Standardized partial regression

The standardized partial regression coefficient t is a measure of the relative importance of the corresponding predictor and is given by... 

The second advantage of the present procedure is that we can now readily obtain the standard error of each partial regression coefficient. We need the residual variance vr which we already have in Table 10.1. To repeat, in the present symbolism, it is 1... 

Regression coefficients, partial, 172 standardized, 168 Regression, linear, 156 multivariate, 171 polynomial, 163 through origin, 162 Residuals, 13 Residuals analysis, 159 Ridge regression, 203 RMS noise, 31 Roots, characteristic, 73... 

With the aim of bringing the regression coefficients a and P into a common scale, Krygowski and Fawcett calculated the partial regression coefficients a and p by the standard method of multi-parameter regression. From these they obtained... 

Fig. 2. Correlation of the a equation-of-state parameter with the structural con bution to the standard partial molal volume of aqueous species at 25 °C and 1 bar (AV ). The symbols represent values obtained from regression of experimental data for aqueous electrolytes and neutral species at elevated temperatures taken from Shock and Helgeson (1988, 1990), Shock et al. (1989), and Shock (1993a). The correlation is given by Eq. (6)...

The author gives an exampie of a study concerning a mixture of ethanol, toluene and ethyl acetate. The case is presented in the form of a Scheffe plan for which choice of compound quantities are not optimised to obtain a good matrix as shown in the matrix of effects correiation there is no point repetition in the middle of the matrix, which thus exciudes the quantification of the level of error of measurement that can only be estimated by the residual standard deviation of the regression. Finaliy, the author uses flashpoints of pure substances from partial experimental data. The available data give 9 to IS C for ethanol (the author 12.8), 2 to 9°C for toluene (5.56) and -4 to -2°C for ethyl acetate. 

Table II gives the temperature interval, the number of laboratories that Battino judges have published reliable solubility data, the number of experimental values used in the linear regression, the linear regression standard deviation at the midpoint temperature, and the temperature of minimum mole fraction solubility (maximum value of Henry s constant) at one atmosphere partial pressure of the gas. For all of the gases except oxygen only a three constant equation was used.

Sivakesava et al. also used Raman (as well as FT-IR and NIR) to perform a simultaneous on-line determination of biomass, glucose, and lactic acid in lactic acid fermentation by Lactobacillus casei.2 Partial least squares (PLS) and principal components regression (PCR) equations were generated after suitable wavelength regions were determined. The best standard errors were found to be glucose, 2.5 g/1 lactic acid, 0.7 g/1 and optical cell density, 0.23. Best numbers were found for FT-IR with NIR and Raman being somewhat less accurate (in this experiment). 

Fourier transform infrared (FTIR) spectroscopy of coal low-temperature ashes was applied to the determination of coal mineralogy and the prediction of ash properties during coal combustion. Analytical methods commonly applied to the mineralogy of coal are critically surveyed. Conventional least-squares analysis of spectra was used to determine coal mineralogy on the basis of forty-two reference mineral spectra. The method described showed several limitations. However, partial least-squares and principal component regression calibrations with the FTIR data permitted prediction of all eight ASTM ash fusion temperatures to within 50 to 78 F and four major elemental oxide concentrations to within 0.74 to 1.79 wt % of the ASTM ash (standard errors of prediction). Factor analysis based methods offer considerable potential in mineral-ogical and ash property applications. 

Table 3 (73) compares the retention coefficients for synthetic peptides from various sources. To ensure comparability, the data has been standardized with respect to lysine and assigned a value of 100. The table shows that there are discrepancies between the results obtained using different chromatographic systems. Predictions of retention times should therefore be made using chromatographic systems similar to those used to calculate the retention coefficients for the amino acids. Casal et al. (75a) have made a comparative study of the prediction of the retention behavior of small peptides in several columns by using partial least squares and multiple linear regression analysis. 

FIGURE 5.6 Partial least-squares regression model showing the correlation between alanine (nmol/g cheese) predicted by GC-FID and FTIR. The model shows a high degree of linear correlation (r-value = 0.99) and a low estimated standard error of prediction (12.70 nmol/g cheese). 

It may be possible to use an array of electrodes containing various enzymes in combination with multivariate statistical analyses (principal component analysis, discriminant analysis, partial least-squares regression) to determine which pesticide(s) the SPCE has been exposed to and possibly even how much, provided sufficient training sets of standards have been measured. The construction methods for such arrays would be the same as described in this protocol, with variations in the amounts of enzyme depending on the inhibition constants of other cholinesterases for the various pesticides of interest. 

Faber, N.M., Song, X.H., and Hopke, P.K., Sample-specific standard error of prediction for partial least squares regression, Trends Anal. Chem., 22, 330-334, 2003. 

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