Standard deviation of the regression

In contr ast to the linear case, there are three degrees of freedom, but there is still only one standard deviation of the regression, s. The reader has the opportunity to try out these ideas in Computer Project 3-4. 

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. 

The slope S may be estimated from the calibration curve of the analyte. The value of may be estimated by (1) calculating the standard deviation of the responses obtained from the measurement of the analytical background response of an appropriate number of blank samples or (2) calculating the residual standard deviation of the regression line from the calibration curve using samples containing the analyte in the range of the QL. 

Vapor pressures obtained by the Knudsen effusion method are shown in Table II. The precision of the determinations is indicated by the 95% confidence interval computed either from an average of experimental determination or from the standard deviation of the regression equation. 

A secondary but no less important goal is to obtain the standard deviations of the regression parameters, and an estimate of the goodness of fit of the data to the model equation. 

As you ve seen, the Solver finds the set of least-squares regression coefficients very quickly and efficiently. However, it does not provide the standard deviations of the coefficients. Without these, the Solver s solution is essentially useless. The following illustrates how to obtain the standard deviations of the regression coefficients after obtaining the coefficients by using the Solver. 

Compare the standard deviations using the forgoing procedure (cells L24 and M24 in Figure 12-9) with those obtained from LI NEST, which are shown in row 32 of Figure 12-10. Once again, it should be made clear that a linear problem was chosen to permit comparison between the standard deviations of the regression coefficients and those from LINEST,... 

For the precision of an analytical method not only the repeatability of single measurements but also the errors in the calibration procedure should be included. This is a complex problem, because according to the nature of the signal noise, the precision obtainable may vary considerably with the concentration level. In the case of a calibration the scattering of the measured Y, values around the calibration curve is given by the standard deviation of the regression ... 

Cell Cl 1 contains the standard error of the estimate (or standard deviation of the regression) and is a measure of the error in estimating values of y. The smaller it is, the closer the numbers are to the line. The other cells contain data we will not consider here Cell B12 is the F value, cell C12 the degrees of freedoin (used for F), cell B13 the sum of squares of the regression, and cell C13 the sum of squares of the residuals. 

Table 10.2 (regression equation Section A) provides the actual h, values, the standard deviation of each h the t-test value for each h and the p-value for each h,. In multiple regression, the i-ratio and p-value have limited use. The standard deviation of the regression equation, Sy xi,x2,x3 = 0.5949, is just more than j log value, and the coefficient of determination, = 86.1%, which means the regression equation... 

When a linear regression is performed, the standard deviation of the regression can be defined ... 

By applying fault length or area to the appropriate empirical relationships, magnitude is obtained in a straightforward manner. Typical examples from Wells and Coppersmith (1994), without the standard deviations of the regression equations, are ... 

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