat et al. / Clinical Neurophysiology 112 (2001) 159±171 163
1 Our method differs from `energy' masking in which the stimulus is
rendered subliminal by following it with a much brighter stimulus.
Andreassi et al. (1976) have reported on the use of energy masking and
its results. In the studies described the index for subliminality was based on
the patient's failure to report the masked stimulus, thus amounting to a
subjective threshold. Our method also differs from pattern masking insofar
as our stimuli are exposed for a very brief duration (1 ms) and are not
followed by a conscious pattern mask. The shortest stimulus duration
generally used in pattern making is one screen refresh of a computer moni-
tor (generally not faster than 10 ms), and stimuli are rendered subliminal by
being followed with a supraliminally presented pattern.
between the single ready tone and participant's `trial
number-ready' response was controlled by the participant.
While most often this response immediately followed the
single tone, sometimes participants would take several
seconds to prepare for a trial. In rare instances participants
took 10 or 15 s. The time between the participants `trial-
number ready' response and the initiation of stimulus deliv-
ery was controlled by the experimenter at the EEG machine.
This time most frequently was within a few seconds, but
sometimes was longer if the EEG re¯ected eye blinking or
excessive muscle movement at which point time was
allowed to elapse until the record appeared free of such
artifacts. Records which were contaminated with artifacts
(from an online visual inspection) were rejected and the trial
presented again in the next cycle. Our reason for being care-
ful to collect artifact free individual ERP trials is that the
stimulus delivery process was slow relative to many compu-
ter based stimulus delivery mechanisms (due to the use of a
tachistoscope needed for 1 ms stimulus durations). This
limited the number of stimulus presentations and thus
made it more important for each trial to be as free of error
as possible. The ®nal source of variance in the ISI was the
time between the double tone and the next single ready tone.
Most frequently this was about 5 s (the time it took for the
card to be changed in the booth).
A 64-item forced-choice detection task was administered
at the end of the experiment under the same 1 ms and 5
footlambert conditions (due to time constraints, one partici-
pant of the 28 did not complete the detection task). Of the 64
stimuli, 16 were the word LEFT, 16 were the word RIGHT,
and 32 were blank. Participants were told that either a word
or a blank card would be presented an equal number of times
in random order. They were asked to state after each presen-
tation whether a word or blank card had been presented, and
to keep their responses approximately equally divided
between the two choices. Sensitivity (d 0) did not differ
from zero (mean � 0:07, SD � 0:344; (t�26� � 1:06,
P , 0:30). These same conditions have been shown to
preclude conscious recognition of stimuli in other studies
(Bernat et al., 2000; Snodgrass et al., 1993; Shevrin et al.,
1996; Wong et al., 1997).
2.5. Physiological measurement apparatus
Recording sites were Fz, Cz, and Pz referenced to linked
ears. Standard Grass instrument silver-silver chloride elec-
trodes were used with impedance kept to less than 5 kV. Eye
activity was monitored by electrodes placed on the outer
canthus and suborbital ridge of the right eye, referenced to
each other (EOG). All signals were collected utilizing a
Grass Model 8-24D polygraph linked to a Macintosh
computer. Signals were digitized at 500 Hz, then stored in
computer ®les for off-line analysis. Signals were analog
®ltered online with a low-pass frequency of 200 Hz, and a
high-pass frequency of 0.1 Hz. ERPs were sampled for 1400
ms, including a 400 ms prestimulus interval. As discussed
earlier, the lack of a conscious visual stimulus, and the
passive oddball design, resulted in small amplitude ERPs
which also tended to contain more alpha. For these reasons,
a digital ®lter at 7 Hz was applied to the raw data before
component peak selection using a 5th order Chebychev
®lter, ripple parameter of 20 dB down, applied using the
Matlab ®lt®lt command.
