Dear Cyril & Giovanni,
Not to hijack Giovanni's thread, but I always get confused by this and wondered if someone could clarify:
> A - make sure this is the correlation between of vectors you are computing, ie compute the cos of their angle (Pearson's > correlation center the data which can give a different result) --> cos_theta = (dot(v1,v2)) / (norm(v1)*norm(v2))
I understand this is how SPM calculates orthogonality of design matrices, but in the absence of centered vectors I don't find the result very meaningful and have always had the suspicion that I must be missing something...
For example, with two vectors pulled out of an SPM design matrix:
The pearson correlation between centered and non centered vectors will be identical (of course). In this case it's -0.237
The cos_theta is -0.237 for centered vectors (also as expected, cosine similarity is same as pearson for centered vectors)
but for non-centered vectors it's 0.024
Thus if I use cosine similarity I would come to very different conclusions about design efficiency. Namely, that the centered design is optimized for contrasts between v1 and v2 (negative correlation is good) but the non-centered design is less so. Even though both designs would give identical beta coefficients.
For this reason I've taken to detrending conditions of interest during estimation so that SPMs orthogonality checks are based on centered regressors and are more meaningful to me. This was the norm in SPM99, but got left out of SPM2 (and ever since).
Is there some way of interpreting the cosine similarity of non-centered vectors (with regards to design efficiency) that I'm not quite catching?
Cheers!
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