Table 4 Two-variate TP tests : implementations and applications

1. Make the two-variate PT test by the method given in Table 3 (a), which provides an improvement on making the two-variate PT test by Hotelling test as suggested in Table 8(1)(a) of Ref. Xu (2015a)

(a) Identify mRNA and lncRNA biomarkers for tumour vs normal in expression analysis

2. Make the FDA-based PT test by Table 3 (b), especially one-tailed z-test or one-tailed t-test by Eq. (32), which is a complementary to the FDA-based PT test listed in Table 2(1) of Ref. Xu (2015a), where a univariate two-tailed t test is made

(b) Identify mRNA and lncRNA biomarkers for 3-year & 5-year survival in expression analysis

3. Find probes that significantly differentiate not only phenotypes but also abnormal vs normal, as well as their common part

(c) Identify mRNA and lncRNA biomarkers for cancer grades I, II, III, & IV in expression analysis

(d) For each of the above cases (a)(b)(c), we use the FDA projection \(s_u(f)= \mathbf{s}^T {\mathop{ {\beta}}\limits^{\rightarrow}}\) to replace the original expression value for painting heatmaps and making the corresponding clustering analysis