chapter  2
116 Pages

Statistical Space for Multivariate Correlations

Abstract ................................................................................................... 78 2.1 Introduction .................................................................................... 78 2.2 Statistical Correlation Analysis ...................................................... 82 2.2.1 Correlation Coefficient ....................................................... 82 2.2.2 Mono-Linear Regression .................................................... 86 2.2.3 Multi-Linear Regression .................................................. 101 2.3 Inferential Analysis of Statistical Samples .................................. 108 2.3.1 Z-Statistics ........................................................................ 108 2.3.2 The Selection Theory ........................................................118 2.3.2.1 From Global (Collectivity) to

Local (Sample) Statistics ....................................118 2.3.2.2 The A-consequence: the chi-square statistics .... 122 2.3.2.3 The B-Consequence: the t-Student Statistics .... 129 2.3.2.4 The C-consequence: The Fisher Statistics......... 134 2.4 Robustness of Correlations: Confidence Intervals ....................... 138 2.4.1 Validation of Statistical Information ................................ 138 2.4.1.1 Synopsis of Statistical Concepts ....................... 138 2.4.1.2 Validation of Statistical Hypothesis for

Median Value in Bilateral Case ......................... 142 2.4.1.3 Validation of Statistical Hypothesis for the

Difference Between the Median Values ............ 147

2.4.1.4 Validation of Statistical Hypothesis By Testing the Similitude of Dispersions .......... 152

2.4.2 Confidence Intervals ........................................................ 154 2.4.2.1 Chi-square Statistics .......................................... 154 2.4.2.2 The Chi-Square-Test for

Normal Concordance ......................................... 159 2.4.2.3 The Chi-Square Test for the Parametrical

Concordances and Likelihood ........................... 166 2.4.3 Robustness of Regression: Multi-Regression

Analysis of Variance (ANOVA) ....................................... 170 2.5 Conclusion ................................................................................... 182 Keywords .............................................................................................. 186 References ............................................................................................. 186 Author’s Main References ........................................................... 186 Specific References ...................................................................... 187 Further Readings .......................................................................... 191

ABSTRACT

Aiming to prepare the conceptual-computational ground for correlating chemical structure with biological activity by the celebrated quantitative structure-activity relationships (QSARs), the fundamental statistical advanced frameworks are here detailed to best understand of the classical multilinear regression analysis to be next generalized by an algebraic (in quantum Hilbert space) reformulation of in terms of data vectors and orthogonal conditions (to be exposed in Chapter 3).