Non-orthogonal regressors: concepts and consequences

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Non-orthogonal regressors concepts and
consequences
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overview

• Problem of non-orthogonal regressors
• Concepts orthogonality and uncorrelatedness
• SPM (1st level)
• covariance matrix
• detrending
• how to deal with correlated regressors
• Example

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design matrix
regressors
Scan number

• Each column in your design matrix represents 1)
  events of interest or 2) a measure that may
  confound your results. Column regressor
• The optimal linear combination of all these
  columns attempts to explain as much variance in
  your dependent variable (the BOLD signal) as
  possible

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error



??1
?2
Time
e
x1
x2
BOLD signal
Source spm course 2010, Stephan http//www.fil.io
n.ucl.ac.uk/spm/course/slides10-zurich/
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• The betas are estimated on a voxel-by-voxel
  basis
• high beta means regressor explains much of BOLD
  signals variance (i.e. strongly covaries with
  signal)

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Problem of non-orthogonal regressors
Y
total variance in BOLD signal
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Orthogonal regressors
every regressor explains unique part of the
variance in the BOLD signal
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Orthogonal regressors
There is only 1 optimal linear combination of
both regressors to explain as much variance as
possible. Assigned betas will be as large as
possible, stats using these betas will have
optimal power
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non-orthogonal regressors
Y
X2
X1


Regressor 1 2 are not orthogonal. Part of the
explained variance can be accounted for by both
regressors and is assigned to neither. Therefore,
betas for both regressors will be suboptimal
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Entirely non-orthogonal
Y
X2
X1


Betas cant be estimated. Variance can not be
assigned to one or the other
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It is always simpler to have orthogonal
regressors and therefore designs. (spm course
2010)
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orthogonality
Regressors can be seen as vectors in
n-dimensional space, where n number of
scans. Suppose now n 2 r1 r2 --------------- 1
2 2 1
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orthogonality

• Two vectors are orthogonal if raw vectors have
• inner product 0
• angle between vectors 90
• cosine of angle 0
• Inner product
• r1 r2 (1 2) (2 1) 4
• ? acos(4 / (r1 r2) about 35 degrees

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orthogonality

• Orthogonalizing one vector wrt another it
  matters which vector you choose! (Gram-Schmidt
  orthogonalization)
• Orthogonalize r1 wrt r2
• u1 r1 projr2(r1)
• u1 1 2 (r1 r2)/(r2 r2)
• u1 -0.6 1.2
• Inner product
• u1 r2 (-0.6 2) (1.2 1) 0

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orthogonality uncorrelatedness
An aside on these two concepts

• Orthogonal is defined as XY 0
• (inner product of two raw vectors 0)
• Uncorrelated is defined as (X mean(X))(Y
  mean(Y)) 0 (inner product of two detrended
  vectors 0)
• Vectors can be orthogonal while being correlated,
  and vice versa!

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• please read Rodgers et al. (1984) Linearly
  independent, orthogonal and uncorrelated
  variables. The American Statistician, 38133-134.
  Will be in the FAM folder as well

Orthogonal because Inner product 15 -51
31 -13 0
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• please read Rodgers et al. (1984) Linearly
  independent, orthogonal and uncorrelated
  variables. The American Statistician, 38133-134.
  Will be in the FAM folder as well

Detrend Mean(X) -0.5 Mean(Y)
2.5 X_det Y_det 1.5 2.5 -4.5 -1.5 3.5 -1.5 -
0.5 0.5 Mean(X_det)
0 Mean(Y_det) 0 Inner
product 5 Orthogonal, but correlated!
3.75 6.75 -5.25 -0.25
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r1_det r2_det -0.9 0.5 0.9 -0.5
r1 r2 -0.6 2 1.2 1
r1
detrend
r2
1
2
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orthogonality uncorrelatedness

• Q So should my regressors be uncorrelated or
  orthogonal?
• A When building your SPM.mat (i.e. running your
  jobfile) all regressors are detrended (except the
  grand mean scaling regressor). This is why
  orthogonal and uncorrelated are both used when
  talking about regressors
• update it is unclear whether all regressors are
  detrended when building an SPM.mat. This seems to
  be the case, but recent SPM mailing list activity
  suggests detrending might not take place in
  versions newer than SPM99.
• Donders batch?

effectively there has been a change between
SPM99 and SPM2 such that regressors were
mean-centered in SPM99 but they are not any more
(this is regressed out by the constant term
anyway). Link
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Your regressors correlate

• Despite scrupulous design, your regressors likely
  still correlate to some extent
• This causes beta estimates to be lower than they
  could be
• You can see correlations using review ? SPM.mat ?
  Design ? design orthogonality

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For detrended data, the cosine of the angle
(black 1, white 0) between two regressors is
the same as the correlation r ! orthogonal
vectors cos(90) 0 r 0 r2 0 correlated
vector cos(81) 0.16 r 0.16 r2
0.0256 r2 indicates how much variance is common
between the two vectors (2.56 in this example).
Note -1 r 1 and 0 r2 1
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• Correlated regressors variance from single
  regressor to shared

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• Correlated regressors variance from single
  regressor to shared
• t-test uses beta, determined by amount of
  variance explained by single regressor.

