Yes, you are quite correct. You can change the Gamma parameter to something larger and it will produce more conservative results. And you are perfectly at liberty to do so.
The default value for the 2nd level inference corresponds to the prior standard deviation of the beta value. Briefly,
2nd level PPMs use a prior over the regression coefficients, beta, with zero mean and (co)variance Cb. The quantity SPM.PPM.Cb is an empirical estimate of Cb. The rest of the formula computes how much of this variance projects onto the contrast you are testing.
You can read more about this in the original paper:
But you don't have to use the above value.
From: SPM (Statistical Parametric Mapping) <[log in to unmask]> on behalf of Paul Glad Mihai <[log in to unmask]>
Sent: 14 March 2014 12:01
To: [log in to unmask]
Subject: [SPM] Thresholding PPMs at the 2nd level
Dear Bayesian inference fans,
I was looking into Bayesian inference using thresholded posterior
probability maps (PPMs). At the first level I find it pretty straight
forward. After estimating the approximate means and covariances using
variational Bayes, PPMs can be obtained with the help of the effect size
threshold (gamma) and the conditional posterior (p_T). According to the
SPM book in chapter 25 I can use an explorative approach to get a
gamma = 0 and p_T = 1-1/(number of voxels)
This is also the default implementation in the SPM script spm_getSPM.m
at line 750
u_default = 1 - 1/SPM.xVol.S
So, for example for p_T = 0.95 and gamma = 0.3 you would get a PPM
showing a 95% probability that the effect size is greater than 0.3% of
the global mean.
What about at the second level, i.e. effects over subjects?
The effect size threshold is by default calculated using the empirical
prior parameter covariance (SPM.PPM.Cb), as taken from spm_getSPM.m line 520
Gamma = full(sqrt(xCon(Ic).c'*SPM.PPM.Cb*xCon(Ic).c));
The comments in the code say:
% The default is one conditional s.d. of the contrast
I can't really wrap my head around it. What does that mean? It has
nothing to do with the per cent global mean. Perhaps I'm not
understanding the covariance computation properly at the second level.
What happens if I change the default Gamma at the second level to
something greater? It should provide a more conservative effect size
I'm looking forward to your answers.