2.6. Component measure
We used a P300 window based on windows reported in
supraliminal studies. Time windows for measuring the P300
in supraliminal studies have varied substantially. Latencies
have been reported as early as 210 ms and ended as late as
900 ms (Pritchard, 1981; Gehring et al., 1992; Cacioppo et
al., 1994; Coles and Rugg, 1995). Donchin et al. (1986)
suggest that endogenous components often occur between
500 and 1000 ms, depending on the experimental task. The
P300 window used in this study was the difference between
baseline and the largest positive going peak between 200
and 900 ms post-stimulus. Component measures were
applied to averages within electrode and participant. Each
participant's 300 ERPs were broken into 5 averages of 60
each (20% of total): one average for the rare stimuli, while
the 240 frequent stimuli were broken down into 4 random
groups of 60 each. This was done to select components from
ERP subaverages with equal numbers of trials.
3. Results
The grand average ERP pro®les are presented for rare and
frequent stimuli in Fig. 3 (frequent curve is the average of
the 4 frequent subaverages used for component selection).
The most obvious difference is a positive going peak
between 500 and 600 ms apparent in all 3 electrodes for
the rare average which is larger than the corresponding
positivity for the frequent average. Statistical analyses to
be presented below will con®rm these impressions.
First, we measured the amplitude and latency of the
largest positive peak between 200 and 900 ms (means
presented in Table 1), and subjected those values to a 2
(oddball: frequent, rare� £ 3 (electrode: Fz, Cz, Pz) analysis
of variance (ANOVA). The hypothesized main oddball
effect was signi®cant for the amplitude measure
(F�1; 27� � 5:75, P , 0:012, h 2 � 0:18; one-tailed2)
where rare stimuli evoked larger amplitudes than did
frequent stimuli. The main effect of electrode was signi®-
cant for amplitude as well (F�2; 54� � 3:72, P , 0:031,
h2 � 0:12), driven by larger frontal values. The interaction
of the oddball and electrode factors for amplitude was non-
signi®cant (F�2; 54� , 1), indicating that even though there
appears to be a greater difference in the amplitude of the
frontal region, the oddball effect does not signi®cantly differ
E. Bernat et al. / Clinical Neurophysiology 112 (2001) 159±171164
2 One-tailed criteria are used for comparisons based on a priori direc-
tional hypotheses (e.g. rare . frequent P300 peak amplitude values).
by region. We then subjected amplitudes and latencies from
each of the 3 electrodes separately to an ANOVA compar-
ing the frequent to rare component values. The rare compo-
nents were signi®cantly greater than the frequent for each of
3 electrodes separately (Fz: F�1; 27� � 4:04, P , 0:027,
h2 � 0:13; Cz: F�1; 27� � 5:21, P , 0:016, h2 � 0:16;
Pz: F�1; 27� � 3:87, P , 0:030, h2 � 0:13; one-tailed).
Analysis of P300 latencies yielded no signi®cant results.
We also assessed activity recorded from the EOG using
the same 200±900 ms window which did show a trend level
difference between the rare and frequent stimuli
(F�1; 27� � 2:90, P , 0:10, h2 � 0:10). This difference
was less signi®cant than ®ndings for the midline electrodes,
suggesting that this activity is more likely a re¯ection than a
source of the component observed at the midline electrodes.
Additionally, results at midline electrodes, do not decrease
in signi®cance in a linear fashion at sites further away from
the EOG (i.e. from frontal to parietal). Finally, the only
observable rare peak in the EOG grand average is near
700 ms, later than the observed rare peak in the midline
electrodes (near 550 ms), suggesting that activity at the
EOG is not the source of the peak observed at the midline
electrodes.
3.1. Principal components analysis
To ®nd convergent validation for the signi®cant oddball
peak in the 200±900 ms window, we conducted a varimax
rotated principal components analysis (PCA). The analysis
was based on a data matrix of 420 averages (3 electrodes £ 5
averages (one rare and 4 frequent� £ 28 participants) by 50
(the 1 s of raw data downsampled to 50 data points using the
Matlab resample function to handle ®ltering before down-
sampling). We did not apply additional ®ltering to the data
in this analysis (i.e. 7 Hz) to allow a closer look at the peaks
and troughs in the individual weighting vectors. Based on a
scree plot, we extracted 3 principal components which
accounted for 76% of the variance (Fig. 4). The ®rst compo-
nent (PC1) corresponded to the late part of the 1 s epoch,
and most closely resembled a late positivity or a slow wave.
