introduction to collinearity: what is it? how does it ... · introduction to collinearity: what is...
TRANSCRIPT
![Page 1: Introduction to collinearity: What is it? How does it ... · Introduction to collinearity: What is it? How does it arise with parametric modulation? Monday, Lecture 5 ... 1.Get sick,](https://reader030.vdocuments.us/reader030/viewer/2022040121/5ecd54e87b8a796bf06b999c/html5/thumbnails/1.jpg)
Introduction to collinearity:What is it? How does it arise with
parametric modulation?
Monday, Lecture 5Jeanette Mumford
University of Wisconsin - Madison
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Collinearity
• Today– Where does it come from?– Is there an easy fix?
• (no)– Collinearity and parametrically modulated
regressors• Tomorrow
– How can we assess the collinearity?– How can we avoid it?
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Collinearity
• Today– Where does it come from?– Is there an easy fix?
• (no)– Collinearity and parametrically modulated
regressors• Tomorrow
– How can we assess the collinearity?– How can we avoid it?
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Have you ever done this?
1. Get sick, so you– Take a vitamin– Take cold-EEZE– Drink a lot of water– Take a nap– Have some chicken soup– Drink green tea
2. You get better! What helped you get better?
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5 10 15 20
−0.2
0.2
0.6
1.0
Time (s)
Behavioral task
• Non-overlapping portions of a trial
• BOLD responses overlap
Stimulus Cue Response Fixation
If the BOLD increases here, what caused it?
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Collinearity
• Each regressor’s p-value (roughly) only reflects its unique variability
Y
X2
X1
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Collinearity
• Each regressor’s p-value (roughly) only reflects its unique variability
Y
X2
X1Less unique variability ->Parameter estimates have larger variability
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Collinearity
• When 1 regressor is correlated with other regressors in a model – Could be a linear combination of other regressors
• If the BOLD signal changes, you cannot attribute the change, easily, to one regressor
• Impact: Power loss
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Collinearity
• Partial way to assess collinearity– Look at pairwise correlations of regressors– This is *not* a thorough way to address
collinearity
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How could this be fixed?
Stimulus Cue Response Fixation
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How could this be fixed?
Stimulus Cue Response Fixation
1s 1s 2s 1s
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How could this be fixed?
Stimulus Cue Response Fixation
1.5s 1.5s 2s 1s
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How could this be fixed?
Stimulus Cue Response Fixation
1.5s 1.5s 2s 1s
F.
.5s
Still not great, but better VIF is a better measure than correlation and we’ll look at that tomorrow!
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Collinearity
• Today– Where does it come from?– Is there an easy fix?
• (no)
– Collinearity and parametrically modulated regressors
• Tomorrow– How can we assess the collinearity?– How can we avoid it?
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What we’re about to cover
• Learn when we need parametrically modulated regressors
• Learn how to make parametrically modulated regressors
• Learn how to interpret the parameter estimates from these regressors
• Learn about orthogonalization– SPM has a confusing orthogonalization step with
PM and you shouldn’t use it!– Simple fix!
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What question does a parametrically modulated regressor answer?
• Show emotional stimuli in scanner– Use post-fMRI behavioral task to estimate valence– Q: Does BOLD activation increase with valence?
• Gambling task– Can win variable amounts for each gamble– Q: Does BOLD activation increase with amount a
person could lose on a gamble?
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What is this orthogonalise option?
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Parametric modulation
• 3 trials• Modulations are 1, 3, 2• How do we make the regressors?
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Our boxcar regressor
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Our boxcar regressor, modulated
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Just add convolution!
0 10 20 30 40 50
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Time (s)
Both of these need to go into the model. Do you know why?
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Interpretation
• Just like a regular old regression slope!• reg 2 = modulated regressor
– How BOLD changes with modulation value• A 1 unit increase in the modulation results in a beta
increase in BOLD (beta is the regression parameter estimate)
• reg 1 = unmodulated regressor• What do you think the interpretation is?
