Bscore

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 onetailed test on this projection score, which may be simply implemented by either onetailed ztest or onetailed ttest. 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 onetailed ztest or onetailed ttest, 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 nondirectional test with the arrow of \({\mathop{ {\beta}}\limits^{\rightarrow}}\) ignored, by using a twotailed ztest or ttest, which is suggested in Table 2(1) of Ref. Xu (2015a) as one example of the boundarybased twosample test or BBT in short

Learning LDA score

We may also perform either a directional test or a nondirectional 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)
