Coefficients of correlation

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) ... 

In the following section on preprocessing of the data we will show that column-centering of X leads to an interpretation of the sums of squares and cross-products in in terms of the variances-covariances of the columns of X. Furthermore, cos djj> then becomes the coefficient of correlation between these columns. 

The seven ways (r, through r7) for calculating correlation as the square root of the ratio of the explained variation over the total variation between X (concentration of analyte data) and Y (measured data) are described using many notational forms. For example, many software packages provide built-in functions capable of calculating the coefficient of correlation directly from a pair of X and Y vectors as given by rx (Equation 59-7). 

As shown in Figure 6.21, excellent linearity was obtained, as represented by the high coefficient of correlation obtained for the least square linear regression. Similar results were obtained for the evaluation of autosampler accuracy when other analytes (propyl paraben and rhodamine 110 chloride) were employed in the determinations. Liu et al.9 conducted similar evaluations for the samples employed in the evaluation of the drug release rate profile of OROS with similar results to those discussed above. 

Coefficient of Correlation Values for Least Squares Regression of Standard Curves... 

The coefficients of correlation again demonstrate the good agreement between the results determined by HPLC and CE. Because of the shorter analysis time and lower solvent consumption, the CE method has been proposed for this type of measurement [186],... 

The coefficients of correlation of the linear relationship between peak area and analyte concentration were high, ranging from 0.9978 to 0.9998 and recoveries varied between 94.7 and 109.0 per cent. It was established that the procedure is reliable and reproducible, therefore in can be applied for the analysis of these synthetic dyes in drinks [111]. 

Again, ranges from 0 to 1. The square root of the coefficient of determination is the coefficient of correlation, r, often called the correlation coefficient. 

It is important to realize that an / or r value (instead of an or value) might give a false sense of how well the factors explain the data. For example, the R value of 0.956 arises because the factors explain 91.4% of the sum of squares corrected for the mean. An R value of 0.60 indicates that only 36% of 55 has been explained by the factors. Although most regression analysis programs will supply both R (or r) and R (or r ) values, researchers seem to prefer to report the coefficients of correlation R and r) simply because they are numerically larger and make the fit of the model look better. 

Matrix composition Kinetic constant № Release exponent (n) (mean SD) Coefficient of correlation ... 

The coefficient of correlation values (R2) were quite low for the estimated friability mathematical models. It seems that this response was independent of the studied parameters and their levels and, in our case, friability was not an important response to be optimized. This response could only be an evaluated property. Lindberg and Holmsquist [6] had also obtained low R2 values for this response (R2=0.57 and 0.68). 

Linear correlation a coefficient of correlation constructed from linear functions of the observations. 

Figure 15 compares the heating value of gas, as expressed in B.t.u. per pound of coal, to the R of the samples analyzed. In general, as the R of vitrinite in coal increases, the heating values decrease (r = —0.72). As indicated by the low coefficient of correlation in Figure 5 (r = —0.72), a close relationship between R and gas heating value does not exist. Therefore, any reference to these data can only be in very general terms. 

Calculate the coefficient of determination, r2, and the coefficient of correlation, r, for the model and data of Exercise 9.6. What is the difference between the coefficient of determination and the coefficient of multiple determination ... 

The linear relation (with coefficient of correlation r = 0.99) agrees with the proposed mechanism. The reaction of a number of primary alcohols with MDI in DMF catalyzed by dibutyltin dilaurate are given in Table II, which also gives the calculated k0 and kcat X K0,5 values. 

The oxygen sensing properties of such electrode systems were measured at first. The reproducibility of the output current versus oxygen tension is shown in Figure 2. The coefficient of correlation was 0.987 using eighteen data points. 

Statistical analysis that calculates the coefficient of correlation (i.e., covariance divided by the product of variances) for a set of variables. Volume 2(2). 

Variables for which coefficients of correlation are inexistent. Volume 2(2). 

Mean maximum observed cetuximab (Cmax) values after first cetuximab administration are displayed as a function of dose in Fig. 14.1. Serum concentrations of cetuximab typically reached maximum levels approximately 1-2 h following the end of the cetuximab infusion, independent of dose. Mean cetuximab Cmax values ranged from 8.7 to 283.8 gg/mL after single cetuximab doses of 20 to 500 mg/m2, respectively. A linear relationship (coefficient of correlation, r2 = 0.98) between cetuximab dose and mean Cmax value was observed, indicating that maximum serum concentrations following infusion are predictable for each dose used. 

The coefficient of correlation, R(r), measures the degree of correlation between the concentrations at two points separated by distance r. It is obtained by randomly throwing a dipole of length r, and is defined as follows ... 

The coefficient of correlation ranges from 1 to —1, denoting, respectively, perfect positive (both points in each couple have the same concentration) and perfect negative (one point is pure minor, the other is pure major) correlations. Figure 7.37 gives some typical correlograms. 

The scale of segregation is the integral of the coefficient of correlation from distance zero, where R(0) = 1, to distance C at which there is no correlation [R(C) = 0]... 

The Coefficient of Correlation of a Two-composition System in Terms of Probabilities Consider a two-composition texture of concentrations x and x2 shown in the figure below. Following a (mental) process of dipole throwing, we find that fen of them fell with both ends on composition x, k.22 fell with both ends on composition X2, and fen fell with one end on composition x and the other on X2-... 

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