Table 2.

Maximum log-likelihood formulations of both univariate and multivariate methods applied in the group-level analysis.

	Univariate	Multivariate
Group analysis model	$\hat{β_{u}} = X_{G} β_{G_{u}} + {\tilde{η}}_{G_{u}}$	$\hat{B} α_{G} = X_{G} β_{G} + \tilde{η_{G}}$
Inputs to group level analysis	Subject-level effects from N subjects $(\hat{β_{u}})$ ; Within-subject variances of the effect (1^st part of $V_{G_{u}}$ );	Subject-level effects of m voxels in a local neighborhood from N subjects $(\hat{B})$ ; Within-subject variances of the effect (1^st part of $V_{G_{u}}$ );
Estimated parameters	Univariate group inference $(β_{G_{u}})$ ; Between-subject variance of the effect (2^nd part of $V_{G_{u}}$ )	Group inference assigned to the center voxel (β_G); Between-subject variance of the effect (2^nd part of V_G)
Objective function	${max}_{β_{G_{u}}, σ^{2}} L (\hat{β_{u}}; β_{G_{u}}, V_{G_{u}}) = - \frac{n}{2} ln (2 π) - \frac{1}{2} ln (\| V_{G_{u}} \|) - \frac{1}{2} {(\hat{β_{u}} - X_{G} β_{G_{u}})}^{T} V_{G_{u}}^{- 1} (\hat{β_{u}} - X_{G} β_{G_{u}})$	${max}_{α_{G}, β_{C}, σ_{G}^{2}} L (\hat{B} α_{G}; β_{G}, V_{G}) = - \frac{n}{2} ln (2 π) - \frac{1}{2} ln (\| V_{G} \|) - \frac{1}{2} {(\hat{B} α_{G} - X_{G} β_{G})}^{T} V_{G}^{- 1} (\hat{B} α_{G} - X_{G} β_{G}), w . r . t {\begin{matrix} α^{(1)} \geq K \sum_{m = 2}^{M} α^{(m)} \\ α^{(1)} \geq 0, \dots, α^{(M)} \geq 0 \\ ‖ \hat{B} α_{G} ‖_{2} = 1 \end{matrix}$