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Application of Linear Mixed-Effects Models in Human Neuroscience Research: A Comparison with Pearson Correlation in Two Auditory Electrophysiology Studies

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Neurophysiological studies are often designed to examine relationships between measures from different testing conditions, time points, or analysis techniques within the same group of participants. Appropriate statistical techniques that can take into account repeated measures and multivariate predictor variables are integral and essential to successful data analysis and interpretation. This work implements and compares conventional Pearson correlations and linear mixed-effects (LME) regression models using data from two recently published auditory electrophysiology studies. For the specific research questions in both studies, the Pearson correlation test is inappropriate for determining strengths between the behavioral responses for speech-in-noise recognition and the multiple neurophysiological measures as the neural responses across listening conditions were simply treated as independent measures. In contrast, the LME models allow a systematic approach to incorporate both fixed-effect and random-effect terms to deal with the categorical grouping factor of listening conditions, between-subject baseline differences in the multiple measures, and the correlational structure among the predictor variables. Together, the comparative data demonstrate the advantages as well as the necessity to apply mixed-effects models to properly account for the built-in relationships among the multiple predictor variables, which has important implications for proper statistical modeling and interpretation of human behavior in terms of neural correlates and biomarkers.
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brain
sciences
Article
Application of Linear Mixed-Effects Models in
Human Neuroscience Research: A Comparison with
Pearson Correlation in Two Auditory
Electrophysiology Studies
Tess K. Koerner 1and Yang Zhang 1,2,3,4,*
1
Department of Speech-Language-Hearing Sciences, University of Minnesota, Minneapolis, MN 55455, USA;
koern030@umn.edu
2Center for Neurobehavioral Development, University of Minnesota, Minneapolis, MN 55455, USA
3Center for Applied and Translational Sensory Science, University of Minnesota, Minneapolis,
MN 55455, USA
4Speech-Language-Hearing Center, School of Foreign Languages, Shanghai Jiao Tong University,
Shanghai 200240, China
*Correspondence: zhanglab@umn.edu; Tel.: +1-612-624-7878
Academic Editor: Heather Bortfeld
Received: 31 December 2016; Accepted: 24 February 2017; Published: 27 February 2017
Abstract:
Neurophysiological studies are often designed to examine relationships between measures
from different testing conditions, time points, or analysis techniques within the same group of
participants. Appropriate statistical techniques that can take into account repeated measures
and multivariate predictor variables are integral and essential to successful data analysis and
interpretation. This work implements and compares conventional Pearson correlations and
linear mixed-effects (LME) regression models using data from two recently published auditory
electrophysiology studies. For the specific research questions in both studies, the Pearson correlation
test is inappropriate for determining strengths between the behavioral responses for speech-in-noise
recognition and the multiple neurophysiological measures as the neural responses across listening
conditions were simply treated as independent measures. In contrast, the LME models allow a
systematic approach to incorporate both fixed-effect and random-effect terms to deal with the
categorical grouping factor of listening conditions, between-subject baseline differences in the
multiple measures, and the correlational structure among the predictor variables. Together, the
comparative data demonstrate the advantages as well as the necessity to apply mixed-effects models
to properly account for the built-in relationships among the multiple predictor variables, which has
important implications for proper statistical modeling and interpretation of human behavior in terms
of neural correlates and biomarkers.
Keywords:
Pearson correlation; linear mixed-effects regression models; repeated measures;
neurophysiology; event-related potential
1. Introduction
Cognitive neuroscience research aims to explore relationships between various neural and
behavioral measures to examine the underlying peripheral/central neural mechanisms in various
testing conditions and subject populations. For this purpose, the bivariate Pearson correlation analysis
is commonly used to examine the strength of the linear relationship between two continuous variables
of interest, which can be graphically represented by fitting a least-squares regression line in a scatter
plot [
1
,
2
]. If the variables do not represent continuous data or if the relationship between the two
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Brain Sci. 2017,7, 26 2 of 11
variables is non-linear, other types of bivariate correlation tests such as Spearman or Point-Biserial
correlations can be used. However, when a study involves multivariate data, the conventional
correlation method only allows for the examination of one predictor and one outcome variable at
a time. Even if the Pearson correlation results are adjusted for multiple comparisons or a simple
multiple regression model is applied, the statistical treatment may not take into account the complex
relationships and categorical grouping terms that likely exist in the multiple within-subject predictor
variables [2].
