(a) Multivariate test per probe If probes are independent, we make a multivariate test on each $$[condition, \ sample]$$ matrix slice per probe. Such a multivariate test can be implemented by Algorithm 1 or one of the methods given in Table 2
(b) LDA-based multivariate test we make the map $$s_u(f)= \mathbf{s}^T {\mathop{ {\beta}}\limits^{\rightarrow}}$$ onto $${\mathop{ {\beta}}\limits^{\rightarrow}}$$ per probe f with $${\mathop{ {\beta}}\limits^{\rightarrow}}$$ obtained by either FDA or learning-based methods in Table 2, and make a multivariate test on $$[ s_u(f_1), \cdots , s_u(f_g)]^T$$ to consider multiple probes $$f_1, \ldots , f_g$$ jointly
(c) Bilinear MDA-based test We make the matrix-variate discriminative analysis (MDA) (see Eq.(33) & Eq.(34) in Ref. Xu (2015a)) to obtain $$\mathbf{v}, {\mathop{ {\beta}}\limits^{\rightarrow}}$$, based on which we test $${\mathop{ {\beta}}\limits^{\rightarrow}}= \mathbf{0}$$ by a multivariate test on $$\mathbf{s}_v= \mathbf{v}^T \mathbf{S}$$ given $$\mathbf{v}$$ fixed and test $$\mathbf{v}= \mathbf{0}$$ by a multivariate test on $$\mathbf{s}_{\mathop{ {\beta}}\limits^{\rightarrow}}= {\mathop{ {\beta}}\limits^{\rightarrow}}^T \mathbf{S} ^T$$ given $${\mathop{ {\beta}}\limits^{\rightarrow}}$$ fixed
Alternatively, we may make test on the scalar statistics $$s_{v{\mathop{ {\beta}}\limits^{\rightarrow}}}= \mathbf{v}^T \mathbf{S} {\mathop{ {\beta}}\limits^{\rightarrow}}$$