Square of the correlation coefficient

A superficial, but common, measure of linearity is the square of the correlation coefficient, R2  

R2 can be used as a diagnostic. It R2 decreases after a method is established, something has gone wrong with the procedure. 

Accuracy is nearness to the truth. Ways to demonstrate accuracy include 

Analyze a Standard Reference Material (Box 3-1) in a matrix similar to that of your unknown. Your method should find the certified value for analyte in the reference material, within the precision (random uncertainty) of your method. 

Compare results from two or more different analytical methods. They should agree within their expected precision. 

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If the T values of Table I are first fitted against EA values, without first fitting with EE(v) values, poor linea correlations result. For example, for tKe one-hour half-life temperatures of reactions 1 and 4, the squares of the correlation coefficients for these linear regression analyses are only 0.51 and 0.55, respectively. 

In QSAR equations, n is the number of data points, r is the correlation coefficient between observed values of the dependent and the values predicted from the equation, is the square of the correlation coefficient and represents the goodness of fit, is the cross-validated (a measure of the quality of the QSAR model), and s is the standard deviation. The cross-validated (q ) is obtained by using leave-one-out (LOO) procedure [33]. Q is the quality factor (quality ratio), where Q = r/s. Chance correlation, due to the excessive number of parameters (which increases the r and s values also), can. 

Fig. 2 Plot of P-Cl distances (in A) vs average P-N distances (in A) for P-chloro-NHPs (diamonds) and for all compounds (R2N)2PC1 (except P-chloro-NHPs) listed in the CSD data base (open squares). The solid and dashed lines represent the result of linear regression analyses. R2 is the square of the correlation coefficient in the regression analysis. (Reproduction with permission from [55])...

X-axis. It presents the coefficients of the linear models (straight lines) fitted to the several curves of Figure 67-1, the coefficients of the quadratic model, the sum-of-squares of the differences between the fitted points from the two models, and the ratio of the sum-of-squares of the differences to the sum-of-squares of the X-data itself, which, as we said above, is the measure of nonlinearity. Table 67-1 also shows the value of the correlation coefficient between the linear fit and the quadratic fit to the data, and the square of the correlation coefficient. 

Because of the large difference in the behavior of the thin plywood and the gypsum board, the type of interior finish was the dominant factor in the statistical analysis of the total heat release data (Table III). Linear regression of the data sets for 5, 10, and 15 min resulted in squares of the correlation coefficients R = 0.88 to 0.91 with the type of interior finish as the sole variable. For the plywood, the average total heat release was 172, 292, and 425 MJ at 5, 10, and 15 min, respectively. For the gypsum board, the average total heat release was 25, 27, and 29 MJ at 5, 10, and 15 min, respectively. 

Calibration is also used to describe the process where several measurements are necessary to establish the relationship between response and concentration. From a set of results of the measurement response at a series of different concentrations, a calibration graph can be constructed (response versus concentration) and a calibration function established, i.e. the equation of the line or curve. The instrument response to an unknown quantity can then be measured and the prepared calibration graph used to determine the value of the unknown quantity. See Figure 5.2 for an example of a calibration graph and the linear equation that describes the relationship between response and concentration. For the line shown, y = 53.22x + 0.286 and the square of the correlation coefficient (r2) is 0.9998. 

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). 

Included in the following table are some data points from a hypothetical enzyme kinetics study. Using a spreadsheet program with graphing abilities (such as Excel), generate a Line-weaver-Burk plot of the data points in the table. Determine the best-fit line for the data along with Vnax, ATm, and r2 (the square of the correlation coefficient of the line). Does this enzyme follow Michaelis-Menten kinetics Why or why not ... 

Particularly in case B where a variable, say y, is assumed to be dependent on the other variable, say x, it is rather interesting to test the square of the correlation coefficient, rxy, which at least in the standard regression model is a measure of the coefficient of determination, i.e. which fraction of the total data variation of y is declared by the mathematical model function of its dependency on x. (1 - rly is called coefficient of nondetermination.)... 

For simple linear regression it holds that COD is related to the square of the correlation coefficient by CODxy = rxy. 