The second principal component (PC2) was most clearly
like the P300 component window we had used; it had
minima at 240 ms and 900 ms, indicating that it corre-
sponded closely to the original 200±900 ms P300 window
we started with. The third principal component (PC3) corre-
sponded to the early part of the 1 s epoch, and is a gross
measure across the entire early epoch. Given the absence of
clearly identi®able peaks in this early time region, we felt a
gross measure was appropriate for representing this early
time epoch in this data.
We then separately multiplied the 3 principal components
by each of the 420 subaverages used to generate the princi-
pal components (Chapman and McCrary, 1995), and used
the mean (component or factor score) from the resulting
curves as data for analysis. We additionally measured the
peak in addition to the mean because P300 is conventionally
measured as a peak. We separately submitted the mean and
peak values to the same 2 (oddball: frequent, rare� £ 3 (elec-
trode: Fz, Cz, Pz) analysis of variance (ANOVA) as we had
E. Bernat et al. / Clinical Neurophysiology 112 (2001) 159±171 165
Table 1
Means and standard deviations for rare and frequent P300 peak amplitudes
and latencies (200±900 ms window) for individual electrodes and the aver-
age across electrodes
P300
Amplitude Latency
Rare Frequent Rare Frequent
Mean SD Mean SD Mean SD Mean SD
Fz 1.96 (2.33) 1.47 (1.81) 552 (210) 556 (184)
Cz 1.35 (1.86) 0.78 (1.24) 552 (206) 488 (142)
Pz 1.30 (1.86) 0.86 (1.39) 578 (206) 530 (142)
Average 1.54 (2.03) 1.04 (1.51) 560 (206) 524 (158)
Fig. 3. Rare (solid line) versus frequent (dashed line) grand averages for
electrodes Fz, Cz, and Pz, digitally ®ltered at 7 Hz..
the 200±900 ms window measures. The numerical statistical
results are presented in Table 2 and summarized below.
First, PC2 (P300) evidences signi®cant frequent-rare
differences, in an omnibus analyses across the electrodes,
for both the peak and the mean measures. As with the 200±
900 ms measure, the oddball effect does not signi®cantly
interact with electrode in this omnibus analysis, while the
main effect of electrode does show trend level differences.
Analysis of individual electrodes con®rms that each of the 3
electrodes evidence signi®cant frequent-rare differentiation
independently. The signi®cance level using the PC2 (P300)
peak measure are stronger than the 200±900 ms peak P300
measure, suggesting that the epoch weighted by the PC2
(P300) component contains more of the differentiating
activity. Second, for the PC1 (late) component the main
oddball effect is signi®cant for the mean and trend level
for the peak. For the individual electrode analyses, Cz
shows signi®cant differentiation for the peak and mean,
while Fz shows trend level for the peak and mean and Pz
show only trend level for the mean measure (Table 2).
Lastly, the PC3 (early) component does not approach signif-
icance for frequent-rare differences, and does show an elec-
trode main effect for the mean measure.
From an examination of the grand average plots, the
observed rare P300 component rises from a negative in¯ec-
tion after 300 ms to a peak after 500 ms, which appears to be
more narrow in time than our broad 200±900 ms window.
