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\begin{document}

\title{Variability of Single Trial ERP Measures Within a Session is Systematic
and Nonlinear}



\author[1]{A. Blythe  LaGasse}%
\author[1]{William J. Gavin}%
\author[1]{Kyle  Wilhelm}%
\author[2]{Ryan  Smith}%
\author[1]{Patricia Davies}%
\affil[1]{Colorado State University}%
\affil[2]{Affiliation not available}%


\vspace{-1em}



  
  \date{June 16, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
Averaging multiple event-related potential (ERP) segments distorts the
brain's response to a stimulus given the false assumptions that the ERP
signal is invariant and hidden by background noise. Our Single Trial
Peaks (STP) procedure measures amplitude and latency of multiple peaks
in each segment based on the peak latencies of the individual's averaged
ERP. This study examined correct trial data from 70 adults performing
two repetitions of a speeded visual flanker task. STP peak data (P1, N1,
P2, N2, and P3) were compared to single-trial averaged voltage in time
windows (STW) and peaks of averaged ERPs. Results indicated that the STP
approach had higher split-half reliability (odd-even, r =.70-.96) and
test-retest reliability (r =.66-.97) than the STW approach. The mean
signal/noise ratios for the amplitude measures for the STP ranged from
1.62 to 2.9 and were larger than the STW approach, although averaged
ERPs had higher ratios than both. Coefficients of variation (CV) for the
STP and STW approaches were similar, and both had smaller CVs than
averaged ERPs. The validity of the STP approach was determined by
regression analyses where mean trial-level peak amplitude, SDs across
trials for amplitude and latency, and noise accounted for a significant
amount of variance in corresponding peaks of the averaged ERPs (R2
=.73-.95). Curve fitting analyses showed nonlinear systematic patterns
in amplitude over trials similar to changes in response time. These
results support the validity of the STP approach which can be used to
investigate systematic changes of brain activity across trial.%
\end{abstract}%



\sloppy


\textbf{Running Head:} VARIABILITY OF SINGLE TRIAL ERP MEASURES

\textbf{Title:} Variability of Single Trial ERP Measures Within a
Session is Systematic and Nonlinear

\textbf{Authors:}

1. A. Blythe LaGasse\textsuperscript{a}

Email: blagasse@colostate.edu

ORCID ID: 0000-0002-1631-3605

2. William J Gavin\textsuperscript{a,b}

Email: william.gavin@colostate.edu

ORCID ID: 0000-0002-2868-477X

3. Kyle Wilhelm\textsuperscript{a}

Email: kyle.wilhelm@colostate.edu

ORCID ID: 0000-0001-9258-520X

4. Ryan Smith\textsuperscript{a}

Email: ryan.smith@state.co.us

ORCID ID: 0000-0002-0152-5162

5. Patricia L. Davies\textsuperscript{b,c}

Email:\emph{patricia.davies@colostate.edu}

ORCID ID: 0000-0003-0189-6809

Affiliations:

\textsuperscript{a} School of Music, Theatre, and Dance

1778 Campus Delivery

Colorado State University

Fort Collins, CO 80523, USA

\textsuperscript{b} Department of Molecular, Cellular \& Integrative
Neuroscience

1680 Campus Delivery

Colorado State University

Fort Collins, CO 80523, USA

\textsuperscript{c} Department of Occupational Therapy

1573 Campus Delivery

Colorado State University

Fort Collins, CO 80523, USA

\textbf{Corresponding author:}

William J Gavin

Department of Molecular, Cellular \& Integrative Neuroscience

Colorado State University

Fort Collins, CO 80523, USA

Email: william.gavin@colostate.edu

Phone: 970-227-9999

Abstract

Averaging multiple event-related potential (ERP) segments distorts the
brain's response to a stimulus given the false assumptions that the ERP
signal is invariant and hidden by background noise. Our Single Trial
Peaks (STP) procedure measures amplitude and latency of multiple peaks
in each segment based on the peak latencies of the individual's averaged
ERP. This study examined correct trial data from 70 adults performing
two repetitions of a speeded visual flanker task. STP peak data (P1, N1,
P2, N2, and P3) were compared to single-trial averaged voltage in time
windows (STW) and peaks of averaged ERPs. Results indicated that the STP
approach had higher split-half reliability (odd-even, \emph{r} =
.70-.96) and test-retest reliability (\emph{r} = .66-.97) than the STW
approach. The mean signal/noise ratios for the amplitude measures for
the STP ranged from 1.62 to 2.9 and were larger than the STW approach,
although averaged ERPs had higher ratios than both. Coefficients of
variation (\emph{CV} ) for the STP and STW approaches were similar, and
both had smaller \emph{CV} s than averaged ERPs. The validity of the STP
approach was determined by regression analyses where mean trial-level
peak amplitude, \emph{SDs} across trials for amplitude and latency, and
noise accounted for a significant amount of variance in corresponding
peaks of the averaged ERPs (\emph{R\textsuperscript{2}} = .73-.95).
Curve fitting analyses showed nonlinear systematic patterns in amplitude
over trials similar to changes in response time. These results support
the validity of the STP approach which can be used to investigate
systematic changes of brain activity across trials.

Variability of Single Trial ERP Measures Within a Session is Systematic
and Nonlinear

\textbf{1.0 Introduction}

For over seven decades, electroencephalography (EEG) has been used to
study brain activity elicited by the presentation of sensory stimuli, or
during performance of motor or cognitive tasks (Kappenman \& Luck,
2016). A cornerstone of the general methodology used to obtain measures
of brain activity has been the processing step of averaging multiple
segments of EEG signals elicited by the same stimulus to create averaged
event-related potentials (ERPs). Then, the resulting peaks and valleys
of the averaged ERP are quantified for a specified time window as
either: 1) the average of the voltage values of the data points in the
window or 2) as the amplitude and latency of the largest peak found in
the time window.

Justification for the processing step of averaging multiple segments is
based on three assumptions. Firstly, EEG obtained at any scalp site is a
continuous mixture of electrical activity from multiple brain regions,
where some of the activity is not related to the stimulus. This
unrelated activity is often referred to as background (brain) noise.
Secondly, the brain's electrical signal elicited by any stimulus (or
response) event is difficult to measure as it is assumed to be smaller
than, and mixed with, the continuous background noise. Thirdly, the
brain's signal in response to an event is invariant in amplitude and
time for each occurrence of the event (Dawson, 1954; Hu et al., 2011;
Quian Quiroga \& Garcia, 2003). While the validity of the second and
third assumptions have received considerable discussion over the history
of ERP-based research, some researchers propose that the best approach
to address these assumptions is using the average voltage in a window
measurement (e.g., De Vos, Thorne, Yovel, \& Debener, 2012; Jongsma et
al., 2006; Kappenman \& Luck, 2016; Kosciessa, Grandy, Garrett, \&
Werkle-Bergner, 2020; Luck, Stewart, Simmons, Rhemtulla., 2021). Yet
even this measurement approach relies on the third assumption, signal
invariance, being true or at least having minimal variations in the
latencies of the various positive and negative peaks across segments
being averaged. While latency variability is often minimal for adults,
research in children has shown that greater latency variability of peaks
leads to decreased amplitude measures, which in turn leads to erroneous
conclusions about brain development (e.g., DuPuis et al., 2015; Gavin,
Lin, \& Davies, 2019).

The present study investigates the feasibility of using a simple
procedure, the Single Trial Peak approach (STP), for obtaining reliable
and valid peak amplitude and latency measures at the trial (i.e.,
segment) level. This procedure does not depend on the validity of the
second and third assumptions traditionally presumed in research using
averaged ERPs; that is, it does not require a signal-to-noise ratio
(SNR) of any value, nor does it assume signal invariance. In contrast,
the STP procedure does assume, allows for, and quantifies not only the
variability between participants representing individual differences in
processing abilities but also the variability across trials within a
paradigm/task completed by the participant. Consequently, the STP
procedure enables researchers to test the hypothesis that some of the
variability in amplitude and latency of peaks across trials may be
systematic rather than random. More importantly, the STP procedure can
produce measures of brain activity at the single trial level with
psychometric properties comparable to or exceeding the commonly used
measure of average voltage in defined time windows. The rationale for
proposing the STP procedure is presented in the following sections.

\textbf{1.1 Multiple Approaches Are Used to Obtain Single Trial ERP
Measures}

Research efforts to derive measures of brain activity from single trial
(ST) ERPs have employed a wide variety of techniques such as
peak-picking the amplitudes of raw EEG or bandpass filtered data (e.g.,
Spencer, 2005), template matching methods including use of Woody filter
(e.g., Fein \& Turetsky, 1989; Gavin et al. 2019; Patterson,
Michalewski, \& Starr, 1988; Pfefferbaum, Ford, Roth, \& Kopell, 1980),
maximum likelihood estimation methods (e.g., Ja\selectlanguage{polish}ś\selectlanguage{english}kowski \& Verleger,
1999; M\selectlanguage{ngerman}öcks et al., 1988), regression-based estimation (e.g., Mayhew et
al., 2006), wavelet filtering or smoothing (Hu, Mouraux, Hu, \&
Iannetti, 2010; Hu et al., 2011; Quian Quiroga, Atienza, Cantero, \&
Jongsma, 2007; Rey, Ahmadi, \& Quian Quiroga, 2015; Roth, Roesch-Ely,
Bender, Weisbrod, \& Kaiser, 2007), decomposition methods such
as~residue iteration decomposition (RIDE; Ouyang, Sommer, \& Zhou,
2015), principal components analysis (PCA) or independent component
analysis (ICA) (e.g., De Vos et al., 2012; Saville et al., 2015), and
graph theory (Dimitriadis, Brindley, Evans, Linden, \& Singh, 2018). For
partial reviews on these techniques, see De Vos et al. (2012) who
compared four of these techniques and Ouyang, Hildebrandt, Sommer, and
Zhou (2017) who compared eight of these techniques. Interestingly, De
Vos et al. (2012) stated ``At present, little information exists as to
which ST estimation method is the more powerful'' (p. 1197). Most of
these techniques were developed to assess a single measure, usually a
mid to late component of the ERP such as the N2 or P3, with some
focusing on measuring amplitude while other techniques focus on
determining latency variability (Spencer, 2005). Though informative,
none of these articles report the psychometric properties of both
amplitude and latency measures of multiple components obtained in single
trials commonly found in the averaged ERP. That is, these articles do
not provide quantified descriptions of ERP components from the viewpoint
of classical measurement and statistical theories for either the
averaged wave forms or averages of ST measures (e.g., measures of
central tendency, variability, and distribution) nor are reliability
coefficients~of split-half or test-retest provided. Consequently, to
date none of the methods cited above have been adopted by the research
community at large.

