[FieldTrip] impact of skewed power distributions on data analysis

[Mike X Cohen]

Interesting discussion here. I think we should take a step back and
distinguish between trivial and nontrivial causes and consequences for the
skewed distribution. To some extent, the non-normal distribution is simply
the result of defining power as a squared distance -- distances are always
positive and squaring them means big values become really big. Consider the
following:

d = randn(10000,1); % random numbers
subplot(311), hist(d,500); % their distribution
subplot(312), hist(d.^2,500); % "power" distribution, also try a log-scaled
y-axis
subplot(313), hist(log(d.^2),500); % log-power distribution

The fact that power values have a power-law-like distribution is therefore
trivial.

But this leads to two non-trivial questions:
1) Is this distribution meaningful for brain function (beyond simply the
result of taking squared values)? People who study "the log-brain" and
fractal-like (or scale-free) organization of brain function would argue
that these distributions reveal meaningful insights into brain function,
and therefore it is not really an artifact for analyses. In other words,
large values are large for a reason; they might not be outliers that we
should attempt to compress.

2) Does it matter for real data analysis? I think this is Teresa's initial
concern. I'm inclined to think that it doesn't really matter, but that's
just based on the idea (hope!) that if it did really matter and the way we
do it is wrong, the field would have discovered this a long time ago. On
the other hand, this wouldn't be the first time that people have gotten
things wrong for decades.

I think the best way to investigate #2 would be with simulated data,
showing that a "true" effect is missed when not first computing log-power,
presumably because the effect was overshadowed by large-amplitude "noise"
(but somehow this would have to be physiological noise that wouldn't get
rejected during data cleaning). In empirical data, all you'd be able to do
is show a difference without knowing the right answer.

btw, make sure to be careful with baselining log-power -- any divisions
(e.g., dB or percent change) will be unstable/run off to infinity when
baseline power is close to zero, i.e., raw power is close to 1. The sign
might also get weird. Probably best to use a baseline subtraction.

Mike





--------------------------------------------><------------------------------------------------

Thanks for the reference.  In return, I also recommend reading Ciuparu and
Mures an's recent rebuttal:

European Journal of Neuroscience, Vol. 43, pp. 861–869, 2016,
doi:10.1111/ejn.13179 <http://dx.doi.org/10.1111/ejn.13179>
TECHNICAL SPOTLIGHT
Sources of bias in single-trial normalization procedures

They demonstrate that the positive bias Grandchamp and Delorme warned about
with single-trial baseline normalization was, in fact, due to the
correlated numerators and denominators in each of the baseline
normalization procedures they tested, which was, in turn, caused by the
skewed distributions of baseline power values.  They demonstrate that this
bias is reduced by using a much longer baseline, ideally incorporated into
the experimental design, but when that's not possible, stitching together
the baselines of many trials.

Neither article addresses my specific question of whether it would be even
better to log-transform the raw power values before averaging, so I suppose
I'll have to test it myself and write my own methods article!  🤓

I will go ahead and incorporate some of these alternative baseline
normalization methods in my git fork of FieldTrip as I go along with my own
analyses, so let me know if anyone else would find them useful, and I'll
submit a pull request to contribute them to the master branch.

Thanks for the fruitful discussion, all!
~Teresa


On Mon, Dec 19, 2016 at 8:01 PM, Alik Widge <alik.widge at gmail.com> wrote:

