Psychophysiology, 52 (2015), 169–181. Wiley Periodicals, Inc. Printed in the USA.
C 2014 Society for Psychophysiological Research
Copyright V
DOI: 10.1111/psyp.12320

Regression-based estimation of ERP waveforms: II. Nonlinear
effects, overlap correction, and practical considerations

NATHANIEL J. SMITHa AND MARTA KUTASbaSchool of Informatics, University of Edinburgh, Edinburgh, Scotland
Departments of Cognitive Science and Neurosciences, University of California, San Diego, San Diego, California, USAb

Abstract
A companion paper introduced the rERP framework, which recasts traditional event-related potential (ERP) averaging
as a special case of a more flexible regression-based approach to estimating ERP waveforms. Here, we build on thisfoundation, showing how rERP analysis can also handle the estimation of nonlinear effects (a generalization of both
the well-known approach of dichotomizing continuous covariates, and also of the ERP image technique), and candisentangle overlapping ERPs to temporally adjacent stimuli. We then consider how the use of rERPs impacts on otherparts of the EEG analysis pipeline, including baselining, filtering, significance testing, and artifact rejection, andprovide practical recommendations. Free software implementing these techniques is available.
Descriptors: Other, Language/Speech, Normal volunteers, EEG/ERP“go” trials.1 Our question is how the scalp potential at a midline
parietal site differs between trials with different response times(RTs). A preliminary analysis using traditional techniques (Figure 1) reveals a complex pattern, in which the three faster bins allshow a positive peak at around 350 ms, but with variable latencies
and different drop-offs, and the slowest bin remains flat from
200 ms onwards. Even after 800 ms, the four waves all remainrather different. In the next two sections, we demonstrate different
ways that rERP analysis can be used to elicit more insight fromsuch messy-seeming data.

As detailed in a companion article, the rERP framework provides aflexible way to adapt well-known regression techniques to the
problem of estimating event-related brain potential (ERP) waveforms. Here, we first examine how this framework can be used toestimate nonlinear relationships between covariates and scalp
potentials, and find that it generalizes both the common approachof dichotomizing/binning continuous covariates, and the ERPimage technique (Jung et al., 2001; Lorig & Urbach, 1995). Next,
we show how regression’s ability to distinguish partially confounded effects can be exploited in order to separate overlapping
ERPs time-locked to temporally adjacent events. Finally, we discuss the practical considerations of baselining, filtering, significance testing, and rejecting artifacts in the rERP context. A list of
free software packages implementing these methods is available at
http://vorpus.org/rERP.
To illustrate these techniques, we use EEG data recorded by
Urbach and Kutas (2008) from seven undergraduate participants
engaging in a speeded go/no-go task, focusing on 673 artifact-free

Discovering Nonlinear Relationships Via Splines
Slope rERPs, as introduced in Smith & Kutas, 2015, provide a natural mechanism for analyzing continuous covariates like RT, butmake a very restrictive assumption: that the scalp potential will ateach latency vary linearly with the covariate. This is unlikely to bestrictly true in most cases, and often we will not know what relationship actually holds. Fortunately, regression provides a standard
set of techniques for estimating nonlinear relationships from data.
These techniques are powerful and, when used appropriately,
can do anything that a simple linear model can do and more. However, this power does come at a cost in conceptual complexity, andtheir use may therefore demand greater sophistication on the partof the analyst, her analysis software, and her audience. So, before
describing their application to EEG analysis, we want to emphasizethat they are likely to add more value in some situations thanothers.

This work was supported in part by NIH grant T32-DC000041 to
the Center for Research in Language, NIH grant T32-MH20002 to theInstitute for Neural Computation, NICHD grant HD22614 to MK, funding from the Army Research Laboratories Cognition & Neuroergonomics Collaborative Technology Alliance, and funding from the European
Union’s 7th Framework Programme under grant agreement No. 270273(XPERIENCE). We thank R. H. Baayen, K. A. DeLong, S. Frank, D.
Groppe, T. F. Jaeger, R. T. Knight, R. Levy, S. Makeig, S. Narayan, M.
Ramscar, M. Steedman, T. Urbach, and two anonymous reviewers forcomments and discussion.
Address correspondence to: Nathaniel Smith, University of Edinburgh, Informatics Forum 3.29, 10 Crichton Street, Edinburgh, EH89AB, Scotland, UK. E-mail: nathaniel.smith@ed.ac.uk

1. Thanks to David Groppe for suggesting this example and preprocessing the data.
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Figure 1. ERP waveforms from our example go/no-go response time task, with go trials broken down into four equal-sized bins by response time(RT), together with no-go trials for comparison.

Do You Need Nonlinear Regression?
The assumption of linearity is not so arbitrary as it may seem, andmay in fact be a viable simplifying approximation in many cases.2
Most relationships seen in the real world are smooth to a greater or
lesser extent. (Smooth means that nearby points have similar values; the more a curve jumps or wiggles, the less smooth it is.)Locally, smooth functions can be reasonably approximated by alinear function. Furthermore, while U-shaped relationships do
occur, they are the exception, rather than the rule. Most relationships are monotonic. If the factor we are studying has a monotonic
effect on the ERP, and if all we care about is detecting whether or
not an effect exists and determining its overall directionality, then alinear model may not be optimal, but it will get the job done.
Even if we are interested in the detailed shape of some relationship, we may not have enough data to estimate it. When we relax
our linearity assumption, we still must make some assumptions
about the range of possible curves. Otherwise, we cannot rule out
pathological curves that, for instance, pass through every single data
point. While there are a variety of techniques used to accomplish
this, they generally take the form of somehow requiring our curves
to be simpler and less wiggly. The fewer data points we have, thestronger these assumptions must be. When our data are sufficiently
limited, we may find ourselves required to use constraints so strongthat we are effectively doing linear regression after all. In such situations, again, we lose little by using conventional linear regression.
On the other hand, there are situations where it’s particularly
valuable to allow the regression model the flexibility to choose anonlinear relationship to match our data. The most obvious is when
discovering the shape of the relationship is itself a target of ouranalysis (e.g., Smith & Levy, 2013). A less obvious case occurs
2. We already make this linearity assumption in other situations
when analyzing ERP data. For instance, when choosing words to use ina language experiment, we often want to choose two sets of words thatare matched on a large number of word properties that would otherwise
become possible confounds, such as cloze probability or frequency.
Ideally, we would choose pairs of words such that each pair was perfectly matched on all of these properties, but in practice we instead settle for ensuring that the average value of these properties is matchedbetween the two groups. The reasoning is that if these properties are thesame on average, then they cannot cause any differences in the ERPs
between our two sets—which is only justified if we also assume thatthese properties have a linear effect on the ERP. If frequency has a nonlinear effect on the ERP, then the average ERP to a set containing one
word with frequency 10 and one word with concreteness score 30 may
not be the same as the average ERP to a set containing two words thatboth have frequency 20, even though the average frequency is the same
in both cases.