In consideration of the violation of the assumed sample independence required of bivariate
Pearson correlations and the like, researchers have long argued for the necessity to apply more
sophisticated statistical techniques to handle repeated measures from the same subjects [
3
5
]. The
use of mixed-effects (or multilevel) models has recently captured attention in longitudinal medical
research [
6
14
], behavioral and social sciences research [
15
19
] (including speech and hearing
research [
20
41
]), and neurophysiological and neuroimaging research [
42
52
]. Its increasing popularity
is shown in the exponential growth over the last three decades in the number of publications in the
scientific literature (Figure 1).
Brain Sci. 2017, 7, 26 2 of 11
the two variables is non-linear, other types of bivariate correlation tests such as Spearman or Point-
Biserial correlations can be used. However, when a study involves multivariate data, the
conventional correlation method only allows for the examination of one predictor and one outcome
variable at a time. Even if the Pearson correlation results are adjusted for multiple comparisons or a
simple multiple regression model is applied, the statistical treatment may not take into account the
complex relationships and categorical grouping terms that likely exist in the multiple within-subject
predictor variables [2].
In consideration of the violation of the assumed sample independence required of bivariate
Pearson correlations and the like, researchers have long argued for the necessity to apply more
sophisticated statistical techniques to handle repeated measures from the same subjects [3–5]. The
use of mixed-effects (or multilevel) models has recently captured attention in longitudinal medical
research [6–14], behavioral and social sciences research [15–19] (including speech and hearing
research [20–41]), and neurophysiological and neuroimaging research [42–52]. Its increasing
popularity is shown in the exponential growth over the last three decades in the number of
publications in the scientific literature (Figure 1).
Figure 1. Number of publication documents (including original articles and reviews) from 1951 to
2016 that contain the keyword “linear mixed-effects model.” Literature search was conducted with
Elsevier’s Scopus database [53].
Data analysis using mixed-effects regression models allows for the examination of how multiple
variables predict an outcome measure of interest beyond what a simple multiple regression model
can handle [2–5]. In addition to the fixed effec ts in a conven tional mul tiple regr ession model, a mixed-
effects model includes random effects associated with individual experimental units that have prior
distributions. Thus mixed-effects models are able to represent the covariance structure that is
inherent in the experimental design. In particular, the linear and generalized linear mixed-effects
models (LME or GLME), as implemented in popular software packages such as R, prove to be a
powerful tool that allows researchers to examine the effects of several predictor variables (or fixed
effects) and their interactions on a particular outcome variable while taking into account grouping
factors and the existing covariance structure in the repeated measures data. For instance, adding
research participants as a random effect in a LME model allows investigators to resolve the issue of
independence among repeated measures by controlling for individual variation among participants.
Essentially, the inclusion of subject as a random effect in the model assumes that each participant has
a unique intercept, or “baseline”, for each variable. Linear mixed-effects models also allow for an
understanding of how changes in an individual predictor variable, among other co-existing variables,
impact the outcome measure. These regression coefficients provide more detailed information about
relationships among predictors and outcome variables than Pearson correlation coefficients as the
Pearson correlation coefficient simply measures the strength of the linear relationship between each
selected pair of variables independent of the others. Additionally, driven by the research questions
and the nature of the independent and dependent variables, researchers can build and compare LME
models differing in complexity to best summarize findings. Many possibilities regarding appropriate
types of models, necessary data transformations to achieve linearity for each variable, and the
inclusion of interaction terms as well as random slopes or intercepts can be considered.
Figure 1.
Number of publication documents (including original articles and reviews) from 1951 to
2016 that contain the keyword “linear mixed-effects model”. Literature search was conducted with
Elsevier’s Scopus database [53].
