Degree of Freedom

AFNI Message Board
1

Dear experts,
I am a little confused regarding the degrees of freedom (DOF) and their relevance for seed-based functional connectivity.

[ol]
[li] After a regression analysis (e.g., the removal of M confound variables in a GLM), what are the DOF of the residual timeseries (i.e., errts)? Are they simply the number of time points or we must subtract to it M, that is, the number of regressors in the model?
[/li][li] Assuming that the latter is true, what happens when we compare timeseries that have been through different models (due to censoring or to different regression matrix)? I would say that the r-values of a correlation analysis should be transformed in z-score = z-fisher * sqrt(dof-3) as it is done in www.pnas.org/cgi/doi/10.1073/pnas.1301725110. However, this does not seem a common approach, surprisingly even when censoring is used!
[/li][/ol]
Thanks
Best
Daniele

2

Daniele,

You do seem to raise a good point! So for seed-based correlation analysis, when there are unequal numbers of confounding effects (including censoring) involved across subjects, it would be more appropriate to perform the following operation (as opposed to the simple Fisher transformation) before group analysis:

(1/2) * ln((1+r)/(1-r)) * sqrt(DOF-2)

Thanks for the suggestion!

3

Gang,

Just to clarify, did you mean to type (1/2) * ln((1+r)/(1-r)) * sqrt(DOF-2) or (1/2) * ln((1+r)/(1-r)) * sqrt(DOF-3) . If it’s the former, could you clarify why it’s not DOF-3 as was suggested in Daniele’s original post?

Thanks!

Phil

4

Phil,

could you clarify why it’s not DOF-3 as was suggested in Daniele’s original post?

The conventional formula of converting Pearson correlation to standardized Z-score is (1/2) * ln((1+r)/(1-r)) * sqrt(N-3), where N is the sample size (number of time points in the time series in FMRI data), or (1/2) * ln((1+r)/(1-r)) * sqrt(DOF-2), in which DOF is N-1 for the typical Pearson correlation computation in a regression model with only one regressor (intercept). I just generalized the latter formula to the situation with multiple regressors.

5

Thank you Gang for the clarification.
Best
Daniele