Euclidean distance (L2D)
As an alternate dimension reduction approach, the null hypotheses associated with the Euclidean or L2 distance (L2D) for ESM can be formulated respectively as (5a) H01L2D:(∑j=1mαj2)1/2=0, (5b) H02L2D:(∑j=1mα1j2)1/2=(∑j=1mα2j2)1/2.
In other words, one captures the overall magnitude for each subject using the L2-distance of the m regression coefficients from no response, and then performs one- or two-sample t-test on the distances.
For ASM, the null hypotheses with the focus on the canonical basis are (6a) H0CAN:α1=0, (6b) H0CAN:α11=α21.
And the null hypotheses for L2D (Calhoun et al., 2004; Steffener et al., 2010) are tested with the first two bases, (7a) H0L2D:sgn(α1)(α12+α22)1/2=0, (7b) H0L2D:sgn(α11)(α112+α122)1/2=sgn(α21)(α212+α222)1/2 or with all the three bases, (8a) H0L2D:sgn(α1)(α12+α22+α32)1/2=0, (8b) H0L2D:sgn(α11)(α112+α122+α132)1/2=sgn(α21)                                                                        (α212+α222+α232)1/2, where sgn is the sign function. That is, the L2D for ASM is similar to the L2D for ESM, but using the two or three weights associated with the two or three basis functions in ASM and assigning the sign of the canonical response to the resultant L2-distance.
Their geometrical interpretations are as follows (Table 1). The data for H01L2D lie on an ℝm−1 iso-sphere, and the associated test for (5a) is executed on the radius of the ℝm−1 iso-sphere, leading to no geometrical distortion (but not necessarily true statistically). On the other hand, the data for H02L2D are on two ℝm−1 iso-sphere surfaces, and the associated test for (5b) acts on the radius difference between the two ℝm−1 iso-spheres, resulting a detection failure when the two HDR curves have roughly the same radius.