Data analysis using mixed-effects regression models allows for the examination of how multiple
variables predict an outcome measure of interest beyond what a simple multiple regression model
can handle [
2
5
]. In addition to the fixed effects in a conventional multiple regression model, a
mixed-effects model includes random effects associated with individual experimental units that have
prior distributions. Thus mixed-effects models are able to represent the covariance structure that is
inherent in the experimental design. In particular, the linear and generalized linear mixed-effects
models (LME or GLME), as implemented in popular software packages such as R, prove to be a
powerful tool that allows researchers to examine the effects of several predictor variables (or fixed
effects) and their interactions on a particular outcome variable while taking into account grouping
factors and the existing covariance structure in the repeated measures data. For instance, adding
research participants as a random effect in a LME model allows investigators to resolve the issue of
independence among repeated measures by controlling for individual variation among participants.
Essentially, the inclusion of subject as a random effect in the model assumes that each participant has
a unique intercept, or “baseline”, for each variable. Linear mixed-effects models also allow for an
understanding of how changes in an individual predictor variable, among other co-existing variables,
impact the outcome measure. These regression coefficients provide more detailed information about
relationships among predictors and outcome variables than Pearson correlation coefficients as the
Pearson correlation coefficient simply measures the strength of the linear relationship between each
selected pair of variables independent of the others. Additionally, driven by the research questions
and the nature of the independent and dependent variables, researchers can build and compare LME
Brain Sci. 2017,7, 26 3 of 11
models differing in complexity to best summarize findings. Many possibilities regarding appropriate
types of models, necessary data transformations to achieve linearity for each variable, and the inclusion
of interaction terms as well as random slopes or intercepts can be considered.
Despite the wide acceptance of the LME method and similar approaches for multivariate data
analysis, researchers do not necessarily take into account the differences between Pearson correlation
and LME models for proper statistical treatment of their data. The current report of side-by-side
comparison was propelled by the successive publication of two recent studies from our lab that
respectively used conventional Pearson correlations and the more sophisticated linear mixed-effects
regression models. In particular, our first study investigated whether noised-induced trial-by-trial
changes in cortical oscillatory rhythms in the ongoing auditory electroencephalography (EEG) signal
could account for the basic evoked response components in the averaged event-related potential (ERP)
waveforms for speech stimuli in quiet and noisy listening conditions [
54
]. When the first study was
submitted, we were not aware of the importance and relevance of the LME approach to the analysis of
our data set. Even though the paper went through two rounds of revisions, the two anonymous peer
reviewers did not raise any concerns for the use of Pearson correlation in our analysis. Our second
study further examined whether the noise-induced changes in trial-by-trial neural phase locking,
as measured by inter-trial phase coherence (ITPC) and spectral EEG power, could predict averaged
mismatch negativity (MMN) responses for detecting a consonant change and a vowel change and
whether the cortical MMN response itself could predict speech perception in noise at both the syllable
and sentence levels [
54
]. In the publication process of the second study, reviewers questioned the
validity of the Pearson correlation analysis for the multiple measures for the same speech stimuli
from the same group of subjects, which led to a major revision adopting the LME regression analysis.
In hindsight, as the trial-by-trial oscillations and the averaged ERPs are different analysis techniques
applied to the same EEG signal, it would have been appropriate to choose the LME models to report
the statistical results in our first publication.
As these two previous publications in auditory neuroscience reported only correlation results
using one statistical approach, a direct comparison of both the Pearson correlation and LME approaches
can be helpful to highlight the differences in the statistical results. Although our examples here are
exclusively focused on speech perception research, the informative comparisons of the statistical
results are presented as a further development to advocate for proper implementation of statistical
modeling and interpretation of multivariate data analysis in future studies of cognitive neuroscience
and experimental psychology.
2. Study 1
Koerner and Zhang [
54
] aimed to determine whether noise-induced changes in trial-by-trial
neural synchrony in delta (0.5–4 Hz), theta (4–8 Hz), and alpha (8–12 Hz) frequency bands in response
to the syllable /bu/ in quiet and in speech babble background noise at a
3 dB SNR (signal-to-noise
ratio) were predictive of variation in the N1–P2 ERPs across participants.