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Reference
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How regression naturally works
• Each regressor’s p-value only reflects its unique variability
YX1
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How regression naturally works
• Each regressor’s p-value only reflects its unique variability
Y
X2
X1
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What is orthogonalization?• Giving that shared portion to one of the
regressors– Say X2 gives it up to X1– We say X2 is orthogonalized with respect to X1
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What is orthogonalization?• Giving that shared portion to one of the
regressors– Say X2 gives it up to X1– We say X2 is orthogonalized with respect to X1
Y
X2
X1
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What happens to estimates?
• Design: 1s stimulus followed by 1s feedback cue
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What happens to estimates?
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What happens to estimates?True activation magnitudes
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What happens to estimates?
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What happens to estimates?
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What happens to estimates?
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Orthogonalization can lead to very misleading results
• Basically “un-controls” one covariate for the other– But that’s the entire reason we model multiple
regressors!
• If you orthogonalize “motion” with respect to task, you are not “fixing” collinearity– Your orthogonalized model behaves, basically, as if
you omitted motion
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Recap
• Orthogonalization is NOT a band-aid for collinearity
Orthogo
nalizat
ion
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Is orthogonalization ever okay?
• Yes!– Mean centering covariates!
– Parametric modulation!• Same idea as mean centering • Unmodulated regressor should represent mean
activation for that task• Modulation regressors should reflect unique variability
of that modulation value
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Is orthogonalization ever okay?
• Yes!– Mean centering covariates!
– Parametric modulation!• Same idea as mean centering • Unmodulated regressor should represent mean
activation for that task• Modulation regressors should reflect unique variability
of that modulation value
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Quick review of mean centering covariates
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Review
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RT = �0 + �1Age+ ✏
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Mean centering X is basically moving the Y-axis
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the Y-axis
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Mean centering is okay
• It does not change the model fit
• It does not fix anything about the mean centered regressor
• It changes the interpretation of the constant regressor
• It never hurts, but it often isn’t doing anything for the part of the model people are studying
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End of mean centering segue
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How does orthogonalization arise with PM?
• Setup: Modeling stimulus intensity and RT as parametric modulators
• Orthogonalization is *okay* if done properly
• Goal: Orthogonalize each of RT and intensity with respect to unmodulated regressor
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Aw no, SPM L
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Aw no, SPM L
• There’s a fix– mean center modulators– Turn off orthogonalization
• What seems like a fix, but isn’t– SPM’s “No orthogonalization” option alone– Okay if you’re not looking at unmodulated
regressor
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Today in lab
• What will happen if you let SPM do the default thing with 2 sets of modulators?– i.e. what’s the interpretation of the parameter
estimates for the unmodulated regressor, first added modulation and second added modulation?
– What would the correlation between the regressors look like?
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Today in lab
• How do we make SPM do what we’d like it to do?
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Let’s recap
• What is the worst case scenario with collinearity?
• Is the removal of collinearity with orthogonalization ever okay?
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Let’s recap
• Why can’t we just use the orthogonalize option in SPM?
• If you have modulations for amount won and amount lost on each trial, how many regressors should be modeled?
![Page 56: Introduction to collinearity: What is it? How does it ... · Introduction to collinearity: What is it? How does it arise with parametric modulation? Monday, Lecture 5 ... 1.Get sick,](https://reader030.vdocuments.us/reader030/viewer/2022040121/5ecd54e87b8a796bf06b999c/html5/thumbnails/56.jpg)
Let’s recap
• What is the most common, acceptable form of orthogonalization?
![Page 57: Introduction to collinearity: What is it? How does it ... · Introduction to collinearity: What is it? How does it arise with parametric modulation? Monday, Lecture 5 ... 1.Get sick,](https://reader030.vdocuments.us/reader030/viewer/2022040121/5ecd54e87b8a796bf06b999c/html5/thumbnails/57.jpg)
Questions?