Regression analysis correlation coefficient

APt = pressure gradient across a train of capsules, Pa/m APc = pressure gradient across a single capsule, Pa/m R = regression analysis correlation coefficient... 

After preparation of a stock solution (0.200 M) of (R)- 1-phenylethyl acetate ((i )-4) and (S)-( 1 -phenylethyl)-1 -13C-acetate ((b1)-l3C-4) in cyclohexane, the solutions are diluted with cyclohexane to concentrations of 0.180, 0.160, 0.140, 0.120, 0.100, 0.080, 0.060, 0.040, and 0.020M (total volume lmL). The absorbance of the resulting samples is measured with a FTIR spectrometer at the corresponding absorption maxima of the carbonyl-stretching vibration ((i )-4 1751 cm-1 (S)-13C-4 1699 cm-1) with a thickness of the layers of 25.0 pm, performing 32 scans at a resolution of 4 cm-1. The molar coefficients of absorbance are determined by linear regression, with correlation coefficients >0.995. Analysis of synthetic mixtures of the pseudo enantiomers of 1-phenylethyl acetate is performed under the same conditions at a concentration of 0.10 M. 

Definition of Linearity The linearity of an analytical method is its ability (within a given range) to elicit test results that are directly, or by a well-defined mathematical transformation, proportional to the concentration of analyte in samples within a given range. Linearity is usually expressed in terms of the variance around the slope of the regression line (correlation coefficient), calculated according to an established mathematical relationship from test results obtained by the analysis of samples with varying concentrations of analyte. 

Multiple linear regression analysis is a widely used method, in this case assuming that a linear relationship exists between solubility and the 18 input variables. The multilinear regression analy.si.s was performed by the SPSS program [30]. The training set was used to build a model, and the test set was used for the prediction of solubility. The MLRA model provided, for the training set, a correlation coefficient r = 0.92 and a standard deviation of, s = 0,78, and for the test set, r = 0.94 and s = 0.68. 

Fuller-Schettler-Giddings The parameters and constants for this correlation were determined by regression analysis of 340 experimental diffusion coefficient values of 153 binary systems. Values of X Vj used in this equation are in Table 5-16. 

The natural and correct form of the isokinetic relationship is eq. (13) or (13a). The plot, AH versus AG , has slope Pf(P - T), from which j3 is easily obtained. If a statistical treatment is needed, the common regression analysis can usually be recommended, with AG (or logK) as the independent and AH as the dependent variable, since errors in the former can be neglected. Then the overall fit is estimated by means of the correlation coefficient, and the standard deviation from the regression line reveals whether the correlation is fulfilled within the experimental errors. 

However, it is not proper to apply the regression analysis in the coordinates AH versus AS or AS versus AG , nor to draw lines in these coordinates. The reasons are the same as in Sec. IV.B., and the problem can likewise be treated as a coordinate transformation. Let us denote rcH as the correlation coefficient in the original (statistically correct) coordinates AH versus AG , in which sq and sh are the standard deviations of the two variables from their averages. After transformation to the coordinates TAS versus AG or AH versus TAS , the new correlation coefficients ros and rsH. respectively, are given by the following equations. (The constant T is without effect on the correlation coefficient.)... 

Linear regression analysis was performed on the relation of G"(s) versus PICO abrasion index. Figure 16.10 plots the correlation coefficient as a function of strain employed in the measurement of loss modulus. The regression results show poor correlation at low strain with increasing correlations at higher strains. These correlations were performed on 189 data points. 

Using regression analysis on a data set of about 50 different molecules, it was found that a. = —4.4,8 = —0.5, Df = 12 cm2/s, and =2.5x 10 5 cm2/s [192], A graphic representation of the effect of relative molecular mass (Mr) and distribution coefficient on corneal permeability is shown in Fig. 13. One observes a rapid reduction in permeability coefficient with decreasing P and increasing Mr. The addition of pores to the model, a mathematical construct, is necessary to account for permeability of polar molecules, such as mannitol and cromolyn. These would also be required for correlating effects of compounds, such as benzalkonium chloride, which may compromise the... 

They include simple statistics (e.g., sums, means, standard deviations, coefficient of variation), error analysis terms (e.g., average error, relative error, standard error of estimate), linear regression analysis, and correlation coefficients. 

For the regression analysis of a mixture design of this type, the NOCONSTANT regression command in MINITAB was used. Because of the constraint that the sum of all components must equal unity, the resultant models are in the form of Scheffe polynomials(13), in which the constant term is included in the other coefficients. However, the calculation of correlation coefficients and F values given by MINITAB are not correct for this situation. Therefore, these values had to be calculated in a separate program. Again, the computer made these repetitive and Involved calculations easily. The correct equations are shown below (13) ... 

Fig. 2 Plot of P-Cl distances (in A) vs average P-N distances (in A) for P-chloro-NHPs (diamonds) and for all compounds (R2N)2PC1 (except P-chloro-NHPs) listed in the CSD data base (open squares). The solid and dashed lines represent the result of linear regression analyses. R2 is the square of the correlation coefficient in the regression analysis. (Reproduction with permission from [55])...

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... 

Because of the large difference in the behavior of the thin plywood and the gypsum board, the type of interior finish was the dominant factor in the statistical analysis of the total heat release data (Table III). Linear regression of the data sets for 5, 10, and 15 min resulted in squares of the correlation coefficients R = 0.88 to 0.91 with the type of interior finish as the sole variable. For the plywood, the average total heat release was 172, 292, and 425 MJ at 5, 10, and 15 min, respectively. For the gypsum board, the average total heat release was 25, 27, and 29 MJ at 5, 10, and 15 min, respectively. 

Partitioning into the CNS will be important for hallucinogens, as for any drug that acts centrally. Correlation between 1-octanol/water partition coefficients and human activity has been reported (13). Regression analysis of log human activity on log P yielded a parabolic fit with an optimum at log P 3.14. The derived equation accounted for only 62% of the variance but included compounds with a variety of substitution patterns and, presumably, qualitative differences in activity. 

Note also that we can use the correlation test statistic (described in the correlation coefficient section) to determine if the regression is significant (and, therefore, valid at a defined level of certainty. A more specific test for significance would be the linear regression analysis of variance (Pollard, 1977). To so we start by developing the appropriate ANOVA table. 

Linearity is evaluated by appropriate statistical methods such as the calculation of a regression line by the method of least squares. The linearity results should include the correlation coefficient, y-intercept, slope of the regression line, and residual sum of squares as well as a plot of the data. Also, it is helpful to include an analysis of the deviation of the actual data points for the regression line to evaluate the degree of linearity. 

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