Introduction to collinearity: What is it? How does it ... · Introduction to collinearity: What is...

Introduction to collinearity:What is it? How does it arise with

parametric modulation?

Monday, Lecture 5Jeanette Mumford

University of Wisconsin - Madison

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?

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?

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?

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?

Collinearity

• Each regressor’s p-value (roughly) only reflects its unique variability

Y

X2

X1

Collinearity

• Each regressor’s p-value (roughly) only reflects its unique variability

Y

X2

X1Less unique variability ->Parameter estimates have larger variability

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

Collinearity

• Partial way to assess collinearity– Look at pairwise correlations of regressors– This is *not* a thorough way to address

collinearity

How could this be fixed?

Stimulus Cue Response Fixation

How could this be fixed?

Stimulus Cue Response Fixation

1s 1s 2s 1s

How could this be fixed?

Stimulus Cue Response Fixation

1.5s 1.5s 2s 1s

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!

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?

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!

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?

What is this orthogonalise option?

Parametric modulation

• 3 trials• Modulations are 1, 3, 2• How do we make the regressors?

Our boxcar regressor

Our boxcar regressor, modulated

Just add convolution!

0 10 20 30 40 50

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Time (s)

Both of these need to go into the model. Do you know why?

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?

Reference

How regression naturally works

• Each regressor’s p-value only reflects its unique variability

YX1

How regression naturally works

• Each regressor’s p-value only reflects its unique variability

Y

X2

X1

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

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

What happens to estimates?

• Design: 1s stimulus followed by 1s feedback cue

What happens to estimates?

What happens to estimates?True activation magnitudes

What happens to estimates?

What happens to estimates?

What happens to estimates?

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

Recap

• Orthogonalization is NOT a band-aid for collinearity

Orthogo

nalizat

ion

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

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

Quick review of mean centering covariates

Review

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

20 30 40 50 60

0.8

0.9

1.0

1.1

1.2

1.3

Age (years)

RT (s

)

RT = �0 + �1Age+ ✏

Mean centering X is basically moving the Y-axis

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

0 10 20 30 40 50 60

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Age (years)

RT (s

)

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

0 10 20 30 40 50 60

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Age (years)

RT (s

)Mean centering X is basically moving

the Y-axis

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

−20 −10 0 10 20

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Mean centered age(years)

RT (s

)Mean centering X is basically moving

the Y-axis

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

End of mean centering segue

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

Aw no, SPM L

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

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?

Today in lab

• How do we make SPM do what we’d like it to do?

Let’s recap

• What is the worst case scenario with collinearity?

• Is the removal of collinearity with orthogonalization ever okay?

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?

Let’s recap

• What is the most common, acceptable form of orthogonalization?

Questions?