Coefficients of determination

Subtracting y —y) from y — T) gives (T — y) which is that portion of the deviation of any data point from the mean which is explained by the correlation. The coefficient of determination, y, a statistical parameter which varies from 0.0 to 1.0, is defined as ... 

Fig. 2. Least-squares plot showing the determinants for the coefficient of determination. A, total deviation y — y) , B, unexplained deviation y — y) , and...

Fig. 5. Commercially available pressing times and relative molecular weights for PF OSB resins [109,110]. Note that the molecular weight values are made relative to fit the scale. The pressing times are for 3/4-inch OSB in s/mm. The coefficient of determination between relative molecular weight and cycle lime is 0.96.

TABLE 17.7 Coefficient of Determination (R ) for Linear Calibration Curves for Four Linear Columns and Four Sets of PEO Standards... 

For all four sets of PEO standards the coefficient of determination (R ) for the linear calibration curves for the four linear columns in water and in water/methanol are better than 0.99, except for the PL PEO standards and the TSK GM-PWxl column in water/methanol. The coefficient of determination for the TSK GM-PWxl column in general is not as good as the other three linear columns. The coefficient of determination for the TSK PEO standards showed the least dependency on columns and mobile phases. The TSK GM-PWxl column has a lower exclusion limit in the high molecular weight range than TSK GM-PW, Shodex SB-806, and SB-806MHQ columns. 

The coefficient of determination is the fraction of the variation that is explained by a linear relationship between two variables and is given by... 

Figure 2.2. Examples of correlations with high and low coefficients of determination. Data were simulated for combinations of various levels of noise (a = 1,5, 25, top to bottom) and sample size (n - 10, 20, 40, left to right). The residual standard deviation follows the noise level (for example, 0.9, 5.7, 24.7, from top to bottom). Note that the coefficient 0.9990 in the top left panel is on the low side for many analytical calibrations where the points so exactly fit the theoretical line that > 0.999 even for low n and small calibration ranges.

Figure 2.8. The slopes and residuals are the same as in Figure 2.4 (50,75,100, 125, and 150% of nominal black squares), but the A -values are more densely clustered 90, 95, 100, 105, and 110% of nominal (gray squares), respectively 96, 98, 100, 102, and 104% of nominal (white squares). The following figures of merit are found for the sequence bottom, middle, top the residual standard deviations +0.00363 in all cases the coefficients of determination 0.9996, 0.9909, 0.9455 the relative confidence intervals of b +3.5%, +17.6%, 44.1%. Obviously the extrapolation penalty increases with decreasing Sx.x, and can be readily influenced by the choice of the calibration concentrations. The difference in Sxx (6250, 250 resp. 40) exerts a very large influence on the estimated confidence limits associated with a, b, Y(x), and X( y ).

Conclusions the residual standard deviation is somewhat improved by the weighting scheme note that the coefficient of determination gives no clue as to the improvements discussed in the following. In this specific case, weighting improves the relative confidence interval associated with the slope b. However, because the smallest absolute standard deviations. v(v) are found near the origin, the center of mass Xmean/ymean moves toward the origin and the estimated limits of detection resp. quantitation, LOD resp. 

Correlation coefficient r or coefficient of determination (these indicators are not overly useful, see Section 2.1, but so well-known that some bureaucrats are unhappy if they do not have them, so they were included here but rounded to four or five decimal places). 

Display key results number of points N, intercept a, slope b, both with 95 % confidence limits, coefficient of determination r, residual standard deviation. 

In this study the reader is introduced to the procedures to be followed in entering parameters into the CA program. For this study we will keep Pm = 1.0. We will first carry out 10 runs of 60 iterations each. The exercise described above will be translated into an actual example using the directions in Chapter 10. After the 10-run simulation is completed, determine (x)6o, y)60, and d )6o, along with their respective standard deviations. Do the results of this small sample bear out the expectations presented above Next, plot d ) versus y/n for = 0, 10,20, 30,40, 50, and 60 iterations. What kind of a plot do you get Determine the trendline equation (showing the slope and y-intercept) and the coefficient of determination (the fraction of the variance accounted for by the model) for this study. Repeat this process using 100 runs. Note that the slope of the trendline should correspond approximately to the step size, 5=1, and the y-intercept should be approximately zero. 

