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Hierarchical Linear Modelling for EEG
Cyril Pernet, PhDNeurobiology Research Unit,
Copenhagen University Hospital, Rigshospitalet
[email protected] @CyrilRPernet30th EEGLAB workshop – 17th June 2021
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Motivation for hierarchical models
• Most often, we compute averages per condition and do statistics on peaklatencies and amplitudes
➢Univariate methods extract information among trials in time and/or frequency across space
➢Multivariate methods extract information across space, time, or both, in individual trials
➢Averages don’t account for trial variability, fixed effect can be biased –these methods allow to get around these problems
Pernet, Sajda & Rousselet – Single trial analyses, why bother? Front. Psychol., 2011, 2, 322
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LIMO EEG Toolbox https://limo-eeg-toolbox.github.io/limo_meeg/
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Framework
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Robust Hierarchical Linear Model Framework
WLS solution – accounts for trial dynamics Trimmed-means: outlier subjects
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Fixed, Random, Mixed and Hierarchical
Fixed effect: Something the experimenter directly manipulates
y=XB+e data = beta * effects + errory=XB+u+e data = beta * effects + constant subject effect + error
Random effect: Source of random variation e.g., individuals drawn (at random) from a population. Mixed effect: Includes both, the fixed effect (estimating the population level coefficients) and random effects to account for individual differences in response to an effect
Y=XB+Zu+e data = beta * effects + zeta * subject variable effect + error
Hierarchical models are a mean to look at mixed effects (with some extra assumptions).
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Hierarchical model = 2-stage LM
For a given effect, the whole group is modelledParameter estimates apply to group effect/s
Each subject’s EEG trials are modelledSingle subject parameter estimates
Single subject
Group/s of subjects
1st
level
2nd
level
Single subject parameter estimates or combinations taken to 2nd level
Group level of 2nd level parameter estimates are used to form statistics
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Fixed effects:
Intra-subjects variation
suggests all these subjects
different from zero
Random effects:
Inter-subjects variation
suggests population
not different from zero
0
2FFX
2RFX
Distributions of each subject’s estimated effect
subj. 1
subj. 2
subj. 3
subj. 4
subj. 5
subj. 6
Distribution of population effect
Fixed vs Random
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Fixed effects
❑Only source of variation (over trials)
is measurement error
❑True response magnitude is fixed
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Random effects
• Two sources of variation
• measurement errors
• response magnitude (over subjects)
• Response magnitude is random
• each subject has random magnitude
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Random effects
• Two sources of variation
• measurement errors
• response magnitude (over subjects)
• Response magnitude is random
• each subject has random magnitude
• but note, population mean magnitude is fixed
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An example
Example: present stimuli fromintensity -5 units to +5 unitsaround the subject perceptualthreshold and measure RT
→ There is a strong positiveeffect of intensity on responses
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Fixed Effect Model 1: average subjects
Fixed effect without subject effect = average without aligning subjects → negative effect
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Fixed Effect Model 2: constant over subjects
Fixed effect with a constant (fixed) subject effect → positive effect but biased result
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HLM: random subject effect
Mixed effect with a random subject effect → positive effect with good estimate of the truth
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MLE: random subject effect
Mixed effect with a random subject effect → positive effect with good estimate of the truth
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T-tests
Simple regression
ANOVA
Multiple regression
General linear model• Mixed effects/hierarchical
• Timeseries models (e.g., autoregressive)
• Robust regression
• Penalized regression (LASSO, Ridge)
Generalized linear models
• Non-normal errors
• Binary/categorical outcomes (logistic regression)
On
e-s
tep s
olu
tion
Itera
tive
so
lutio
ns (e
.g., IW
LS
)
The GLM Family
Tor Wager’s slide
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HLM and designsBringing Robustness
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Factorial Designs: 3 (objects) * 2 (spectrum)
Bienek, et al (2012). Phase vs Amplitude Spectrum. Journal of Vision 12
ob
ject
s
Amplitude spectrum
Level 1: Y = X β +e → each parameter estimate (β) reflect a condition = 6 parameter + subjects’ adj. meanLevel 2: Y = X β +e → 1st level βs are split into factors (3*2) and interaction (Repeated Measure ANOVA)
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Factorial Designs: 3 (objects) * N*N …To investigate how phase coherence and amplitude spectra of images modulate the ERP, you will need to create many levels of phase coherence and amplitude spectra
With 30 trials per conditions (2*7*10) = 4200 trials !
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Regression based designs (1)
• Test the effect of a continuous (or set of) variable(s)
• Variable level of noise for each stimulus – noise here is in fact a given amount of phase coherence in the stimulus
Rousselet, Pernet, Bennet, Sekuler (2008). Face phase processing. BMC Neuroscience 9:98
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Regression based designs (1)
2nd level analysis on parameters
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Regression based designs (2)Control the category (face vs house) and test with continuous of low-level physical properties
PhaseAmplitude
P x A
Bienek, Pernet, Rousselet (2012). Phase vs Amplitude Spectrum. Journal of Vision 12(13), 1–24
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Regression based designs (3)
Fully Parametric designs: study the effect of a group variable (e.g. age) on continuous variables (e.g.phase coherence) within subjects
Rousselet, Gaspar, Pernet, Husk, Bennett, Sekuler (2010). Aging and face perception. Front Psy
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Conclusion
• HLM allows you to model any designs
• Not just categorical designs, also model confounds (e.g. stimulus properties) or test continuous variables
• 1st level is like getting averages for each condition but better because (i) it removes subjects effect (ii) accounts for trial variability
• GLM is just your usual statistics but using generic approach, i.e. it’s better because more flexible
• LIMO provides robust estimates (against outlier trials and outlier subjects – not reviewed here) which increases population inferential power over OLS