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Hi SPM users,

I was wondering how the regressor entered for a parametric modulation is
scaled (I know that it is mean-corrected to zero, but then what?).

In other words, how is the data transformed between
SPM.Sess(i).U(j).P.P and SPM.Sess(i).U(j).u


Thanks in advance for any help!

~Cary
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Hi Cary,

The regressor entered for a parametric modulation is scaled via a Euclidean
normalization and then mean-centered to create the final parametric values.
 The Euclidean normalization transforms the set of parametric values entered
by the user into a set of new values, which are chosen such that if you
square each of them and them add the resultant squared values the result
will always be 1.

You can see how this works by creating a row vector of values and feeding it
to spm_en (the bit of SPM code that actually performs the normalization).
For example, first create a row vector with the numbers 1-8 by typing
a=[1:8]' in the MATLAB window.  Next, type b=spm_en(a).  The values of b
will turn out to be 0.0700, 0.1400, 0.2100, 0.2801, 0.3501, 0.4201, 0.4901,
and 0.5601.  Now square each of these and add them all up by typing
c=sum(b.*b) in the MATLAB window.  See how the result equals 1?

In the next step, the values of b are mean-centered. You can see what they
are by typing d=b-mean(b), the result of which is -0.2450, -0.1750, -0.1050,
-0.0350, 0.0350, 0.1050, 0.1750, 0.2450.

If you are modeling using a basis function of 1s (i.e., an FIR model), you
will actually see the mean-centered values above (i.e., d) in your design
matrix.  They will be entered in columns that are distinct from the columns
that code for the "average" response for a particular trial type.  Each of
the values in d will be associated all of the FIR parametric regressors for
a particular trial.

What if you are modeling with a canonical HRF?  In this case, I believe that
the values in the single parametric regressor column will be composed by
multiplying the canonical HRF with the first value in d for trial 1, the
second trial in d for trial 2, etc.  Basically, you multiply the transformed
parametric values by whatever basis function is being used to model the data
   (I think).

****The use of a Euclidean normalization raises some important issues that I
wouldn't mind getting some feedback about.*****

First, a Euclidean normalization will transform the values 10, 20, 30, 40,
50, 60, 70, 80 in exactly the same way as it transforms the values 1, 2, 3,
4, 5, 6, 7, 8.  In both cases, you end up with new values that are 0.0700,
0.1400, 0.2100, 0.2801, 0.3501, 0.4201, 0.4901, and 0.5601.  Thus, using
such a transformation "compresses" the variability present in one set of
parametric values more than it "compresses" the variability present in the
other set of parametric values, such that after the transformation both sets
of parametric values have exactly the same variance.

Now, what if the two sets of parametric regressors are RTs for two different
trial types?  And, let's say that one wants to determine whether the
relationship between RT and MR signal is stronger for one trial type than
for another?  If MR signal scales linearly with some absolute measure of RT,
then we'll get a higher beta value for the parametric regressor whose
original, non-transformed values are more variable (e.g., 10, 20, 30, etc.)
than for the parametric regressor whose original values were less variable
(e.g., 1, 2, 3, etc.), even though MR signal scales with RT in exactly the
same way for both of these regressors!  Thus, there appears to be an
assumption that MR signals scales with some kind of "normalized" measure of
RT rather than an absolute measure (e.g., milliseconds).  Could someone
please comment on this?

Second, regarding normalized measures, would there be a big difference
between using a Euclidean normalization (followed by mean-centering) to
transform the original parametric values and using a z-transformation?  Both
methods will accomplish mean-centering and both will involve unequal
"compression" of the variability present in each of the original sets of
parametric values (i.e., there will be more compression for the sets of
parametric values that have greater variance).  So, is there any reason for
preferring a Euclidean normalization (followed by mean-centering) to a
Z-transformation of the original parateric values?

Third, if one assumes that MR signal varies linearly with some absolute
measure of a parametric value (e.g., milliseconds for RT), then would one
want to use a mean-centered version of the original parametric values
without performing a Euclidean transformation (e.g., for the values 1,2,3
enter -1, 0, and 1)?  In this case, one would get the same betas for two
parametric regressors that differed in the variability of their original
values (e.g., 1, 2, 3 versus 10, 20, 30)because the variability in MR signal
would be proportionally greater for the more variable regressor than for the
less variable regressor.  However, there might be a magnitude difference
between the columns that could be problematic.  Any thoughts?


Yours,

Daniel Weissman
Center for Cognitive Neuroscience
Duke University
Durham, NC 27705