Correlation coefficient square

If the parameters enter the equation linearly, then the minimization problem reduces to a set of linear equations which are solved easily by Excel and MATLAB. The effectiveness of the curve fit is often reported as values of the linear correlation coefficient squared, r. The linear correlation coefficient is defined as (Press et al., 1986, p. 484) ... 

The selection was based on measures of the goodness of fit the standard deviation (a), the correlation coefficient squared (r ), and the coefficient of determination (Q). It is noteworthy that the range of variables selected is much wider than that experienced by a single polymer. Parameters for the goodness of fit for Eqs. (6.43) and (6.44) are listed in Table 6.2. Similarly, the scaled CED and S = S(T, P) were approximated by... 

A second reason to fit data to a function is to test whether the data are consistent with a model or to use a theoretical function to extrapolate experimental results to conditions otherwise not attainable. The goodness of fit of the data to a theoretical function can be gauged by the coefficient of determination EP), which is the correlation coefficient squared. B is often interpreted as the fraction of the variability of the response variable that is explained by its functional relationship to the independent variable. For example, if R = 0.8 then 80% of the variation in the response variable is explained by the model and 20% of the variation is the result of factors, including random error, that are not part of the model. Some geochemical processes occur under conditions or over time intervals that are difficult to simulate by experiments, making it necessary to use a theoretical model to predict their behavior. This approach uses data from experiments performed under easily attainable conditions to quantify parameters that are used to calibrate a theoretical model. These parameters are then used to predict the value of the variable at... 

The use of the function (DR)", rather than the use of DR per se, was to let the relationship with DR be either direct or reciprocal and to allow the relationship to be curvilinear. The value of n was selected, in each case, by trial and error, such that the value of squared (correlation coefficient squared) would be minimized. 

The correlation coefficient square (r ) increased from 0.53 to 0.75 by the inclusion of structural parameters, while the standard error of estimate has fallen from 0.53 to 0.40. Note that the largest residual (i.e., the difference between the measured value and the estimate from the regression equation) is substantially smaller for the QSAR model. 

TABLES Summary of number of compounds (n), correlation coefficient squares (r ), standard error of estimate (s), and Constantsa, b, c, d, and e for multiple linear correlation analyses of data set according to general equation 3. 

These pairs correspond to a linear fit with correlation coefficient squared equal to 1.000. 

Figure 11.4 shows the graph of the data points and the least-squares line. The correlation coefficient squared was equal to 1.000. 

Volume-based thermodynamics was applied by Glasser [227] to model the heat capacities, based on the microscopic ionic volumes VIN at an unspecified temperature. It was applied to 11 l-alkyl-3-methylimidazolium RTlLs (C2, C4, and Ce) with a variety of anions and to Ci4mim NTF2 and C4-2,3-dimethylimidazolium+ PFs with a linear correlation coefficient squared of 0.980 when forced through the origin. When data for 28 further imidazohum salts were added and the microscopic volumes from Tables 2.4 and 6.9 (rather than V/Va) were employed, the slope was slightly larger than in [227] and the data followed the expression ... 

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