# Time frequency statistics - comparing 3 conditions

Michael Wibral michael.wibral at WEB.DE
Wed Nov 28 12:56:50 CET 2007

```Hi Helen,

you wrote:" The other alternative seems to be using a statistic based on linear regression; that is, to evaluate whether the dual task activity is a mix of both single tasks..."

comparing TFRs of this kind of triple task sets is actually a fundamental mathematical problem.
Say you have task 3 that is a combination of tasks 1and 2 and you want to test wether activity(3) is more or less than the sum of activity(1) and activity(2).
The problem is that when you compute TFRs you are adding/avaraging the amplitude/power values over trials or tasks which is phase INSENSITIVE as is everything that you do afterwards (i.e. summing or regressing). This is fundamentally different in task 3: here the activties induced by both subtasks are added phase sensitively within each trials, i.e. the signals are added physically  inside the head, no matter how you later analyse them. Later power/amplitude analysis will not get rid of any phase sensitive effects in this "in-the-head addition" first step.

In mathematical terms:
While it is true that  FFT(a+b) = FFT(a) + FFT(b) this holds only for the complex values (linearity of the fourier transform);
for the amplitudes (or power) the following is true:
|FFT(a+b)| <= |FFT(a)| + |FFT(b)|  (i.e. this is your null hypothesis)
Hence, you can only test whether this expectation is violated by an interaction of the two tasks in terms of power INCREASES: |FFT(task3)|>|FFT(task1)|+|FFT(task2)| ? (alternative hypothesis)
Any differences in the opposite direction (i.e. power decreases) are meaningless!

You can of course sum the full (complex valued) FFTs in each trial of task 1 and 2 over first  to mimick the phase sensitivity of the "in-the-head" summation when both task elements are performed in task3. You could subsequently take the average (of the comlex valued summation results) over trials and compare this to the average of the FFTs (complex valued) of task3.  For the comparison you would of course have to take something like the real part, (imaginary part, amplitude, power,...) where '>' or '<' make sense.
However, in the end this would be more or less the same as comparing the ERPs (because they are just the phase sensitive sums):

I hope the above made some sense. If not don't not hesitate to ask any further questions. If any list member would like to comment on this I'd be happy to hear some feedback.

Best Regards,
Michael

P.S.: The math of the problem is the same that prevents the separation of time-frequency power in evoked (possible) and 'purely induced' activity (impossible) that people sometimes ask for.

> -----Ursprüngliche Nachricht-----
> Von: FieldTrip discussion list <FIELDTRIP at NIC.SURFNET.NL>
> Gesendet: 28.11.07 11:33:05
> An: FIELDTRIP at NIC.SURFNET.NL
> Betreff: [FIELDTRIP] Time frequency statistics - comparing 3 conditions

>
> Hi,
> I wonder if anyone can give me some advice about the appropriate type of
> analysis for my data set.
> I am currently analysing data from a repeated-measures design, in which
> there are three conditions: two are single tasks, and the third is a dual
> task consisting of both single tasks. I have obtained TFRs (using the
> multitaper method) for each condition, showing relative increase/decrease in
> power. However, I am uncertain which type of statistical analysis to use for
> this data. The single tasks seem to produce a different pattern of activity,
> and I have compared them with a cluster-based permutation test using a
> t-statistic (using 'freqstatistics' with the 'montecarlo' method). However,
> the dual task is expected to consist of a combination of the activity
> patterns for both single tasks (and possibly some kind of additional
> integration process). so comparing this to each of the two single tasks
> using a t-statistic may not be suitable. An F test does not provide enough
> information and would need to be followed by t-tests anyway. The other
> alternative seems to be using a statistic based on linear regression; that
> is, to evaluate whether the dual task activity is a mix of both single tasks.
> Can anyone recommend an unbiased way to analyse this kind of data using the
> functions implemented in fieldtrip? I would be very grateful for any ideas.
> Thanks,
> Helen
>
> ----------------------------------
> The aim of this list is to facilitate the discussion between users of the FieldTrip  toolbox, to share experiences and to discuss new ideas for MEG and EEG analysis. See also http://listserv.surfnet.nl/archives/fieldtrip.html and http://www.ru.nl/fcdonders/fieldtrip.
>

----------------------------------
The aim of this list is to facilitate the discussion between users of the FieldTrip  toolbox, to share experiences and to discuss new ideas for MEG and EEG analysis. See also http://listserv.surfnet.nl/archives/fieldtrip.html and http://www.ru.nl/fcdonders/fieldtrip.
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