> Indeed, in a separate thread with Michael Cohen several months back he
> suggested precisely that paper.
>
> On Dec 19, 2016 5:07 PM, "Nicholas A. Peatfield" <nick.peatfield at
gmail.com>
> wrote:
>
>> I think this paper is relevant to this discussion.
>>
>> Grandchamp, R., & Delorme, A. (2011). Single-Trial Normalization for
>> Event-Related Spectral Decomposition Reduces Sensitivity to Noisy
Trials. *Frontiers
>> in Psychology*, *2*, 236. http://doi.org/10.3389/fpsyg.2011.00236
>>
>> https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3183439/
>>
>>
>>
>> On 19 December 2016 at 13:08, Teresa Madsen <tmadsen at emory.edu> wrote:
>>
>>> I appreciate everyone's feedback, but I still wonder if something is
>>> being missed.  I understand that the non-normally distributed power
values
>>> may be less of an issue when performing non-parametric stats or even a
>>> paired-samples t-test that looks at difference values which may be
normal
>>> even when the raw data isn't.  However, my concern comes into play even
>>> before these statistical comparisons are made, whenever any averaging is
>>> done to freq-type data across times, frequencies, trials, electrodes,
>>> subjects, etc.  That means any time any of these configuration options
are
>>> used for any of these functions, and probably more:
>>>
>>> ft_freqanalysis:          cfg.keeptrials or cfg.keeptapers = 'no';
>>> ft_freqgrandaverage:   cfg.keepindividual = 'no';
>>> ft_freqstatistics:         cfg.avgoverchan, cfg.avgovertime, or
>>> cfg.avgoverfreq = 'yes';
>>> ft_freqbaseline:          cfg.baseline = anything but 'no'
>>>
>>> In each case, if raw power values are averaged, the result will be
>>> positively skewed.  Maybe it's not a huge problem if all of the data is
>>> treated identically, but the specific case that triggered my concern
was in
>>> ft_freqbaseline, where the individual time-frequency bins are compared
to
>>> the mean over time for the baseline period.  For example, when using
>>> cfg.baselinetype = 'db', as Giuseppe Pellizzer suggested, the output
freq
>>> data does indeed have a more normal distribution over time, but the mean
>>> over the baseline time period is performed *before* the log transform,
when
>>> the distribution is still highly skewed:
>>>
>>>   meanVals = repmat(nanmean(data(:,:,baselineTimes), 3), [1 1
>>> size(data, 3)]);
>>>   data = 10*log10(data ./ meanVals);
>>>
>>> That's what I had originally done when analyzing data for my SfN poster,
>>> when I realized the background noise that shouldn't have changed much
from
>>> baseline was mostly showing a decrease from baseline of about -3dB.
>>>
>>> Now, I've realized I'm seeing this as more of a problem than others
>>> because of another tweak I made, which was to use a long, separate
baseline
>>> recording to normalize my trial data, rather than a short pre-trial
period
>>> as ft_freqbaseline is designed to do.  Averaging a few hundred
milliseconds
>>> for a baseline power estimate might be okay because overlapping time
points
>>> in the original data are used to calculate those power values anyway,
>>> probably making them less skewed, but also (it seems to me) more
arbitrary
>>> and prone to error.  I already offered my custom function BLnorm.m to
one
>>> person who was asking about this issue of normalizing to a separate
>>> baseline recording, and I would be happy to contribute it to FieldTrip
if
>>> others would appreciate it.
>>>
>>> Since a few people suggested using the median, and it is also suggested
>>> in Cohen's textbook
>>> <https://mitpress.mit.edu/books/analyzing-neural-time-series-data> as
>>> an alternative measure of the central tendency for skewed raw power
values,
>>> I wonder if the simplest fix might be to add an option to select mean or
>>> median in each of the functions listed above.  Another possibility
would be
>>> adding an option to transform the power values upon output from
>>> ft_freqanalysis.
>>>
>>> Would anyone else find such changes useful?
>>>
>>> Thanks,
>>> Teresa
>>>
>>>
