Table 1.

Schematic comparisons among various testing methods.

One-sample
Methoda	MVT/LME	AUC	L2D	EXC (XUV and XMV)
H0	α1 = … = αm = 0	α1 + … + αm = 0	${({\alpha }_1^2+\dots +{\alpha }_m^2)}^{1∕2}=0$	α1 = … = αm
Dimensions in ℝm	0	m−1	m−1	1
DFs for F-statisticb	m, n−m−q+1	1, n−q	1, n−q	m−1, (m−1)(n−q)
Geometric representationc of H0 and H1 (m = 2)
Geometric representationd of HDR when detection failure occurs due to improper H0 formulation	no		no
Two-sample or paired
Method	MVT	AUC	L2D	EXC (XUV and XMV)
H0	α11 = α21, …, α1m = α2m	$\sum _{j=1}^m{\alpha }_{1j}=\sum _{j=1}^m{\alpha }_{2j}$	${(\sum _{j=1}^m{\alpha }_{1j}^2)}^{1∕2}={(\sum _{j=1}^m{\alpha }_{2j}^2)}^{1∕2}$	α11 − α21 = … = α1m − α2m
Dimensions in ℝm	0	m−1	m−1	1
DFs for F-statistic	m, n−m−q+1	1, n−q	1, n−q	m−1, (m−1)(n−q)
Geometric representatione of H0 and H1
Geometric representationf of HDR when detection failure occurs due to improper H0 formulation	no

^aThe table is meant to show the dimensions of each null hypothesis and an instantiation in the rejection domain while the whole rejection domain is not represented here. For example, the reject region of one-sample Hotelling T²-test for MVT (2a) is outside of an m-dimensional ellipse.

^bAn interesting fact is that the numerator degrees of freedom for the F-statistic under MVT and UVT are the dimensions of the complementary space to the associated null hypothesis H₀, or the dimensions of the alternative hypothesis H₁.

^cThe two axes represent the two weights associated with the two basis functions. The whole rejection regions are not shown here, and the shaded (gray) and solid (black) areas correspond respectively to the null hypothesis H₀ space and an instantiation (and its dimension) in the alternative hypothesis H₁ space. Detection failure occurs when the group centroid falls on the diagonal line other than the origin under AUC and EXC.

^dThe horizontal and vertical axes represent time and the amplitude of HDR curve (dashed line).

^eThe two axes represent the two weights associated with the two basis functions. The whole rejection regions are not shown here, and the shaded and sold areas correspond respectively to the null hypothesis H₀ space and an instantiation (and its dimension) in the alternative hypothesis H₁ space. The two types of line thickness (or dot size) differentiate the two groups (or conditions).

^fThe horizontal and vertical axes represent time and the amplitude of HDR curves. The two line types, dashed and dotted, differentiate the two groups or conditions.