Table 2.

Expected values of the DSE table under different nominal models. First two rows show expected mean squared (MS) values under the separable noise model, for whole and global sum of squares. Third and fourth rows show expected MS normalized to the total variability A-var for the separable model. Final two rows show the expected normalized MS under a naive, default model of independent and identically distributed (IID) data in time and space. ${\overline{\sigma }}^2$ is the average of the I voxel-wise variances, ρ is the common lag-1 autocorrelation, and ${\overline{\overline{\sigma }}}^2$ is the average of the $I^2$ elements of the voxels-by-voxels spatial covariance matrix. This shows that D-var and S-var are equal under independence but, when normalized, differ by about ρ; this is a general result that doesn't depend on the separable noise model used here (see Appendix D.8).

	A-var	D-var	S-var	E-var
Separable Model: Whole	${\overline{\sigma }}^2$	$\frac{1}{2}\frac{T-1}{T}\left(1-\rho \right){\overline{\sigma }}^2$	$\frac{1}{2}\frac{T-1}{T}\left(1+\rho \right){\overline{\sigma }}^2$	$\frac{1}{T}{\overline{\sigma }}^2$
Separable Model: Global	${\overline{\overline{\sigma }}}^2$	$\frac{1}{2}\frac{T-1}{T}\left(1-\rho \right){\overline{\overline{\sigma }}}^2$	$\frac{1}{2}\frac{T-1}{T}\left(1+\rho \right){\overline{\overline{\sigma }}}^2$	$\frac{1}{T}{\overline{\overline{\sigma }}}^2$
Separable Model: Whole, % of A	1	$\frac{1}{2}\frac{T-1}{T}\left(1-\rho \right)$	$\frac{1}{2}\frac{T-1}{T}\left(1+\rho \right)$	$\frac{1}{T}$
Separable Model: Global, % of A	${\overline{\overline{\sigma }}}^2/{\overline{\sigma }}^2$	$\frac{1}{2}\frac{T-1}{T}\left(1-\rho \right){\overline{\overline{\sigma }}}^2/{\overline{\sigma }}^2$	$\frac{1}{2}\frac{T-1}{T}\left(1+\rho \right){\overline{\overline{\sigma }}}^2/{\overline{\sigma }}^2$	$\frac{1}{T}{\overline{\overline{\sigma }}}^2/{\overline{\sigma }}^2$
IID Model: Whole, % of A	1	$\frac{1}{2}\frac{T-1}{T}$	$\frac{1}{2}\frac{T-1}{T}$	$\frac{1}{T}$
IID Model: Global, % of A	$\frac{1}{I}$	$\frac{1}{2}\frac{1}{I}\frac{T-1}{T}$	$\frac{1}{2}\frac{1}{I}\frac{T-1}{T}$	$\frac{1}{I}\frac{1}{T}$