Determination multiple, coefficient

Here Fcaic is the mean square (explained by regression) for the remainder variance ratio and i is a multiple determination coefficient characterizing the portion of total scatter of data in relation to the mean value of the dependent variable explained by regression. For example, the last equation accounts for 98% of the total scatter. 

Note that a number of complicating factors have been left out for clarity For instance, in the EMF equation, activities instead of concentrations should be used. Activities are related to concentrations by a multiplicative activity coefficient that itself is sensitive to the concentrations of all ions in the solution. The reference electrode necessary to close the circuit also generates a (diffusion) potential that is a complex function of activities and ion mobilities. Furthermore, the slope S of the electrode function is an experimentally determined parameter subject to error. The essential point, though, is that the DVM-clipped voltages appear in the exponent and that cheap equipment extracts a heavy price in terms of accuracy and precision (viz. quantization noise such an instrument typically displays the result in a 1 mV, 0.1 mV, 0.01 mV, or 0.001 mV format a two-decimal instrument clips a 345.678. .. mV result to 345.67 mV, that is it does not round up ... 78 to ... 8 ). 

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. 

The coefficient of multiple determination, / is a measure of how much of SS, is accounted for by the factor effects. 

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. 

Figure 9.4 Relationships among SS, SS and SS for calculating both the coefficient of multiple determination, R, and the variance-ratio for the significance of the factor effects,...

Calculate the coefficient of determination, /-j. and the coefficient of cdrrelation, r, for the model and data of Problem 9.6. What is the difference between the coefficient of determination and the coefficient of multiple determination ... 

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 ... 

Source Schwarzenbach et al. (10) quoting Abraham et al. (11,12). This material is used by permission of John Wiley Sons, Inc. and reproduced by permission of The Royal Society of Chemistry, n is the number of compounds used in this regression, and is the regression s coefficient of multiple determinations. 

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. 

R is the multiple regression coefficient, also called the coefficient of determination. It is the square of the coefficient which is always calculated and used. 

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. 

As noted earlier, the multiple partial coefficient of determination, r, usually is more of interest than the multiple partial correlation coefficient, because of its direct applicability. That is, an = 0.83 explains 83% of the variability. An r = 0.83 cannot be directly interpreted, except the closer to 0 the value is, the smaller the association the closer r is to 1, the greater the association. The coefficient of determination computation is straightforward. From the model, F = /3o + IB Xi + 2 2 + 3X3 + 4X4, suppose that the researcher wants to compute x x, X2>y or the joint contributiOTi of X3 and X4 to the model with Xi and X2 held constant. The form would be... 

However, the multiple partial coefficient of determination generally is not as useful as the F-test. If the multiple partial F-test is used to evaluate the multiple contribution of independent predictor values, while holding the others (in this case Xi, X2) constant, the general formula is... 

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