2.1. Statistical Methods
In the published data [
54
], Pearson correlations were used to examine the strength of linear
relationships between ITPC and the N1–P2 amplitude and latency measures pooled across the two
listening conditions for each participant and frequency band, resulting in 12 correlations. The reported
p-values were adjusted for multiple comparisons. Prior to this analysis, scatterplots were used to check
the linearity of each pair of continuous variables. Separate repeated measures analysis of variance
(ANOVA) were also used to examine the effects of background noise on ITPC and N1–P2 latency and
amplitude measures. The ITPC values ranged from 0 to 1, where 1 represents perfect synchronization
across trials and 0 represents absolutely no synchronization across trials. Resulting p-values were
adjusted for multiple comparisons. For the current comparative report, linear mixed-effects models
were developed using R [
55
] and the nlme package [
56
]. Participants were used as a “by-subject”
Brain Sci. 2017,7, 26 4 of 11
random effect and listening condition (quiet vs. noise) was included as a blocking variable in each
linear mixed-effect model. ITPC values at time points associated with the N1 and P2 responses in
delta, theta, and alpha frequency bands were included as fixed effects. For each Pearson correlation
and linear mixed-effects model, the significance of each variable in predicting behavioral performance
was assessed with the significance level at 0.05.
2.2. Results
Koerner and Zhang [
54
] provided detailed results from repeated measures ANOVAs and the
Pearson correlations (see replicated Table 1for summary of correlation coefficients). The repeated
measures ANOVA revealed significant noise-induced delays in N1 (F(1, 10) = 53.71, p< 0.001) and P2
(F(1, 10) = 22.27, p< 0.001) latency as well as a significant reduction in N1 amplitude (
F(1, 10) = 13.85
,
p< 0.01
). Additionally, the repeated measures ANOVA revealed significant noise-induced reductions
in ITPC for N1 in delta (F(1, 10) = 20.68, p< 0.01), theta (F(1, 10) = 18.51, p< 0.01), and alpha
(
F(1, 10) = 23.45
,p< 0.001) frequency bands as well as for P2 in delta (F(1, 10) = 13.27, p< 0.01), theta
(F(1, 10) = 14.86, p< 0.01), and alpha (F(1, 10) = 14.57, p< 0.001) frequency bands.
Results from the Pearson correlation tests showed that ITPC was significantly correlated with
N1 latency in delta (r=
0.586, p< 0.01), theta (r=
0.521, p< 0.05), and alpha (r=
0.510, p< 0.05)
frequency bands. Similarly, significant correlations were found between ITPC and N1 amplitude in
delta (r= 0.780, p< 0.001), theta (r=
0.765, p< 0.001), and alpha (r=
0.720, p< 0.001) frequency
bands. Correlational analysis also revealed significant correlations between ITPC and P2 latency in
delta (r=
0.468, p< 0.05), theta (r=
0.575, p< 0.01), and alpha (r=
0.586, p< 0.01) frequency bands
as well as between ITPC and P2 amplitude in delta (r= 0.666, p< 0.01), theta (r= 0.612, p< 0.01), and
alpha (r= 0.599, p< 0.01) frequency bands.
Table 1.
Correlation coefficients for relationship between-phase locking values and N1 and P2 latency
and amplitude values in response to the CV syllable /bu/ at electrode Cz as reported in Koerner and
Zhang [54].
N1 P2
Frequency Band Latency Amplitude Latency Amplitude
Delta 0.586 ** 0.780 *** 0.468 * 0.666 **
Theta 0.521 * 0.765 *** 0.575 ** 0.612 **
Alpha 0.510 * 0.720 *** 0.586 ** 0.599 **
*** p< 0.001; ** p< 0.01; * p< 0.05.
Results from the linear mixed-effects models showed that ITPC in the delta frequency band was a
significant predictor of N1 (F(1, 7) = 16.12, p< 0.01) and P2 amplitude (F(1, 7) = 10.72, p< 0.05) across
listening conditions. Neural synchrony in the alpha frequency band was a significant predictor of
N1 latency (F(1, 7) = 12.51, p< 0.05) across listening conditions. Potential interaction effects were
statistically nonsignificant when examined in a full LME model and were therefore removed from the
report. An examination of regression coefficients allows for an interpretation of how each fixed effect
is related to the outcome measure of interest. For example, a one-point decrease in ITPC in the delta
frequency band is associated with a 1.05 unit increase in the N1 amplitude (see Table 2for a summary
of F-statistics and correlation coefficients (B)). The residual plots from each linear mixed-effects model
were normally distributed and did not reveal heteroscedasticity or significant trends. Therefore, it is
not expected that generalized linear models would provide better results.
