Hello,
some days ago I posted a question about partialling out covariates in a
first vs. second-level model. In particular, I was interested in
partialling out covariates both within an experimental condition (as can
be done easily with the parametric modulator approach at the first
level) and, most importantly, between experimental conditions. I worked
out an apparently useful first-level approach. Donald McLaren gave some
feedback on this, so I am sharing the procedure with the list.
I am attaching a figure that illustrates the procedure with simulated
block-design and event-related data (three conditions + baseline). In
these simulated data each of the conditions have a different mean !=
baseline (see top-left panel), and the one covariate is a slightly noisy
version of the simulated EPI i.e., it explains almost all of the within-
and between-condition differences.
The procedure requires carrying out two steps.
Step 1: fit a first-level model with one single condition for all of the
non-baseline trials, and include your covariates as parametric
modulators. The parametric modulators need to be mean centered (i.e.,
subtract mean). Mean centering is not strictly necessary: AFAIK SPM
automatically mean-centers your parametric modulators (and assumes that
parametric modulator = 0 for trials not part of condition, otherwise
estimates for non-condition trials will be affected).
Step 2: fit your regular first level model with all of the experimental
conditions but use as dependent variable the residuals of the covariate
prediction from Step 1. In particular, to create your dependent data
extract the beta estimated for the parametric modulators at Step 1, and
use them (along with the global scaling factor stored here: SPM.xGX.gSF,
and the covariate-related regressors from SPM.xX.X) to estimate the EPI
data based on the covariate alone (don't include intercept term here).
Finally, subtract from your EPI data the covariate-based prediction. The
condition-specific estimates based on this residual data will be
independent of any variance explained by your covariate (both within and
between conditions).
Two nice features of this procedure:
- it does not alter the beta estimate for the baseline (verified with
real data);
- it works with N experimental conditions >=2 (with N = 2 experimental
conditions + baseline a one-step procedure can be devised, omitted here
for brevity).
- it does not require averaging covariates across trials from the same
condition (as would be necessary with a second-level approach).
One practical consideration:
- with real data include the head-motion parameters both at Step 1 and
at Step2. In this way, your beta estimates for the covariates (Step 1)
will not be "influenced" by head motion, and you will not subtract head
motion on each of Steps 1 and 2.
Feedback more than welcome.
Best,
Bruno
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Bruno L. Giordano, PhD
Voice Neurocognition Laboratory (CCNi, Glasgow Univ) &
Affiliate member of Music Research Dept. (McGill Univ, Montréal)
URL: http://www.brunolgiordano.net
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