Multiple correlation coefficient determination

Here, the notation (, I C, X2) stands for the squared multiple correlation coefficient (or coefficient of determination) of the multiple regression of y, on Xj and X2. The improvement is quite modest, suggesting once more that there is only a weak (linear) relation between the two sets of data. 

Such a high correlation coefficient indicates that the regression model describes the experimental data extremely well. Apart from the mentioned multiple correlation coefficient the following partial coefficient of determination ... 

In Equations 4 and 5, r is the multiple correlation coefficient, r2 is the percent correlation, SE is the standard error of the equation (i.e the error in the calculated error squares removed by regression to the mean sum of squares of the error residuals not removed by regression. The F-values were routinely used in statistical tests to determine the goodness of fit of the above and following equations. The numbers in parentheses beneath the fit parameters in each equation denote the standard error in the respective pa-... 

Coefficient of determination, Bf. The squared multiple correlation coefficient that is the percent of total variance of the response explained by a regression model. It can be calculated from the model sum of squares MSS or from the residual sum of squares RSS ... 

Usually, the linearity of a NIR spectroscopic method is determined from the multiple correlation coefficient (R) of the NIR predicted values of either the calibration or validation set with respect to the HPLC reference values. It may be argued that this is an insufficient proof of linearity since linearity (in this example) is not an independent test of instrument signal response to the concentration of the analyte. The analyst is comparing information from two separate instrumental methods, and thus simple linearity correlation of NIR data through regression versus some primary method is largely inappropriate without other supporting statistics. 

It is often less ambiguous to denote the multiple correlation coefficient, as per Kleinbaum et al. (1998), as simply the square root of the multiple coefficient of determination. 

If the regression parameters are estimated by the least squares method, the square root of the coefficient of determination and the multiple correlation coefficient will be identical. 

The coefficient of determination describes the fraction of the sum of squares due to the factors in relation to the sum of squares corrected for the mean. The square root of the coefficient of determination reveals the multiple correlation coefficient ... 

Adopting the unsupervised option initially, the first two variables to be selected are those with the lowest pairwise correlation. The next variable selected has the smallest multiple squared correlation with those first two variables. This process is continued until the preset maximum level of multicolinearity (determined by the squared multiple correlation coefficient) is reached. Whitley et al. refer to this procedure as unsupervised forward selection (UFS). UFS can also be performed with a minimum variance criterion where only variables with variance above this minimum will be selected. These two criteria can be used by scientists simultaneously. With supervised variable selection, only those variables having a sufficiently high correlation with the response are considered for what effectively is UFS on this reduced set of variables. We will term this latter process, supervised forward selection (SFS). To see how these options work and to examine the effect they have on the model produced, we performed PLS on the data with both UFS and SFS configured to run with a range of response variable correlations (Table 8). 

R In organic chemistry, symbol for a general attached group or radical, frequently an aliphatic or aromatic hydrocarbon group, that may take on any of various specific identities. Abbreviation for Roentgen. Abbreviation for generalized of multiple correlation coefficient, the square root of the coefficient of determination. °R, Abbreviation for degree of Rankine. Symbol for electrical resistance. 

The heat-transfer coefficient as a function of coolant flow (Fc) was ailso determined experimentally. A fifth-order polynomial with a multiple correlation coefficient of 0.9975 was fit to the data and is... 

High correlations were observed between the protein percentages determined by the Kjeldahl method and those estimated by the InfraAlyzer 400 (Table 20.6). The square multiple correlation coefficients of 0.97 to 0.999 with SEC of 0.18 to 0.34% are acceptable compared with the standard deviation of the reference method of about 0.15% for milk powders. 

Canonical Correlation Analysis (CCA) is perhaps the oldest truly multivariate method for studying the relation between two measurement tables X and Y [5]. It generalizes the concept of squared multiple correlation or coefficient of determination, R. In Chapter 10 on multiple linear regression we found that is a measure for the linear association between a univeiriate y and a multivariate X. This R tells how much of the variance of y is explained by X = y y/yV = IlylP/llylP. Now, we extend this notion to a set of response variables collected in the multivariate data set Y. 

Recently the data concerning to interaction of propanthiole with chlorine dioxide in 8 solvents have been published [1], In this work it was shown, that the dependence of process rate from solvents properties is satisfactory described for seven solvents, after the exclusion of data for ethyl acetate, by the Koppel-Palm four parameters equation (coefficient of multiple correlation R 0,96) at determining role of medium polarity (coefficient of pair correlation between lg(k) and (s - l)/(2e + 1) - r 0.90). Chemical mechanism of the reaction including the formation of ion-radical RS H and radical RS has been proposed by authors [ ] ... 

The coefficient of multiple determination ranges from 0 (indicating that the factors, as they appear in the model, have no effect on the response) to 1 (indicating that the factors, as they appear in the model, explain the data perfectly ). The square root of the coefficient of multiple determination is the coefficient of multiple correlation, R. 

---