Multivariate testing (MVT)
As the analogs of one- and two-sample or paired t-tests under UVT, the two prototypes can be expressed with the following null hypotheses, (2a)  H01MVT:α1=0,α2=0,...,αm=0, (2b) H02MVT:α11=α21,α12=α22,...,α1m=α2m.
In other words, the m regression coefficients associated with the m basis functions from each subject are brought to the group level and treated as the instantiated values of m simultaneous variables. When the effect estimates associated with the basis functions of ESM or ASM are treated as the values of m simultaneous response variables, the hypothesis (2a) or (2b) can be analyzed through MVT under the model (1). Geometrically, the data for H01MVT represent the group centroid (α1, α2, …, αm) in the m-dimensional real coordinate space ℝm (Table 1), and the associated one-sample Hotelling T2-test is performed to reveal whether the group centroid lies in the rejection region (outside of an m-dimensional ellipse centering around the origin in the case of H01MVT). Similarly, the data for H02MVT are expressed as two group centroids, (α11, α12, …, α1m) and (α21, α22, …, α2m), and the corresponding two-sample Hotelling T2-test is conducted to see if the hypothesis (2b) about the two centroids can be rejected. The hypothesis (2b) can be easily generalized to the situation with more than two groups of subjects (e.g., three genotypes) as well as more than one subject-grouping variable (e.g., sex, genotypes, and handedness) through the formulation of general linear testing (Chen et al., 2014). One noteworthy feature of MVT is that it allows those simultaneous effects to have different scales or units, unlike the traditional AN(C)OVA or univariate GLM in which all the levels of a factor are usually of the same dimension.
Table 1  Schematic comparisons among various testing methods.
One-sample
Methoda   MVT/LME   AUC   L2D   EXC (XUV and XMV)
H 0  α1 = … = αm = 0  α1 + … + αm = 0  ( α  1  2   + … + α  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)
H 0  α11 = α21, …, α1m = α2m  ∑  j = 1  m   α  1 j   = ∑  j = 1  m   α  2 j  ( ∑  j = 1  m   α  1 j  2    )   1 ∕ 2   = ( ∑  j = 1  m   α  2 j  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
a  The 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.
b  An 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.
c  The 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.
d  The horizontal and vertical axes represent time and the amplitude of HDR curve (dashed line).
e  The 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).
f  The 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.