\textbf{1.2 Justification for an Alternate Method for Quantifying Single
Trial Measurements: A Peak-Picking Approach}

Peak picking is defined as the identification of the largest positive or
negative peak amplitude in the averaged ERP in a pre-determined time
window (Weeda et al., 2012; Zhang \& Luo, 2011). The time window used to
define a given ERP component can be based on time windows used in the
extant literature. If prior studies using similar stimulus paradigms are
unavailable, the morphology depicted in the grand averaged ERP (i.e.,
the average of all averaged ERPs of all participants in the study) can
be used to determine a time window that captures the component of
interest for the peak picking of the averaged ERPs of each individual
participant. The advantages of the peak-picking methodology are that it
allows one to 1) obtain an amplitude and latency measure of the
component of interest at its maximum point, 2) measure differences in
latency between individual participants or the conditions in the study,
and 3) attain a greater sensitivity (i.e., lower measurement error) in
the peak amplitude measurements when compared to averaging voltages in
the window. Additionally, if each major peak of an averaged ERP is
measured, then the obtained amplitude and latency measures define a
generalized morphology of brain processing to the event for each
participant.

Once a peak is identified in the peak-picking approach described above,
there are two ways that the amplitude can be calculated. One approach to
peak picking has been to measure peak amplitude from the baseline
(a.k.a. baseline-to-peak measures) after performing the processing step
of baseline correction. While baseline correction aids in normalizing
the segmented trial data to changes in voltage from a zero-voltage
referent, this procedural step can also make the peak sensitive to
background noise, especially in single trials (Weeda et al., 2012). The
second approach, measuring amplitudes from peak-to-peak may be less
sensitive to baseline noise and, therefore, a better way to measure
amplitudes with single trials (Ribeiro, Paiva, \& Castelo-Branco, 2016;
Unsal \& Segalowitz, 1995). However, it has yet to be identified how
peak-picking single trials compares with an averaged voltage within a
time window.

The ST approach used in this study to identify the principal peaks in a
single trial is based on the common method of peak picking to identify
component peaks and latencies \emph{in the averaged ERP of each
individual participant} in a study. When developing the STP approach,
peak picking \emph{at trial level} was viewed as a logical extension of
peak picking at the averaged ERP level. That is, any given peak
identified in the averaged ERP is a product of the peaks in the single
trials occurring on or near the time point (latency) of the averaged ERP
component of interest. Furthermore, when the general morphology of the
averaged ERP is captured as successive measures of the major components
(e.g., the N1, P2, N2, P3, N3 peaks), the time window for finding the ST
peaks of a component of interest is defined by the time points of the
two surrounding components in the averaged ERP (i.e., the latencies of
the preceding component and following component). For example, to obtain
a measurement of the P2 for a single trial one can determine the maximum
voltage between the latencies of the N1 and the N2 components of the
averaged ERP. The latency for this maximum voltage in the single trial
will most likely be close to the latency of the P2 of the averaged ERP
but will vary from trial-to-trial. The use of the prior and post
latencies for determining a peak at the ST level is logical, as every
data point in the single trial contributes to the general morphology of
the data points in the averaged ERP.

\textbf{1.3 Peak Amplitudes and Latencies Are Measurable Independent of
the SNR}

A common argument against measuring peak amplitude and latency at the ST
level is that the SNR is too low to identify the components of interest
(Blankertz, Lemm, Treder, Haufe \& Müller, 2011; Chamanzar,
Malekmohammadi, Bahrani, \& Shabany, 2015; Ja\selectlanguage{polish}ś\selectlanguage{english}kowski \& Verleger, 1999).
In other words, there is the assumption that the signal in ST ERPs
representing the brain activity elicited by a stimulus event is
generally smaller than the random background brain noise and, therefore,
cannot be directly seen or measured in the single trial. Although viewed
as a descriptive indication of the quality of the data, there is little
information in the literature specifying the minimum SNR needed for
obtaining reliable and valid measures of brain activity in the averaged
ERP or in single trials. This may be due to the fact that efforts to
understand SNR have often been conducted by combining synthesized
signals with recordings of resting brain signals, then studying
computational methods retrieving the synthesized signal (Hu et al.,
2011; Ja\selectlanguage{polish}ś\selectlanguage{english}kowski \& Verleger, 1999).

Furthermore, SNR has been calculated through a variety of methods (Luck,
2005; Spencer, 2005). For example, one method is to calculate the ratio
of the largest evoked potential peak-to-peak amplitude and the mean
amplitude of the baseline period prior to the presentation of the
stimulus (Pfurtscheller et al., 1987). Another method is to divide the
standard deviation of the averaged ERP by the standard deviation of the
noise as estimated from amplitudes of oscillatory signals recorded
during the resting states (Ouyang et al., 2015). Alternatively, SNR has
been defined as the signal being the amplitude of a peak in ST ERP,
while the noise, or measurement error, defined as uninformative
intra-subject variability created by neural activity not activated by
the stimulus as well as non-neural artifacts (Blankertz et al., 2011;
Ribeiro et al., 2016). While this latter approach can be viewed as more
compatible with test and measurement theory, this definition of noise
may contain elements of meaningful variability across single trials.

Independent of how SNR is calculated, a low SNR may not be problematic
if the effect of the baseline noise can be shown to be relevant to brain
processing post-stimulus (for review see Rey et al., 2015). Pre-stimulus
brain activity has been shown to consistently influence later ERP
components after the presentation of a task-relevant stimulus in both
young adults and older adults, even when significant age-related changes
in post-stimulus neural processing are found. For example, a greater
pre-stimulus gamma power correlated with larger P300 amplitudes and
longer response times (Reinhart, Mathalon, Roach, \& Ford, 2011), and
pre-stimulus alpha power influenced errors on a task (Shou, Dasari, \&
Ding, 2015). In another study, McNair, Kayser, and Kayser (2019) found
that the amplitudes of P1 and N1 were significantly larger and the
latencies of N1 and P2 were greater in the older adults compared to
young adults, while the pre-stimulus rhythmic brain activity influence
on post-stimulus brain activity remained constant across age groups.

\textbf{1.4 Variability in Single Trials - It Can Be Systematic}

While often dismissed as measurement error, the EEG/ERP research
community has long recognized that information inherent within
trial-to-trial variation is lost through averaging (Jung et al., 2001;
Walsh, Gunzelmann, \& Anderson, 2017; Yajing et al., 2020). When
trial-to-trial variability in either latency or amplitude is systematic
rather than random, analysis of single trials can show valuable
information such as the state of the participant, the presence of a
disability or disorder, or the ability to learn over time (Jongsma et
al. 2006; Pfefferbaum, Ford, Wenegrat, Roth, \& Kopell, 1984;
Pfefferbaum, Wenegrat, Ford, Roth, \& Kopell, 1984; Saville et al.,
2015). Several researchers have explored trial-to-trial variability
(latency jitter) using ST techniques and have shown individual or group
differences, especially in children or individuals with disabilities
(Gavin et al., 2019; Milne, 2011; Unsal \& Segalowitz, 1995). Roth and
colleagues (2007) found increased latency and amplitude variability in
participants with schizophrenia compared to typical adults. ST analyses
have also been used to examine systematic changes in amplitude or
latency across the duration of the ERP task to identify informative
patterns that may represent learning over time (Quian Quiroga et al.,
2007; Rey et al., 2015). For example, Jongsma et al. (2006) observed
slightly faster response times with frequently presented targets versus
infrequently presented targets showing learning in healthy adults.

\textbf{1.5 The Aims of the Present Study}

Most studies looking at single-trial data analyzed one or two
components, typically the P300 and N200. By looking at multiple
components, including baseline, researchers can determine if the
variability is random or systematic (Gaspar, Rousselet, \& Pernet,
2011). If within-subject variability is systematic, it would be helpful
to find a reliable method to analyze single-trial data that preserves
individual differences and can demonstrate systematic changes during
task performance within and across individuals. Therefore, the purpose
of this study was to determine the feasibility and validity of measuring
multiple ERP components (e.g., the N1, P2, N2, and P3 peaks) using a
simple computational strategy to obtain ST measurements. The feasibility
of these ST measures was considered by computing the SNR for the
traditional averaged ERP, the STP approach, and the Single Trial Average
Time Window (STW) approach. Reliability was determined through the
comparison of split-half and test-retest reliability measures of
amplitude and latency computed for single trials to those computed for
measures of amplitude for the STW approach and amplitude and latencies
obtained from the traditional averaged ERP. We expect that both the SNR
and correlations for reliability measures will demonstrate similarities
in the different ST approaches. Validity was considered by comparing the
standard measurement error (SME) of the STP and STW approaches. The
validity of the ST measures was further determined by computing how well
an individual's mean ST amplitude, ST amplitude variability, ST latency
variability, and mean ST noise predict the individual's peak amplitude
measures of the averaged ERP. Given issues of SNR, we expect measures of
reliability and validity to improve across the phases of cognitive
processing, reflected by successive ERP components. Furthermore, we used
curve fitting to look for the presence of systematic changes across
trials. If the process of identifying ERP components within ST data can
be shown to be reliable and valid, the methodology can then be used to
demonstrate that systematic changes do occur within a paradigm from one
trial to the next.

