# cluster statistic on one sample

Eric Maris e.maris at DONDERS.RU.NL
Tue Oct 21 11:03:50 CEST 2008

```Dear Floris and others,

> If I understand what SnPM (for fMRI) does well, it uses a permutation
> test on one sample, creating a probability distribution by randomly
> changing the sign of the observations. I quote (taken from
> http://www.sph.umich.edu/ni-stat/SnPM/)
>
> "Under the null hypothesis we can permute the labels of the effects of
> interest. One way of implementing this with contrast images is to
> randomly change the sign of each subject's contrast. This
> sign-flipping approach can be justified by a symmetric distribution
> for each voxel's data under the null hypothesis. While symmetry may
> sound like a strong assumption, it is weaker than Normality, and can
> be justified by a subtraction of two sample means with the same
> (arbitrary) distribution.

This quote is about a study involving TWO experimental conditions that are
manipulated within every subject. This is also called a paired-samples
design. So, it is NOT a one-sample study. The confusion probably arises from
the fact that a paired-samples (dependent-samples) T-test is identical to a
one-sample T-test on the difference scores. For a paired-samples T-test it
makes sense to construct a reference distribution by randomly switching the
signs of the difference scores. For a one-sample T-test on the data of a
single condition (e.g., the IQ-scores of a group of students), however, it
does not make sense to think about such an operation. (This is a brief,
intuitive explanation, which could be made more precise, but only at the
expense of more text.)

In your quote, also the concepts "symmetry of the distribution" and
"exchangeability" are mentioned. To keep things clear in our minds, it is
good to know that

1. Some statisticians (e.g., Tom Nichols, the author of your quote) motivate
permutation tests from considerations involving the shape of the probability
distribution (such as symmetry). Other statisticians (like Fortunato
Pesarin, and I) motivate permutation tests from considerations about
exchangeability between experimental conditions.

2. "Exchangeability" is a very general concept that can be used in very
different contexts. Some statisticians (e.g., Tom Nichols) use the term
"exchangeability" to denote a property of the probability distribution from
which the subjects are drawn, not mentioning the fact that these subjects
were observed in two experimental conditions. Other statisticians (like I)
use "exchangeability" with explicit reference to the data observed in the
two experimental conditions. For me, exchangeability involves that the
probability distribution of paired observations is invariant under random
permutations of the members of these pairs. This assumption of
exchangeability implies that the data in the two experimental conditions
have the same marginal probability distribution.

(If you like this explanation, Floris, you could join Jan-Mathijs in his
effort to make a Wiki-tutorial about the statistical rationale of
permutation tests.)

Don't start cursing the statisticians now!

Eric Maris

>
> Hence the null hypothesis here is:
>     H0: The symmetric distribution of (the voxel values of the)
> subjects' contrast images have zero mean.
>
> And some more detail on the assumptions:
>
> (..) to analyze a group of subjects for population inference, we need
> to only assume exchangeability of subjects. The conventional
> assumption of independent subjects implies exchangeability, and hence
> a single exchangeability block (EB) consisting of all subjects.
>
> (On a technical note, the assumption of exchangeability can actually
> be relaxed for the one-sample case considered here. A sufficient
> assumption for the contrast data to have a symmetric distribution, is
> for each subject's contrast data to have a symmetric but possibly
> different distribution. Such differences between subjects violates
> exchangeability of all the data; however, since the null distribution
> of the statistic of interest is invariant with respect to
> sign-flipping, the test is valid.)
>
> I don't see why this approach wouldn't be applicable for MEG data?
> As a side note, comparing my regression weights with a condition of
> all zeros with a dependent sample T-test works well, and is
> mathematically equivalent to a one-sample T-test as far as I can see,
> at least in the parametric domain?
>
>
> Best wishes,
> Floris
>
> --
> --
> Floris de Lange
> http://www.florisdelange.com
>
> ----------------------------------
> 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
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