when our analysis includes control covariates—covariates whose
effect we don’t actually care about, but which are correlated withcovariates that we do care about. For such covariates, we aren’t
particularly worried about giving them too much flexibility andallowing them to overfit the data—we don’t plan to interpret them,and the worst that could happen is that they soak up more variancethan they really deserve. This should be done in moderation, ofcourse, but just as it’s better to err on the side of including control
covariates you don’t need, it’s better to have a model that is a little
too flexible than a model that is a little too inflexible. Too inflexible a model may fail to fully account for the control factor’s realeffect, leaving some portion of it to be spuriously explained by thefactors we do care about. Similar remarks apply to analyses ofinteractions between multiple covariates. In particular, a term representing the quadratic (nonlinear) effect of one covariate will be
partially collinear with terms representing linear interactions
between this covariate and other covariates that are correlated with
it. If our model includes only the interaction terms but not the nonlinear terms, then we may get spurious results.
The third case where a nonlinear model is crucial is when we
have reason to believe that our covariate may show a U-shaped
effect. This is obvious, but in the rERP context, there is a lessobvious consequence: if some covariate affects the latency of acomponent, then that covariate will effectively produce a U-shaped
effect at some time points. An example occurs in Figure 1: ataround 400–600 ms in the ERP, we see more positivity formedium-length RTs as compared to both short or long RTs. This isa U-shaped effect of RT on ERP amplitude, and so using a sloperERP to analyze the effect of RT here would be inappropriate.
There is a large literature on nonlinear regression; our goal is toshow specifically how the basic idea of nonlinear regression (however implemented) applies to ERP analysis. Therefore, we give
only a brief introduction here; for more information, see, for example, Wahba (1990) or Wood (2006).
The Spline Trick
In order to fit nonlinear models with linear least squares regression,
we need a trick. That trick is to pick some particular set of basis
functions, f1(x), . . . , fk(x), which are selected so that taking
weighted sums of them, a1f1(x) 1  1 akfk(x), lets us construct awide variety of nonlinear curves. Different sets of basis functions
lead to different spaces of possible curves; the idea is that our
regression fit will choose from these curves to pick the one thatbest matches the data. There are some good standard basis sets thatare generally used. We must also decide how many basis functions

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a

b

c

d

Figure 2. Two possible sets of basis functions for use in fitting nonlinear curves to data. a: A step-function (zero-degree B-spline) basis with fourfunctions. By taking weighted sums of these functions, we can construct a variety of piecewise-constant curves. b: Two examples of such curves, as
might result from fitting a regression model. c: A cubic (third-degree) B-spline basis with four functions. By taking weighted sums of these functions,
we can construct a variety of smooth curves. d: Two examples of such curves, as might result from fitting a regression model.

we wish to use, since the more we use, the more flexible our spaceof possible curves—and the more data we will need to use to getsensible estimates. Two possible basis sets are shown in Figure 2,
along with examples of the curves they can produce.
But having selected our basis functions, how do we convince
our regression model to actually do the fitting? It’s surprisingly
simple: we define a set of k different predictors, one for each basis
function:x1i 5 f1 ðRT on trial iÞ
x2i 5 f2 ðRT on trial iÞ⯗
xki 5 fk ðRT on trial iÞ
Now the least squares fitting will, as usual, find whatever set ofweights b1, . . . , bk causes these transformed predictors to work
together to best match our data. As far as the fitting process is concerned, these are b coefficients like any others. But these individualbs are generally not very interpretable on their own; instead, weinterpret them as together selecting the single curve in our spacethat best fits the data: the curve b1f1(x) 1  1 bkfk(x). We’ve
fooled a linear technique into fitting a nonlinear curve.
What basis functions should we use? The simplest option is todivide up the possible values for our factor into several nonoverlapping ranges. For each range, we define one basis function as(
fj ðxÞ5

1; if x falls into the jth range of values
0; otherwise(Figure 2a). By taking weighted sums of these functions, ourregression model is able to construct any function that ispiecewise-constant on these ranges (Figure 2b). Notice that when
we then use these basis functions to make predictors xji, these predictors will be very similar to dummy- or treatment-coded categorical predictors; we can think of them as indicating categoricallywhether our factor of interest falls into a particular range on eachtrial. Using this set of basis functions is therefore equivalent to thecommon practice of splitting a continuous covariate into two or
more bins (e.g., a median split) and then averaging across each bin,that is, dichotomizing, as we did in Figure 1.3

3. And just as with treatment-coded categorical predictors, in practice
we generally drop one spline basis function from our analysis in order
to avoid creating perfect collinearity with the intercept term.

But while a step-function basis is simple, it produces rather
odd-looking staircase functions, which seem unlikely to actually
represent any real brain response. A more standard way of constructing a basis for this kind of regression would be to use higherorder spline functions. There are a number of different families of
splines (which you choose usually matters little in practice), but
cubic B-splines are one of the most popular (Figure 2c). Splines
have two desirable features: first, like the step functions we saw
earlier, each basis function covers a fairly short range of the data,with minimal overlap. (In fact, our step functions themselves are a
special case of splines, but built out of zero-degree constant functions instead of the cubic functions that are more commonly used.)
As a result, the behavior of our fitted curve near any particularvalue will primarily be determined by the data that we actuallyobserved near that value, since only those data points are able toaffect the relevant b coefficients. But, unlike the step functions,they produce more realistic—and thus, potentially more accurate—smooth curves (Figure 2d).4 As with traditional dichotomization,
we can vary the size of the bin covered by each basis function byadjusting parameters called knots; in the case of the step functions,the knots are simply the break points in our dichotomization.
rERP Spline Regression
Figure 3 shows a nonlinear rERP analysis of the go trials fromour go/no-go data. With standard ERPs or slope rERPs, eachcovariate has a single b value associated with each latency, andwe can plot these values as a waveform. Nonlinear rERPs are
harder to visualize, because at each latency we are now estimating an arbitrary curve representing the nonlinear effect ourcovariate has on the scalp potential. Or, put another way, foreach of the infinitely many possible values of our covariate, wehave a potentially distinct ERP waveform. So, to visualize nonlinear rERPs, instead of plotting individual waveforms, we use
false-color images. In these figures, latency relative to the timelocking event runs along the x axis, and response time varies
along the y axis. Each horizontal slice of the figure represents
the predicted ERP waveform for a particular RT (compare

4. Another common practice is to use a polynomial basis, i.e., one inwhich f1(x) is a linear function, f2(x) is a quadratic function, etc. Thisapproach does produce smooth curves, but it fails on our first criterion:
aberrations in the data in one localized region might cause the entire fitted curve to change radically. This is a significant obstacle to interpreting and trusting the results of polynomial regression; splines are
generally preferred.

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a

b

c

d

Figure 3. Four ways of analyzing the nonlinear effect of response time on scalp potential. In these plots, each horizontal strip represents the predicted
ERP for a given RT, while each vertical strip shows how the predicted ERP at a particular latency varies with RT or trial number. Black lines indicate
where the response (button press) occurs within each epoch, and a histogram on the right shows how many trials were observed with each RT. a:
Regression using a step-function basis set containing four step functions. This is equivalent to the standard approach of splitting the data into four
equal-sized bins and calculating the average over each; this figure is simply a different way of visualizing the go curves from Figure 1. b: Regression
using a cubic B-spline basis set with four basis functions. c: Predictions from simple linear regression (slope rERPs). d: An ERP image of the samedata for comparison (trials are sorted by RT, then vertically smoothed by a Gaussian-weighted moving average with r 5 100 trials). (a), (b), and (c)are rERP analyses; (a) and (d) are previously described methods from the literature.

Figure 3a to Figure 1; these two figures depict the same analysis
in different ways). Each vertical slice of the figure represents
the estimated nonlinear relationship between RT and scalp
potential at one particular latency (comparable to Figure 2b,d).
Figure 3a,b demonstrate nonlinear rERPs using step-function
and cubic B-spline bases, respectively. In these figures, there is adiagonal pattern visible in the 400–600 ms range, running roughly
parallel to the black RT line. This diagonal pattern is what a latency
effect looks like. But unsurprisingly, this is absent in the linear(slope) rERP predictions plotted in Figure 3c. As discussed above,
latency effects create U-shaped effects at specific latencies, andbecause a slope rERP cannot capture U-shaped effects, the estimated slope coefficient during this period is essentially zero. Thisis another example of how slope rERPs are analogous to difference
waves: if we had divided our RTs into just two bins and computed
their difference, then the difference wave in this region would havebeen near zero.

