Multiple correlation coefficient

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. 

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. 

R = multiple correlation coefficient the other symbols have the same meanings as in connection with equations 12 and 13 in Section III.D.)... 

Multiple correlation coefficient an index which measures the joint effect of several variables on some response. 

Multiple regression programs also calculate auxiliary statistics, designed to help decide how well the calibration fits the data, and how well it can be expected to predict future samples. For example, two of these statistics are the standard error of calibration (SEC) and the multiple correlation coefficient (R). The SEC (also called standard error of estimate, or residual standard deviation) and the multiple correlation coefficient indicate how well the calibration equation fits the data. Their formulas are given in Table 3. 

The multiple correlation coefficient R is a dimensionless measure of how well the calibration fits the data. R can have values between -1 and +1, but in a calibration situation only positive values exist. A value close to zero indicates that the calibration fails to relate the spectra to the reference values. As the correlation coefficient increases, the spectra become better and better predictors of the reference values. Because the multiple correlation coefficient is dimensionless, it is a useful way of comparing data or results with different units, and that are difficult to compare in other ways. However, its value gives no indication of how well the calibration equation can be expected to perform on future samples. 

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 ... 

It is evident that the contribution of each latent factor to the portion of mass for each chemical parameter varies according to the different impact of the source on the concentration. For instance, the chloride concentration is distributed between the anthropogenic factor (88.1%) and the biological factor (11.9%), but the anthropogenic impact is much higher. Similar conclusions can be drawn for any chemical parameter involved in water quality. The last column of the table shows the multiple correlation coefficient R1. This gives an idea of the suitability of the respective models for each of the chemical parameters. The nonsignificant coefficients are underlined. As a whole, most of the models are statistically appropriate and can be used for predictive purposes.1112... 

Ouantitative Structure-Activity Relationship Ouantitative Structure-Biodegradability Relationship Ouantitative Structure-Pharmacokinetic Relationship Ouantitative Structure-Property Relationship Multiple correlation coefficient and its square Relative Binding Affinity... 

The statistics reported for the tit are the number of compounds used in the model (n), the squared multiple correlation coefficient (R2), the cross-validated multiple correlation coefficient (R2Cv) the standard error of the fit (s), and the F statistic. The squared multiple correlation coefficient can take values between 0 (no fit at all) and 1 (a perfect fit) and when multiplied by 100 gives the percentage of variance in the response explained by the model (here 83%). This equation is actually quite a good fit to the data as can be seen by the plot of predicted against observed values shown in Figure 7.6. 

Multiple Correlation Coefficient =. 994 Standard Error of the Estimate - 2.6%... 

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-... 

In conclusion, the solvation equation could be applied to describe several liquid chromatographic partition systems in terms of their sensitivity towards molecular properties. The standard error for estimating the retention data was low and the multiple correlation coefficients of the solvation equations were high. The parameters of the solvation equations help us to understand and describe the different selectivity of the stationary phases, and also to understand the retention of the compounds in a given system based on its molecular properties. 

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 ... 

A related quantity is the multiple correlation coefficient R defined as the square root of R. It is a measure of linear association between the observed response and the estimated response, i.e. the response obtained by a linear combination of the predictor variables in a linear regression model. A quantity complementary to is the coefficient of nondetermination defined as ... 

Note. A quality factor (Q) was also proposed [Pogliani, 1994a] for measuring the quality of the regression models and defined as the ratio between the multiple correlation coefficient R and the standard deviation s. However, s being dependent on the measurement unit used for the response, it should be not used to measure the quality of the regression models. 

A necessary condition for the validity of a regression model is that the multiple correlation coefficient is as close as possible to one and the standard error of the estimate s small. However, this condition (fitting ability) is not sufficient for model validity as the models give a closer fit (smaller s and larger R ) the larger the number of parameters and variables in the models. Moreover, unfortunately, these parameters are not related to the capability of the model to make reliable predictions on future data. 

Variable selection is performed by checking the squared multiple correlation coefficient or the corresponding cross-validated quantity from univariate regression models y = ho + the selection being made separately for each /th variable of the p variables xi, X2,..., Xp. 

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. 

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