[FieldTrip] Z-transformed coherence equation after spectral decomposition with Hanning taper

[Schoffelen, J.M. (Jan Mathijs)]

Ciao Sara,

TLDR:
The formula from https://doi.org/10.1016/j.jneumeth.2007.02.011<https://eur01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fdoi.org%2F10.1016%2Fj.jneumeth.2007.02.011&data=05%7C02%7Cfieldtrip%40science.ru.nl%7C69259190c22c4d7efd6008dd3c48dabc%7C084578d9400d4a5aa7c7e76ca47af400%7C1%7C0%7C638733005784420762%7CUnknown%7CTWFpbGZsb3d8eyJFbXB0eU1hcGkiOnRydWUsIlYiOiIwLjAuMDAwMCIsIlAiOiJXaW4zMiIsIkFOIjoiTWFpbCIsIldUIjoyfQ%3D%3D%7C0%7C%7C%7C&sdata=EfpwDS2cG%2F5%2Fib4Ns%2BKEP24X0%2B%2FgFUwQbHkRpWrt07g%3D&reserved=0>, contains, next to the inverse hyperbolic tangent transforms, bias-correction terms (subtraction of the per condition 1/(df-2)), and a normalization (scaling term). If the number of observations is the same in both conditions, then the subtraction will not have an effect (they drop out of the equation). Either or not applying the df’s in the denominator will only change the distribution of the values but will not change the outcome of a permutation test (when the statistical inference is performed across trials within a subject). When using the test statistic in a group level analysis, the denominator will ’stabilise’ the coherence values a bit across subjects if there’s a large variability in bias across subjects (i.e. a large range in terms of the difference in number of trials per subject).

The short answer is: the df corresponds to the total number of (officially) independent observations, i.e. 2*(the sum of the total number of tapers applied). In your case, with a Hanning taper, this amounts to 2*(# trials). The formula is also described here: https://doi.org/10.1016/j.jneumeth.2006.07.011<https://eur01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fdoi.org%2F10.1016%2Fj.jneumeth.2006.07.011&data=05%7C02%7Cfieldtrip%40science.ru.nl%7C69259190c22c4d7efd6008dd3c48dabc%7C084578d9400d4a5aa7c7e76ca47af400%7C1%7C0%7C638733005784443440%7CUnknown%7CTWFpbGZsb3d8eyJFbXB0eU1hcGkiOnRydWUsIlYiOiIwLjAuMDAwMCIsIlAiOiJXaW4zMiIsIkFOIjoiTWFpbCIsIldUIjoyfQ%3D%3D%7C0%7C%7C%7C&sdata=cyYspxwo%2Fqk0K1zvaD4QZuDv1lmcHrnT46gUBQh%2BGkE%3D&reserved=0>

Best wishes,
Jan-Mathijs


On 23 Jan 2025, at 18:14, Mazzini, Sara via fieldtrip <fieldtrip at science.ru.nl> wrote:

Dear Fieldtrip community,

I have a question with regards to the Z-transformed coherence, as explained in Schoffelen et al. (2005, 2011).
In the articles, Z-spectra difference in coherence between a channel pair is calculated after computing the spectral decomposition with a multitaper method and the coherence between the channel pair.
I was wondering how the Z-transformed coherence can be calculated when spectral decomposition is computed using a Hanning taper instead.
Would it then only be necessary to compute the inverse hyperbolic tangent of the coherence value for each condition and then subtract these (i.e. atanh(coherence_condition1) – atanh(coherence_condition2))?

Thank you in advance for any help with this question!

Best wishes,

Sara

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