Coefficient of multiple correlations

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. 

Coefficient of multiple correlations is used as a level of linear connection between the measuring variable and a composition of the independent variables [11] ... 

Rydberg constant coefficient of multiple correlation SAMS... 

The experimental data for PC-PMMA. blends, taken from the literature >1, are shown in Table 3, as are the calculated values of volume fractions v and reinforcement factors e. The plot of e vs. v is shown in Fig. 3 (evidently outstanding points are omitted). Thus three types of blend, A,B and C, prepared and measured under different conditions, can be compared. We have assumed, in the first approximation, that the differences between the types are in the limits of experimental error. Hance the parallel (additive) model can be applied and the least-squares method gives e=0.192 v, with coefficient of multiple correlation being equal to 0.94 (cf. Table 4). However,... 

It is evident from the examples described above that a 4-th order equation for the suspension model fits the experimental data for PP-PC blends best. In the case of PC-PMMA blends, however, several equations of different order can be consistent with the experimental data, and the choice of the best one is more difficult. Coefficients of multiple correlation seem to be a rather inadequate criterion here. 

Calibration and validation statistics for each regression include standard error of calibration, coefficient of multiple correlation, standard error of prediction. 

MSC, multiple scatter correction SNV, standard normal variate BC, baseline correction ID, first-derivative transformation 2D, second-derivative transformation SEC, standard error of calibration R, coefficient of multiple correlation SEP, standard error of prediction r, validation correlation coefficient VCC, %, variation coefficient for calibration set-(SEC/Mean value) x 100 VCV, %, variation coefficient fa- validation set-(SEP/Mean value) x 100. 

Values of the coefficient of multiple correlation are higher than 0.9 in all cases.That means that the degree of association existing between experimental data and fitted values obtained from relations (6)-(9) is high. 

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 ... 

Foaming properties can be quantitatively related to surfactant chemical structure, surfactant physical properties, and test conditions using the technique of multiple correlation analysis.(11) The current studies were restricted to linear correlation equations to permit the analyses to be performed on a small microcomputer. While non-linear equations having higher correlation coefficients than obtained herein can be developed, theoretical insights are often limited due to the complexity of the various terms of such equations. The quality of the correlations were assessed using the correlation coefficient (r ) criteria of Jaffe (12)... 

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. 

From the values of dependent variables yj obtained from experimental results (Table 2), the values of the coefficients of the regressive function (1) were calculated (Table 3). As the values of the indexes of multiple correlation I y show, the mathematical model describes the effects of the system composition on its characteristics in an extremely fitting manner. An exception is the case of elongation at break (yg) where the value of I, y is... 

The least-squares technique can be extended to any number of variables as long as the equation is linear in its coefficients. The linear correlation ofj vs X can be extended to the correlation ofj vs multiple independent variables generating an equation of the form ... 

Then vkt is calculated from the vX values as (-ln(l-vX)). The independent function Temperature vx is expressed as 1000 K/vT for the Arrhenius function. Finally the independent variable vy is calculated as In(vkt). Next a linear regression is executed and results are presented as y plotted against Xi The results of regression are printed next. The slope and intercept values are given as a, and b. The multiple correlation coefficient is given as c. 

The a s are dimension constants, with a value of 1. is the multiple correlation coefficient, the fraction of total variance in the data accounted for by the model. 

R r Multiple correlation coefficient. R indicates the percentage of the variability of the relative biological response that can be accounted for by the selected independent variables. 

The study is based on four iinear hydrocarbons (in Ci, Ce to Ca) and the model uses Antoine and Clapeyron s equations. The flashpoints used by the author do not take into account all experimental values that are currently available the correlation coefficients obtained during multiple linear regression adjustments between experimental and estimated values are very bad (0.90 to 0.98 see the huge errors obtained from a correlation study concerning flashpoints for which the present writer still has a coefficient of 0.9966). The modei can be used if differences between pure cmpounds are still low regarding boiling and flashpoints. 

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. 

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. 

A complication arises. We learn from considerations of multiple regression analysis that when two (or more) variables are correlated, the standard error of both variables is increased over what would be obtained if equivalent but uncorrelated variables are used. This is discussed by Daniel and Wood (see p. 55 in [9]), who show that the variance of the estimates of coefficients (their standard errors) is increased by a factor of... 

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