Pearson correlation

The relationship between renal impairment and the absorption and disposition of HMR1964 (insulin glulisine) will be assessed by regressing pharmacokinetic parameters onto CLcr. Regression parameter estimates ( standard error) with confidence intervals and coefficients of correlation (Pearson) with p-values for test of difference from zero will be reported. Scatter plots of the concentration time profiles and pharmacokinetic parameters against creatinine clearance will be produced. 

Eighty-three women in the study also provided breast milk samples within 6 months of the birth of their child. The C17-C19 homologues in breast milk and cord blood were moderately correlated (Pearson s r=0.29 p<0.05), while no correlation was found for the light- and moderately-chlorinated homologues. Actual values of PCBs in milk were not provided. 

Correlation (product-moment correlation, Pearson correlation) is a statistical measure for the relation between two or more sets of variables. 

Table 2.21. Correlation (Pearson) coefficient matrices between the hydrous metal contents and grain proportion of the core sediments in the Dagu Estuary (Qin et al., 2006) (With permission from Qin YW)...

Figure 4.3 Bivariate correlations (Pearson) of processes influencing adaptation in scenario 1 (bold) and scenario 2 (normal font)...

The magnitude of dependencies in the variables is determined by the correlation coefficient. The correlation coefficient according to Pearson is given by Eq. (1). 

It is also important to check for correlations between the descriptors. Highly correlated descriptors could lead to the information that they encode being over-represented. A straightforward way to determine the degree of correlation between two properties is to calculate a correlation coefficient. Pearson s correlation coefficient is given by ... 

There were two schools of thought concerning attempts to extend Hammett s treatment of substituent effects to electrophilic substitutions. It was felt by some that the effects of substituents in electrophilic aromatic substitutions were particularly susceptible to the specific demands of the reagent, and that the variability of the polarizibility effects, or direct resonance interactions, would render impossible any attempted correlation using a two-parameter equation. - o This view was not universally accepted, for Pearson, Baxter and Martin suggested that, by choosing a different model reaction, in which the direct resonance effects of substituents participated, an equation, formally similar to Hammett s equation, might be devised to correlate the rates of electrophilic aromatic and electrophilic side chain reactions. We shall now consider attempts which have been made to do this. 

Efforts to establish a theoretical explanation of the reactivity of nucleophilic reagents have centered on correlations with intrinsic electron-donor properties which are the fundamental basis of nucleophilicity. According to Edwards and Pearson, in general, such properties include basicity, polarizability, and the presence of unshared electron pairs on the atom adjacent to the nucleophilic atom of the reagent. When only the first two of these properties are operative, Eq. (8), which was proposed by Edwards, has proved successful in... 

Correlation analysis quantifies the degree to which the value of one variable can be used to predict the value of another. The most frequently used method is the Pearson product-moment correlation coefficient. 

Correlation coefficient. In order to establish whether there is a linear relationship between two variables xx and the Pearson s correlation coefficient r is used. 

For substituted anilines (Thompson and Williams, 1977) and for 1-naphthylamine and a series of derivatives thereof (Castro et al., 1986a), k2 and the ratio Ar 2/Ar3 have been determined for nucleophilic catalysis with Cl-, Br-, SCN-, and SC(NH2)2. The values of k2 correspond fairly well to those found for the diazotization of aniline, but those of Ar 2/Ar3 increase markedly in the above sequence (Table 3-1). As k3 is expected to be independent of the presence of Cl- or Br- and to show little dependence on that of SCN- or thiourea, the increase in k 2/k3 for this series must be due mainly to 2. Indeed, the value of log(Ar 2/Ar3) shows a linear correlation with Pearson s nucleophilicity parameter n (Pearson et al., 1968). This parameter is based on nucleophilic substitution of iodine (as I-) in methyl iodide by various nucleophiles. The three investigations on nucleophilic catalysis of diazotization demonstrate that Pearson s criteria for bimolecular nucleophilic substitution at sp3 carbon atoms are also applicable to substitution at nitrogen atoms. 