On the right of each image, we plot a histogram showing how
much data were available in different parts of the range of RT values. In general, the nonlinear curves will be more trustworthy inareas where more data are available, and we accordingly arrange
our basis functions to allow more flexibility in these regions. Thisis equivalent to the standard practice of choosing dichotomization
boundaries so as to produce equal-sized bins, rather than spacing
boundaries equally in covariate space.
Finally, Figure 3d demonstrates for comparison the ERP image
technique (Jung et al., 2001; Lorig & Urbach, 1995). An ERP
image is produced by sorting trials according to some factor ofinterest (RT, in this case), and stacking the raw data from these trials vertically. Then, we use a moving average to smooth out theraw data measured at each latency on adjacent trials, and plot theresults as a two-dimensional pseudocolor image. This smoothing
serves exactly the same purpose as our use of a limited set of basis
functions, in that it allows us to pool data from trials with similar

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RTs, to reduce the effect of noise while still being able to see whatever nonlinearities are present. In fact, while we focus here on thebasis function approach for its simplicity, the use of such nonparametric smoothers is a well-studied alternative technique for
peforming nonlinear regression (Hastie & Tibshirani, 1990). Conceptually, the main difference here between the ERP image and therERP estimates is the y axis. For the ERP image, this is the trial
number; we can think of the ERP image as a nonlinear rERP thathas been stretched and squished vertically in order to flatten out thelatency distribution over the y axis. Thus, if we had seen exactly
the same brain responses but a different distribution of RTs (perhaps in another condition), the ERP image might look very different. For the rERP estimate, the y axis is the RT itself; a different
distribution of RTs would potentially make our estimates more orless accurate at different points on the y axis, but the image itself
would remain comparable across conditions.5
The rERP approach is also more general than ERP images. Figures 3a–3c were all produced by the same software routine; all that
had to be changed was a single line specifying the predictors to be
used. We could just as easily have added additional controls, or
simultaneously estimated the nonlinear effects of multiple partiallyconfounded covariates like word expectancy and word frequency,
neither of which are possible with traditional ERP images.
To keep this example manageable, we’ve used only a small number of basis functions, which is equivalent to assuming that theunderlying relation between RT and scalp potential is relatively
smooth. In general, though, this is a parameter of our analysis that
we can vary, and it results in a trade-off. As we add more flexibility
to our model, we find ourselves trying to extract more informationfrom a fixed amount of data. At one extreme, one could enter a different dummy-coded predictor for every stimulus that appears in theexperiment. If we had unlimited data, this might even be the best
option; in some experiments, every stimulus is, in fact, different(Laszlo & Federmeier, 2011). But in practice, such an analysis will
rarely be useful. Instead, we have to strike a balance: as we weaken
our model’s assumptions, we increase the chance that it can accurately represent reality—but these assumptions about the form of therelationship between stimulus properties and brain response areexactly what allows the model to compare data across multiple trials
to distinguish signal from noise. So, we have to strike some trade-off
that allows us to pool data across multiple trials without introducing
unacceptable biases. One of the great advantages of regression models is that they let us move mindfully between these extremes, and incases where we do have a large amount of data, they allow us to use
relatively mild assumptions and let the data speak.
More sophisticated regression strategies can make this trade-off
by automatically optimizing some measure of overfitting based on
the data itself; see Wood (2006) for details. These techniques havepreviously been applied to electroencephalography (EEG) by Hendrix (2009), Hendrix, Bolger, and Baayen (2014), Kryuchkova,
Tucker, Wurm, and Baayen (2012), Tremblay (2009), and Tremblay and Baayen (2010).
Nonlinear rERPs are potentially applicable to any continuous
covariate: RT, word expectancy or frequency, visual eccentricity,
tone loudness, etc. In Figure 3, though, the parallelism between theblack RT line and the dark red diagonal of increased positivity suggest that what we have in this case may be two overlapping effects:
one time-locked to the stimulus, and the other time-locked to the
5. If we really wanted to reproduce the traditional ERP image’s
stretching and squishing, we also have the option of computing a nonlinear rERP, but entering rank RT as our predictor instead of raw RT.

173
response (Jung et al., 2001). So this analysis suggests that an evenbetter approach in this instance might be not to use a generic nonlinear rERP, but to model this overlap explicitly. In the next section, we demonstrate how this can be done.
Overlap Correction
ERP activity often lasts for a second or more beyond the timelocking event. A lot can happen in a second. In a response timetask, participants may respond; in a language comprehension study,
several more words may be presented; in ordinary life, the average
second contains a rich, multidimensional stream of sensory stimulation, multiple visual saccades, and other complex motor actions.
If there are multiple events happening within a single second, and
ERPs to these events last for a second or more, then these ERPs
must be active simultaneously—they overlap.
There are often practical advantages to presenting stimuli withshort interstimulus intervals (ISIs). Short ISIs allow investigations
into interactions between the processing of successive stimuli (e.g.,priming and habituation effects), improve performance on selective
attention tasks, and make it possible to collect more data in a limited amount of time (Woldorff, 1993). But the traditional averaging
technique provides no reliable way to isolate and reconstruct the
ERPs attributable to temporally adjacent events. Particularly pernicious are situations in which adjacent stimuli have properties thatare correlated with each other, creating a new kind of confounding
problem. Previously, we’ve discussed situations where there is confounding between the different properties of a single event (e.g.,word frequency is correlated with word expectancy), and thus
effects that are caused by one may be incorrectly attributed to theother. But equally problematic is confounding between propertiesof adjacent events. For example, in naturalistic English text, thefrequency of word n is correlated with that of word n 1 1. If we’re
not careful, then in a sentence reading experiment we may conclude that we have found an ERP effect of word frequency at, say,
100 ms poststimulus onset, when in fact the component in questionis driven by the frequency of the previous word at a later latency.
Woldorff (1993) discusses analogous problems with confoundingthat arise in selective attention experiments, where, for example, ina design with “attend to visual flash” and “attend to auditory click”
conditions, the estimated ERP for unattended auditory events will
be contaminated by overlap with attended visual events, while theestimated ERP for attended auditory events will be contaminatedby overlap with unattended visual events. An apparent effect ofattention on auditory processing thus might actually arise from aneffect of attention on visual processing, and vice versa.
The simplest method to reduce the effect of overlap is to randomly vary (jitter) ISIs, which “smears” the contributions of adjacent events to the ERP estimate. This provides only a partial
solution at best. The best known technique for further correcting
ERP estimates for overlap is Adjar (Woldorff, 1993), which is arelatively complex procedure requiring the manual computation of
a number of successive approximations with uncertain accuracy.
The technique proposed here is similar to Adjar in spirit, but lets
the computer do the work of finding exact solutions automatically,
and allows us to use results from the regression literature to guarantee theoretical rigor and characterize the conditions required toachieve accurate results. And, because it is part of the generalrERP framework, the technique applies equally well to standard
ERPs, slope rERPs, nonlinear rERPs, and combinations of these.
Techniques related to the one proposed here have previously
been used for estimating hemodynamic response functions in

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Figure 4. An illustration of the procedure for converting a set of separate regression models for different latencies into a single combined model for
all latencies, as required for overlap correction. On the left, we see three of the models that previously we would have fit to different subsets of ourdata. (The particular latencies shown reflect a 250 Hz sampling rate.) On the right, we see how they are combined by, for each model, replacing eachpredictor by a new one that takes on the same values on the subset of data that its model was previously fit to, and zero elsewhere.

event-related fMRI (Hinrichs et al., 2000), and analyzing skin conductance responses (Bach & Friston, 2013).
Encoding Overlap in a Regression Equation
From by-epoch regression to continuous-time regression.
Previously, we calculated rERPs by fitting a separate regression
model for each latency, and then collected the resulting b coefficients from each model together to form individual waveforms.
The first step to accomplishing overlap correction is to combine
these many regression models into a single, giant model. To dothis, we use the same “zero trick” that we used in the companion
article (Smith & Kutas, 2015) to combine two intercept-only“classic” ERP models into a single model with dummy coding.
There, this trick was used to replace the binning process, so thatinstead of dividing our events into separate groups before analysis,
the division effectively happened as part of the regression fitting.
Now we’ll replace epoching in a similar manner, giving a regression model that is applied directly to continuous-time EEG.
If we have data for N trials, P predictors, and L distinct latencies, then before, we would fit L different regression models, where
each regression model had entries of the form:

ytrial 1;latency 0 5 b1 xpredictor 1@trial 1 1    1 bP xpredictor P@trial 1⯗
ytrial N;latency 0 5 b1 xpredictor 1@trial N 1    1 bP xpredictor P@trial N
To coalesce these models into one, we have to replace each of our
P predictors xpredictor p by L new predictors xpredictor p, latency l, giving P 3 L predictors in total. The b value for xpredictor p was in themodel for latency l. Since we’ll be fitting our new model on all of
the original ytrial i, latency l data points, without epoching, we have todefine what values these predictors take for each of N 3 L data
points, as follows:(
xpredictor p;latency l1 @trial i;latency l2 5

xpredictor p@trial i ; if l1 5l20;otherwiseThis looks complicated, but all it’s doing is saying in regression
language that, when we want to predict the scalp potential at