\textbf{2. Method}

\textbf{2.1 Participants}

A total of 74 neurotypical adults, aged 18 to 30 years (\emph{M} = 22.3
years, \emph{SD} = 2.5; 31 males, 43 females) were recruited from the
local community and university through campus emails, flyers, research
subject pool of the Psychology department, and word of mouth. These
participants were part of a larger data set described in Lin, Davies,
Stephens, \& Gavin. (2020). All participants were scheduled for two
laboratory visits, however, only data from the first session are being
used in this study. Participants were screened for use of
psychopharmaceutical drugs (e.g., antidepressants) and neurological
disorders by self-report. Three adults were excluded from data analysis
due to disability and one adult was excluded due to failure to complete
both sessions. No participants were excluded due to error rate. Thus, a
total of 70 participants are included in this study.

\textbf{2.2 Procedure}

Participants underwent two sessions, each including 2.5 hours that were
divided as follows - 1.5 hours of EEG tasks, then 1 hour of behavioral
testing with a 10-15 minute break between the EEG and behavioral
testing. Participants were prepared for the EEG recording by two
research assistants. Participants underwent a 3-minute artifact training
period, then performed three separate ERP paradigms (a novelty odd-ball,
a speeded visual flanker task, and a visually-evoked contingent negative
variation, CNV) in an electrically shielded, sound attenuated recording
room. Only the data from the speeded visual flanker task will be
presented here. Procedures for this study were approved by the
University's human research committee. Written informed consent was
obtained from all participants.

\textbf{2.3 Electrophysiological paradigm}

Participants completed two blocks of 240 trials (480 trials total) of
the speeded visual flanker task (Eriksen \& Eriksen, 1974). The CNV
paradigm was conducted between the two flanker task blocks. For the
flanker task, the visual stimuli consisted of character arrays,
combinations of the letters H or S, with two congruent arrays (HHHHH and
SSSSS; 80 trials each), and two incongruent arrays (SSHSS and HHSHH; 160
trials each). One of four stimuli were randomly presented for each trial
using E-Prime software version 2.0 (Psychology Software Tools,
Pittsburgh). Participants were instructed to press a button with their
right index finger if the middle letter was an S or a button with their
left index finger if the middle letter was an H. Participants were told
the stimuli would be presented quickly and asked to try to not make
errors and perform the task as accurately as possible. The stimulus
duration was 250 ms and the inter-trial interval (ITI) was initially set
at 1400 ms at the beginning of each block. The E-Prime program evaluated
the overall error rate after each set of 30 trials and adjusted the ITI
by 100 ms in accordance with performance; increasing it if the error
rate was greater than 30\% or decreasing it if fewer than 10\%. To allow
for brain processing time and the brain activity of the response to
resolve, a minimum ITI was set at 800 ms. Because correct responses are
considerably more numerous than incorrect responses and significantly
differ from incorrect responses (Lin, Davies, Taylor, Prince, \& Gavin
2021), only trials with correct responses are used in the single-trial
analyses for this study.

\textbf{2.4 Electrophysiological Recording and Data Processing}

The EEG was recorded from the scalp using either 33 or 64 channels from
the Active-Two BioSemi system (BioSemi, Inc., Amsterdam, the
Netherlands) using a modified 10-20 electrode placement system (American
Electroencephalographic Society, 1994). Additional electrodes were
placed on the left and right earlobes and two bipolar electrooculograms
(EOG) electrodes were placed at the supra- and infraorbital regions of
the left eye to measure vertical eye movements as well as on the left
and right outer canthi to measure the horizontal eye movements. All EEG
data were collected using a sampling rate of 1024Hz.

The EEG data were analyzed offline using the software BrainVision
Analyzer2 (Brain Products GmbH, Munich, Germany). The data were
referenced to the averaged signals of bilateral earlobes and then
filtered with a bandpass filter of 0.1-30 Hz with 24 dB/oct. The
continuous data were then segmented into 1200 ms epochs, spanning from
200 ms before stimulus onset to 1000 ms after stimulus onset. Segments
were then baseline-corrected based on the average voltage of -200 to 0
ms of stimulus onset, and eye movement artifacts were removed via a
regression approach based on the vertical EOG (VEOG) channel
(Segalowitz, 1996) then baseline-corrected again using the period of
-200 to 0 ms of stimulus onset. Segments containing voltage greater than
\selectlanguage{ngerman}± 100 \selectlanguage{greek}µ\selectlanguage{english}V in the midline (e.g. Fz, FCz, Cz, Pz) and VEOG channels were
rejected. The segments were then averaged to create averaged ERPs for
each participant. The number of segments in the averaged ERPs ranged
from 220 to 473. The averaged ERPs obtained for each participant were
scored using a custom-built peak-picking computer program (Gavin, 2013)
written in MATLAB (Mathworks, Natick, MA). This program performs
automatic scoring and visual inspection of ERP components, and, when
necessary, allows for manual marking of components to adjust for any
values that were scored on a slope instead of a peak. In a previous
study (Lin, et al., 2020), for each participant, the averaged ERP
components were measured from baseline-to-peak using the following time
windows: P1 between 0 and 100 ms; N1 between 70 and 150 ms; P2 between
110 and 240 ms; N2 between 170 and 350 ms; and P3 between 320 and 575
ms. The N3 was not measured. All components were measured at the site
Cz. These data were used as the input data for ST measurement for this
current study.

\textbf{2.5 Measurement of Components using Single Trial Peaks Approach}

The Single Trial Peaks (STP) approach involved passing the
stimulus-locked ERP trials through a custom-built MATLAB program created
by the authors to capture the trial-by-trial amplitude and latency data
for each component of interest. The trial-by-trial peak-picking process
for each trial segment for each participant was conducted in two steps,
a forward and a backward pass, to optimize the accuracy in the detection
and measurement of each component peak. During the \emph{forward pass} ,
all components measured at averaged ERP level were measured at the trial
level beginning with the component closest to the stimulus onset time
(i.e., P1), and proceeding in successive order until the component
furthest from onset time was reached. For each trial segment, successive
time windows for the peak-picking of each component were developed in a
recursive manner. For the first time window for the first component, P1,
the left boundary of the time window was time of stimulus onset (i.e., 0
ms) and the right boundary of the time window was the peak latency of
the second component, N1, obtained from the participant's averaged ERP.
Then the next component at the trial-level, N1 was measured in the time
window with the left boundary defined by the peak latency of P1 from the
segment being scored and the right boundary defined by peak latency
measure of P2 from the participant's averaged ERP. This recursive
procedure continued until all components measured in the averaged ERP
were measured in the trial segment. In order to measure the P3 at the
trial-level, the latency of N3 was estimated by adding 250 ms to the
participant's averaged ERP latency for P3. If the right boundary latency
was equal to the averaged ERP latency, then an adjustment was completed
to narrow the window. This ensured that the picked amplitudes were not
on a slope and were the min/max of the averaged window. Within a time
window, the maximum voltage for positive components or the minimum
voltage for negative components in the window was recorded, along with
the corresponding latency.

The second processing step in the STP procedure, the
\emph{backwards}pass, was designed for the purpose of finding individual
peaks that lay outside the window obtained from the averaged
stimulus-locked ERPs that was used in the forward pass. Starting with
the P3 and moving backwards to P1, the program used the obtained latency
values from the forward pass to recalculate the max/min value between
the two obtained latencies for that trial. For the special case of the
P1 latency being found to equal to 0 ms after the first pass, the left
boundary of the window was adjusted to 50 ms with the right boundary
being the latency of N1. This adjustment was made to capture a peak in
the most common window in the literature (Gruber, Klimesch, Sauseng,
Doppelmaryr, 2004; Zhang \& Luo, 2011). If P1 did not equal 0, then the
window was adjusted to 10 ms for the left boundary in the backward pass
check.

Additionally, peak-to-peak (p2p) measures for N1 \textsubscript{p2p},
P2\textsubscript{p2p}, N2 \textsubscript{p2p}, and P3
\textsubscript{p2p}were determined at each trial the STP by subtracting
the baseline-to-peak value of the component preceding from the
baseline-to-peak (b2p) value component of interest. For example,
N1\textsubscript{p2p} was determined by subtracting
P1\textsubscript{b2p} from N1\textsubscript{b2p} (i.e.,
N1\textsubscript{p2p} = N1\textsubscript{b2p} - P1\textsubscript{b2p}).
Peak-to-peak measures have been found to be a more stable component in
the single-trial ERP analyses (Jongsma, et al., 2006) as well as in
averaged ERPs (Gavin, et al. 2019).

\textbf{2.6 Measurement of Components Using the Average Time Window
Approach at the Single Trial Level}

For each participant, each single trial was also measured using the
Average Time Window (STW) approach to provide a comparison to the STP
approach. A customized MATLAB program (Gavin, 2013) was used to find the
average amplitude for each component in each trial segment, using
pre-defined windows as follows: P1 was 20-70 ms, N1 was 70-130 ms, P2
was 150-250 ms, N2 was 250-320 ms, and P3 was 350-500 ms. Component
latencies are not obtained using this approach, as the voltages of all
data points in the entire window are averaged.

\textbf{2.7 Determining Signal-to-Noise Ratios at the Single Trial
Level}

The signal-to-noise ratios were calculated for each component in the
STP, STW, and Averaged ERP approaches in order to determine if the SNR
for the STP approach would be similar to the other approaches. Given
that the baseline period is not a flat line in single trials and rarely
so even in averaged ERPs, ``noise'' was calculated by obtaining the
standard deviation of the amplitude of all data points in the baseline
period (i.e., -200 ms to 0 ms). The ``signal'' was the peak amplitude of
each component. This procedure is similar to the approach for
determining a SNR value for components in averaged ERPs (Debener,
Ullsperger, Siegel, \& Engel, 2006; Hu et al., 2010; Spencer, 2005).