A measure called the coefficient of determination can be calculated as the proportion of the variability in one variable that is accounted for by variability in the other. The coefficient of determination is simply the square of the correlation coefficient. In the case of a perfect correlation, either positive (1.0) or negative (minus 1.0), the coefficient of determination will be identical to the correlation coefficient. This is also true for a correlation coefficient of zero. In any other case, the coefficient of determination will always be smaller than the correlation coefficient, since a value of less than 1 multiplied by itself results in a smaller value. 

One might try to improve the least accurate parameter estimate, or minimize the correlation between the parameters by minimizing the sum of squares of the correlation coefficients between the parameters which arc also calculated from the elements of (XTX) I ... 

The correlation coefficient has a value between —1 and +1. If close to +1, the two variables are perfectly correlated. In many applications, correlation coefficients of — 1 also indicate a perfect relationship. Under such circumstances, the value of y can be exactly predicted if we know x. The closer the correlation coefficients are to zero, the harder it is to use one variable to predict another. Some people prefer to use the square of the correlation coefficient which varies between 0 and 1. 

The results observed with a representative set of test-liquids are recorded in Fig. 11. The square of the correlation coefficient (r2), determined by linear regression analysis, was in every case >0.99 when the slope was >1, but r2 for those with slopes less than 1 were sometimes less than 0.98, owing to decrease in accuracy as a result of decrease in the difference (S — v) where v is the porosity characteristic of the respective composite film samples. In such cases the determination was repeated to obtain sets of averaged values, which typically exhibited r2 > 0.99. 

The mono- and di-substituted benzenes, for which the relative swelling power (C as defined in Eq. 14) have been determined thus far, are listed in Tables 2-4. In every case the volume (S) of sorbed liquid per gram of polymer increased linearly with X1/3 as noted in Figs. 9-11, and the square of the correlation coefficient (r) to the line of best fit, determined by linear regression analysis of the set of six S-data points, was in every case r2 > 0.99, and q13 was equal to 1.84 0.09. The corresponding C and oc (calculated therefrom Eq. 15) are also collected in Tables 2-4. These data show clearly that the value a = 2.50 for benzene is greater than that for any benzene derivative listed therein. Any substituent in place of... 

The square of the correlation coefficient (r2 Table 19) to the line of best fit through each set of 5 or more data points is >0.99 when v is >0.2, 0.965 when v is 0.2 Figure 62 also shows that Xo ar>d A, as defined in Eq. 40, both increase with v. The correlations of Xo and A with v (Table 19 Fig. 63) show that both relationships are linear as expressed respectively by Eqs. 41 and 42 ... 

The relative contributions to the variance in 1/t for each factor are given in the last column of Table VIII. The values are the squares of the correlation coefficients for each variable in Equations 7 and 8 since the factors are orthogonal variables. The sums of contributions are S % in the 30-point case and 80% in the 26-point case. That these values are less than 100 states that the three factors are sufficient to account for 85 and 80 , respectively, of the variance in 1/t. 

At the least, a chart of the data should be produced for visual inspection of the fit, as illustrated in Figure 11-2. You can also use the F SQ(known ys, known xs) worksheet function to return the square of the correlation coefficient, to provide information about the goodness of fit to the straight line. 

In QSAR equations, n is the number of data points, r is the correlation coefficient between observed values of the dependent and the values predicted from the equation, is the square of the correlation coefficient and represents the goodness of fit, is the cross-vahdated (a measure of the quality... 

The top row of the block will now show the values of the slope and intercept, the second row the corresponding standard deviations, the third will contain the square of the correlation coefficient and the variance in y, then follow the value of the statistical function F and the number of degrees of freedom and, in the bottom row, the sums of squares of the residuals. The function LINEST provides a lot of information very quickly, albeit in a rather cryptic, unlabeled format. 

According to the rules of statistical analysis, the fraction of the variation of a property accounted for by a correlation is equal to the square of the correlation coefficient. The square of the correlation coefficient is hence sometimes referred to as the "coefficient of determination". A correlation coefficient of 0.9957 indicates that Equation 3.9 accounts for approximately 99.15% of the variation of the Vw values. The addition of extra terms proportional to °%v and... 

A more conservative measure of closeness of fit is the square of the correlation coefficient, r, and this is what most statistical programs calculate (including Excel—see Figure 3.8). An r value of 0.90 corresponds to an value of only 0.81,... 

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