100 ms after some time-locking event, we should multiply the predictor values for the event we’re looking at by the b values that are
100 ms into our rERP waveform, and ignore the other b values.
The overall process is shown schematically in Figure 4. By itself,
this transformation has no effect whatsoever: fitting this model
once will give exactly the same results as fitting each of the L original models one at a time. The only differences are that we get our
entire waveforms in one go, instead of having to assemble them
one point at a time, and that our computer will have to work a bit
harder—fitting these models takes a few seconds instead of tens of
milliseconds.
One way to interpret our new, expanded predictors is as interactions between our original per-event predictors, and a second
dummy-coded predictor that is 1 for data points measured at a certain latency from the time-locking event, and 0 otherwise:
xpredictor p;latency l@data point d 5 xpredictor p;event i 3 xlatency to event i is l@data point dLike any dummy coding scheme, this approach allows our model
to make independent estimates for each value of our predictor.
When our predictor is latency, this means we are allowing the
waveform to vary arbitrarily from one time point another, constrained only by the data. Now that our model is written in thisform, though, we can consider using other codings instead. For
example, we could impose a smoothness constraint on our rERP
waveforms by encoding the latency predictor using a spline
basis—and doing exactly this is one component of the generalized
additive modelling approach used by Hendrix (2009), Hendrix et al.
(2014), Kryuchkova et al. (2012), Tremblay (2009), and Tremblay
and Baayen (2010). However, this is not as advantageous as it
might at first appear. Encoding latency with a spline basis is essentially a way of applying noncausal smoothing to our ERP waveforms a priori before estimating them, which is not generally
recommended (Luck, 2005). Like all smoothing, it will introduce
distortions; but, unlike the use of post hoc low-pass filtering, whichallows us to directly compare before and after waveforms, the consequences of spline-based smoothing may be hard to understand,especially when the degree of smoothing is selected by a methodthat is automatic, and thus opaque. And the usual justification for
spline smoothing—that it allows us to pool data from similar measurements, amplifying the signal while reducing the noise—applies

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only weakly here. In EEG recordings, the noise at adjacent time
points may be just as similar as the signal is; to really increase oursignal-to-noise ratio we have to pool data from multiple events thatare more widely separated in time.6 Therefore, for now we recommend sticking with the dummy-coded latency approach describedabove, which is the direct analogue to traditional ERP averaging in
the continuous regression setting.
Overlap as confounding. Once we have combined our regression
models into this expanded form, it’s easy to see how to correct foroverlap. Overlap occurs when a single EEG measurement (y value)
is affected by multiple events—say, event 1 occurred 500 ms
before this measurement, and event 2 occurred 100 ms before thismeasurement. If we assume that the ERPs to event 1 and event 2
sum, then this means that measurement y is

y 5 ERP to event 1500 ms 1 ERP to event 2100 ms 1 noise:
But the ERP to event 1 at 500 ms is simply the b values for 500 ms
multiplied by the predictors for event 1, and similarly for the ERP
to event 2 at 100 ms. To represent this overlap, in the combinedmodel we allow the combined-model predictors for 500 ms to takeon the appropriate values for event 1 at the same time as the
combined-model predictors for 100 ms take on the appropriate values for event 2. More formally, we adjust our predictor definition
like so:

xpredictor p;latency l@data point d
8< xpredictor p@event i ; if d was measured l ms after some event i
5:0;otherwiseThis transforms the problem of disentangling overlapping ERPs
into the problem of disentangling partially confounded predictors
in a regression model, and, as discussed in the companion article(Smith & Kutas, 2015), least squares fitting solves such problems
automatically. In essence, the fitting process will consider all possible b waveforms, and pick the ones that best match the data after
being time-locked to successive events and summed.
Under what circumstances should we expect this to work? We
do not currently have a way to calculate precise variance inflation
factor (VIF) values for continuous-time regression, because to bedone properly this will have to take into account the temporaldependencies in the EEG background noise.7 Nonetheless, the fundamental rule of partial collinearity still holds: the higher the correlations between our predictors, the more data are needed. In
practice, there are two mechanisms that reduce collinearity
between our expanded predictors: variation in ISI, and differences
between stimuli. This is intuitive. If our experiment always uses an
ISI of 500 ms exactly, and our stimuli are identical, then no method
will ever be able to tell whether some component in the EEG
occurs at 100-ms latency to one event versus 600-ms latency toanother, because these latencies always co-occur. In the regression
6. Correlations between the EEG noise at nearby time points will
also tend to thwart automatic smoothness selection methods (Wood,2006), reducing their ability to prevent overfitting.
7. A useful target for further research would be to derive a VIF-likeformula that will predict ahead of time how badly overlap impacts anarbitrary experimental design, as compared to a similar design without
overlap, while taking into account the 1/f structure of the EEG background noise (Smith, 2011).

175
model, this means the predictors xpredictor p, 100 ms and xpredictor p,
600 ms will have identical values for every data point, creating per-fect collinearity (aside from the first and last trials). But if we addsome trials that use different ISIs, then on those trials there will bedata points where one of these predictors is zero and the other isnot, breaking the collinearity. The more variation there is in our
ISI, the less correlated our predictors will be, and the better leastsquares will be at correcting for overlap.
But even if we have a fixed ISI, all is not lost. A fixed ISImeans that the expanded predictors for the 100-ms and 600-ms
latencies will always be zero or nonzero at the same times. But
when they are nonzero, their values are determined by the corresponding events. If our predictor reflects word expectancy, forexample, then the 100-ms predictor will reflect the expectancy for
word n at the same time the 600-ms predictor reflects the expectancy for word n 2 1, and these values will generally be different,
again breaking the perfect collinearity. The extent to which thisform of variability will help, of course, depends on the predictors
in question. If the original predictors are entirely uncorrelated
between nearby events, then our expanded predictors will beorthogonal; in this case, applying overlap correction is harmless
but unnecessary. If the predictors are perfectly correlated, then jitteris our only hope. But if we have partial correlation between nearby
events, then overlap correction can potentially be both effective
and useful even in the absence of ISI variability. This may differ
on a predictor-by-predictor basis within a single model; intercept
terms, for example, never vary across stimuli, and thus require jitterto estimate. But as we saw before, collinearity problems affect only
the collinear predictors,8 and for some questions it may not matter
how poorly estimated the intercept bs are so long as the other bs
are estimated reliably.
Best of all, for overlap correction purposes, is to have as muchvariability as possible in both the ISI and the stimuli. Naturally,
this must be balanced against other experimental considerations,and as long as there is enough variability somewhere, and we haveenough data, then overlap correction can work. In any case, theleast squares fitting process will automatically take full advantageof whatever variability is present, without any need for humanintervention.
Simulation
At least, that’s what the theory says. To verify this, we simulated
three experiments involving high degrees of overlap. In the first, acontinuous train of identical stimuli is presented, with each ISI randomly selected to fall between 200 ms and 400 ms in increments of
16.7 ms (the achievable ISIs for visual presentation using a monitor
with a 60 Hz refresh rate). We time-locked a known ERP waveform (1,100 ms long) to each stimulus, and summed these timeshifted ERPs with phase-shuffled EEG background noise to produce simulated recordings. We then attempted to estimate the ERP
from these simulated data, both with and without overlap correction. In this experiment, each data point has an average of 3.8
simultaneous ERPs overlapping with it.
The second simulated experiment is similar to the first, exceptthat we used a fixed ISI of 300 ms, and now the stimuli themselves
vary along an arbitrary continuous dimension (a stand-in for properties like visual contrast or expectancy), and this variation has alinear effect on the evoked potential:
8. A demonstration on simulated data is given in the supporting
information.