Metallation and oxymetallation reactions have been observed with the salts of only a few metals, namely mercury(II) (66, 67), thallium(III) (66,67), lead(IV) (66, 67), palladium(II) (100), gold(III) (63), and platinum-(II) (29). These facts correlate well with what Chatt (1) has termed class b, and Pearson (130) has called "soft acid character. Soft acids are characterized by low charge, large size, and, often, d electrons in their outer shell. No class b metal is known, in fact, which contains fewer than five d... 

FIGURE 3.3 Pearson correlation coefficients between fish (Gambusia) Hg concentration and MeHg concentrations in various environmental media sediment, porewater, surface water, and suspended particulate matter (SPM) from the Florida Everglades (1995-1998). 

Note that in data analysis we divide by n in the definition of standard deviation rather than by the factor n - 1 which is customary in statistical inference. Likewise we can relate the product-moment (or Pearson) coefficient of correlation r (Section 8.3.1) to the scalar product of the vectors (x - x) and (y - y) ... 

Fig. 8. Correlation between Pearson s hardness parameter (7P) derived from gas-phase enthalpies of formation of halide compounds of Lewis acids (19), and the hardness parameter in aqueous solution (/A), derived from formation constants of fluoride and hydroxide complexes in aqueous solution (17). The Lewis acids are segregated by charge into separate correlations for monopositive ( ), dipositive (O), and tripositive ( ) cations, with a single tetrapositive ion (Zr4+, ). The /P value for Tl3+ was not reported, but the point is included in parentheses to show the relative ionicity of Tl(III) to ligand bonds. 

On occasion you need to obtain correlation coefficients between two variables. Correlation coefficients are a way of measuring linear relationships between two variables. A correlation coefficient of 1 or -1 indicates a perfect linear relationship, and a coefficient of 0 indicates no strong linear relationship. Pearson correlation coefficients are useful for continuous variables, while Spearman correlation coefficients are useful for ordinal variables. For example, look at the following SAS code ... 

The first PROC CORR sends the Pearson correlation coefficients to a data set called pearson for the continuous variables Age and Weight, while the second PROC CORR sends the Spearman correlation coefficients to a data set called spearman for the categorical variables Race and Treatment Success. The correlation coefficients are found where the TYPE variable is equal to CORR in the pearson and spearman data sets. 

Fig. 3. Scatterplots of the Nh residual dipolar couplings of A131A measured in 2, 4, 6, or 8 M urea (y-axes) plotted against the same couplings measured in the absence of urea (x-axis) ris the Pearson correlation coefficient. Alignment was achieved with alkyl PEG bicelles (Ackerman and Shortle, 2002.)...

Equation 16.12 expresses a relation between q and B.This is not a universal relation, but it does apply to the sp-bonded elements of the first four columns of the Periodic Table. Using chemical hardness values given by Parr and Yang (1989), and atomic volumes from Kittel (1996), it has been shown that the bulk moduli of the Group I, II, III, and IV elements are proportional to the chemical hardness density (CH/atomic volume) (Gilman, 1997). The correlation lines pass nearly through the coordinate origins with correlation coefficients, r = 0.999. Thus physical hardness is proportional to chemical hardness (Pearson, 2004). 

It is shown that the stabilities of solids can be related to Parr s physical hardness parameter for solids, and that this is proportional to Pearson s chemical hardness parameter for molecules. For sp-bonded metals, the bulk moduli correlate with the chemical hardness density (CffD), and for covalently bonded crystals, the octahedral shear moduli correlate with CHD. By analogy with molecules, the chemical hardness is related to the gap in the spectrum of bonding energies. This is verified for the Group IV elements and the isoelec-tronic III-V compounds. Since polarization requires excitation of the valence electrons, polarizability is related to band-gaps, and thence to chemical hardness and elastic moduli. Another measure of stability is indentation hardness, and it is shown that this correlates linearly with reciprocal polarizability. Finally, it is shown that theoretical values of critical transformation pressures correlate linearly with indentation hardness numbers, so the latter are a good measure of phase stability. 

Reproducibility testing demonstrated that protein abundance measured using the spectral counting method exhibited a Pearson correlation R2 value >0.99 and a coefficient of variance of 14% (Fig. 20.6). 

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