\textbf{2.8 Data Analysis}

Descriptive data including means and standard deviations are reported.
To provide a better comparison between means with differing standard
deviations, coefficients of variation (\emph{CV} ), a measure of
relative variability calculated as the ratio of the standard deviation
to the mean, are reported.

The reliability of the STP approach was evaluated for each component by
conducting both split-half and test-retest reliability tests on each
participant's trial-by-trial baseline-to-peak data using the Pearson's
Product Moment correlation procedures in MATLAB (R2020b). For split-half
reliability, the mean baseline-to-peak amplitudes for each participant's
even trials were correlated to the mean baseline-to-peak amplitudes of
their odd trials separately for Block One (trials 1 -- 240) and for
Block Two (trials 241-480). A similar process was utilized for the
peak-to-peak and STW data. For test-retest reliability, the mean
baseline-to-peak amplitudes of trials in Block One (trials 1 -- 240) for
each participant were correlated to their mean baseline-to-peak
amplitudes of trials in Block Two (trials 241 -- 480). The test-retest
reliability for the peak-to-peak data of the STP and for STW data were
conducted in a similar manner. When deriving the mean of the amplitudes
for each component in all approaches, trials with scores of null and 0
values were eliminated, resulting in data loss of 0.2\% to 1.6\% of all
trials (See Table 6).

Two statistical approaches were used to provide evidence of the validity
of the STP approach. The first approach used the Standard Error of
Measurement (SEM) to compare the variability of the trial-by-trail STP
data to the variability of the trial-by-trail STW data. The SEM of each
component was calculated across all trials for each participant (Luck et
al., 2021). The SEM was calculated by dividing the standard deviation of
the baseline-to-peak single trial ERP amplitudes by the square root of
the number of trials. A Pearson's correlation coefficient was calculated
in SPSS 26 to compare the STP and STW approaches, where an \emph{r}value
can be interpreted as: \textless{} 0.5 poor, 0.5-0.75 moderate, .075 -
0.9 good, \textgreater{} 0.9 excellent (Koo \& Li, 2016; Portney \&
Watkins, 2009). The STW approach does not yield latency information;
therefore, SEMs for latency were only calculated for the STP approach.

To provide further evidence of the validity of the STP approach,
separate linear regression analyses across all participants were
completed for each of the components (P1, N1, P2, N2 and P3) to
determine the extent to which measures derived from the STP approach
accounted for the variance of the component's amplitude measures derived
from the averaged ERP. Analyses of each component began by determining
for each participant their mean and the standard deviation of
trial-by-trial baseline-to-peak amplitudes from the STP approach, their
mean and standard deviation of the trial-by-trial peak latencies from
the STP approach, and their mean of the trial-by-trial noise measures.
Then the regression analysis was conducted using the general model of
four input variables entered in three steps. In the first step the mean
of the baseline-to-peak amplitudes was the sole predictor variable. The
second step added the two standard deviations, one for the
baseline-to-peak amplitudes and one for the latencies. The third step
added the mean of the noise measure.

The possibility of systematic change across the single trial data was
evaluated using an exploratory curve-fitting approach implemented via
the Curve Fitting app in MATLAB (2020b). Curve fitting for each
component within each trial began by computing the mean of the ST
baseline-to-peak ERP amplitudes across all participants. Scatterplots of
these mean amplitudes (y-axis) as a function of trial number (x-axis)
were then created and then followed by curve fits performed using sum of
sines models where the number of terms was varied from one to four until
overfitting was observed (e.g., Figure 1). For purposes of comparison,
this curve fitting approach was also applied to peak-to-peak ERP
amplitudes and latencies from the STP approach, amplitudes from the STW
approach, and response time data. The degree of fit for each curve is
reported as \emph{R} \textsuperscript{2}, the portion of variance of the
amplitude measures accounted for by the curve plotted across trials.

\textbf{3.0 Results}

\textbf{3.1 The Feasibility of Single Trial Measures}

\textbf{Signal to Noise Ratios:} SNR values for each component in a
participant's averaged ERP were calculated in a manner recommended by
Spencer (2005), Debener et al. (2006) and Hu et al. (2010) where the
``signal'' was the peak amplitude of the component, and the ``noise''
was the standard deviation of the amplitude of all data points in the
baseline period (i.e., -200 ms to 0 ms). For both the STP and STW
approaches, SNR values were first calculated at the trail level using
the same definitions of ``signal'' and ``noise''. Additionally, for
direct comparison purposes to the SNR derived each individual
participant's averaged ERPs, the SNRs at the trial level were then
averaged for each participant. For a brief summary, means across
participants and their corresponding standard deviations for each
component are shown in Table 1.

As expected, and in keeping with the justification for averaging ERPs
often found in the literature, the mean SNR for all components was the
largest when the components were measured from averaged ERPs. However,
the general assumption that the ``signal'' of interest is too small to
be measured in single trials may not be true for at least the paradigm
used in this study as the mean SNRs for the STP approach, while lower
than SNRs of the averaged ERPs, did range from 1.62 to 2.9 and were
larger than the SNRs of the STW approach. More importantly, the
within-group variability (i.e., \emph{SD} s) of the SNRs for the STP
approach is considerably smaller than the variability of the SNR of the
averaged ERPs but larger than those for the STW approach. A more direct
comparison of the three approaches can be made by computing their
coefficient of variation (\emph{CV} ). The lower the coefficient of
variation value the better, as it denotes that the spread of data values
are low relative to the mean of the values. When the differences in the
variability of the approaches are viewed in relation to their mean
amplitudes as shown in Table 1, other than for P1, the STP and STW
approaches are nearly the same regarding finding a ``signal'' within
``noise'', and both have much smaller \emph{CV} s than the averaged
ERPs.

\textbf{Comparison of Measurement Approaches:} To allow for direct
comparison between the STP (baseline-to-peak), STW, and averaged ERP
approaches, the ST data for each participant were first averaged across
trials then these averages were summarized across participants to
produce descriptive statistics; the means, standard deviations, minimum
and maximum values. These descriptive statistics for the amplitude
measures derived from the averaged ERP, STW, and STP approaches are
provided in Table 2. Descriptive data for the latencies for the averaged
ERP and STP approach are provided in Table 3. In general, the STP
approach produced the largest within-group mean amplitudes along with
having the largest standard deviation across all of the components
measured. With the exception of P1, the averaged ERP approach produced
considerably larger with-group means but only slightly larger standard
deviations than the STW approach. However, when these approaches are
viewed in relation to their coefficient of variations (Table 4), the STP
approach had considerably smaller \emph{CV} values than the other two
approaches for all components. The STW approach had larger
absolute\emph{CV} values than the averaged ERP approach for all
components except for the P1 component.

\textbf{3.2 Reliability}

The second research aim of this study was to determine if a simple
procedure could produce reliable measures of amplitude and latency. To
address this question, a Pearson's \emph{r} correlation coefficient was
completed for split-half reliability comparing the mean amplitude of the
even numbered trials to mean amplitude of the odd numbered trials
separately for Block One (trials 1 -- 240) and Block Two (trials 241 --
480). Reliability was further assessed by conducting Pearson's
correlation analyses to examine the test-retest reliability for mean
amplitudes between the two blocks for the STP approach compared to the
STW approach. A Pearson's correlation was also conducted to examine
test-retest reliability for latencies in the STP approach. Analyses for
reliability are shown in the next two sections.

\textbf{3.2.1 Split-half Reliability}

Measures of reliability derived from split-half reliability testing
indicated that the STP approach produced reliable measures of amplitude
and latency across all components (Tables 5 -- 7). In Block one,
the\emph{r} values for the mean baseline-to-peak amplitude of even
trials compared to odd numbered trials were considerably higher for the
STP approach than the STW approach for all components except P3, which
was close in value (\emph{r} = .74 for STP and \emph{r} = .79 for STW;
Table 5). However, the differences between the coefficients are much
less in block two, particularly for the later components.

Correlations for STP peak-to-peak measures are shown in Table 6, with
moderately strong to very strong correlations for all components.
Components P2 and P3 increased in strength from Block One to Block Two.
Table 7 shows that coefficients for latency measures are very high for
all components in both blocks with all coefficients exceeding .93 except
for the P1 component where \emph{r} = .79 and .89, respectively for
block one and two.

\textbf{3.2.2 Test-Retest Reliability}

Measures of reliability derived from test-retest reliability indicated
that the STP (baseline-to-peak) approach produced reliable measures of
amplitude, with coefficient values between .87 and .91 compared to .60
and .94 for the STW approach. The values were higher for the STP
approach than the STW approach for all components except P2 and N2,
which were both close in value (Table 8). Latencies for the STP approach
yielded values between \emph{r} = .64 and \emph{r} = .74.

\textbf{3.3. Validity of STP Approach to Single Trial Measurement}

The third research aim was to determine if a simple procedure for
measuring ERP peaks could be shown to be a valid approach. To evaluate
validity of the STP approach, standard error of measurement (SEMs) for
the STP approach were compared to the SEMs of the STW approach.
Regressions were then used to determine if the single trial amplitude
and latency measurement would significantly predict the averaged ERP
amplitudes. These results are shown in the next two sections.

\textbf{3.3.1 Standard Error of Measurement}

The SEMs for the STP (baseline-to-peak) approach ranged from \emph{M} =
.44 (\emph{SD} = .095) to \emph{M} = .68 (\emph{SD} = .132), with more
variability shown in the later components (see Table 9). The STW
approach ranged between \emph{M} = .43 (\emph{SD} = .094) and \emph{M} =
.55 (\emph{SD} = .117). Pearson's correlations between the STP and the
STW approaches yielded very strong values at or above \emph{r =} .93.
These data indicate that the SEMs were comparable for the STP and STW
approaches across participants.