Brain Sci. 2017,7, 26 5 of 11
Table 2.
F-statistics and regression coefficients (
β
) for each fixed effect from linear mixed-effects
regression models for N1–P2 latencies and amplitudes.
Variable N1 Latency N1 Amplitude P2 Latency P2 Amplitude
FβFβFβFβ
Intercept 964.79 *** - 155.62 *** - 568.62 *** - 31.64 *** -
Condition 106.88 *** - 16.58 ** - 31.93 *** - 4.13 -
Delta 0.06 0.30 16.12 ** 1.05 0.46 0.48 10.72 * 0.96
Theta 0.46 0.45 0.17 1.82 4.01 0.23 0.00 0.11
Alpha 12.51 ** 0.80 3.24 2.01 0.68 0.41 0.00 0.09
*** p< 0.001; ** p< 0.01; * p< 0.05.
3. Study 2
Koerner et al. [
57
] aimed to examine whether noise-induced changes in the MMN and spectral
power in the theta frequency band in response to a consonant change (/ba/ to /da/) and vowel change
(/ba/ to /bu/) in a double-oddball paradigm were predictive of speech perception in noise at the
syllable and sentence levels.
3.1. Statistical Methods
For a direct comparison, Pearson correlations were used to examine correlations between
the objective MMN (latency, amplitude, and EEG theta power) in response to /da/ and /bu/
and behavioral responses (percent correct phoneme detection, reaction time, and percent correct
sentence recognition) pooled across quiet and speech babble noise listening conditions, resulting
in 18 correlations. A check of linearity was performed on each pair of continuous variables using
scatterplots. Final p-values for each correlation coefficient were adjusted to account for multiple
comparisons. As reported in Koerner et al. [
57
], repeated measures ANOVAs were used to examine the
effects of background noise on MMN latency, amplitude, and EEG theta power. Linear mixed-effects
models were developed to determine whether these objective neural measures were able to predict
behavioral performance. Participant was included as a “by-subject” random effect in each linear
mixed-effect model while listening condition (quiet vs. noise) and stimulus (/da/ vs. /bu/) were
included as blocking (or grouping) variables in each linear mixed-effect model. MMN latency,
amplitude, and theta power were added as fixed effects in models with percent correct phoneme
detection or reaction time as outcome variables. Similar models were developed to examine whether
MMN latency, amplitude, and theta power in response to /da/ or /bu/ were able to predict
sentence-level perception using listening condition as a blocking variable. Data transformations
for the linear mixed-effects models included re-scaling the MMN latency and behavioral reaction times
for phoneme detection as well as log-transforming the percent correct phoneme detection and sentence
recognition scores to account for skewness in the data. The significance of each correlation coefficient
from the Pearson correlation analysis as well as each fixed effect from the linear mixed-effects models
for predicting each behavioral outcome measure was assessed at α= 0.05.
3.2. Results
In the Pearson tests, significant correlations were found between MMN latency recorded in
response to the vowel-change and percent correct phoneme detection (r= 0.53, p< 0.05) for /bu/ as
well as percent correct sentence recognition (r=
0.40, p< 0.05) across the quiet and noise listening
conditions. Significant correlations were also found between MMN amplitude recorded in response
to the vowel-change and percent correct phoneme detection (r=
0.50, p< 0.05) and reaction time
(
r= 0.56
,p< 0.01) for /bu/, as well as percent correct sentence recognition (r=
0.66, p< 0.01) across
listening conditions. Similar trends were found between theta power in response to the vowel-change
and percent correct phoneme detection (r= 0.41, p< 0.05) and behavioral reaction time (r=
0.49,
Brain Sci. 2017,7, 26 6 of 11
p< 0.05
) in response to the CV syllable /bu/, as well as behavioral sentence recognition (r= 0.59,
p< 0.01) across listening conditions. Additionally, results revealed significant correlations between
MMN latency recorded in response to the consonant-change and percent correct phoneme detection
(r=0.47, p< 0.05) for /da/ as well as sentence recognition (r=0.53, p< 0.01) across the quiet and
noise listening conditions (see Table 3for a summary of correlation coefficients).
