Correlation coefficient description

The description of the degree of retention data correlation is more complicated than it appears. For example, the 2D retention maps cannot be characterized by a simple correlation coefficient (Slonecker et al., 1996) since it fails to describe the datasets with apparent clustering (Fig. 12.2f). Several mathematical approaches have been developed to define the data spread in 2D separation space (Gray et al., 2002 Liu et al., 1995 Slonecker et al., 1996), but they are nonintuitive, complex, and use multiple descriptors to define the degree of orthogonality. 

The following description and corresponding MathCad Worksheet allows the user to test if two correlation coefficients are significantly different based on the number of sample pairs (N) used to compute each correlation. For the Worksheet, the user enters the confidence level for the test (e.g., 0.95), two comparative correlation coefficients, r, and r2, and the respective number of paired (X, Y) samples as N and N2. The desired confidence level is entered and the corresponding z statistic and hypothesis test is performed. A Test result of 0 indicates a significant difference between the correlation coefficients a Test result of 1 indicates no significant difference in the correlation coefficients at the selected confidence level. 

The square of the correlation coefficient is a satisfactory 0.90. This equation gives a rather reliable and quantitative description of the influence on the amide-induced replacement of the leaving group in 2-X-4-phenylpyrim-idines. The factor +0.55 reflects the fraction of the molecules of 4-phenylpyrimidine R and F are zero) that undergoes an Sn(ANRORC) aminodehydrogenation. This has indeed been experimentally established (see Section II,C,l,g). 

It is important to know the effectiveness of the model for predicting values however, it is also important to know the strength of the linear relationship between the two variables (known and predicted) being studied. This is achieved using the linear correlation coefficient (Pearson s product moment correlation coefficient), r, as a descriptive measure for the strength of the linear relationship (straight line) between the two variables ... 

The description of large data tables by the usual univariate statistics (mean, standard deviation, range,. ..) and by histograms is still used in recent literature. Comparison between categories is made by the use of category means and ran s. Sometimes, the correlation coefficients are considered. The discussion of the extracted information can be wide-ranging and difficult to understand immediately. 

Cotton-Mouton effect), NMR chemical shift and coupling constants, the optical rotation of polarized light and correlation coefficients between different properties. Extensions to incorporate long-range interactions have also been elaborated11 and it has even been possible to adapt RIS theory for the description of the dynamics of transitions between rotational isomeric states.12,13... 

Carbon atoms are classified depending on their hybridization and whether their neighbors are carbon atoms or heteroatoms. Halogen atoms are classified by the hybridization and oxidation state of the C atom to which they are attached. O, S, Se, N, and P are classified in different ways. The model uses 120 different atom-type descriptions and has been developed with a training set of 893 compounds. Observed versus calculated log Kow showed a correlation coefficient of 0.926 and a standard deviation of 0.496. This method has been implemented in the Toolkit. Applications are shown in Figures 13.4.5 and 13.4.6 for the same compounds used to illustrate the Broto et al. method (Figs. 13.4.2 and 13.4.3). 

A principal component analysis is reasonable only when the intrinsic dimensionality is much smaller than the dimensionality of the original data. This is the case for features related by high absolute values of the correlation coefficients. Whenever correlation between features is small, a significant direction of maximum variance cannot be found (Fig. 3.7) all principal components participate in the description of the data structure hence a reduction of data by principal component analysis is not possible. 

The next step in the description of the DNA sequence is the analysis of the pair-wise nearest-neighbor correlations Cy, for example, the normalized probabilities to And successive pairs of the nucleotides i and j. Eor all 16 possible successive dinucleotides in the coding strand of DNA, only the functions Cag and CcT correlate with OGT. The excess probabilities to find ApG and CpT pairs in the coding DNA are increasing significantly with OGT, correlation coefficient R = 0.68 and 0.601 (Fig. 2, upper row). Remarkably, the codon bias explains the observed sequence correlations in the coding parts of DNA. Eirst,... 

The correlation coefficient and the standard deviation indicate that the plot provides an excellent description of the data... 

Cox and Clifford (1982) have proposed a way of presenting correlation coefficient data for a suite of rocks in a diagrammatic form. Their method, which is purely descriptive, uses the Pearson product-moment coefficient of correlation and is an attempt to utilize and display graphically the large amount of information contained in a correlation matrix, without resorting to plotting the enormous number of... 

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