Our work benefited significantly from the statistical computational language and environment R, its many packages, and the great support of the R community. All the plots were created in R with the base graphics library. Special thanks are due to Helios de Rosario for his help in technical details of using the R package phia. The research and writing of the paper were supported by the NIMH and NINDS Intramural Research Programs of the NIH/HHS, USA. 1The efficiency in the statistics context measures the optimality of a testing method. A more efficient test requires a smaller sample size to attain a fixed power level.

Appendix A

List of acronyms used in the paper
AN(C)OVA  Analysis of (co)variance
ASM  Adjusted-shape method
AUC  Are under the curve
ESM  Estimated-shape method
EXC  Effect-by-component interaction
FPR  False positive rate
FSM  Fixed-shape method
GLM  General linear model
HDR  Hemodynamic response
IRF  Impulse response function
L2D  Euclidian (L2) distance
LME  Linear mixed-effects
MAN(C)OVA  Multivariate analysis of (co)variance
MVM  Multivariate modeling
MVT  Multivariate testing
UVM  Univariate modeling
UVT  Univariate testing
XMV  Multivariate testing for interaction
XUV  Univariate testing for interaction.

Appendix B

Formulation of multivariate testing in the presence of one or more within-subject factors
As discussed in Chen et al. (2014), all the within-subject factors are flattened into ℝ1 under the multivariate model (MVM) formulation (1). Once the regression coefficient matrix A is estimated through solving the MVM system (1) with the least squares principle, each general linear test (GLT) can be expressed as a function of A, (A1) H0:Lu×q Aq×m Rm×v=0u×v, where the hypothesis matrix L, through premultiplying, specifies the weights among the rows of A that are associated with the between-subjects variables (groups or subject-specific quantitative covariates), and the response transformation matrix R, through postmultiplying, formulates the weighting among the columns of A that correspond to the m response variables. It is assumed that L and R are full of row- and column-rank respectively, and u ≤ q, v ≤ m. The matrix L (or R) plays a role of contrasting or weighted averaging among the groups of a between-subjects factor (or the levels of a within-subject factor).
The conventional multivariate test (MVT) can be performed through any of the four multivariate statistics (Wilks' λ, Pillai-Bartlett trace, Lawley-Hotelling trace, and Roy's largest root) with R = Im once the hypothesis matrix L in (A1) is constructed (Appendix B in Chen et al., 2014). For instance, suppose that we consider an m-variate model with the following explanatory variables: three genotypes of subjects, age and their interactions. Via effect coding with the first genotype as reference, the model matrix X in (1) is of q = 6 columns: one for the intercept, two for the three genotypes, one for age, and two for their interactions. Accordingly, the q = 6 rows in A represent the overall mean, the respective effects for the second and third genotypes relative to the overall mean, the age effect associated with the overall mean, and the respective age effects for the second and third genotypes relative the average age effect. MVT for the main effect of genotypes, the genotype-by-age interaction, and the age effect for the first genotype can be obtained under (A1) respectively with L1=[0 1 0 0 0 00 0 1 0 0 0],L2=[0 0 0 01 00 0 0 0 0 1],L3=[0 0 0 1 −1 −1],R1=R2=R3=Im.
Similarly, both univariate and within-subject multivariate tests can be formulated by obtaining both the hypothesis matrix L and the response transformation matrix R in (A1) (Appendix C in Chen et al., 2014). In addition, all the post-hoc t- and F-tests (options -gltCode and -glfCode respectively in 3dMVM) are also constructed as MVT under the platform (A1). For instance, the effect under a specific level and the contrast between two levels of a within-subject factor through -gltCode are evaluated essentially by a one-sample and a paired t-test respectively, while the main effect of a within-subject factor through -glfCode is assessed by a within-subject multivariate test.
When R = 1m×1, the hypothesis (A1) solely focuses on the between-subjects explanatory variables (columns in the model matrix X of MVM; 1) while the effects among the levels of the within-subject factors are averaged (or collapsed). Therefore, the AUC approach (4) can be conceptually tested under the multivariate framework (A1), respectively for one group, L4=1,R4=1m×1, and two groups, L5=(0,1),R5=1m×1, even though they would be readily performed through the conventional one- and two-sample t-tests.
When applied to the effect-by-component interaction (9a or 9b) with ESM (EXC in Table 1), the MVM framework offers both univariate (XUV) and multivariate (XMV) approaches, which are tested under the same formulation, respectively for one group (A1), H0:α1=α2=...=αm,L6=1,R6=[Im−1−11×(m−1)],xma and two groups, H0:α11−α21=α12−α22=...=α1m−α2m,L7=(0,1),R7=R6.
For XMV, standard multivariate testing statistics (Wilks' λ, Pillai-Bartlett trace, Lawley-Hotelling trace, Roy's largest root) are constructed through the eigenvalues of the “ratio” H(H + E)−1 between the SSPH matrix H for the hypothesis (A1) against the SSPE matrix E for the errors in the full model (Rencher and Christensen, 2012). In contrast, the univariate approach XUV is tested through the formulation of an F-statistic with the numerator and denominator sums of squares being as tr(H(RT R)−1) and tr(E(RTR)−1) under the sphericity assumption (Fox et al., 2013), and the F-value can be adjusted through the Greenhouse and Geisser (1959) or Huynh and Feldt (1976) correction if the sphericity assumption is violated.
All the applications so far in the literature have been focused on either MVT or UVT. In other words, a strict MVT applies to the situations of truly multivariate nature while a purely UVT is adopted to the conventional AN(C)OVA or GLM. However, if we treat the components from ESM as simultaneous response variables, the presence of one or more within-subject factors (e.g., two task conditions in the experimental data of this paper) necessitates a partial MVT. Here we demonstrate a strategy to formulate partial MVT with the construction of L and R using a template of two-way within-subject ANOVA with factors A and B of a and b levels respectively. Suppose that we want to model the levels of factor A as a simultaneous response variables (e.g., components or effect estimates from ESM) while factor B is considered as an explanatory variable (e.g., conditions). MVT for the effect of B can be achieved through the following specifications in (A1), L=Iq,R=Ia⊗R(B).
Similarly, if the levels of factor B are modeled as b simultaneous response variables while factor A is considered as an explanatory variable, we have the following MVT specifications for the effect of A, L=Iq,R=R(A)⊗Ib.
The notations R(A)=[Ia−1−11×(a−1)] and R(B)=[Ib−1−11×(b−1)] above are conveniently the effect coding matrices for factors A and B respectively.