B-score We get a projection score $$B_{ \tilde{\mathbf{s}}_u }(\mathbf{s})$$ by Eq. (28), namely, the projection of statistics $$\mathbf{s}_u- \tilde{\mathbf{s}}_u$$ onto a particular direction $$S_j^{-0.5} {\mathop{ {\beta}}\limits^{\rightarrow}}$$. Actually, we are lead to a univariate one-tailed test on this projection score, which may be simply implemented by either one-tailed z-test or one-tailed t-test. Also, we may estimate the univariate distribution of the score, and then compute $$\underline{p}_{h,j}^{\omega }$$ based on the estimated distribution
FDA score We get $${\mathop{ {\beta}}\limits^{\rightarrow}}$$ by making the Fisher discriminative analysis (FDA) on the control samples and the case samples, and then obtain the projection score $${\mathop{ {\beta}}\limits^{\rightarrow}}^T\mathbf{s}$$ with $$\mathbf{s}$$ given by Eq. (4), where the arrow of $${\mathop{ {\beta}}\limits^{\rightarrow}}$$ points from the control to the case, as illustrated in Fig. 1b, while the classical FDA does not care about which direction of two choices is taken as the arrow. A directional test can be made by either one-tailed z-test or one-tailed t-test, using the statistics
$$\quad \quad \quad \quad \quad \quad \quad \quad t_{\mathop{ {\beta}}\limits^{\rightarrow}} = \frac{{\mathop{ {\beta}}\limits^{\rightarrow}}^T\mathbf{s} }{{ \sigma }}, \ \sigma ^2=\alpha _0 \sigma ^2_0+ \alpha _1 \sigma ^2_1,\quad \quad \quad \quad \quad \quad \quad \quad$$(32)
where $$\sigma ^2_0, \sigma ^2_1$$ are the sample variances of the projections of control–case samples onto $${\mathop{ {\beta}}\limits^{\rightarrow}}$$, respectively, and $$\alpha _0, \alpha _1$$ are corresponding proportions
We may also perform a non-directional test with the arrow of $${\mathop{ {\beta}}\limits^{\rightarrow}}$$ ignored, by using a two-tailed z-test or t-test, which is suggested in Table 2(1) of Ref. Xu (2015a) as one example of the boundary-based two-sample test or BBT in short
Learning LDA score We may also perform either a directional test or a non-directional test as above, but with $${\mathop{ {\beta}}\limits^{\rightarrow}}$$ obtained by