\textbf{3.3.1 Regression}

To further consider the validity of the STP approach, linear regression
analyses were completed to determine the extent to which the component
measurements obtained from this approach accounted for the
between-participant variance of components obtained the from averaged
ERPs. A separate regression analysis was conducted for each component
(i.e., P1, N1, P2, N2 and P3) using the same general model of four input
variables entered in three steps. Step 1 included the participants' mean
amplitude of the STP (baseline-to-peak) component; in Step 2,
participants' the standard deviations of both the latency and amplitude
of the STP component were added; in Step 3, participants' mean noise
value was added. For all components, the regression models showed a
significant amount of variance accounted for in each successive step.
Therefore, \emph{R} \textsuperscript{2}, \emph{R}
\textsuperscript{2}Change, and the standardized coefficients only after
the third step are reported in Table 10.

In the models of each component, the results after the third step
indicate that of the four input variables, the mean amplitude variable
from the STP approach had the highest beta weight and accounted for a
significant amount of variability in its corresponding component
measurement from the averaged ERP. Noise accounted for a significant
amount of variability in the early components (P1 and N1); however, the
beta weight values decreased suggesting it accounted for a smaller
portion of the variability in the later components. Although the beta
weights of the standard deviation of latencies were significant
indicating that the latencies obtained in the STP approach accounted for
some variability, this was much less so than the mean amplitude or
noise. Across all components, the beta weights of the standard deviation
of the amplitude were not significant. Interestingly, the results of the
second step, the step prior to the addition of the noise factor, the
beta weights of the standard deviation of the amplitude were significant
in each of the regression analyses. The fact the amplitude variability
is no longer a significant predictor of the averaged ERP component when
noise is introduced to the prediction model suggest that these two
variables are colinear, i.e., they share a common portion of the
variance. Taken together, these results of these regression analyses
indicate that the single trial peak measures are strong predictors of
the stimulus-locked averaged ERPs, and thus, may be considered valid
measures of brain activity.

\textbf{3.4 Do trial-to-trial measures display systematic change?}

The Curve Fitting app and functions of the Curve Fitting Toolbox in
MATLAB (2020b) were used to perform exploratory analyses to determine if
any of the components' ST amplitude or latency variability exhibit
systematic change across successive trials. The curve fitting process
for each component measure (e.g., ST baseline-to-peak P1 amplitude)
started with averaging the measure across those participants with
correct responses for the first trial and then averaging separately for
each successive trial in a similar manner through trial number 480. This
averaging procedure was also completed for participant response times
(RT) for each correct trial. Scatterplots were then made with
cross-participant mean amplitude (or latency) plotted on the y-axis as a
function of trial number on the x-axis separately for each block (i.e.,
Trials 1-240 and Trials 241-480) and then with both blocks combined
(i.e., Trials 1-480). Then to determine if these distributions of means
contained any change in values as a function of trial number, the
scatterplot of each ST component was evaluated using four Sum of Sine
models where in each successive model the phase constant for the model's
sine wave term was incremented from 1 to 4 unless there was evidence of
overfitting of the curve as depicted in Figure 1. Overlays of the single
trial mean amplitudes derived from STP baseline-to-peak, STP
peak-to-peak, and STW approaches are shown in Figure 2 for each
component. For comparison purposes, the curve fit of the RT are also
shown in Figure 2. Non-linear patterns were found in all components of
STP baseline-to-peak and STW approaches with the most dramatic changes
in those from the STP peak-to-peak approach. The single trial mean
latencies for STP approach are shown in Figure 3 along with mean RT.

The amount of variance (i.e., \emph{R} \textsuperscript{2} values)
accounted for by each fitted curve depicted in the figures are reported
in Table 11. For the baseline-to-peak and peak-to-peak measures from STP
approach, all were successfully fitted to sine wave value of 4 except
for one. The\emph{R\textsuperscript{2}} values of the peak-to-peak
measures were generally larger than those of the baseline-to-peak
measures. In contrast, only two of the amplitude measures from the STW
approach could be fitted to sine wave value of 4 and all but two of
the\emph{R\textsuperscript{2}} values of measures from the STW approach
were lower than those from the STP approach.

\textbf{4.0 Discussion}

The purpose of this study was to establish the feasibility of using the
Single Trial Peaks (STP) procedure to derive amplitude and latency
measures of multiple components in single ERP segments. The reliability
of the STP procedure was assessed by calculating split-half and
test-retest reliability coefficients for each component, and compared
these to the coefficients derived from the Single Trial Windows (STW)
approach. For P1, N1, P2, N2 and P3 components, the split-half
coefficients obtained for STP amplitude measures ranged from .62 to .87
for Block 1 and .93 to .95 for Block 2. These reliability coefficients
were generally greater than their corresponding coefficients for the STW
approach that ranged from .33 to .79 for Block 1 and .59 to .95 for
Block 2. The test-retest reliability coefficients for STP amplitude
measures ranged from .87 to .91 and were less variable across the
components than the STW approach that ranged from .60 to .94.
Furthermore, the STP approach produced highly reliable latency measures
at the trial level -- measures that the STW approach cannot produce.

As part of the assessment of the validity of the STP procedure,
descriptive comparisons between the STP and STW approaches were made for
the SNR and the SEM measures. For all components, the grand means of the
SNR were greater for the STP approach compared to those obtained from
the STW approach. This difference in signal to noise ratio suggests that
the STP procedure is better at indicating that the signals are distinct
from baseline activity and are therefore valid measures of component
peak amplitude. The comparisons of the grand means of the SEM showed
that the two approaches were comparable and were very strongly
correlated (\emph{r} = .93 - .97), indicating both approaches had nearly
identical degrees of measurement error across participants. The validity
of the STP approach was further demonstrated by using multiple
regression analyses. Four variables from each participant were used in
each analysis; the participant's mean and standard deviation of single
trial amplitude measures across trials, the participant's standard
deviation of single trial latency measures, and the participant's mean
of the ``noise'' measures. These analyses revealed that these four
variables accounted for 73\% (P1) to 95\% (P3) of the variance of the
corresponding component derived from the individuals' averaged ERPs.

Interestingly, when the average noise was added in the third step of the
predictive models, its beta value was significant and its inclusion in
the model reduced the beta weight of the standard deviation of the
amplitude such that it was \emph{no longer significant} in four of the
five of the components examined. These results suggest a collinearity
exists between the standard deviation of the peak amplitude and the
amplitude variability during the baseline period, the noise measure used
as the denominator in calculating SNR values. Taken together, the above
results indicate that the STP approach produced highly reliable and
valid peak amplitude and latency measures at the trial level.

\textbf{4.1} \textbf{Revisiting Assumptions Regarding Measuring Peaks in
Single Trial ERPs}

In EEG research, it is commonly assumed that the electrical activity in
an ERP elicited by a single stimulus event is unique to the stimulus and
is assumed to be invariant in amplitude and latency for each occurrence
of the stimulus (Dawson, 1954; Hu et al., 2011; Quian Quiroga \& Garcia,
2003). Furthermore, the elicited signal is assumed to be smaller than
the electrical activity from unrelated noise sources (i.e., the elicited
signal has a low SNR) and, consequently, is hidden in the mixture of
activity from multiple brain sources not associated with the stimulus
(Dawson, 1954; Quian Quiroga \& Garcia, 2003). The STP approach was
designed and implemented to be independent of these assumptions.

After filtering and segmenting the continuous EEG for a given stimulus,
visual inspection of trial-level ERPs revealed multiple fluctuations in
voltage, both large and small, that do not appear in the individual's
averaged ERPs. Despite the multiplicity of fluctuations, many of the
positive and negative peaks in the trial-level ERP coincided to high
degree with the amplitude and latency of the peaks in the averaged ERP.
Thus, the general morphology of the averaged ERP was visible in almost
all trial-level ERPs with the most maximal peaks (positive or negative)
occurring at time periods closely related to the latencies of the
successive components of the averaged ERP. As the averaged ERP is
determined mathematically incorporating each time point, the general
morphology of the averaged ERP must be present in each trial-level ERP
to some degree. If this was not the case, the plot of an individual's
averaged ERP would begin to approximate a straight line as the number of
segments in the average increases.

Using the morphology of the individual's averaged ERP to set the time
windows to select the maximal positive or negative amplitudes for
successive components in each of the individual's trial-level ERPs is
the cornerstone of the STP approach. Additionally, this approach assumes
and allows for the latency of the maximal peak to vary within the time
window that defines the component. By considering the entire morphology
rather than an isolated peak, a better understanding of how brain
processing influences behavior is obtained (Gaspar et al., 2011; Hu et
al., 2011; Lin et al., 2021). If change across the entire morphology is
systematic rather than random, analysis of single trials can show
valuable information, which could be important to understanding human
function such as the state of the participant, the presence of a
disability or disorder, or the ability to learn over time (Pernet,
Sajda, \& Rousselet, 2011; Rey, Ajmadi, \& Quian Quiroga, 2015).

Thus, choosing peaks in an ERP for measurement at the trial-level based
on matching an approximation of the morphology of the averaged ERP is
independent of the degree of background brain activity in the ERP,
though a portion of the amplitude value of the peak may represent the
background activity to some degree. From the viewpoint of test and
measurement framework, the best estimate of the brain ``signal''
elicited by the stimulus would be the mean of all trial-level peak
amplitude measures and the variability of the trial-to-trial amplitude
measures that represent the measurement error of the signal due to
variations of background activity.

Though not a specific aim of the present study, the results of the STP
approach provided evidence that \emph{the assumptions of signal
invariance and low SNRs} may be misleading, at least for ERPs obtained
from adults while performing a speeded version of the flanker
task.\emph{Related to signal invariance} , the peak amplitude and
latency measures of each component as defined by their maximal/minimum
values in respective time windows were found to vary trial-to-trial
within and across participants (Tables 2 and 3). Thus, one cannot assume
that a brain's electrical activity elicited by a unique stimulus is
invariant. In addition, one cannot assume that the amplitude and latency
variability represents only the ``noise'' from the electrical activity
of unrelated brain sources as the curve fitting results suggest that as
much as 20 -- 25\% of the variance across trials represent systematic
change in brain activity that leads to performance changes in RT.