The shape of the PC2 (P300) also supports the conclusion of
a component centered on this peak, particularly in view of
E. Bernat et al. / Clinical Neurophysiology 112 (2001) 159±171166
Table 2
Signi®cant F ratios, P values, and h 2 values for omnibus and individual electrode analyses for the peak and mean of each of the 3 extracted principal
componentsa
Effect d.f. Principal component
Early (PC3) P300 (PC2) Late (PC1)
F P h 2 F P h 2 F P h 2
Peak
Omnibus analysis
Oddball 1,27 9.48 0.003 0.26 3.87 0.060 0.13
Electrode 2,54
Oddball £ Electrode 2,54
Electrodes separately
Oddball (Fz) 1,27 7.44 0.006 0.22 2.93 0.098 0.10
Oddball (Cz) 1,27 7.04 0.007 0.21 4.67 0.040 0.15
Oddball (Pz) 1,27 8.64 0.004 0.24
Mean
Omnibus analysis
Oddball 1,27 4.23 0.025 0.14 4.43 0.045 0.14
Electrode 2,54 5.07 0.010 0.16 2.82 0.069 0.09 2.73 0.075 0.09
Oddball £ Electrode 2,54
Electrodes separately
Oddball (Fz) 1,27 3.96 0.029 0.13 3.05 0.092 0.10
Oddball (Cz) 1,27 3.14 0.044 0.10 4.41 0.045 0.14
Oddball (Pz) 1,27 3.33 0.040 0.11 3.71 0.065 0.12
a Oddball main effect P values for P300 (PCA2) in omnibus and individual electrode analyses are one-tailed.
Fig. 4. Principal components derived using data from electrodes Fz, Cz, and
Pz.
the fact that ®ndings for the PC2 (P300) peak measure were
stronger than the 200±900 ms peak measure. However, the
grand average of the frequent stimuli, because an obvious
peak is lacking, offers little guidance for the rational selec-
tion of a more narrow component window to apply
uniformly to both rare and frequent stimulus responses.
Perhaps the amplitude of the responses to frequent stimuli
was too small to be detected, variability was too great,
substantial habituation occurred, or there simply was no
measurable response to subliminal frequent stimuli. The
dif®culty in ®nding a P300 peak for subliminal frequent
stimuli was also reported by Devrim et al. (1997), as well
as for supraliminal frequent auditory stimuli when using a
passive oddball design (Polich, 1989).
To further investigate whether a closer ®t for the P300
component could be found, we selected a more narrow
window based on the PC2 (P300) component, using the
local minima at 400 ms and 760 ms as a window de®nition
(Fig. 4). These minima were chosen as window boundaries
because they included the strongest weightings in the center
portion of the principal component and excluded the parts
which overlapped with the strongest portions of the other
two principal components. We then applied this more
narrow window to the raw ERPs (not ERPs transformed
by PCA weighting vectors) in the same manner as the
200±900 ms windows (same level of subaveraging, 7 Hz
®ltering, etc.). Results from analysis of the peak amplitude
in this 400±760 ms window were considerably stronger than
those from the 200±900 ms window both for the main effect
of oddball in the omnibus analysis including electrode as a
factor (oddball (one-tailed): F�1; 27� � 10:10, P , 0:002,
h2 � 0:27; electrode: F�1; 27� � 2:35, P , 0:105,
h2 � 0:08; oddball by electrode: F , 1), and the oddball
effect in the electrodes separately (Fz: F�1; 27� � 6:73,
P , 0:008, h2 � 0:20; Cz: F�1; 27� � 7:37, P , 0:006,
h2 � 0:21; Pz: F�1; 27� � 11:42, P , 0:001, h2 � 0:30;
one-tailed). These ®ndings provide further supportive
evidence that indeed more of the differential activity
happens in a this narrow window, which is more similar
in latency and duration to many P300 components evoked
with supraliminal stimuli than the 200±900 ms window.
3.2. Relationship between d 0 and the oddball effect
As discussed above, a skeptical conscious perception
explanation of putatively subliminal ®ndings makes several
testable predictions. First, it predicts that participants should
be able to detect the stimuli above chance; this was not the
case in the current experiment. However, because the grand
detection mean was numerically above chance, one might
wonder if the current non-signi®cant results might have
been due to insuf®cient power. Fortunately, further tests
are possible to test the alternative conscious perception
hypothesis. Recall that (1) a conscious perception explana-
tion would predict a positive relationship between detection
and the experimental effect, and relatedly (2) that no experi-
mental effect should be present when detection d 0 � 0. In
contrast, we predicted that (1) the oddball effect would
correlate negatively with detection, and (2) that the experi-
mental effect would not only be present at d 0 � 0 but at or
near its maximum.