Repeated measures ANOVA results from Koerner et al. [
57
] showed significant effects of
background noise on MMN latency (F(1, 14) = 29.43, p< 0.001), amplitude (F(1, 14) = 32.52, p< 0.001),
and EEG theta power (F(1, 14) = 19.37, p< 0.001). Koerner et al. [
57
] also provided detailed results
from the linear mixed-effects regression analysis (see replicated Table 4for summary of regression
model results). Linear mixed-effects models showed that both MMN latency (F(1, 40) = 7.86, p< 0.01)
and spectral power in the theta band (F(1, 40) = 6.61, p< 0.05) were significant predictors of percent
correct phoneme detection across listening conditions and stimuli. Additionally, MMN amplitude
in response to the syllable /bu/ was a significant predictor of sentence recognition across listening
conditions (F(1, 11) = 7.21, p< 0.05). As all residual plots from each linear mixed-effects model revealed
that residuals were normally distributed without any signs of heteroscedastic variance or significant
trends, we do not expect that generalized linear models would improve the results. Interactions were
tested in previous models and were subsequently removed due to a lack of statistical significance.
Table 3.
Correlation coefficients for brain-behavior correlations between neural MMN latency,
amplitude, and theta power for /bu/ and /da/ at electrode Cz and behavioral phoneme detection
percent correct, reaction time, and percent correct sentence recognition scores.
Latency (ms) Amplitude (µV) Power (dB)
/bu/ /da/ /bu/ /da/ /bu/ /da/
Phoneme Detection (%) 0.53 * 0.47 * 0.50 * 0.17 0.41 * 0.13
Reaction Time (ms) 0.34 0.39 0.56 ** 0.02 0.49 * 0.01
Sentence Recognition (%) 0.40 * 0.53 ** 0.66 ** 0.07 0.59 ** 0.18
*** p< 0.001; ** p< 0.01; * p< 0.05.
Table 4.
F-statistics and regression coefficients (
β
) for fixed effects from linear mixed-effects regression
models for each behavioral measure (Koerner et al. [57]).
Variable Percent Correct
Phoneme Detection
Phoneme Detection
Reaction Time
Percent Correct Sentence
Recognition (/bu/)
Percent Correct Sentence
Recognition (/da/)
FβFβFβFβ
Intercept 161.51 *** - 4199.98 *** - 431.41 *** - 335.12 *** -
Condition 131.68 *** - 61.92 *** - 291.32 *** - 247.69 *** -
Stimulus 114.20 *** - 21.05 *** - - - - -
Latency 7.86 ** 0.61 0.000 0.03 1.24 0.19 0.44 0.21
Amplitude 3.10 0.09 0.002 0.02 7.21 * 0.24 0.41 0.05
Theta Power 6.61 * 0.05 0.368 0.01 0.46 0.01 1.50 0.02
*** p< 0.001; ** p< 0.01; * p< 0.05.
4. Discussion
The current report compared results from Pearson correlations and linear mixed-effects regression
models using data from two published ERP studies. It was determined that Pearson correlations were
not appropriate for examining relationships in our data, which contained built-in differences across
within-subject repeated measures. The results showed how linear mixed-effects regression models
(after verification of normality of residuals and homogeneity of variance) are able to depict relationships
between the predictor and outcome variables while taking into account repeated measures across
participants. While the LME models were able to confirm basic conclusions gained from the Pearson
Brain Sci. 2017,7, 26 7 of 11
correlation analyses for both studies [
54
,
57
], a comparison of methods and results for each model
highlighted differences between the two approaches.
The repeated measures ANOVA indicated that background noise had a significant effect on N1
and P2 latencies as well as N1 amplitudes in response to the syllable /bu/ [
54
]. Similarly, the repeated
measures ANOVA revealed that MMN latency, amplitude, and spectral power were significantly
impacted by background noise [
57
]. These results support the possibility that pooling data from
quiet and noise listening conditions created a built-in contrast and bias between data points when
Pearson correlations were used, which partly led to the overestimation of the association strength in
the reported results (Tables 1and 3). In other words, the Pearson correlation analysis ignores these
built-in differences and treats this type of data as if each variable in the repeated measures design
were independent and normally distributed across the two listening conditions. The resulting p-values
represent the probability of observing an effect that is as large, or larger, than what would be observed
if there was no covariance structure in the repeated measures. In contrast, LME regression analysis
was able to account for the covariance structure and grouping factors for the repeated measures.