N.J. Smith and M. Kutas

176

Figure 5. First row: An intercept ERP from a simulated experiment with ISIs jittered randomly between 200 and 400 ms. Second row: A slope rERP
from a simulated experiment with fixed 300 ms ISI, and r 5 0.9 trial-to-trial correlation in the simulated stimulus property (measured in arbitrary units,a.u.). Third row: Same as the second row, but with ISIs varying between 200 and 400 ms. Each plot shows estimates from five simulated runs of theexperiment, along with the true (r)ERP for comparison (dotted line).

Stimulus ERPn 5 Intercept rERP
1 Stimulus propertyn 3 Slope rERP
The simulated stimulus properties were selected so that on average
across the experiment they are centered around zero, but on adjacent trials are highly correlated (r 5 0.9); thus, the effect of event n
is confounded with the effect of event n 2 1. This is similar to what
occurs when, for example, trying to estimate the effect of word frequency for word-by-word presentation of naturalistic text (though
our simulation has much higher item-to-item correlation, making
the overlap correction problem more challenging). As discussed
above, with a fixed ISI it is impossible to reliably estimate theintercept rERP, and our estimates of this waveform were highly
noisy (see online supporting information), but the slope rERP in
the same model can be estimated reliably, even with this highdegree of correlation and high degree of overlap.
The third simulated experiment is identical to the second,
except that we replaced the fixed ISI with one that varied between
200–400 ms as in the first experiment.
The results, using 2,000 trials (10 min) of simulated data, areshown in Figure 5. As expected, we find that if we omit overlap
correction our (r)ERP estimates are entirely wrong. With overlap
correction enabled, our method is able to recover the correct waveform even under these rather adverse conditions, so long as jitter or
stimulus variation is present—and having both is best of all. (Ofcourse, 2,000 trials is an unusually large number of trials to present
in 10 min, but the high trial density is exactly what makes this achallenging case for overlap correction.) The supplementary information contains a more detailed set of simulations, showing theintercept rERPs for Experiments 2 and 3, as well as performancefor different data set sizes.
Response-Time Effect Revisited
Our nonlinear analysis of the go trials from the go/no-go data
above gave us reason to suspect that there might be two ERPs

occurring simultaneously: one time-locked to the stimulus, and asecond, overlapping ERP time-locked to the button press response.
Our overlap correction technique allows us to estimate the true
forms of these two ERPs separately, as shown in Figure 6. The two
curves in Figure 6a are chosen by the fitting process so as to maketheir overlapped predictions (Figure 6b) match the actual trial-bytrial data as closely as possible in the least squares sense—compare
Figure 6b to Figure 3a,b. As suspected, we find that the best explanation involves a large response-locked positivity, which begins
somewhat before the button press is registered.
Note that in Figure 6a we use fairly long time windows, with all
curves returning to zero before the edge of the analysis window.
When not using overlap correction, it doesn’t much matter how
large an analysis window one uses; if it’s too short, then it merelymeans that we can’t see all of the curve. But when fitting the overlap correction model, all ERPs are assumed to be exactly zero
everywhere outside of our analysis window. If this is not true, thenany effects outside this window will not be subject to overlap control, and their overlap may contaminate the parts of the waveformthat we do analyze. (This is another example of the principle thatleaving predictors out of one’s model may cause their effects to be
spuriously attributed to whichever predictors remain.) So whenusing overlap correction to estimate waveforms, it’s important to
err on the side of using a long time window.
Validation
This approach to overlap correction treats the brain as a linear
time-invariant (LTI) system; that is, it assumes that the brain
response to seeing two items in quick succession is simply the sumof the brain response to seeing each of those items in isolation,with an appropriate temporal shift to account for time-locking.
Theory and simulations confirm that, if the LTI assumption is true,
then our method is capable of recovering the true underlying ERPs.
But does this assumption accurately describe neural processing?
There are two ways we can respond to this concern. One is toperform various model-checking procedures, such as running the

rERPS IIa177b

Figure 6. a: rERP estimates time-locked separately to stimulus presentation and button-press response, estimated with overlap correction to go trials
from Urbach and Kutas (2008). b: The predictions made by this model. Compare to Figures 3a,b.

same experiment with both short and long ISIs, and checking
whether the estimated rERPs are the same in the two cases, or simply checking whether overlap correction improves our model’s fit.
Burns, Bigdely-Shamlo, Smith, Kreutz-Delgado, and Makeig(2013) performed the latter experiment, analyzing EEG from a task
where images were presented at high-speed (12 per second), andfound that, when performing cross-validation, the overlapcorrected rERPs explained a higher proportion of the variance inthe EEG signal than traditional ERPs from averaging, while using
the same number of parameters. This doesn’t tell us that the LTI
assumption is correct, but it suggests that, for this task, using thisform of overlap correction brings us closer to the truth than using
no overlap correction at all. In addition, studies in fMRI suggestthat the LTI assumption holds approximately for the hemodynamic
response (Boynton, Engel, Glover, & Heeger, 1996; Cohen, 1997;
Dale & Buckner, 1997), and to the extent that the hemodynamic
response reflects the same generators as EEG, this suggests that it
will be a useful approximation in this context as well.
In the long run, though, focusing on whether the brain violates
the LTI assumption is probably not the most productive
approach—it surely does. Instead, we should ask how the brain violates the LTI assumption. The overlap correction method proposed
here should be viewed as a method for determining precisely whichaspects of our data can be explained as arising from “mere overlap”—and thus which aspects of it cannot. If we find that overlapcorrected rERPs estimated from short-ISI and long-ISI conditions
differ, then this would mean that we have successfully learned
something new: that in our task ISI has effects that go above andbeyond mere overlap. And the rERP approach is flexible enough to
let us integrate covariates like ISI or the identity of the previous
item (relevant for priming or habituation effects) directly into ouranalysis of the event-locked waveform, while still correcting foroverlap with waveforms time-locked to other events.
In our go/no-go task, there must be something different about
the neural processing between slow trials and fast trials that makes
them slow or fast. So we expect there to be effects of RT that go
beyond the timing of the response—but previously, it was difficult
to look for these, since the massive positivity time-locked to the
response overwhelms any other effects. Using our model of overlap
correction, though, we can now subtract off the model predictions

from the actual data (producing residuals), and then look to see if
there are any patterns that remain unexplained. Figure 7a shows anonlinear rERP analysis of the residuals from the overlap-correctedrERPs in Figure 6. Two things might strike us here.
First, all the amplitudes are much smaller than before; the overlap correction model has successfully accounted for most of thesystematic variability in this data (or at least, that which can be
captured by a nonlinear rERP). But there are some intriguing regularities in what’s left: at around 200 ms, we see a roughly linear
effect where slow trials are more positive than fast trials (red aboveblue); at around 300 ms, this switches to the opposite pattern (blueabove red). A somewhat crude way to visualize this in more familiar terms is to subtract off only the response-locked rERP as estimated by our model, and then compute ERP estimates fordichotomized RT bins (see Figure 7b). This graph is directly comparable to Figure 1.
Figure 1 showed a RT-related latency shift in the positivity
peaking near 350 ms, and sustained differences between our four
RT bins extending to 800 ms and beyond. Figure 7b shows thatboth of these apparent effects can be accounted for by mere overlap
with a response-locked positivity. But even after correcting for the
response-locked ERP, we see residual amplitude differences
between fast and slow trials in the 150–250 ms and 250–400 ms
time windows, which call for further examination. In addition, we
can now see that the no-go trials in fact deviate substantially fromthe go trials, especially in the 400–700 ms range; previously, they
appeared similar, but this turns out to have been an artifact of the
response-locked positivity in the go trials coincidentally matchinga stimulus-locked positivity in the no-go trials.
But Figure 7b is (in statistical terms) a somewhat naughty
approach, because the response-locked rERP in Figure 6a was calculated on the assumption that the stimulus-locked ERP did notvary with RT; if the stimulus-locked rERP had been allowed tovary, then the response-locked rERP might have come out differently. We’re effectively assuming that all the unexplained variancein our original overlap-corrected model can be attributed to thestimulus-locked ERP, rather than the response. A more theoretically sound method of doing this analysis would be to allow thestimulus-locked rERP to vary in a nonlinear fashion (e.g., by using
a step-function basis), and estimating it simultaneously with the

