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Non-orthogonal regressors: concepts and consequences

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

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design matrix 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 Scan number regressors

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BOLD signal Time = 1 22 + + error x1x1 x2x2 e Source: spm course 2010, Stephan

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The beta’s are estimated on a voxel-by-voxel basis high beta means regressor explains much of BOLD signal’s variance (i.e. strongly covaries with signal)

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Problem of non-orthogonal regressors total variance in BOLD signal Y

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Orthogonal regressors total variance in BOLD signal X1 X2 every regressor explains unique part of the variance in the BOLD signal Y X2 =+ X1

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Orthogonal regressors total variance in BOLD signal X1 X2 There is only 1 optimal linear combination of both regressors to explain as much variance as possible. Assigned beta’s will be as large as possible, stats using these beta’s will have optimal power Y X2 =+ X1

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non-orthogonal regressors 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 Y X2 =+ X1

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Entirely non-orthogonal total variance in BOLD signal regressor 2 regressor 1 Betas can’t be estimated. Variance can not be assigned to one or the other Y X2 =+ X1

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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 r1r r1 r2

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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 θ = acos(4 / (|r1| * |r2|) = about 35 degrees r1 r2 35

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Orthogonalizing one vector wrt another: it matters which vector you choose! (Gram-Schmidt orthogonalization) Orthogonalize r1 wrt r2: u1 = r1 – proj r2 (r1) u1 = [1 2] – (r1 r2)/(r2 r2) u1 = [ ] Inner product: u1 r2 = (-0.6 * 2) + (1.2 * 1) = 0 orthogonality 1 2 r1 r2 u1

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Orthogonal is defined as: X’Y = 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! orthogonality & uncorrelatedness An aside on these two concepts

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please read Rodgers et al. (1984) Linearly independent, orthogonal and uncorrelated variables. The American Statistician, 38: Will be in the FAM folder as well Orthogonal because: Inner product 1*5 + -5*1 + 3*1 + -1*3 = 0

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please read Rodgers et al. (1984) Linearly independent, orthogonal and uncorrelated variables. The American Statistician, 38: Will be in the FAM folder as well Detrend: Mean(X) = -0.5 Mean(Y) = 2.5 X_detY_det ================== Mean(X_det) = 0 Mean(Y_det) = 0 ================== Inner product: 5 Orthogonal, but correlated!

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1 2 r1 r2 detrend r1_detr2_det r1r

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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? orthogonality & uncorrelatedness “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).” LinkLink

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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 Your regressors correlate

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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 = 0r 2 = 0 correlated vector cos(81) = 0.16 r = 0.16r 2 = r 2 indicates how much variance is common between the two vectors (2.56% in this example). Note: -1 ≤ r ≤ 1 and 0 ≤ r 2 ≤ 1

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Correlated regressors: variance from single regressor to shared

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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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-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 How to deal with correlated regressors?

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Orthogonalize regressor A wrt regressor B: all shared variance will now be assigned to B. How to deal with correlations between contrast and rest of design matrix?

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orthogonality r1 r2

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orthogonality 1 2 r1 r2 total variance in BOLD signal regressor 1 regressor 2

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

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

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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) Voon et al. (2010) Mechanisms underlying dopamine-mediated reward bias in compulsive behaviors. Neuron Highly correlated!

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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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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, don’t 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 Take home messages

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f8a efb2196bdc90fc419 Combines SPM book and Rik Henson’s 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, 38: minute read that describes three basic concepts in statistics/algebra Interesting reads

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regressors

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Inner product: 54 Non-orthogonal Same vectors, but detrended: xy inner product: 0 uncorrelated xy Raw vectors: But!

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