Dear Xiang,

Wu Xiang wrote:
> Thanks so much Jan, it was like a detailed manual for parametric modeling
> and help me much. There remain several questions needing further help.
>
> Q1. Previously I simplified the experiment for discussion, actrually it has
> 8 blocks. Is 8 blocks usually sufficient for quadratic trend? If so, since
> quadratic trend includes linear trend in spm, I would like to do it. I have
> never done quadratic trend before, for 8 blocks, should it be [-29.5 -13.5
> -4.5 -0.5 0.5 4.5 13.5 29.5], that, the differences are [4^2, 3^2, 2^2, 1^2,
> 2^2, 3^2, 4^2]?

I *think* that your contrast weights should be

[1:8].^2 - repmat(mean([1:8].^2),1,8)

which is the mean-corrected version of vector 1:8 raised to the power of 2.
But this is without the linear trend.

If you want to include both linear and the orthogonalized quadratic trends
manually (as parametric modulators) you can derive the values using this:

X = [1:8;[1:8].^2]';
X = spm_orth(X);

>
> (btw, I checked the designed metrixs with "Time Modulation" and "Parametric
> Modulation" by adding 2st order trend. As you said, they both add two
> columns, one for linear and one for quadratic, although the values in the
> columns are different between "Time Modulation" and "Parametric Modulation",
> but it would doesn't matter. I feel "Time Modulation" is more convenient
> because I do not need to input values:))

Yes, the "Time Modulation" is the more convenient way in your case.

> Q2. I am confused by the below words. Given model 1 is without parametric
> regressor, and the first beta value is task; model 2 is with linear
> parametric regressor, and the first and second beta value are task and
> linear trend respectively. Do you mean adding parametric regressor would
> change the beta value of task, in other word, the first beta value is
> different in model 1 and model 2? What I care is, with model 1, the first
> beta value reflects the difference between task and rest. So, in model 2,
> what's task versus rest? (btw, the rest is not modeled explicitly)
> ------------------
> [1:4] increments by 1 and will get you the same effect, but the size of the
> beta will be different (it should be half as large). (The size of the beta
> scales with the regressors.)
> ------------------

I was referring to the difference in betas for the parametric modulators,
when you enter either [1:4] or [2:2:8]. I did not mean the beta of the task
(the simple onset). When you scale your parametric value differently (e.g.
change the increment), then your beta will change as well.

You can convince yourself on this very easily by doing the following:

y = [1:8]' + rand(8,1)*2; % some data (linear trend plus noise)
x1 = [ones(8,1) [1:8]'-repmat(mean(1:8),8,1)]; %const+linear (incr 1)
x2 = [ones(8,1) [2:2:16]'-repmat(mean(2:2:16),8,1)]; %const+linear (incr 2)
beta1 = x1\y
beta2 = x2\y

In this example, you will find that the first beta is equal in both models,
but that the second beta in model 2 (beta2(2)) is half as large as that in
model 1 (beta1(2)).

Nevertheless, including a parametric modulator (pm) will probably change
the beta of the task compared to a model without the pm.

In the model with the pm, the task regressors will capture the average
activation of the events in that regressor minus the linear trend that you
are modeling in the pm (which will be capture by the beta of the pm).

In the model without the pm, the beta of the task has to accommodate both
the average activation and the linear trend (if there is one). Hence, in
that case it is likely that the beta of this model is different that the
beta of the task in the model with the pm.

If you have a substantial linear trend in your data, then the model with
the pm is more sensitive, because it fits better to the data. In addition,
the interpretability of the effects (betas) is increased as you can then
clearly separate the average activation from the linear trend (because your
modeled them separately).

In the model with the pm, the beta of your task regressor (the one with the
plain onsets) will model the effect of task vs. rest.

Cheers,
Jan

>
> Xiang
>
>

--
Jan Gläscher, Ph.D. Div. Humanities & Social Sciences
+1 (626) 395-3898 (office) Caltech, Broad Center, M/C 114-96
+1 (626) 395-2000 (fax) 1200 California Blvd
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