The results of the present study also question the validity of
the\emph{assumption of low SNR} in the EEG signal post onset of a
stimulus. While the SNRs of the STP approach were lower than the
traditional averaged ERP approach, the SNRs ranged from 1.6 for P1 to
2.9 for P3 and all were higher than the best SNR of the STW approach
which was 1.5 for P3. These results suggest that single-trial peaks
(i.e., the signal) are sufficiently large to differentiate from the
noise. The findings shown in Table 4, the coefficients of variation, a
measure of relative variability, were the lowest for measures from the
STP approach compared to the other two approaches. The low coefficients
of variation found for the peaks from the STP approach also support the
conclusion that the elicited signals from stimulus are large enough to
allow ERP components to be measured at the single-trial level.

Given that this study's SNR values are at or exceed 1.6 for all
components, procedures such as ICA and wavelet transform were deemed not
necessary to further increase the SNR. While these procedures have been
used to enhance SNR in other single-trial studies, they may also
manipulate the raw signal in a manner that may increase measurement
error. The data used for measurement of the ST peaks were the same
segments used in producing the averaged ERP. Thus, these segments were
pre-processed using the same steps and parameters used to obtain
averaged ERPs, and no other manipulations of the raw signal were
employed. By avoiding additional processing steps when the mean of the
peak amplitudes across trials is obtained for a given component, the
functionality of the component may be directly interpreted based on the
literature associated with the respective component as measured from
averaged ERPs. And, given that it is has less measurement error (i.e.,
not confounded with latency jitter) it is more likely a truer
representation of the functionality of brain processing within
participants than the measurements taken from an averaged ERP.

\textbf{4.2 Peak-to-peak Amplitude Measurement in Single Trials}

Using a peak-to-peak measure to report the amplitude of an ERP component
has several advantages compared to reporting baseline-to-peak measures
or averaged amplitude within a window (i.e., STW approach). Firstly,
baseline-to-peak and STW measures are centered on a baseline obtained in
a step in the preprocessing stage for creating ERPs. This processing
step may not always be successful in producing a constant zero-voltage
baseline prior to the stimulus onset. In these cases, measuring in
relation to the baseline can be problematic as the deflection of
interest may move more positive or more negative than a true zero
baseline would guarantee. Secondly, previous research demonstrates that
measuring amplitude using a peak-to-peak approach produces a more
reliable measure of amplitude compared to baseline-to-peak or average
voltage in a window, suggesting that there is less measurement error in
the peak-to-peak amplitude measure (Gavin et al., 2019; Lin et al.,
2020; Taylor et al., 2016). Finally, previous studies have shown that in
a stream of processing from stimulus presentation to behavior (e.g.,
response time) each phase is not independent from the next, i.e.,
successive phases are interrelated (Lin, et al., 2021; Taylor, 2019).
These interrelationships are consistent with the concept of phase
sequences proposed by connectionist theory (Bear \& Cooper, 1990; Hebb,
1949; Lowe et al., 2016; McNaughton \& Nadel, 1990). Thus, using a
peak-to-peak measure preserves the interrelationship between brain
process reflected by each component. However, it should be noted that
using multiple peak-to-peak measures from single ST ERPs or averaged
ERPs can lead to multicollinearity, so caution needs to be heeded when
using these measures in statistical analyses.

In the current study, the use of the STP peak-to-peak amplitude measures
demonstrated enhanced reliability and measurement sensitivity compared
to the STP baseline-to-peak and the STW amplitude measures. The
split-half correlation coefficients were greater when examining STP
peak-to-peak amplitudes (see Table 6) compared to the STP
baseline-to-peak and STW amplitudes (see Table 5). This suggests that
there was more stability in the peak-to-peak measure. In addition, the
changes within a session were more pronounced for each component when
using the STP peak-to-peak measure compared to the STP baseline-to-peak
and STW amplitude measures (see Figure 2). This suggests that the
smaller measurement error in the STP peak-to-peak measure increases the
sensitivity to voltage fluctuations across trials demonstrating changes
within a session, at least when examining the ST ERPs during the speeded
flanker task. The interrelationship between processing phases required
to successfully complete the speeded flanker task are captured when
using the peak-to-peak measure when observing changes across a session.

\textbf{4.3 Latency Variability Within and Across Individuals}

Amplitude and latency each provide unique information about brain
processing. Amplitude measures are thought to relate to efficiency,
resources allocation, capacity, and effort while latency relates to
speed of processing (Polich, 2007; Riggins \& Scott, 2020; van Dinteren,
Arns, Jongsma, \& Kessels, 2014). However, studies measuring averaged
ERPs often report significant differences in amplitude when comparing
task conditions or groups, but few show significant latency differences
(Schubert, L\selectlanguage{ngerman}öffler, Hagemann, \& Sadus, 2022). While healthy adults
often show more or less consistent latencies, greater latency
variability has been shown in children and individuals with a
disability. For instance, decreases in latency variability have been
shown to be characteristic of brain maturation (Gavin et al., 2019;
Milne, 2011); whereas, when compared to neurotypical participants,
greater latency variability have been shown in neurodivergent children
(Magnuson et al., 2020), older adults with dementia (Patterson et al.,
1988), individuals with schizophrenia (Reinhart et al., 2011; Roth et
al., 2007), and individuals who have had a traumatic brain injury (Unsal
\& Segalowitz, 1995).

Difficulty in obtaining significant mean latency differences between
groups may be due to greater measurement error in peak latency measures
(Schubert et al., 2022). Peak latencies for most components derived from
averaged ERPs have proven to be less reliable than peak amplitudes of
the same components in the averaged ERPs and even less so for children
compared to adults (Lin et al., 2020; Taylor et al., 2016). This may be
because peak latency measures taken from averaged ERPs are not
independent from peak amplitude variability in a similar manner as peak
amplitude measures from averaged ERPs are not independent of
trial-to-trial latency variability (Dupuis et al., 2015; Gavin et al.,
2019). That is, in averaged ERPs the latency measures are confounded by
fluctuating amplitudes as well as fluctuations of ST latency. An
important benefit of using the STP approach is that when the peak
amplitude and latency measures are obtained at the trial level for a
component, each measure (i.e., amplitude and latency) is independent of
the other. Consequently, ST measures of peak latency are shown in this
study to be valid and reliable in addition to peak amplitude.

\textbf{4.4 Systematic Changes Demonstrated Across Single-Trial
Amplitude and Latency Measures}

Previous studies using ST analyses have examined systematic changes in
latency across the duration of the ERP task to identify informative
patterns that may represent learning over time (Jongsma et al., 2006,
Quian Quiroga et al., 2007; Rey et al., 2015). In this current study, an
exploratory curve fitting procedure was used to determine whether
changes in amplitude, latency, or RT across trials in a single EEG/ERP
session of the speeded version of the flanker task were random or
systematic. Curve fitting revealed systematic changes that were similar
across all components, for both amplitude and latency, as well as RT.

These systematic changes suggest that for the speeded flanker task
participants require some time at the start of a session to develop a
strategy for successful goal directed behavior. This is reflected by the
increasing of the components' amplitudes, increasing latencies, and
longer RTs across the initial trials, peaking around 50 to 100 trials.
Following these initial peaks, there was a systematic trend of
successively smaller component amplitudes, decreasing component
latencies, and shorter RTs. This suggests that once a strategy was
developed, participants were able to sustain their behavioral
performance more efficiently, more quickly, and with fewer allocated
resources. At the end of the block, there were slight increases in
component amplitudes and latencies with concomitant RT increases which
may reflect fatigue or boredom. Similar systematic changes were repeated
in the 2\textsuperscript{nd} block (i.e., after the hashed line in
Figures 2 and 3) although blunted, indicating that the strategy was
already developed during block one and less time was needed to
re-establish and less effort to sustain the behavioral pattern. These
systematic changes were most prominent for peak-to-peak data from the
STP approach and much less so in the data from the STW approach.

The observed non-linear trends in the peak-to-peak amplitude measures
accounted for 12\% to 25\% of variance across all trials depending on
the component. Curve fitting models for latency revealed non-linear
systematic change similar to component amplitudes and response times.
These findings are consistent with other ST analyses which also have
examined systematic changes in latency across the duration of the ERP
task to identify informative patterns that may represent learning over
time (Jongsma et al., 2006, Quian Quiroga et al., 2007; Rey et al.,
2015). Thus, the consistency of the trends in the trial-to-trial changes
in amplitude and latency across components and RT suggest that the
participants ``learn'' to implement a strategy for task performance
which improves with the accumulation of experience, i.e., successive
trials. Similarly, trends of increased amplitudes and latencies at the
end of blocks suggests a time-dependent effect such as boredom or
fatigue.

\textbf{4.5 Conclusion}

The STP approach is a reliable and valid method to measure peak
amplitude and latencies in single trials in the speeded flanker task.
The averaged ERP approach is useful in certain situations such as when
the components of interest are stable across individuals, and when there
is little interest in individual differences or change over time.
However, if there is variability across individuals or when trying to
examine changes over time, the STP approach would provide valuable
information about brain activity. Contrary to the assumptions often made
to justify averaging across trials, the present study shows that the
peaks in the single trial are large enough to be measured, that the
brain's response to a stimulus is not invariant, and systematically
changes occur over successive trials. While the STP approach is not the
only way to measure single trial data, it is a parsimonious way of
measuring ERP components at the trial level that can be easily
replicated and interpreted by ERP researchers across labs.

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\textbf{Author Notes}

This study was funded in part by NIH/NICHD (R03HD046512) to PLD and WJG,
and the Department of Occupational Therapy and the College of Liberal
Arts at the Colorado State University (CSU). The authors acknowledge the
students in the Brainwaves Research Lab who assisted in collecting data
and scoring the ERPs, with a special acknowledgement going to Sam Ek who
was involved with discussions at various parts of this project. There
are no known conflicts of interest or disclosures to report.

