8. Cancer Gene Diagnosis of Tian et al. Microarray

We developed the New Theory of Discriminant Analysis after R. A. Fisher (theory). Although there are five severe problems of discriminant analysis, theory solves five problems completely. Especially, Revised IP-OLDF (RIP) based on MNM and Method2 firstly succeed in the cancer gene analysis (Problem5) from 1970. RIP decomposes six microarrays into the many SMs those are signals (MNM = 0) explained in Chap. 1. Although Revised LP-OLDF decomposes the microarray into many SMs as same as RIP, we find the defect of Revised LP-OLDF that cannot find all SMs from the microarray in Chap. 4. However, Revised LP-OLDF can find many SMs faster than RIP. It may be convenient for many researchers to analyze SMs found by Revised LP-OLDF. Tian’s microarray consists of 173 subjects (36 False subjects and 137 True patients) and 12,625 genes. In this chapter, Revised LP-OLDF decomposes Tian’s microarray into the 104 SMs. We analyze 104 SMs by the standard statistical method such as one-way ANOVA, t-test, Ward cluster analysis, PCA, logistic regression, and Fisher’s LDF. Although we expected standard statistical methods were useful for cancer gene diagnosis, only logistic regression could discriminate 104 SMs correctly, and other methods did not show the linear separable facts. Because Revised LP-OLDF discriminates 104 SMs, and the range of 104 RatioSVs is [8.34%, 22.79%], we make signal data by 104 Revised LP-OLDF discriminant scores (LpDSs) instead of 12,625 genes. By this breakthrough, hierarchical cluster methods can separate two classes as two clusters entirely. In addition to these results, the Prin1 axis of PCA indicates proper malignancy indexes as same as 104 malignancy indexes. Thus, we reconsider the signal data is the signal. Moreover, we examine the characteristic of 104 LpDSs precisely as same as Chap. 7 using the correlation analysis.