>>> On Wed, Dec 14, 2016 at 4:22 AM, Herring, J.D. (Jim) <
>>> J.Herring at donders.ru.nl> wrote:
>>>
>>>> In terms of statistics it is the distribution of values that you do the
>>>> statistics on that matters. In case of a paired-samples t-test when
>>>> comparing two conditions, it is the distribution of difference values
that
>>>> has to be normally distributed. The distribution of difference values
is
>>>> often normal given two similarly non-normal distributions, offering no
>>>> complications for a regular parametric test.
>>>>
>>>>
>>>>
>>>> The non-parametric tests offered in fieldtrip indeed do not assume
>>>> normality, so you should have no problem there either.
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> *From:* fieldtrip-bounces at science.ru.nl [mailto:fieldtrip-bounces
at scie
>>>> nce.ru.nl] *On Behalf Of *Alik Widge
>>>> *Sent:* Tuesday, December 13, 2016 3:10 PM
>>>> *To:* FieldTrip discussion list <fieldtrip at science.ru.nl>
>>>> *Subject:* Re: [FieldTrip] impact of skewed power distributions on
>>>> data analysis
>>>>
>>>>
>>>>
>>>> In this, Teresa is right and we have observed this in our own EEG data
>>>> -- depending on one's level of noise and number of trials/patients, the
>>>> mean can be a very poor estimator of central tendency. My students are
>>>> still arguing about what we really want to do with it, but at least
one of
>>>> them has shifted to using the median as a matter of course for baseline
>>>> normalization.
>>>>
>>>>
>>>> Alik Widge
>>>> alik.widge at gmail.com
>>>> (206) 866-5435
>>>>
>>>>
>>>>
>>>> On Mon, Dec 12, 2016 at 6:45 PM, Teresa Madsen <tmadsen at emory.edu>
>>>> wrote:
>>>>
>>>> That may very well be true; to be honest, I haven't looked that deeply
>>>> into the stats offerings yet. However, my plan is to express each
>>>> electrode's experimental data in terms of change from their respective
>>>> baseline recordings before attempting any group averaging or
statistical
>>>> testing, and this problem shows up first in the baseline correction
step,
>>>> where FieldTrip averages raw power over time.
>>>>
>>>> ~Teresa
>>>>
>>>>
>>>>
>>>> On Mon, Dec 12, 2016 at 4:56 PM Nicholas A. Peatfield <
>>>> nick.peatfield at gmail.com> wrote:
>>>>
>>>> Correct me if I'm wrong, but, if you are using the non-parametric
>>>> statistics implemented by fieldtrip, the data does not need to be
normally
>>>> distributed.
>>>>
>>>>
>>>>
>>>> On 12 December 2016 at 13:39, Teresa Madsen <tmadsen at emory.edu>
wrote:
>>>>
>>>> No, sorry, that's not what I meant, but thanks for giving me the
>>>> opportunity to clarify. Of course everyone is familiar with the 1/f
pattern
>>>> across frequencies, but the distribution across time (and according to
the
>>>> poster, also across space), also has an extremely skewed, negative
>>>> exponential distribution. I probably confused everyone by trying to
show
>>>> too much data in my figure, but each color represents the distribution
of
>>>> power values for a single frequency over time, using a histogram and a
line
>>>> above with circles at the mean +/- one standard deviation.
>>>>
>>>> My main point was that the mean is not representative of the central
>>>> tendency of such an asymmetrical distribution of power values over
time.
>>>> It's even more obvious which is more representative of their actual
>>>> distributions when I plot e^mean(logpower) on the raw plot and
>>>> log(mean(rawpower)) on the log plot, but that made the figure even more
>>>> busy and confusing.
>>>>
>>>> I hope that helps,
>>>> Teresa
>>>>
>>>>
>>>>
>>>> On Mon, Dec 12, 2016 at 3:47 PM Nicholas A. Peatfield <
>>>> nick.peatfield at gmail.com> wrote:
>>>>
>>>> Hi Teresa,
>>>>
>>>>
>>>>
>>>> I think what you are discussing is the 1/f power scaling of the power
>>>> spectrum. This is one of the reasons that comparisons are made within
>>>> a band (i.e. alpha to alpha) and not between bands (i.e. alpha to
gamma),
>>>> as such the assumption is that within bands there should be a relative
>>>> change against baseline and this is what the statistics are performed
