Squared correlation coefficient

For the data the squared correlation coefficient was 0.93 with a root mean square error of 2.2. The graph of predicted versus actual observed MS(1 +4) along with the summary of fit statistics and parameter estimates is shown in Figure 16.7. 

Figure 8. Fit of PCOMP-resolved absorption spectra of TIN (solid line) to the absorption spectrum of TIN in acetonitrile (squares) (correlation coefficient 0.997).

A number of performance criteria are not primarily dedicated to the users of a model but are applied in model generation and optimization. For instance, the mean squared error (MSE) or similar measures are considered for optimization of the number of components in PLS or PC A. For variable selection, the models to be compared have different numbers of variables in this case—and especially if a fit criterion is used—the performance measure must consider the number of variables appropriate measures are the adjusted squared correlation coefficient, adjR, or the Akaike S information criterion (AIC) see Section 4.2.3. 

As the model complexity increases, R2 becomes larger. In linear regression, R2 is the squared correlation coefficient between y and y, and d R2 is called the adjusted squared correlation coefficient. Thus adjR2 is a modification of the R2 that penalizes larger models. A model with a large value of adj 2 is preferable. Another, equivalent representation for adj 2 is... 

FIGURE 4.12 Linear latent variable with maximum variance of scores (PCA) and maximum correlation coefficient between y and scores (OLS). Scatter plot of a demo data set with 10 objects and two variables (x1 x2, mean-centered) the diameter of the symbols is proportional to a property y R2 denotes the squared correlation coefficients between y and x1 y and x2, y and PCI scores, y and y from OLS. 

The best correlation equations obtained for the Km values in the presence of carboxylesterase (CE) or human plasma (HP) are given below as Eqns. 8.3 and 8.4, respectively. The statistical quality of the equations can be assessed by r2, the squared correlation coefficient, and q2, the cross-validated correlation coefficient (a measure of the predictive power of the equation, which is considered as acceptable when q2> 0.4). Both equations are statistically sound and have acceptable predictive power. 

The main characteristics of these applications are good linearity, with squared correlation coefficient above 0.999, and repeatability with RSD below 2% when no internal standard is used and below 1% with an internal standard. Depending on the buffer and injection optimization, the different authors found an POD and limit of quantitation (LOQ) close or below 1 pg/mL. Using a conductivity detector, the POD and POQ decreased by a factor 100. Many authors also report the accuracy and recovery of the method. 

Average monthly morning concentrations of radon in air increased with average monthly values of atmospheric stability during the 12 hours before filter collection, but there was appreciable scatter of values about lines of best fit. The squared correlation coefficient was 0.53 for a linear relation between radon concentration and stability. Lower correlations applied to stability values averaged over periods longer or shorter than 12 hours. Precipitation during the 12-hour period before collection decreased radon concentrations. 

Average monthly morning radon concentration varied inversely with soil moisture content, expressed in terms of Thomthwaite s total moisture detention. The two variables were highly correlated, the squared correlation coefficient being 0.84. By relating radon concentration in air to both TMD and stability, the squared correlation coefficient increased to 0.89. An alternative mathematical relation, the logarithm of the radon concentration vs. TMD and stability, showed identical correlation. 

Note Problems 3 through 7 and 10 through 14 are, m part, given at two levels of sophistication and expectation The best" answers involve doing the part marked with an asterisk ( ) instead of the immediately preceding part The parts with an involve the use of the method of least squares, correlation coefficients, and confidence intervals for the slopes and intercepts, the preceding parts do not... 

The fourth recommendation of the OECD experts is related to appropriate measuring and reporting goodness-of-fit, robustness, and predictivity of the model. The main intention was to clearly distinguish, whether a measure was derived only from the training set, from the internal validation (i.e., cross-validation, where the same chemicals are used for training and validation, but not at the same time) or from validation with use of an external set of compounds, not previously engaged in model optimization and/or calibration (external validation). A widely applied measure of fit is the squared correlation coefficient R2-1 — (RSS/ TSS), where RSS is the residual sum of squares and TSS is the total sum of squares... 

Model evaluation. Before a QSAR model can be applied to predict the potency of new molecules, model evaluation and validation are essential. Initially, a squared correlation coefficient, r1, is usually calculated to determine how well the actual linear function/model predicts the potency of training set compounds. 

