Table 2 Three typical projection scores

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

(a)  Support vector machine (SVM) (Suykens 1999; Suykens et al. 2002), as suggested in Table 4(c) of Ref. Xu (2015a)

(b)  Sparse logistic regression (Shevade and Keerthi 2003; Koh et al. 2007)