N.J. Smith and M. Kutas

178
a

b

c

d

Figure 7. a: A nonlinear rERP (cubic spline with 4 df) analysis of the residual amplitudes left by subtracting the overlap model’s predictions (Figure 6b) from the single-trial EEG data. b: ERPs formed by subtracting out Figure 6a’s response-locked rERP (but not the stimulus-locked rERP) fromthe single-trial data, and then averaging across four RT bins. Compare to Figure 1. c: A more sophisticated overlap-corrected rERP model, in which
both stimulus- and response-locked ERPs are assumed to vary linearly with RT. Because the RT predictor was centered, intercept rERPs here estimate
ERP activity on trials with the mean RT (460 ms), while RT slope rERPs estimate how this changes as the RT varies above or below this. Tick marks
on RT slope graphs indicate 1 lV change in scalp amplitude per 100 ms change in RT. d: Nonlinear rERP analysis of residual amplitudes from thismore sophisticated model. Notice that images (a) and (d) share a different color bar from that used in previous figures.

response-locked rERP. This is perfectly doable with our method,
but it turns out that, if we try on these data, the stimulus- andresponse-locked rERPs are too highly collinear to produce usefulestimates. This is unsurprising, since, as Figure 3a shows, thestimulus-locked step-function basis is already able to capture a significant portion of the response-locked effect. One way to deal
with this is to first fit one model—allowing it to capture all the variance that it can, resolving all ambiguities in its favor—and thenexamine what’s left, as we did in Figure 7a (Kryuchkova et al.,2012; Laszlo & Federmeier, 2014). Another would be to gather
more data. And a third is to fit a model that is flexible enough tocapture the effects we care about, but not so flexible as to create
massive collinearity problems. (When interpreting our results, however, we must keep in mind that no data analysis alone, no matter
how clever, can rule out the interpretation where the entire pattern
we observe is due to an extremely complicated nonlinear stimuluslocked ERP—this is the fundamental problem that the collinearity
here is warning us about.)
Fortunately, as shown in Figure 2c, stimulus-locked linear(slope) rERPs are unable to capture the response-locked effect,
which means that we can safely use them in our overlap analysis toanalyze the residual effects of RT without creating too much collinearity. Further support for this strategy comes from our discussionof Figure 7a, in which the major unmodeled patterns appear to beroughly linear. Therefore, we fit a more sophisticated model, in
which both the stimulus- and response-locked ERPs are allowed tovary linearly with RT (Figure 7c–d.9 ) Consistent with the cruder
analysis in Figure 7b, we find that, in the stimulus-locked rERPs,
slower responses are more negative than faster responses at around
200 ms, but from 300–400 ms the pattern reverses. Turning to the
response-locked rERPs, we also now see that faster responses seemto be associated with an overall greater positivity than slower ones.
In a real study, we would want to use statistical tests to confirm
that these patterns were real, or we might continue by examining
the residuals from this more sophisticated model for more unmodeled patterns (Figure 7d). Our goal here is not to provide a definitive analysis of these go/no-go data, but to illustrate how thesemodels can be used and interpreted.
Practical Considerations
Having completed our discussion of the core rERP methods, we
now turn to a number of practical issues that arise when using themas part of a full analysis pipeline.

9. Notice that to compute these rERPs, we have combined one intercept 1 slope rERP analysis for the stimulus-locked events with a second,
separate intercept 1 slope rERP analysis for the response-locked events.
This is accomplished by writing down the standard predictors for eachof these two models, and then using the zero trick yet again to combine
the two into a single model, which we fit using overlap correction as
described above.

rERPS II
Baselining and Temporal Filtering
Baselining corrects for drift in the EEG signal by subtracting off
the average potential measured over some prestimulus baseline
period; temporal filtering smoothes away high-frequency noise that
may be known to be irrelevant to some analysis of interest. Bothare commonly used in ERP research. When working with traditional ERPs, these operations can be applied either to the individual
epochs before averaging, or to the estimated ERP waveforms after
averaging. It turns out that with averaging-based ERPs the results
come out identical whichever order you use (Luck, 2005). Therefore, it’s common to do averaging first—it’s faster to baseline orfilter an averaged waveform than many individual epochs, and,because filtering in particular has the potential to create harmful
distortions, saving it for the end is useful because it allows one totry multiple filter settings, and apply it only where necessary (e.g.,
for display, but not statistical analysis; Luck, 2005).
Remarkably, in this respect, least squares rERPs estimated without overlap correction turn out to act just like those derived byaveraging: you can baseline or filter either your original epochs or
your rERP waveforms, and you get the same results either way (seesupporting information for proof). Therefore, the same recommendations apply.
When using overlap correction, the equivalence theorem does
not hold, but there still doesn’t seem to be any reason to apply
baselining or filtering first. (And since the continuous-time regression model replaces epoching, it’s not clear how baselining before
regressing would even work.) The simplest and safest approach still
seems to be to apply these to estimated rERP waveforms only.
Baselining makes just as much sense for slope and nonlinear
rERPs as it does for categorical rERPs; these estimates are just as
likely to be thrown off by outliers caused by low-frequency drift.
Spatial Filtering
A number of useful transformations on EEG data can be interpreted
as the application of linear spatial filters: spatial principal component analysis (PCA) or independent component analysis (ICA)
decomposition, rereferencing, the Laplacian transformation, someforms of blink correction, etc. Similar to baselining and filtering,these have an identical effect whether they are applied to EEG data
before regressing or afterwards to derived rERP waveforms. Unlike
baselining and temporal filtering, this is true regardless of whether
you use overlap correction. (See supporting information.)
Significance Testing
After estimating an ERP waveform, we must determine whether
the patterns we think we see are real, or whether we’re just looking
at noise and fooling ourselves. Thus, we need ways to perform statistical significance testing.
Regression models come with a rich set of tools for testing
hypotheses about b coefficients and combinations of b coefficients,
and in rERP analysis, all our estimated and predicted waveforms
are b coefficients and combinations of b coefficients. In principle,
therefore, we should be able to use our rERP model to directly testnull hypotheses like “at least one point within this latency window
is nonzero.” But, the most straightforward way of applying thesestatistical tests to rERP analyses turns out not to work. Leastsquares regression itself is an unbiased technique regardless of how
our noise is distributed. But the textbook statistical tests for regression models require much stronger assumptions, including in par-179
ticular the assumption that noise is uncorrelated across data points;
that is, they require that knowing the amplitude of the EEG background activity at time t tells you nothing about its amplitude attime t 1 1. This is very far from true, and thus more work will beneeded before we can use rERP models to directly derive trustworthy parametric p values for activity spread over multiple time
points (Smith, 2011). Another natural extension to the approachdescribed here would be to estimate rERP waveforms using mixed
effects regression to handle crossed random effects of both items
and participants (Baayen, Davidson, & Bates, 2008; Clark, 1973),
but this runs into a similar problem: these methods require distributional assumptions about how waveforms vary across items andparticipants, and understanding what assumptions are appropriate
will require further investigation.
In the meantime, the traditional methods for testing significanceof features in ERP waveforms remain just as valid for rERPs. Specifically, we can estimate per-participant rERPs, extract some feature from each waveform—such as average amplitude over awindow, or peak latency—and then submit it to an analysis of variance (ANOVA) across participants. The waveforms used can be
individual b waveforms, or combined prediction waveforms. Thiscan handle arbitrary rERP designs (including those with overlap
correction) and arbitrary features. It cannot account for crossed random effects, but neither can current ERP analysis methods.
In the special case where one wants to test mean amplitude overa latency window in an rERP design and is not using overlap correction, then there are three approaches that might seemreasonable:
1. As above, calculate rERP waveforms for each participant andelectrode, baseline them, compute mean rERP amplitude over
the window, then enter these into an ANOVA.
2. On a trial-by-trial basis within each participant and electrode,baseline each trial, calculate mean amplitude over the window,
then perform a repeated measures regression across participantsand electrodes (Dambacher, Kliegl, Hofmann, & Jacobs, 2006;
Lorch & Myers, 1990).
3. On a trial-by-trial basis within each participant and electrode,baseline each trial, calculate mean amplitude over the window,
then submit these to a mixed effects regression (Amsel, 2011;
Frank, Otten, Galli, & Vigliocco, 2013).
Options 1 and 2 turn out to be mathematically identical (see supporting information). Options 2 and 3 are conceptually very similar: repeated measures regression/ANCOVA (analysis ofcovariance) and mixed effects models are two different statisticaltechniques for answering the same questions, with the latter being
more modern, powerful, and prone to complications (Barr, Levy,
Scheepers, & Tily, 2013). And, in this case, using only single data
points derived from each baselined epoch minimizes the problems
described above involving waveform covariation and correlationsin the background noise. Thus, if handling crossed random effects
is important and appropriate sphericity correction is available, there
may be cases where it makes sense to use least squares to estimaterERPs for plotting but then use Option 3 for testing. In all othercases, the best bet for the moment is to stick to the traditional
Option 1.
Artifact Rejection
A common problem in EEG analysis is the presence of artifacts
such as blinks or muscle activity, which create large bursts of noisethat may contaminate ERP/rERP estimates. Thus, it’s common toN.J. Smith and M. Kutas