Tests of significance from the LME models examined whether each predictor variable, or fixed effect,
was significantly different than zero while taking into account the other fixed or random effects in
the model.
One issue common to regression analysis concerns the possible existence of multi-collinearity
(or the existence of high correlations) among the predictor variables and how it may inflate the
results with unstable estimates of regression coefficients such as an overall significant model with
no significant predictors [
2
5
]. In the mixed-effects (or multilevel) models, the implementation of
fixed and random effects allows control of the within-subject factor for repeated measures, and
the additional stepwise approach allows removal of predictor variables in a systematic fashion, for
instance, calculating a variance inflation factor (VIF) to identify collinear predictors to aid the stepwise
removal of predictors from the LME models. The VIF represents the proportion of variance in one
predictor variable accounted for by all the other predictors in the model. Estimation of VIFs for each
predictor and progressive dropping of the predictor with the largest VIF beyond the cutoff criterion
can be helpful in dealing with the collinearity of interaction terms. By contrast, Pearson correlation
analysis assumes independence of the variables, and only fixed effects are directly examined piecewise
without elaborate procedures to take into account how the existing associations/differences among
the predictor variables may contribute to (oftentimes inflate) the correlation coefficients. The bivariate
Pearson correlation analysis disregards potential correlations and data groupings among variables,
which makes it inappropriate for research questions that aim to examine associations between variables
that contain built-in differences between experimental conditions or subject groups.
Although the flexibility in model selection can be considered a strength of LME regression analysis,
the number of educated choices a researcher must make while developing and implementing models
can be a challenge. For instance, the inclusion of interactions or random effects in LME models affects
the regression coefficients and interpretation of fixed effects, which cannot properly be taken into
account in the bivariate Pearson correlation analysis. Although stepwise regression methods are
available as a systematic approach to choose an appropriate model, it is important for researchers to
think deeply about the subject matter in order to determine whether the inclusion and interpretation of
specific fixed and random effects are appropriate for the specific research question and study objective.
While the two ERP studies reported here are clearly limited in scope and depth of analysis, the
side-by-side comparisons clearly demonstrate the limitations and inappropriateness of the Pearson
approach as well as its inflated correlation estimation results for the data sets. Given that multiple
analysis techniques (for example, waveform analysis, source localization, time-frequency analysis)
can be applied to the same neurophysiological data in cognitive neuroscience research [
54
,
57
59
],
a cautionary note against the convenient use of the simple Pearson correlation test is necessary when
selecting and applying statistical models to interpret brain-behavior correlations (e.g., biomarkers
Brain Sci. 2017,7, 26 8 of 11
of various diseases and disorders) or correlations among the various brain measures with prior
distributions and covariance structure for repeated measures.
5. Conclusions
In sum, this report compared conventional Pearson correlations and linear mixed-effects (LME)
regression models using data from two published auditory electrophysiology studies. The Pearson
correlation test is inappropriate for the specific research questions in both studies as the neural
responses across listening conditions were simply treated as independent measures. Although our
comparative analysis is limited in its scope and depth, this technical note demonstrates the advantages
as well as the necessity to apply mixed-effects models to properly account for the built-in relationships
among the multiple predictor variables, which has important implications for proper modeling and
interpretation of human behavior in terms of neural correlates and biomarkers.
Acknowledgments:
This work was supported in part by the Charles E. Speaks Graduate Fellowship (T.K.K.), the
Bryng Bryngelson Research Fund (T.K.K. and Y.Z.), the Capita Foundation (Y.Z.), the Brain Imaging Research
Project award and the single semester leave award (Y.Z.) from the College of Liberal Arts, and the University of
Minnesota Grand Challenges Exploratory Research Project Grant (Y.Z.). We would like to thank Boxiang Wang,
Hui Zou, Peggy Nelson, and Edward Carney for their assistance.
Author Contributions:
Y.Z. conceived the study; T.K.K. and Y.Z. designed the experiments, performed the
experiments, and analyzed the data; T.K.K. and Y.Z. wrote the paper.
Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this manuscript:
LME Linear mixed-effects
GLME Generalized linear mixed-effects
EEG Electroencephalography
ERP Event related potential
SNR Signal-to-noise ratio
N1 Negative-going ERP response that peaks at approximately 100 ms after auditory stimulus onset
P2 Positive-going ERP response that follows the N1
ANOVA Analysis of variance
nlme A R package for linear and nonlinear mixed effects models
CV Syllable structure consisting of a consonant and vowel (e.g., /ba/)
F
F statistic, a value from ANOVA or regression analysis to indicate differences between two means
βVector of fixed effect slopes in the linear mixed effects models
µV Microvolt
dB Decibel
VIF Variance inflation factor
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2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons Attribution
(CC BY) license (http://creativecommons.org/licenses/by/4.0/).
... These patterns are consistent with findings from healthy [22,28] and clinical [23,25,27,74] populations. Although previous studies have examined whether ITPC is able to predict variations in the obligatory N1-P2 complex response to speech sounds [101], very few studies have investigated whether measures of trial-by-trial neural synchrony are potential indicators of auditory ERP responses (especially late components) using emotional speech stimuli. Therefore, our novel finding has added to the extant literature in showing that stimulus-evoked phase alignment of cortical oscillations contributes to the neural generation of auditory ERPs in early and late emotional speech processing [24,30]. ...
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Background and Objective Functional connectivity (FC) measures can be used to differentiate epileptogenic zones (EZs) from non-EZs in patients with medically refractory epilepsy. However, little work has been done to evaluate stability of stereoelectroencephalography (SEEG) FC measures over time and their relationship with anti-seizure medication (ASM) use – a critical confounder in epilepsy FC studies to date. In this study, we aim to answer the following questions: 1) are SEEG FC measures stable over time, 2) are they influenced by ASMs, and 3) are they impacted by patient data collection state? Methods In 32 patients with medically-refractory focal epilepsy, we collected a single two-minute prospective SEEG resting-state (awake, eyes closed) data set, and consecutive two-minute retrospective pseudo-rest (awake, eyes open) data sets for days 1-7 post-implantation. ASM dosages were recorded for days 1-7 post-implantation and drug load score (DLS) per day was calculated to standardize and compare across patients. FC was evaluated using directed and nondirected measures. Standard clinical interpretation of ictal SEEG was used to classify brain regions as EZs and non-EZs. Results Over seven days, presumed EZs consistently had higher FC than non-EZs when using Between Imaginary Coherence (ImCoh) and Partial Directed Coherence (PDC) Inward strength, without accounting for DLS. These measures were demonstrated to be stable over a short-term period of three consecutive days with the same DLS. Between ImCoh FC differences between EZs and non-EZs were reduced with DLS decreases, whereas other measures were not affected by DLS. FC differences between EZs and non-EZs were seen during both resting-state and pseudo-rest conditions; ImCoh values were strongly correlated between the two conditions, whereas PDC values were not. Discussion Inward and non-directed SEEG FC is higher in presumed EZs versus non-EZs, and measures are stable over time. However, certain measures may be impacted by ASM dose, as Between ImCoh differences between EZs and non-EZs are less pronounced with lower doses, and other measures such as PDC are poorly correlated across recording conditions. These findings allow novel insight into how SEEG FC measures may aid surgical localization, and how they are influenced by ASMs and other factors.
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Statistical Rethinking: A Bayesian Course with Examples in R and Stan builds readers’ knowledge of and confidence in statistical modeling. Reflecting the need for even minor programming in today’s model-based statistics, the book pushes readers to perform step-by-step calculations that are usually automated. This unique computational approach ensures that readers understand enough of the details to make reasonable choices and interpretations in their own modeling work. The text presents generalized linear multilevel models from a Bayesian perspective, relying on a simple logical interpretation of Bayesian probability and maximum entropy. It covers from the basics of regression to multilevel models. The author also discusses measurement error, missing data, and Gaussian process models for spatial and network autocorrelation. By using complete R code examples throughout, this book provides a practical foundation for performing statistical inference. Designed for both PhD students and seasoned professionals in the natural and social sciences, it prepares them for more advanced or specialized statistical modeling. Web Resource The book is accompanied by an R package (rethinking) that is available on the author’s website and GitHub. The two core functions (map and map2stan) of this package allow a variety of statistical models to be constructed from standard model formulas.
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