on.
>>>> That is, baseline correction is assumed to be the mean for a specific
>>>> frequency and not a mean across frequencies.
>>>>
>>>>
>>>>
>>>>  And this leads to another point that when you are selecting a
>>>> frequency range to do the non-parametric statistics on you should not
do
>>>> 1-64 Hz but break it up based on the bands.
>>>>
>>>>
>>>>
>>>> Hope my interpretation of your point is correct. I sent in
>>>> individually, as I wanted to ensure I followed your point.
>>>>
>>>>
>>>>
>>>> Cheers,
>>>>
>>>>
>>>>
>>>> Nick
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> On 12 December 2016 at 08:23, Teresa Madsen <tmadsen at emory.edu>
wrote:
>>>>
>>>> FieldTrippers,
>>>>
>>>>
>>>>
>>>> While analyzing my data for the annual Society for Neuroscience
>>>> meeting, I developed a concern that was quickly validated by another
poster
>>>> (full abstract copied and linked below) focusing on the root of the
>>>> problem:  neural oscillatory power is not normally distributed across
time,
>>>> frequency, or space.  The specific problem I had encountered was in
>>>> baseline-correcting my experimental data, where, regardless of
>>>> cfg.baselinetype, ft_freqbaseline depends on the mean power over time.
>>>> However, I found that the distribution of raw power over time is so
skewed
>>>> that the mean was not a reasonable approximation of the central
tendency of
>>>> the baseline power, so it made most of my experimental data look like
it
>>>> had decreased power compared to baseline.  The more I think about it,
the
>>>> more I realize that averaging is everywhere in the way we analyze
neural
>>>> oscillations (across time points, frequency bins, electrodes, trials,
>>>> subjects, etc.), and many of the standard statistics people use also
rely
>>>> on assumptions of normality.
>>>>
>>>>
>>>>
>>>> The most obvious solution for me was to log transform the data first,
>>>> as it appears to be fairly log normal, and I always use log-scale
>>>> visualizations anyway.  Erik Peterson, middle author on the poster,
agreed
>>>> that this would at least "restore (some) symmetry to the error
>>>> distribution."  I used a natural log transform, sort of arbitrarily to
>>>> differentiate from the standard decibel transform included in
FieldTrip as
>>>> cfg.baselinetype = 'db'.  The following figures compare the 2
distributions
>>>> across several frequency bands (using power values from a wavelet
>>>> spectrogram obtained from a baseline LFP recorded in rat prelimbic
>>>> cortex).  The lines at the top represent the mean +/- one standard
>>>> deviation for each frequency band, and you can see how those
descriptive
>>>> stats are much more representative of the actual distributions in the
log
>>>> scale.
>>>>
>>>>
>>>>
>>>>
>>>> ​​
>>>>
>>>> For my analysis, I also calculated a z-score on the log transformed
>>>> power to assess how my experimental data compared to the variability
of the
>>>> noise in a long baseline recording from before conditioning, rather
than a
>>>> short pre-trial baseline period, since I find that more informative
than
>>>> any of FieldTrip's built-in baseline types.  I'm happy to share the
custom
>>>> functions I wrote for this if people think it would be a useful
addition to
>>>> FieldTrip.  I can also share more about my analysis and/or a copy of
the
>>>> poster, if anyone wants more detail - I just didn't want to make this
email
>>>> too big.
>>>>
>>>>
>>>>
>>>> Mostly, I'm just hoping to start some discussion here as to how to
>>>> address this.  I searched the wiki
>>>> <http://www.fieldtriptoolbox.org/development/zscores>, listserv
>>>> <
https://mailman.science.ru.nl/pipermail/fieldtrip/2006-December/000773.html>
>>>>  archives
>>>> <
https://mailman.science.ru.nl/pipermail/fieldtrip/2010-March/002718.html>,
>>>> and bugzilla
>>>> <http://bugzilla.fieldtriptoolbox.org/show_bug.cgi?id=1574> for
>>>> anything related and came up with a few topics surrounding
normalization
>>>> and baseline correction, but only skirting this issue.  It seems
important,