180

Figure 8. A schematic depiction of the challenge when combining artifact rejection and overlap correction. Top: EEG data we measured. Bottom: ERPs embedded in this EEG that we are trying to reconstruct.
When the data contains an (uncorrectable) artifact, we must reject theartifact from our analysis. But if we reject event 2 as a whole, we mustalso reject the portions of its data that overlap with other events.

use some sort of method to identify and discard data containing
such artifacts. When not using overlap correction, rERP analysis
takes epoched data as input, and we can apply artifact rejection tothese epochs just as we do with traditional averaging. When using
overlap correction, things are more complicated.
Consider the situation depicted in Figure 8, where one epoch
contains an artifact and also overlaps with other artifact-freeepochs. The usual way of handling this in traditional ERP analysis
would be to drop event 2, while keeping events 1 and 3. But wecan’t do that here; if we analyzed this data as if event 2 had notoccurred, then it would cause event 2’s effects in the marked
regions to contaminate our analysis of events 1 and 3. (This is
another example of how dropping a control variable creates inconsistent estimates.) Traditionally, rejecting an event and rejecting its
data are the same thing. Here, the mapping between events anddata is not so simple, and artifact rejection must be applied to data,
not to events.
One option is to reject just the portion of the data where the artifact actually occurs (the red box in Figure 8), and continue to analyze the remainder of event 2’s epoch, along with all of the epochs
for events 1 and 3. Technically, this is easy to do; when using thecontinuous-time regression model for overlap correction, every
data point enters as its own row in the regression, so we can rejectan arbitrary subset of data points by simply dropping them fromthe regression. However, some might find it bothersome to reject
part of an epoch while keeping the rest, and it may cause difficulties for baseline correction if different portions of our waveformare estimated from different subsets of our data and thus exposed
to different levels of drift. If we require every epoch to either be inor out as a whole, then we must reject all of event 2’s epoch, andall of the epochs that overlap it, and all of the epochs that overlap
those, and so on. In some cases, like our go/no-go data, this is notbothersome, since we never have more than two events overlapping. In others, for example, a sentence-reading study where events
are successive words presented in rapid serial visual presentation(RSVP), it would be unfortunate to have to throw out the entiretyof every sentence that contains a blink anywhere within it, and wemight prefer to reject partial epochs instead.
When working with data sets or software packages not originally intended for use with overlap correction, we may be able todetermine which epochs contain artifacts, but not where within theepoch they occur. In this case, the best available option is probably
to treat the whole epoch as if it were artifactual, and reject all thedata it contains from all the epochs that it overlaps.
In general, these issues mean that the use of continuous regression will benefit from the use of artifact correction techniques suchas blink correction (Joyce, Gorodnitsky, & Kutas, 2004) or ICA(Jung et al., 2000), the adoption of regression methods that are
more robust to outliers (Maronna, Martin, & Yohai, 2006) and thus
depend less on explicit artifact rejection, and the development oftechniques for precisely detecting the location and extent of artifacts in continuous, nonepoched data.
Conclusion
Smith & Kutas (2015) introduced the fundamentals of the rERP
framework, examined its relation to classic ERP averaging, andreviewed the theoretical results that guide us in considering tradeoffs between different designs. Here, we’ve built on this foundation, and found that, in addition to handling classic ERP designs,
the rERP framework straightforwardly generalizes to even complex
analyses involving nonlinear effects and overlapping brain
responses to temporally adjacent events. It also—with some minor
adjustments—fits naturally into a traditional ERP analysis pipeline.
In the process, we’ve found that the concepts introduced to understand trade-offs in simple linear designs—like the effects of partial
and perfect collinearity—are also sufficient to understand evencomplicated situations—like the interplay between overlap correction and nonlinear effects in a combined analysis. rERP analysis
unifies classic ERPs, difference ERPs, linear slope ERPs, dichotomization, ERP images, single-trial analysis of mean window amplitudes, and overlap correction into a single flexible andconceptually coherent system.
A list of free software packages implementing these methods isavailable at http://vorpus.org/rERP.

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Supporting Information
Additional Supporting Information may be found in the online
verson of this article:
Appendix S1: Detailed simulations of overlap correction
method.
Appendix S2: Derivations for baselining, filtering, windowing.
Figure S1: Intercept-only model, with jitter.
Figure S2: Intercept 1 slope model, no jitter.
Figure S3: Intercept 1 slope model, with jitter.
Table S1: Root mean squared error between estimated rERP
and true rERP.

Supplementary information for Regression-based
estimation of ERP waveforms: II. Non-linear effects,
overlap correction, and practical considerations
Nathaniel J. Smith and Marta Kutas

S1

Detailed simulations of overlap correction method

The main text describes three simulated experiments used to test the overlap correction technique.
In the first (Intercept-only, with jitter), a series of identical stimuli are presented, with ISI jittered
between 200 and 400 ms in 16.7 ms intervals. In the second (Intercept + slope, no jitter), a fixed
300 ms ISI is used, and the ERP response to the stimuli varies according to a linear model:
Stimulus ERPn = Intercept rERP + Stimulus propertyn × Slope rERP
The stimulus properties are sampled from an AR(1) model:
Stimulus propertyn = 0.9 × Stimulus propertyn−1 + Normal(µ = 0, σ = 1)
The resulting stimulus properties have a mean value of zero, and strong trial-to-trial correlations;the correlation between trial n and trial n + k is r = 0.9k . The third experiment (Intercept + slope,with jitter), combines these manipulations; it uses the slope rERP model of experiment 2 and thejittered stimulus presentation schedule of experiment 1.
Fig. S1-S3 show the results of analyzing these simulated experiments on varying amounts ofstimuli, both with and without overlap correction. Table S1 reports the root mean squared error(RMSE) between the estimated rERP and the true rERP for each of these conditions. To visualize
sampling noise, the figures show results from five randomly selected simulation runs. RMSE
calculations are based on estimates from 10,000 simulation runs. All waveforms were baselined
before plotting or RMSE calculation.
In the Intercept + slope, no jitter condition (Fig. S2) we see that, as discussed in the main text,
the intercept term cannot be reliably estimated with overlap correction enabled; we get extremely
noisy results. But these waveforms are not entirely arbitrary. What’s happening is, essentially, thatthe model knows very well what the steady-state response ought to look like, because it has plentyof data showing this. Then out of all the waveforms that, when overlapped, produce the samesteady-state response, it chooses the ones which best match the first and last trials in our simulated
experiment, since these are the only trials that are observed with only partial overlap. So while themodel can identify the steady-state response, when it comes to uniquely picking out which of the
waveforms that correspond to this steady-state response was actually present, we effectively havean N of ∼1. In the companion article, we discussed how removing confounded predictors from1