To test these predictions, we employed Greenwald and
associates' regression approach (e.g. Greenwald et al.,
1995). In this method, the experimental effect (i.e. the
oddball effect in this study) is regressed onto direct measure
performance (detection d 0). Of primary interest to us was
not only the y-intercept (which estimates the oddball effect
when d 0 � 0, and which was Greenwald's primary interest),
but the direction of the relationship (i.e. positive or negative
slope). A potential problem with the regression approach is
bias in estimating the y-intercept. There is good reason to
expect such bias because regression assumes perfect
measurement of the predictor(s), whereas experimentally
measured detection performance clearly possesses some
measurement error. It is well-known (see, e.g. Dosher,
1998; Klauer et al., 1998a,b) that such measurement error
will systematically decrease the absolute value of the slope
estimate, which will in turn systematically bias the y-inter-
cept estimate. Crucially, however, the direction of this bias
is different depending on whether the slope is positive or
negative. If the true slope is positive, the `¯attening' effect
of measurement error will bias y-intercept estimates
upward, thus potentially producing artifactually signi®cant
above-zero y-intercepts. Importantly, however, if the true
slope is negative, measurement error will bias the y-inter-
cept estimate downward producing a conservative bias.
Measurement error is only problematic for the regression
approach, then, when positive slopes are obtained (Snod-
grass, 2000). If our prediction of a negative slope was
con®rmed, then, the associated y-intercepts will be conser-
vative estimates of the true oddball effect when d 0 � 0.
For the regression, the oddball effect was measured as the
difference between the P300 amplitude to the rare stimuli
and the frequent stimuli, using the narrow component
window (400±760 ms) based on the PCA. Separately for
each electrode, and again for the average of the 3 electrodes,
we computed a regression analyses between the oddball
effect and d 0. A discordancy test for single outliers (Barnett,
1994) indicated that all P300 difference and d 0 scores were
within an expectable chance distribution (individual parti-
cipant's P300 and d 0 scores are listed in Table 3). Scatter
plots of this relationship for the 3 electrodes are presented in
Fig. 5, with a linear regression line displayed. As hypothe-
sized, the regression b coef®cient and the corresponding
correlation are negative for the average across the 3 electro-
des (b � 21:61, r � 20:44, t�25� � 22:45, P , 0:011;
one-tailed) and for all 3 electrode locations separately (Fz:
b � 22:03, r � 20:48, t�25� � 22:73, P , 0:006; Cz:
b � 21:50, r � 20:34, t�25� � 21:79, P , 0:043; Pz:
b � 21:30, r � 20:37, t�25� � 22:01, P , 0:028; one-
tailed). Thus, the extent to which a participant is above d 0 �0 is associated with a decreased oddball effect. Interpreted
E. Bernat et al. / Clinical Neurophysiology 112 (2001) 159±171 167
in accord with the Snodgrass hypothesis, this ®nding means
that the more a participant might have detected (been
conscious of), the less the subliminal oddball effect, a coun-
terintuitive result based on a single process (conscious)
explanation.
Perhaps more important, however, in establishing subli-
minality at the objective threshold is to determine what is
happening when d 0 � 0. As discussed earlier, a signi®cant
effect when d 0 � 0 (i.e. the y-intercept) would support the
presence of unconscious processing (Greenwald et al., 1995;
Snodgrass, 2000). As hypothesized, the y-intercept is signif-
icantly above zero for the average of the 3 electrodes
(t�25� � 4:08, P , 0:0002, one-tailed), and at all 3 electro-
des separately (Fz: t�25� � 3:72, P , 0:0005; Cz:
t�25� � 3:16, P , 0:002; Pz: t�25� � 4:02, P , 0:0003;
one-tailed).
4. Discussion
Our results provide evidence that a late endogenous
component can be elicited by subliminal stimuli presented
in an oddball paradigm with all stimuli meeting criteria for
the objective detection threshold. Although Brazdil et al.