Table 1. Means, standard deviations and corresponding coefficient of
variations (\emph{CV} ) of the SNR across participants for each
component for STP (baseline-to-peak), STW, and averaged ERPs approaches.\selectlanguage{english}
\begin{longtable}[]{@{}lllllll@{}}
\toprule
& Averaged ERP & Averaged ERP & STP & STP & STW & STW\tabularnewline
\midrule
\endhead
Components & \emph{M(SD)} & \emph{CV} & \emph{M(SD)} & \emph{CV} &
\emph{M(SD)} & \emph{CV}\tabularnewline
P1 & 2.06 ( 2.72) & 1.32 & 1.62 (.21) & 0.13 & 1.00 (.10) &
0.1\tabularnewline
N1 & 8.15 (5.62) & 0.69 & 1.95 (.30) & 0.15 & 1.10 (.16) &
0.15\tabularnewline
P2 & 9.81 (7.50) & 0.76 & 2.10 (.41) & 0.2 & 1.19 (.23) &
0.19\tabularnewline
N2 & 7.75 (6.40) & 0.83 & 2.03 (.42) & 0.21 & 1.41(.30) &
0.21\tabularnewline
P3 & 20.77 (12.50) & 0.60 & 2.90 (.73) & 0.25 & 1.50 (.37) &
0.25\tabularnewline
\bottomrule
\end{longtable}

Note: STP = Single Trial Peaks, STW = Single Trial Average Time
Window,\emph{M} = Means, \emph{SD} =Standard deviation,\emph{CV} =
Coefficient of Variation, a measure of relative variability calculated
as the ratio of the standard deviation (\emph{SD} ) to the mean
(\emph{M} ).

Table 2. Descriptive data for amplitudes of each component for averaged
ERP, STW, and STP (baseline-to-peak) approaches (\emph{N} = 70). Means
were calculated across trials for each individual and then averaged
across participants for each component.\selectlanguage{english}
\begin{longtable}[]{@{}llllllllll@{}}
\toprule
& Averaged ERP & Averaged ERP & Averaged ERP & STP & STP & STP & STW &
STW & STW\tabularnewline
\midrule
\endhead
Components & \emph{M (SD)} & Min & Max & \emph{M (SD)} & Min & Max &
\emph{M (SD)} & Min & Max\tabularnewline
P1 & 0.24 (1.66) & -2.11 & 6.91 & 10.47 (3.18) & 4.91 & 19.03 & -0.53
(1.06) & -2.33 & 2.07\tabularnewline
N1 & -3.60 (2.42) & -10.25 & 5.28 & -11.83 (3.83) & -22.63 & -0.29 &
-1.22 (2.21) & -4.58 & 4.67\tabularnewline
P2 & 4.19 (3.25) & -2.48 & 11.24 & 12.51 (4.04) & 4.01 & 22.21 & 0.77
(3.13) & -7.34 & 7.14\tabularnewline
N2 & -2.18 (3.82) & -11.58 & 5.53 & -11.72 (4.56) & -22.28 & -2.34 &
0.67 (4.41) & -8.87 & 13.42\tabularnewline
P3 & 9.45 (4.75) & -2.4 & 21.48 & 17.7 (5.65) & 5.08 & 29.34 & 6.01
(3.62) & -2.9 & 16.1\tabularnewline
\bottomrule
\end{longtable}

Note: STP = Single Trial Peaks, STW = Single Trial Average Time
Window,\emph{M} = Means, \emph{SD} =Standard deviation, Min = minimum
value, Max = maximum value

Table 3. Descriptive data for latencies of each component for averaged
ERPs and STP (baseline-to-peak) approach (\emph{N} = 70).\selectlanguage{english}
\begin{longtable}[]{@{}lllllll@{}}
\toprule
& Averaged ERP & Averaged ERP & Averaged ERP & STP & STP &
STP\tabularnewline
\midrule
\endhead
Components & \emph{M (SD)} & Min & Max & \emph{M (SD)} & Min &
Max\tabularnewline
P1 & 49.25 (27.29) & 4.88 & 142.58 & 61.93 (10.96) & 44.28 &
104.72\tabularnewline
N1 & 112.42 (26.10) & 66.41 & 163.09 & 127.84 (18.67) & 82.02 &
163.74\tabularnewline
P2 & 195.08 (36.52) & 113.28 & 279.30 & 204.53 (25.90) & 131.79 &
260.94\tabularnewline
N2 & 268.72 (36.65) & 165.04 & 325.20 & 280.31 (25.52) & 215.50 &
327.88\tabularnewline
P3 & 380.12 (31.05) & 324.22 & 456.05 & 368.19 (26.82) & 320.71 &
434.26\tabularnewline
\bottomrule
\end{longtable}

Note: STP = Single Trial Peaks, STW = Single Trial Average Time Window,
M = Means, SD =Standard deviation, Min = minimum value, Max = maximum
value

Table 4. Comparison of the three approaches viewed in relation to their
coefficient of variations (\emph{CV} ).\selectlanguage{english}
\begin{longtable}[]{@{}llll@{}}
\toprule
Components & Averaged ERP & STP Approach & STW Approach\tabularnewline
\midrule
\endhead
Amplitude & \emph{CV} & \emph{CV} & \emph{CV}\tabularnewline
P1 & 6.92 & .30 & -2.00\tabularnewline
N1 & .67 & .32 & -1.81\tabularnewline
P2 & .84 & .32 & 4.06\tabularnewline
N2 & 1.75 & .39 & 6.58\tabularnewline
P3 & .50 & .32 & .60\tabularnewline
Latency & & &\tabularnewline
P1 & .55 & .18 &\tabularnewline
N1 & .59 & .15 &\tabularnewline
P2 & .19 & .13 &\tabularnewline
N2 & .14 & .09 &\tabularnewline
P3 & .08 & .07 &\tabularnewline
\bottomrule
\end{longtable}

Note: STP = Single Trial Peaks, STW = Single Trial Average Time
Window,\emph{CV} = Coefficient of Variation, a measure of relative
variability calculated as the ratio of the standard deviation (\emph{SD}
) to the mean (\emph{M} ).

Table 5. Split-half reliability for mean amplitude measures from STP
(baseline-to-peak) and STW approaches for even and odd numbered trials
(\emph{N} = 70).\selectlanguage{english}
\begin{longtable}[]{@{}lllllllll@{}}
\toprule
& Block One & Block One & Block One & Block One & Block Two & Block Two
& Block Two & Block Two\tabularnewline
\midrule
\endhead
Component & STP & STP & STW & STW & STP & STP & STW & STW\tabularnewline
& \emph{r} & \emph{p} & \emph{r} & \emph{p} & \emph{r} & \emph{p} &
\emph{r} & \emph{p}\tabularnewline
P1 & 0.83 & .008\textsuperscript{e-16} & 0.46 &
.006\textsuperscript{e-02} & 0.94 & .003\textsuperscript{e-31} & 0.59 &
.006\textsuperscript{e-05}\tabularnewline
N1 & 0.62 & .008\textsuperscript{e-06} & 0.33 & 0.005 & 0.93 &
.002\textsuperscript{e-27} & 0.87 &
.009\textsuperscript{e-20}\tabularnewline
P2 & 0.81 & .003\textsuperscript{e-14} & 0.74 &
.002\textsuperscript{e-10} & 0.95 & .006\textsuperscript{e-34} & 0.92 &
.006\textsuperscript{e-26}\tabularnewline
N2 & 0.87 & .001\textsuperscript{e-19} & 0.78 & .002\textsuperscript{e
-12} & 0.95 & .004\textsuperscript{e-32} & 0.95 &
.002\textsuperscript{e-32}\tabularnewline
P3 & 0.74 & .003\textsuperscript{e-10} & 0.79 & .007\textsuperscript{e
-13} & 0.95 & .004\textsuperscript{e-33} & 0.93 &
.004\textsuperscript{e-29}\tabularnewline
\bottomrule
\end{longtable}

Table 6. Split-half reliability for mean peak-to-peak amplitude measures
of odd compared to even numbered trials using the STP approach
(\emph{N}= 70).\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
Component & Block One & Block One & Block Two & Block Two\tabularnewline
\midrule
\endhead
& \emph{r} & \emph{p} & \emph{r} & \emph{p}\tabularnewline
N1 & 0.89 & .006\textsuperscript{e-22} & 0.97 &
.003\textsuperscript{e-41}\tabularnewline
P2 & 0.70 & .001\textsuperscript{e-08} & 0.97 &
.002\textsuperscript{e-42}\tabularnewline
N2 & 0.96 & .002\textsuperscript{e-37} & 0.97 &
.003\textsuperscript{e-40}\tabularnewline
P3 & 0.66 & .005\textsuperscript{e-07} & 0.96 &
.004\textsuperscript{e-35}\tabularnewline
\bottomrule
\end{longtable}

Note: Zero values were removed from peak-to-peak measure data, resulting
in the following percent of available trials used in peak-to-peak
calculations (of 30038): N1 = 29556 (98.4\%), P2 = 29867 (99.4\%), N2 =
29801 (99.2\%), and P3 = 29973 (99.8\%).