At first sight these may appear to be reasonably diverse in terms of their chemistry even though there are three halogens in the set. The it values are nicely spread out from hydrophilic (-1.23) to hydrophobic (1.44) but, unfortunately, so are the o values. In general there is no correlation between the hydrophobic substituent constant it and the electronic substituent constant o, but for this set the squared correlation coefficient between these two physicochemical properties is 0.95. In much the same way as the methyl-futile set is uninformative, so would this set be in terms of distinguishing electronic effects from hydrophobic effects in any QSAR model derived from it. 

The estimated analyte concentrations in each of the three mixtures, with added random errors of a standard deviation equal to 10% of the mean value of S2, are summarized in Table 12.1. The results are presented for prediction with each of the three standards for each of the three mixtures utilizing the GRAM algorithm with concatenated (Table 12.1a) and added (Table 12.1b) samples. Similarly, the squared correlation coefficients between the true and estimated X-way and Y-way profiles are presented in Table 12.2. Figure 12.5 presents the best-case and worst-case examples for estimating the X-way and Y-way profiles with GRAM. In the worst-case scenario (p2x = 0.9938, p2y = 0.9987), there is very little discernible variation between the true and estimated profiles. In most cases, the true and estimated profiles... 

Squared Correlation Coefficient (angle cosign) of Predicted and True Y-Way Spectra from GRAM with Concatenated Matrices... 

DTLD allows estimation of the interferent X-way and Y-way profiles, as well estimation of the analyte profiles. The squared correlation coefficients of these estimated profiles with the true, noiseless profiles are presented in Table 12.4. For the analyte, which is present in all six samples, the true and estimated profiles are indistinguishable. The interferent profiles, which are present in only half of the samples, are comparable in accuracy with the estimated analyte profiles from GRAM. 

Table 12.3 compares the estimated analyte concentrations for DIED, PARAFAC, and PARAFAC x 3 noise (PARAFAC with the addition of a factor of three greater random errors) applied to the same calibration problem. Table 12.4 is analogous to Table 12.3, except that it also presents the squared correlation coefficients between the true and estimated X-way and Y-way profiles for all three species present in the six samples. It is first evident that PARAFAC slightly outperforms DTLD when applied to the same calibration problem. However, the improvement often lies in the third or fourth decimal place and is hardly significant when compared with the overall precision of the data. This near equivalence of DTLD and PARAFAC is rooted in the fact that DTLD performs admirably, and there is little room for... 

The quality of a multiple linear regression is usually assessed using the squared correlation coefficient, written r2 [44]. r2 is calculated using the following formula ... 

With each QSAR, the following statistics are given the 95% confidence limits for each term in parentheses the correlation coefficient r between observed values of the dependent and the values predicted from the equation the squared correlation coefficient s the standard deviation defines the cross-validated (indication of the quality of the fit) and the F-values are given for... 

The relationship between the concentration of an element in "whole" coal and the ash content can be used as a guide to the affinity of that element for, or incorporation in, the mineral matter or the carbonaceous material. If the concentration of an element increases with increasing ash content that element is presumed to be associated with the inorganic species that form ash, or in other words may be said to have an inorganic affinity. If the concentration shows no correlation with ash content, that element would be said to have an organic affinity. Linear least squares correlation coefficients were calculated for the concentrations of 39 elements... 

The squared correlation coefficient R, the coefficient of determination, represents the proportion of common variation in the two variables (i.e., the magnitude of the relationship). 

The resulting networks with dataset A showed an average squared correlation coefficient (r ) of 0.96 [standard deviation (SD) 0.56 logS units] for the training set, and 0.95 (standard deviation 0.57 logS) for the test set. However, the result for the val-... 

Figure 1 A QSAR equation (cf. Eq. (14) recalculated from original values) should contain 95% confidence intervals of all regression coefficients (preferable to standard deviations of regression coefficients), the number of objects ( = compounds) included in the analysis, the correlation coefficient r or a squared correlation coefficient r2, the standard deviation s, the Fisher F value (optional), the crossvalidation correlation coefficient Q 2, and the crossvalidation standard deviation press (both optional but recommended).

A significant difference between Q2 and the normal squared correlation coefficient (R2) is that the former may also assume negative values, indicating that the model has worse predictive ability than using the mean value as predicted value for each compound. Q1 should be >0.5 for the model to be considered to have reasonable practical predictive performance. 

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