(1998) used a design similar to ours, they could not rule
E. Bernat et al. / Clinical Neurophysiology 112 (2001) 159±171168
Fig. 5. Scatter plot of d 0 versus rare-frequent peak amplitude difference
(400±760 ms window), for the average amplitude across electrodes Fz,
Cz, and Pz (top), and separately for each electrode, with a linear regression
line displayed in each plot.
Table 3
Listing of individual participant d 0 scores and P300 peak amplitude values
(400±760 ms window) averaged across Fz, Cz, and Pz for rare, frequent,
and the rare-frequent difference, sorted by rare-frequent difference
Rare-freq. Rare Freq. d 0
2.92 0.78 22.14 20.39
2.91 4.20 1.29 0.24
2.34 3.90 1.55 0.16
2.20 2.89 0.68 0.00
1.74 3.12 1.38 20.47
1.72 4.06 2.34 20.16
1.71 3.18 1.47 20.16
1.52 2.98 1.46 0.00
1.45 2.97 1.51 20.16
1.36 3.20 1.84 0.16
1.21 3.70 2.49 0.08
1.12 0.86 20.26 0.24
1.07 21.20 22.27 20.75
1.05 0.39 20.65 20.16
0.97 2.89 1.92 0.4
0.67 0.38 20.29 0.00
0.66 1.10 0.44 0.16
0.44 0.75 0.31 20.24
0.34 21.08 21.42 0.64
0.26 0.97 0.71 0.39
0.20 1.06 0.86 0.00
0.10 1.47 1.37 0.4
20.05 1.14 1.19 Not available
20.38 0.73 1.12 0.39
20.91 21.91 21.00 0.00
21.32 22.00 21.00 0.24
21.76 21.00 0.53 0.00
21.77 21.49 0.28 0.91
out conscious perception as an explanatory factor because of
their use of a subjective threshold. Devrim et al. (1997)
found that subliminal frequent stimuli increased P300
amplitude to a supraliminal rare stimulus in an oddball
task, although it should also be reiterated that their study
relied on a problematic implementation of signal detection
theory procedures and thus conscious perception of the
stimuli can not be ruled out. Our design made it possible
to obtain a P300 to the rare stimuli presented without
conscious awareness. Notably, some have theorized that
P300 is directly associated with conscious registration.
Libet (1977) has suggested that the later ERP components
(after 100 ms in his research) might be directly associated
with consciousness. Similarly, Posner and Boies (1971)
have suggested that P300 might be a marker for conscious-
ness, as has Pritchard (1981) who cites evidence suggesting
that the P300 is absent until d 0 exceeds zero. Our results do
not support this understanding of P300. Based on much
®rmer methodological grounds for ruling out consciousness,
our results strongly suggest that ERPs to subthreshold
stimuli are markers for some of the same functional proper-
ties as ERPs to above threshold stimuli, and thus linking
®ndings on subliminal ERPs to a considerable body of
ERP research and theory.
Our results show no signi®cant electrode differences for
the frequent-rare comparisons, although the Fz amplitude is
greater than Cz and Pz. Whether a P300 component is fron-
tally or parietally maximal has been an important considera-
tion in P300 studies. Perhaps the clearest examples are the
functional differences found for P3a and P3b. The more
frequently observed P3b shows a parietal maximum while
P3a shows a frontal maximum (Polich and Kok, 1995). The
failure to ®nd a clear frontal or parietal maximum is consis-
tent with the ®ndings of Polich (1989) showing decreased
parietal maximum, or less difference between these regions,
using a passive auditory paradigm. It is worth speculating
that a number of factors might have reduced any frontal
effect, given the smaller amplitude of the subliminal
ERPs, and the resulting greater contribution of any level
of artifact such as any eye movement or muscle potentials.
While the question of frontal or parietal maximum will go
without a de®nitive answer in this report, the critical main
®nding is the signi®cantly higher amplitude for a late endo-
genous component occurring within the P300 window to the
rare as compared to the frequent stimuli when objective
detection d 0 is at zero.
