Dear Emma,

Good to hear from you.  I've been puzzling over this since Steve & I discussed last week.  

Firstly, let me state that post-hoc power analyses are perilous.  See [1].

Second, the best way to find out the scale factor needed to increase your observed TFCE to a target level is to use fslmaths.  E.g. fslmaths  with your T-image and "-tfce 2 0.5 6" should give you the TFCE image randomise produces.  Then, playing around with different multiplication factors, scaling your T-image up or down with fslmaths -mul, will change the TFCE image by some amount.  This will require something like a binary search, but it will give you a concrete answer to "by how much did my data have to increase in order to reach a TFCE value of X.

Finally, if we make a bunch of assumptions, we can use some algebra to estimate what the scale factor is.  Steve correctly predicted the cubic relationship with height, but this neglected the extent term.  Below is a sketch of a proof, but if we assume that the statistic has a parabolic shape around the peak, then e(h) will scale with a 3/2-rd power term; the E=1/2 power makes this a 3/4-rd power, and so the overall approximate scaling factor is 3+3/4 or 4.25.  (Note that in the TFCE paper, RFT argues for H=2/3, in which case the power would be exactly 4).  So, if you require TFCE to increase by a multiplicative factor M, then the statistic image would need to be scaled by M^(1/4.25).  For your case of M=3, this is 1.295.


So, I'd try the empirical approach with fslmaths; after checking that you can reproduce your TFCE image exactly with no scaling, try a scaling of 1.3, and see how close that gets you to your target.

-Tom


Derivation:

For a statistic image Z, at a given voxel with intensity z, define TFCE as
    TFCE(z) = \int_0^z e(h)^1/2 h^2 dh
in pseudo-LaTeX, where e(h) is the extent of the cluster containing the given voxel based on a cluster-forming threshold h.

For a peak of interest, assume the TFCE score is dominated by the local cluster, and that the form of the statistic image about the peak is parabolic.  Let K>1 be the scalefactor that transforms the existing statistic image Z into a new, hypothetical dataset, i.e.
	Z* = K Z
Then, based on a parabolic assumption (see Eqn 5 in [2]), around the peak voxel
	e*(z_new) = e*(K z) = K^3/2 e(z)
Thus
	TFCE*(z*) = \int_0^z* e*(h*)^1/2 h*^2 dh*
with a change of variables (h*=K h, dh* = K dh) we have
	                              = \int_0^z* e*(K h)^1/2 K^2 h^2 K dh
and with the parabolic approximation 
	                              = \int_0^z* K^3/4 e(h)^1/2 K^2 h^2 dh
and re-arranging 
	                              = K^{3+3/4} \int_0^z* e(h)^1/2 h^2 dh
Finally, note that at the peak voxel with value z*_peak, e(h) is zero for h>z_peak, so we only need to consider the integral up to z_peak.  This gives peak TFCE value in the new image as
	TFCE_new(z*) = K^4.25 TFCE(z)

So, based on the local parabolic approximation, to obtain a M-fold increase in TFCE you'll need an M^(1/4.25) increase in z.


[1] Hoenig, J. M., & Heisey, D. M. (2001). The Abuse of Power: The Pervasive Fallacy of Power Calculations for Data Analysis. The American Statistician, 55(1), 1-6.

[2] Zhang, H., Nichols, T. E., & Johnson, T. D. (2009). Cluster mass inference via random field theory. Neuroimage, 44(1), 51–61. Elsevier. doi: 10.1016/j.neuroimage.2008.08.017.


On 11 May 2011, at 13:19, Emma Sprooten wrote:

> 
> 
> 
> -----------------------------------------------------------------------
> 
> source: https://www.jiscmail.ac.uk/cgi-bin/webadmin?A2=FSL;33b6d1f5.1105
> 
> On Tue, May 3, 2011 at 1:07 AM, Stephen Smith <[log in to unmask]> wrote:
> 
>     Hi - interesting, this hasn't come up before.    Probably Tom
> should check what I say below?..
> 
> 
>     On 2 May 2011, at 13:23, Emma Sprooten wrote:
> 
>>    Hello,
>> 
>>    I'm trying to do a post-hoc power analysis for a negative result  I obtained with TBSS using TFCE, ie. checking that the negative  result isn't due to sample size. For example, I would like to  estimate the sample size required to get a significant result, using  the critical TFCE value derived from another (positive) analysis  with the same sample.
>> 
>>    I was thinking of calculating (somehow) the raw T-stat required  to obtain a critical TFCE (180,000) given the current "extent/e(h)"  for the voxel with the maximum effect size. From there I was going  to do a normal power analysis to estimate the required sample size  to get this T.
> 
>     Yes - this is the way to go I think and is pretty easy.  So -
> find the best TFCE voxel and work out how much you would have to
> multiply it by to reach the critical value (let's say 1.8).
> 
>     Then convert this to a scaling factor for the raw t-stat
> values??given that the E parameter is 2, then looking at Eq1 in the
> TFCE paper, I think after integration over h (meaning t), you'll have
> a cubic relationship between the max t value and the TFCE statistic -
> though this could also depend on how the shape e(h) interacts with
> this.  So a rough estimate would be that you take the cube-root of 1.8
> and say that you would have needed that much larger peak effect size
> to get significance.
> 
>     It might well be that this rough answer is changed (hopefully not
> too much) by the effect of e(h)??
> 
>     Cheers.
> 
> 
> 
> 
> 
>>    And/or do it the other way: finding out what the extent  (supporting area) would need to be to obtain the critical TFCE given  the maximum raw T ("h") of my negative analysis.
>> 
>>    I can't get my head around this for several reasons. First I'm  not sure what "h" represents in the TFCE equasion. I'm guessing in  my case it is the raw T-statistic but I'm not sure. Also, following  the TFCE equasion, in reality e(h) changes according to h, so I'm  wondering whether it is right to assume e(h) remains constant while  caluclating the required h?
>> 
>>    I'm partly hoping I'm making this more difficult than it reallu  is.... Is there a simpler solution? Has anyone else done this  already (can't find anything so far). Or is there an existing  equation describing TFCE as a function of sample size? Or otherwise,  ideas on how to check for power issues in a less empirical way are  also very welcome!
>> 
>>    Thanks!
>>    Emma
>> 
>>    p.s. I'm not mathematician 8-/
>> 
> 
> 
> 
> ---------------------------------------------------------------------------
>     Stephen M. Smith, Professor of Biomedical Engineering
>     Associate Director,  Oxford University FMRIB Centre
> 
>     FMRIB, JR Hospital, Headington, Oxford  OX3 9DU, UK
>     +44 (0) 1865 222726  (fax 222717)
>     [log in to unmask]    http://www.fmrib.ox.ac.uk/~steve
> 
> ---------------------------------------------------------------------------
> 
> 
> 
> 
> -- 
> The University of Edinburgh is a charitable body, registered in
> Scotland, with registration number SC005336.
> 
> <TFCE_equations.pdf><TFCE_equations.doc>

____________________________________________
Thomas Nichols, PhD
Principal Research Fellow, Head of Neuroimaging Statistics
Department of Statistics & Warwick Manufacturing Group
University of Warwick
Coventry  CV4 7AL
United Kingdom

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