Table 7. Split-half reliability for mean latencies for even and odd
numbered trials using the STP approach (\emph{N} = 70).\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
Component & Block One & Block One & Block Two & Block Two\tabularnewline
\midrule
\endhead
& \emph{r} & \emph{p} & \emph{r} & \emph{p}\tabularnewline
P1 & 0.79 & .006\textsuperscript{e-13} & 0.89 &
.009\textsuperscript{e-22}\tabularnewline
N1 & 0.95 & .005\textsuperscript{e-32} & 0.96 &
.003\textsuperscript{e-39}\tabularnewline
P2 & 0.93 & .001\textsuperscript{e-28} & 0.97 &
.006\textsuperscript{e-39}\tabularnewline
N2 & 0.95 & .008\textsuperscript{e-33} & 0.97 &
.003\textsuperscript{e-41}\tabularnewline
P3 & 0.98 & .002\textsuperscript{e-50} & 0.99 &
.008\textsuperscript{e-52}\tabularnewline
\bottomrule
\end{longtable}

Table 8. Test-retest reliability of mean amplitudes for the STP
(baseline-to-peak) and STW approaches. Test-retest reliability for
latencies in the STP approach (\emph{N} = 70).\selectlanguage{english}
\begin{longtable}[]{@{}lllllll@{}}
\toprule
& STP Amplitude & STP Amplitude & STW Amplitude & STW Amplitude & STP
Latency & STP Latency\tabularnewline
\midrule
\endhead
Component & \emph{r} & \emph{p} & \emph{r} & \emph{p} & \emph{r} &
\emph{p}\tabularnewline
P1 & 0.89 & .003\textsuperscript{e-22} & 0.60 &
.003\textsuperscript{e-05} & 0.74 &
.002\textsuperscript{e-10}\tabularnewline
N1 & 0.91 & .002\textsuperscript{e-25} & 0.88 &
.007\textsuperscript{e-21} & 0.74 &
.003\textsuperscript{e-10}\tabularnewline
P2 & 0.90 & .009\textsuperscript{e-23} & 0.91 &
.001\textsuperscript{e-24} & 0.74 &
.002\textsuperscript{e-10}\tabularnewline
N2 & 0.91 & .006\textsuperscript{e-25} & 0.94 &
.003\textsuperscript{e-31} & 0.68 &
.001\textsuperscript{e-07}\tabularnewline
p3 & 0.87 & .001\textsuperscript{e-19} & 0.85 &
.002\textsuperscript{e-17} & 0.64 &
.002\textsuperscript{e-06}\tabularnewline
\bottomrule
\end{longtable}

Note: Latencies cannot be calculated using the STW approach.

Table 9. SEM means and standard deviations for the amplitude measures of
the STP (baseline-to-peak) and STW approaches. Correlation between STP
and STW approaches (\emph{N} = 70).\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
& STP & STW & Correlations & Correlations\tabularnewline
\midrule
\endhead
Component & \emph{M (SD)} & \emph{M (SD)} & \emph{r} &
\emph{p}\tabularnewline
P1 & 0.44 (0.10) & 0.43 (0.09) & 0.97 & .002 e-41\tabularnewline
N1 & 0.52 (0.12) & 0.45 (0.10) & 0.94 & .003 e-31\tabularnewline
P2 & 0.54 (0.11) & 0.47 (0.10) & 0.95 & .002 e-33\tabularnewline
N2 & 0.56 (0.12) & 0.55 (0.12) & 0.95 & .005 e-33\tabularnewline
P3 & 0.68 (0.13) & 0.52 (0.10) & 0.93 & .007 e-28\tabularnewline
\bottomrule
\end{longtable}

Table 10. Summary of regression models of ST amplitude and latency
measures predicting the averaged ERP amplitude. The degrees of freedom
associated \emph{F} values are 4 and 65.\selectlanguage{english}
\begin{longtable}[]{@{}llllllllll@{}}
\toprule
Model Statistics & Model Statistics & Model Statistics & Model
Statistics & Model Statistics & Model Statistics & Standardized
coefficient & Standardized coefficient & Standardized coefficient &
Standardized coefficient\tabularnewline
\midrule
\endhead
DV Averaged ERP & Model & \emph{R}\textsuperscript{2} &
\emph{R}\textsuperscript{2} Change & \emph{F} & \emph{p} & IVs &
\selectlanguage{greek}\emph{β} & \emph{t} & \emph{p}\tabularnewline
P1 Amp & 1 & 0.27 & 0.27 & & & & & &\tabularnewline
& 2 & 0.40 & 0.13 & & & & & &\tabularnewline
& 3 & 0.73 & 0.33 & 43.52 & \textless{}.0005 & \emph{M} P1 Amp & 1.57 &
12.89 & \textless{}.0005\tabularnewline
& & & & & & \emph{SD} P1 Amp & 0.01 & 0.04 & 0.967\tabularnewline
& & & & & & \emph{SD} P1 Lat & -0.23 & -3.36 & 0.001\tabularnewline
& & & & & & Noise & -1.26 & -8.87 & \textless{}.0005\tabularnewline
N1 Amp & 1 & 0.54 & 0.54 & & & & & &\tabularnewline
& 2 & 0.66 & 0.11 & & & & & &\tabularnewline
& 3 & 0.80 & 0.14 & 63.72 & \textless{}.0005 & \emph{M} N1 Amp & 1.37 &
15.00 & \textless{}.0005\tabularnewline
& & & & & & \emph{SD} N1 Amp & -0.21 & -2.27 & 0.027\tabularnewline
& & & & & & \emph{SD} N1 Lat & 0.47 & 7.11 &
\textless{}.0005\tabularnewline
& & & & & & Noise & 0.80 & 6.74 & \textless{}.0005\tabularnewline
P2 Amp & 1 & 0.64 & 0.64 & & & & & &\tabularnewline
& 2 & 0.81 & 0.17 & & & & & &\tabularnewline
& 3 & 0.88 & 0.08 & 121.65 & \textless{}.0005 & \emph{M} P2 Amp & 1.01 &
21.27 & \textless{}.0005\tabularnewline
& & & & & & \emph{SD} P2 Amp & -0.06 & -1.14 & 0.259\tabularnewline
& & & & & & \emph{SD} P2 Lat & -0.29 & -6.59 &
\textless{}.0005\tabularnewline
& & & & & & Noise & -0.38 & -6.54 & \textless{}.0005\tabularnewline
N2 Amp & 1 & 0.67 & 0.67 & & & & & &\tabularnewline
& 2 & 0.74 & 0.08 & & & & & &\tabularnewline
& 3 & 0.80 & 0.06 & 65.35 & \textless{}.0005 & \emph{M} N2 Amp & 1.11 &
15.07 & \textless{}.0005\tabularnewline
& & & & & & \emph{SD} N2 Amp & 0.13 & 1.87 & 0.066\tabularnewline
& & & & & & \emph{SD} N2 Lat & 0.17 & 2.76 & 0.008\tabularnewline
& & & & & & Noise & 0.36 & 4.37 & \textless{}.0005\tabularnewline
P3 Amp & 1 & 0.89 & 0.89 & & & & & &\tabularnewline
& 2 & 0.93 & 0.04 & & & & & &\tabularnewline
& 3 & 0.95 & 0.03 & 326.00 & \textless{}.0005 & \emph{M} P3 Amp & 1.02 &
32.73 & \textless{}.0005\tabularnewline
& & & & & & \emph{SD} P3 Amp & -0.06 & -1.62 & 0.111\tabularnewline
& & & & & & \emph{SD} P3 Lat & -0.18 & -6.49 &
\textless{}.0005\tabularnewline
& & & & & & Baseline Noise & -0.19 & -5.87 &
\textless{}.0005\tabularnewline
\bottomrule
\end{longtable}

Table 11.\emph{R} \textsuperscript{2}values indicating the amount of
variance accounted for by curve fitting for each component's mean
amplitudes and mean latencies for trials of Block One only, trials of
Block Two only, and all trials of Bock One and Two together.\selectlanguage{english}
\begin{longtable}[]{@{}llllllllll@{}}
\toprule
& Block One & Block One & Block One & Block Two & Block Two & Block Two
& All Trials & All Trials & All Trials\tabularnewline
\midrule
\endhead
Component & STP B2P & STP P2P & STW & STP B2P & STP P2P & STW & STP B2P
& STP P2P & STW\tabularnewline
Amplitude & & & & & & & & &\tabularnewline
P1 & 0.09 & n/a & 0.02\textsuperscript{c} & 0.09 & n/a &
0.08\textsuperscript{b} & 0.07 & n/a &
0.10\textsuperscript{b}\tabularnewline
N1 & 0.06 & 0.12 & 0.02\textsuperscript{b} & 0.14 & 0.20 &
0.03\textsuperscript{b} & 0.15 & 0.20 &
0.07\textsuperscript{b}\tabularnewline
P2 & 0.07 & 0.08 & 0.07\textsuperscript{c} & 0.05 & 0.16 &
0.04\textsuperscript{b} & 0.09 & 0.22 &
0.06\textsuperscript{b}\tabularnewline
N2 & 0.19 & 0.15 & 0.05\textsuperscript{b} & 0.10\textsuperscript{b} &
0.11 & 0.03\textsuperscript{b} & 0.11 & 0.12 & 0.09
\textsuperscript{c}\tabularnewline
P3 & 0.14 & 0.29 & 0.04\textsuperscript{a} & 0.08 & 0.19 &
0.06\textsuperscript{a} & 0.16 & 0.25 & 0.03\tabularnewline
Latency & & & & & & & & &\tabularnewline
P1 & 0.10\textsuperscript{a} & & & 0.10 & & & 0.13 & &\tabularnewline
N1 & 0.11 & & & 0.06 & & & 0.14 & &\tabularnewline
P2 & 0.08 & & & 0.05 & & & 0.19 & &\tabularnewline
N2 & 0.12 & & & 0.08 & & & 0.19 & &\tabularnewline
P3 & 0.18 & & & 0.12 & & & 0.26 & &\tabularnewline
\bottomrule
\end{longtable}

Note: Numbers without superscript indicate \emph{R\textsuperscript{2}}at
sine 4, a = \emph{R\textsuperscript{2}} at sine 3, b
=\emph{R\textsuperscript{2}} at sine 2, c =
\emph{R\textsuperscript{2}}at sine1.

Figure 1. Example of over-fitting for N2 for Block One. The plot of Sine
2 is considered an appropriately fitted function while Sine 3 is
considered overfitted by the appearance of the fit line with high
frequencies being plotted.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

Figure 2. \emph{Curve Fitting Plots for Each Component for the STW and
STP Approaches} .\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

Figure 3. \emph{Cure Fit of Latencies Measures for Single Trials}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\selectlanguage{english}
\FloatBarrier
\end{document}