The negative relationship between values of d 0 (equal to
or above zero) and the oddball effect is a powerful argument
against the hypothesis that consciousness can account for
the effect. Interestingly, a non-signi®cant relationship would
also argue against the consciousness hypothesis, but a nega-
tive relationship raises the provocative possibility,
suggested by the Snodgrass hypothesis, that some conscious
awareness of the stimuli counteracts subliminal in¯uences
(Snodgrass, 2000). In that sense, a little conscious percep-
tion may actually inhibit a strong subliminal effect. If d 0
were considerably above zero and the stimuli were all
clearly in consciousness then an oddball effect would be
obtained (as it has been many times), but the correlation
between d 0 and the oddball effect would be positive. It is
this type of reversal in direction of the correlation which
Snodgrass (2000; see also Dagenbach et al., 1989) refers to
as the non-monotonicity of the relationship between detec-
tion d 0 and the subliminal effect.
As a ®nal consideration, one could argue that because
direct detection measures are often measured with fewer
trials than indirect subliminal measures, indirect-without-
direct effects may simply be an artifact of undersampling
the direct measure. Our study could be subject to this criti-
cism because we did sample d 0 with fewer trials than the
oddball effect (64 vs. 300 trials, respectively). In this
scenario, if enough trials were administered the true extent
of consciousness on the direct measure would be revealed.
The underlying logic in this argument, however, is that the
true relationship is positive and that if detection were
adequately sampled a positive correlation would be found
between d 0 and the apparent subliminal effect, thus under-
mining the claim that the indirect task was eliciting an
unconscious effect. However, in our case the sampling
size proved suf®cient to ®nd a relationship, but as hypothe-
sized it was negative (and not positive or non-signi®cant). It
would be hard to account for this ®nding with a hypothesis
based solely on undersampling which would predict no rela-
tionship. The ®nding of a negative correlation also applies to
criticisms that detection tasks may simply be less sensitive
to even the same conscious processes than other measures
(e.g. physiological measures may be intrinsically more
sensitive than verbal report), even if well sampled (e.g.
Eriksen, 1960). This criticism would again predict no rela-
tionship, not a signi®cant negative relationship.
A demonstrable P300 component to subliminal stimuli is
a potentially important contribution to the study of
consciousness and P300 related information processing.
First, we can begin to speculate about the role of conscious
awareness in the context updating processes. Sommer and
Matt (1996) have provided evidence suggesting that parti-
cipants can have a conscious index of the amplitude of their
P300 when evoked in a standard supraliminal oddball para-
digm. Devrim et al. (1997) offer suggestive evidence that
subliminal frequent stimuli enhance P300 responses to
supraliminal rare stimuli in an oddball paradigm. Results
from this study now suggest that a P300 response can be
evoked even when both rare and frequent stimuli are
presented outside conscious awareness. Indeed, the P300
phenomenon appears to span the range from unconscious
processes to conscious awareness. Because subjective prob-
ability is given fundamental importance in interpreting P300
®ndings (Donchin et al., 1986), we are left to speculate that
either subjective probability is somehow not fundamental to
the current ®ndings (e.g. perhaps unconscious processes
operate under qualitatively different rules), or perhaps
subjective probabilities can occur without conscious aware-
E. Bernat et al. / Clinical Neurophysiology 112 (2001) 159±171 169
ness. This would suggest that expectancies can be both
modi®ed and evoked without conscious awareness. In
other words, a mental process which is dynamically inter-
active with the environment can happen without the bene®t
of consciousness. This is compatible with a growing body of
literature demonstrating that learning in the form of aversive
conditioning can occur without bene®t of conscious aware-
ness (Ohman et al., 1995; Wong et al., 1997). There may
well be other such processes operating without bene®t of
conscious awareness, the presence and nature of which
remain for future research to investigate.
Acknowledgements
Portions of this data were presented at the third meeting
of the Association for the Scienti®c Study of Consciousness,
in London, Ontario, Canada, June, 1999. The research
reported in this study was supported in part by gifts from
Robert H. Berry, the Department of Psychiatry, Department
of Psychology, and the Rackham Graduate School of the
University of Michigan. We thank William Williams,
William Gehring, and Philip Wong for assistance and feed-
back on many aspects of this manuscript.
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