Hi Eliza,
> Do you know if it is necessary to standardize (scale to unit std dev)
> any user-defined regressors, such as motion, before entering them into
> the SPM stats analysis?
My understanding of this, in case no-one else has replied, is that the
short answer is no. My longer answer follows:
Rescaling a regressor will change the beta for that regressor alone*,
so if they are nuisance regressors that are not tested with any
contrasts, then it doesn't matter how they are scaled (in mathematical
terms, it is the vector space spanned by the nuisance regressors that
matters, and this doesn't change by scaling them; or adding one
nuisance regressor to another -- which includes the special case of
de-meaning/centring nuisance covariates, assuming that your constant
term is a nuisance variable)
Statistical tests of the regressors (i.e. they are not just nuisance
covariates, but are included in contrasts) will have identical
t/F-values (and hence p-values) under rescaling. This is because both
t and F statistics will be affected equally by the scaling in their
numerator and denominator.
The "contrast" images (numerators of t-statistics) will change for
contrasts that include the rescaled covariates (contrasts are simply
linearly weighted combinations of the betas, which have changed). This
can be important for multi-subject studies, if the different subjects
had different scaling for equivalent regressors that were included in
contrasts at a lower level, being analysed at the second. Usually,
motion etc are nuisance covariates, so this doesn't matter.
If you are taking the covariates up to a higher level analysis, then
they should be comparable between subjects. I think this is a little
subtle though, as one could argue that "comparable" would suggest
re-scaling the covariates to have similar variances (for example); or
one could argue that comparable would mean the same physical units.
E.g. if one subject had age in years and another in months, the
covariates should be rescaled to be both on the same scale, but if two
subjects both have movement parameters in mm and radians, then these
should be left as they are. I think the latter makes sense.
*(This post is already probably too long and complicated, but the
starred statement above isn't quite true... for rank-deficient
designs, where a pseudo-inverse solution is used, the betas for the
other regressors can change, but only in a way such that the contrast
results for "estimable contrasts" will remain the same. I've had a
quick check that this holds in practice, but I don't think I can
clearly explain why in theory... probably again due to the
vector-space nature of the linear model. Anyway, again, the short
answer is rescaling doesn't matter.)
Phew, sorry the long answer turned out to be so long, I hope it (or
more likely the short version!) was helpful,
Ged
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