>>>> so I want to find out whether others agree with my approach or already
have
>>>> other ways of avoiding the problem, and whether FieldTrip's code needs
to
>>>> be changed or just documentation added, or what?
>>>>
>>>>
>>>>
>>>> Thanks for any insights,
>>>>
>>>> Teresa
>>>>
>>>>
>>>>
>>>>
>>>> 271.03 / LLL17 - Neural oscillatory power is not Gaussian distributed
>>>> across time
>>>> <http://www.abstractsonline.com/pp8/#!/4071/presentation/24150>
>>>>
>>>> *Authors*
>>>>
>>>> **L. IZHIKEVICH*, E. PETERSON, B. VOYTEK;
>>>> Cognitive Sci., UCSD, San Diego, CA
>>>>
>>>> *Disclosures*
>>>>
>>>>  *L. Izhikevich:* None. *E. Peterson:* None. *B. Voytek:* None.
>>>>
>>>> *Abstract*
>>>>
>>>> Neural oscillations are important in organizing activity across the
>>>> human brain in healthy cognition, while oscillatory disruptions are
linked
>>>> to numerous disease states. Oscillations are known to vary by
frequency and
>>>> amplitude across time and between different brain regions; however,
this
>>>> variability has never been well characterized. We examined human and
animal
>>>> EEG, LFP, MEG, and ECoG data from over 100 subjects to analyze the
>>>> distribution of power and frequency across time, space and species. We
>>>> report that between data types, subjects, frequencies, electrodes, and
>>>> time, an inverse power law, or negative exponential distribution, is
>>>> present in all recordings. This is contrary to, and not compatible
with,
>>>> the Gaussian noise assumption made in many digital signal processing
>>>> techniques. The statistical assumptions underlying common algorithms
for
>>>> power spectral estimation, such as Welch's method, are being violated
>>>> resulting in non-trivial misestimates of oscillatory power. Different
>>>> statistical approaches are warranted.
>>>>
>>>>
>>>>
>>>> --
>>>>
>>>> Teresa E. Madsen, PhD
>>>> Research Technical Specialist:  *in vivo *electrophysiology & data
>>>> analysis
>>>>
>>>> Division of Behavioral Neuroscience and Psychiatric Disorders
>>>> Yerkes National Primate Research Center
>>>>
>>>> Emory University
>>>>
>>>> Rainnie Lab, NSB 5233
>>>> 954 Gatewood Rd. NE
>>>> Atlanta, GA 30329
>>>>
>>>> (770) 296-9119
>>>>
>>>> braingirl at gmail.com
>>>>
>>>> https://www.linkedin.com/in/temadsen
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> _______________________________________________
>>>> fieldtrip mailing list
>>>> fieldtrip at donders.ru.nl
>>>> https://mailman.science.ru.nl/mailman/listinfo/fieldtrip
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> --
>>>>
>>>> Nicholas Peatfield, PhD
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> --
>>>>
>>>> Nicholas Peatfield, PhD
>>>>
>>>>
>>>>
>>>>
>>>> _______________________________________________
>>>> fieldtrip mailing list
>>>> fieldtrip at donders.ru.nl
>>>> https://mailman.science.ru.nl/mailman/listinfo/fieldtrip
>>>>
>>>>
>>>>
>>>> _______________________________________________
>>>> fieldtrip mailing list
>>>> fieldtrip at donders.ru.nl
>>>> https://mailman.science.ru.nl/mailman/listinfo/fieldtrip
>>>>
>>>
>>>
>>>
>>> --
>>> Teresa E. Madsen, PhD
>>> Division of Behavioral Neuroscience and Psychiatric Disorders
>>> Yerkes National Primate Research Center
>>> Emory University
>>> Rainnie Lab, NSB 5233
>>> 954 Gatewood Rd. NE
>>> Atlanta, GA 30329
>>> (770) 296-9119
>>>
>>> _______________________________________________
>>> fieldtrip mailing list
>>> fieldtrip at donders.ru.nl
>>> https://mailman.science.ru.nl/mailman/listinfo/fieldtrip
>>>
>>
>>
>>
>> --
>> Nicholas Peatfield, PhD
>>
>>
>> _______________________________________________
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>>
>
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>



-- 
Teresa E. Madsen, PhD
Division of Behavioral Neuroscience and Psychiatric Disorders
Yerkes National Primate Research Center
Emory University
Rainnie Lab, NSB 5233
954 Gatewood Rd. NE
Atlanta, GA 30329
(770) 296-9119

-- 
Mike X Cohen, PhD
mikexcohen.com
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