Data
TrialsTime (min)

100

0.5

500

2.5

2000

10

No overlap correction

5 µV

Truth

With overlap correction

5 µV

400 ms

400 msFigure S1: Intercept-only model, with jitter: Comparison between estimated waveforms from fivesimulated runs, and true waveforms.
an analysis may increase your power, but it does this by increasing your power to detect spurious
effects. This is an example: when not using overlap corection, the intercept rERP is stable andreliably wrong. When using overlap correction (and thus correcting for the confounding effects
of temporally adjacent events), the rERP estimate is highly variable, thus correctly and reliably
indicating that the true underlying rERP cannot be estimated from this data.
Within this simulation, the intercept and slope rERPs are estimated simultaneously together
within each model fit. Notice how these fits produce correct estimates of the slope rERPs, evenwhile failing to estimate the intercept rERP. This gives an extreme example of how collinearity
only affects estimates for the collinear predictors, and does not ‘infect’ other predictors within thesame model.
Finally, notice that while our technique can use stimulus variability to successfully estimate theslope rERP even with a fixed ISI (Fig. S2), it does even better when it has both stimulus variability
and jitter together (Fig. S3; see also Table S1, where in the right column, the with jitter values are
systematically lower than no jitter values). This demonstrates how the least-squares fitting process
automatically integrates all data available.

S2

Derivations for baselining, filtering, windowing

Here we give the mathematical derivations which show that many standard operations — including
baselining, temporal and spatial filtering, and taking averages over latency windows — can be
performed either on raw EEG before using least-squares regression to estimate rERPs, or else
2

3
5 µV
400 msIntercept rERP

5 µV/a.u.
400 msSlope rERP

No overlap correction

5 µV

400 msIntercept rERP

5 µV/a.u.
400 msSlope rERP

With overlap correction

Figure S2: Intercept + slope model, no jitter: Comparison between estimated waveforms from five simulated runs, and true waveforms.

Truth

2.5

500

10

0.5

100

2000

Time (min)Trials

Data

4
5 µV
400 msIntercept rERP

5 µV/a.u.
400 msSlope rERP

No overlap correction

5 µV

400 msIntercept rERP

5 µV/a.u.
400 msSlope rERP

With overlap correction

Figure S3: Intercept + slope model, with jitter: Comparison between estimated waveforms from five simulated runs and true
waveforms.

Truth

2.5

500

10

0.5

100

2000

Time (min)TrialsData

Table S1Root mean squared error between estimated rERP and true rERP
No overlap correction
TrialsTime (min)Intercept rERP

Slope rERP

With overlap correction
Intercept rERP

Slope rERPIntercept-only model, with jitter (Fig. S1)
100
500
2000

0.5
2.5
10

5.22
5.13
5.09

5.34
1.79
0.78

Intercept + slope model, no jitter (Fig. S2)
100
500
2000

0.5
2.5
10

5.87
5.76
5.73

2.35
2.19
2.18

23.20
21.31
25.80

2.79
1.23
0.79

Intercept + slope model, with jitter (Fig. S3)
100
0.5
5.25
2.15
5.14
1.99
500
2.5
2000
10
5.09
1.99
Note. Darker shading corresponds to larger error values.

5.71
1.80
0.81

1.83
0.70
0.43

on the resulting rERPs themselves, and the exact same result will be obtained either way. This
theorem underlies the discussion of baselining and filtering in the main text. It also justifies theequivalence between ‘option 1’ and ‘option 2’ in the discussion of statistical significance testing,because repeated-measures regression works by first running a standard regression model acrossmean window amplitudes, and then submitting these to an ANOVA.
This is most clear if we switch to matrix notation for writing our regression model. Instead ofyi = β1 x1i + β2 x2i + · · · + noisei
we can write the mathematically identical expression
β + noise.
y = Xβ(∗)

Here we use the convention that column vectors are written in bold, while matrices are written inuppercase; the vectors are
y =< y1 , . . . , yn >T
β =< β1 , . . . , βn >Tnoise =< noise1 , . . . , noisen >T
and X is the design matrix, i.e., a matrix formed by putting the values of x1i into the first column,
x2i into the second, and so on:
Xi j = x ji .
5The least-squares solution to equation (∗) can now be expressed using the so-called ‘normal equations’:βˆ = (X T X)−1 X T y.
(Here the hat on the βˆ is to remind us that this value is our estimate of β , and probably has adifferent value than the actual β that generated our data.)
One useful thing about this notation is that if we are performing m regressions with the same
predictors but different data, then we can combine these into a single formula. Each regression has
a different column vector of data y . If we combine these to be the columns of a single large matrix
Y , then we can write
Y = XB + Noise.
where B (the uppercase form of β ) and Noise are now matrices. There’s nothing particularly deep
about combining the regression models in this way; it’s basically just a notational trick, where
we use the standard rules of linear algebra to express that we want to fit m regression models inparallel. And here, too, we can write a single formula for the least-squares solution to all of these
regressions at once:B̂ = (X T X)−1 X T Y.
Again, if you work through the matrix multiplications, this is just saying that the single-regression
equation above should be applied to each column of Y separately.
But this final expression is very useful when it comes to thinking about how the rERP is affectedby these operations. Recall that y =< y1 , . . . , yn >T are the values of the EEG at a single electrodeand latency but on different trials. Within a single electrode we have one y vector for each latency,
and we can collect these together to make a single Y matrix for that electrode. If we order thesecolumn vectors by latency then — this is critical — each row of Y will contain the raw EEG datafrom a single epoch.
Baselining, temporal filtering, and taking the average over a window all have a critical attributein common: they transform each epoch of data by applying a linear transformation to that epochalone. So they take our matrix Y , and replace each row by a linear transformation of that row. Thatmeans that they can be written as a right-multiplication of our Y matrix by another matrix. For
averaging over a window containing k points, this matrix would have a single column, with entries< 0, . . . , 0, 1/k, . . . , 1/k, 0, . . . , 0 >T . We can construct the matrix that performs baseline correction
by starting with a matrix that has n columns, each of which computes the average of the baseline
window, and subtracting it off from an n × n identity matrix. And filtering, which operates by replacing each point in an epoch with some linear combination of the points in that epoch, would use
a matrix in which each column contained a shifted copy of the filter’s impulse response function.
In any case, let’s call the matrix which performs the desired operation F.
If we first perform this transformation on the raw data and then enter these transformed values— Y F, in matrix notation — into a regression, we get a matrix of new β values, which we denote
B̂F :B̂F = (X T X)−1 X T (Y F).
6

Alternatively, we could first compute the matrix of rERP values, B̂, and then apply our transformation to these β values directly, which in matrix notation would be written B̂F.
So the claim is that it doesn’t matter whether we average/filter our raw data, or instead average/filter our resulting rERP; that is, we claim that B̂F = B̂F. But this is now easy to see, because
matrix multiplication is associative, which allows us to rearrange parentheses:B̂F = (X T X)−1 X T (Y F) = (X T X)−1 X T Y F = B̂F.
If we are using the continuous-time regression model necessary for overlap correction (andactually using it in a non-trivial way, i.e., not just doing an analysis that could be equally welldone by epoched regression), then the above proof does not go through, because now all the time
points from each epoch go into a single column of Y . However, even in this case, we still regresseach electrode separately. Therefore, the above proof still applies to spatial linear transformations,which can be treated as right multiplication of our Y matrix of data, where each column in Y hasdata from a single electrode.

7