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• Correlated regressors variance from single
  regressor to shared
• t-test uses beta, determined by amount of
  variance explained by single regressor.
• Large shared variance low statistical power

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• Correlated regressors variance from single
  regressor to shared
• t-test uses beta, determined by amount of
  variance explained by single regressor.
• Large shared variance low statistical power
• Not necessarily a problem if you do not intend to
  test these two regressors!

Movement regressor 1
Movement regressor 2
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How to deal with correlated regressors?

• Strong correlations between regressors are not
  necessarily a problem. What is relevant is
  correlation between contrasts of interest
  relative to the rest of the design matrix
• Example lights on vs lights off. If movement
  regressors correlate with these conditions
  (contrast of interest not orthogonal to rest of
  design matrix), there is a problem.
• If nuisance regressors only correlate with each
  other, no problem!
• Grand mean scaling is not centered around 0 (i.e.
  not detrended), these correlations are not
  informative

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How to deal with correlations between contrast
and rest of design matrix?

• Orthogonalize regressor A wrt regressor B all
  shared variance will now be assigned to B.

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orthogonality
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orthogonality
r1
r2
1
2
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How to deal with correlations between contrast
and rest of design matrix?

• Orthogonalize regressor A wrt regressor B all
  shared variance will now be assigned to B.
• Only permissible given a priori reason to do
  this hardly ever the case

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How to deal with correlations between contrast
and rest of design matrix?

• do an F-test to test overall significance of your
  model. For example, to see if adding a regressor
  will significantly improve your model. Shared
  variance is taken along to determine significance
  then.
• In the case where a number of regressors
  represent the same manipulation (e.g. switch
  activity, convolved with different hrfs) you can
  serially orthogonalize the regressors before
  estimating betas.

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Example how not to do it

• 2 types of trials gain and loss

Voon et al. (2010) Mechanisms underlying
dopamine-mediated reward bias in compulsive
behaviors. Neuron
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Example how not to do it

• 4 regressors
• Gain predicted outcome
• Positive prediction error (gain trials)
• Loss predicted outcome
• Negative prediction error (loss trials)

Highly correlated!
Highly correlated!
Voon et al. (2010) Mechanisms underlying
dopamine-mediated reward bias in compulsive
behaviors. Neuron
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Example how not to do it

• Performed 6 separate analyses (GLMs)
• Shared variance is attributed to single regressor
  in all GLMs
• Amazing! Similar patterns of activation!

Voon et al. (2010) Mechanisms underlying
dopamine-mediated reward bias in compulsive
behaviors. Neuron
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Take home messages

• If regressors correlate, explained variance in
  your BOLD signal will be assigned to neither,
  which reduces power on t-tests
• If you orthogonalize regressor A with respect to
  regressor B, values of A will be changed and A
  will have equal uniquely explained variance. B,
  the unchanged variable, will come to explain all
  variance shared by A and B. However, dont do
  this unless you have a valid reason.
• Orthogonality and uncorrelatedness are only the
  same thing if your data is centered around 0
  (detrended, spm_detrend)
• SPM does (NOT?) detrend your regressors the
  moment you go from job.mat to SPM.mat

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Interesting reads

• http//imaging.mrc-cbu.cam.ac.uk/imaging/DesignEff
  iciencyhead-525685650466f8a27531975efb2196bdc90fc
  419
• Combines SPM book and Rik Hensons own attempt at
  explaining design efficiency and the issue of
  correlated regressors.
• Rodgers et al. (1984) Linearly independent,
  orthogonal and uncorrelated variables. The
  American Statistician, 38133-134
• 15-minute read that describes three basic
  concepts in statistics/algebra

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regressors
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Same vectors, but detrended x y -3 3 0 -6 3 3
Raw vectors
x y 3 6 6 -3 9 6
Inner product 54 Non-orthogonal
inner product 0 uncorrelated
? But! ?