To benchmark our learning methodology with alternative conventional approaches, we used the same 500 (x, y) data records and followed the usual regression analysis steps (including stepwise variable selection, examination of residuals, and variable transformations) to find an approximate empirical model, / (x), with a coefficient of determination = 0.79. This model is given by... 

It is usual to have the coefficient of determination, r, and the standard deviation or RMSE, reported for such QSPR models, where the latter two are essentially identical. The value indicates how well the model fits the data. Given an r value close to 1, most of the variahon in the original data is accounted for. However, even an of 1 provides no indication of the predictive properties of the model. Therefore, leave-one-out tests of the predictivity are often reported with a QSAR, where sequentially all but one descriptor are used to generate a model and the remaining one is predicted. The analogous statistical measures resulting from such leave-one-out cross-validation often are denoted as and SpR ss- Nevertheless, care must be taken even with respect to such predictivity measures, because they can be considerably misleading if clusters of similar compounds are in the dataset. 

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. 

Regression correlation coefficient Regression coefficient of determination Rolling circle amplification Water solubility Sodium dodecyl sulfate Supercritical fluid extraction Standard operating procedure Solid-phase extraction Surface plasmon resonance Thymine... 

Linearity is often assessed by examining the correlation coefficient (r) [or the coefficient of determination (r )] of the least-squares regression line of the detector response versus analyte concentration. A value of r = 0.995 (r = 0.99) is generally considered evidence of acceptable fit of the data to the regression line. Although the use of r or is a practical way of evaluating linearity, these parameters, by... 

Acid dissociation constant Regression correlation coefficient Regression coefficient of determination Water solubility... 

Note The coefficient of determination ranges in value from 0 to 1. If it is equal to one then there is a perfect correlation in the sample. If on the other hand it has a value of zero then the model is not useful in calculating a y-value. 

Alternatively, NIR spectroscopy has been applied to relate NIR data to mechanical properties [4], A multivariate data analysis was performed on a series of commercial ethene copolymers with 1-butene and 1-octene. For the density correlation, a coefficient of determination better than 99% was obtained, whereas this was 97.7% for the flexural modulus, and only 85% for the tensile strength. 

Y as a function of a change in X. These include, but are not limited to correlation (r), the coefficient of determination (R2), the slope (, ), intercept (K0), the z-statistic, and of course the respective confidence limits for these statistical parameters. The use of graphical representation is also a powerful tool for discerning the relationships between X and Y paired data sets. 

For a graphical comparison of the coefficient of determination (R2) and the standard deviation of the calibration samples (Sr), a value is entered for the SEE for a specified range of Sr. The resultant graphic displays the Sr (abscissa) versus R2 (ordinate). From this graph it can be seen how the coefficient of determination increases as the standard deviation of the data. The SEE is set at 0.10 as in the examples shown in Graphics 59-la and 59-lb. Note that the same recommendation holds whether using r or R2, that being... 

The attached worksheet from MathCad ( 1986-2001 MathSoft Engineering Education, Inc., 101 Main Street Cambridge, MA 02142-1521) is used for computing the statistical parameters and graphics discussed in Chapters 58 through 61, in references [b-l-b-4]. It is recommended that the statistics incorporated into this series of Worksheets be used for evaluations of goodness of fit statistics such as the correlation coefficient, the coefficient of determination, the standard error of estimate and the useful range of calibration standards used in method development. If you would like this Worksheet sent to you, please request this by e-mail from the authors. 

A variety of statistical parameters have been reported in the QSAR literature to reflect the quality of the model. These measures give indications about how well the model fits existing data, i.e., they measure the explained variance of the target parameter y in the biological data. Some of the most common measures of regression are root mean squares error (rmse), standard error of estimates (s), and coefficient of determination (R2). 

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