Testing regression coefficient

FIGURE 26.38 Correlation and regression coefficients between road test ratings obtained on three different wet road test tracks and laboratory ratings obtained on a wet Alumina 60 laboratory surface as function of... 

Correlation, Regression Coefficients and Intercept of a Linear Relation between the Road Test Ratings from Table 26.4 and the Laboratory Ratings Obtained for Each Cell... 

FIGURE 26.71 Correlation and regression coefficients between road test ratings of seven truck tire compounds 80 SBR/20 NR differing by type and amount of filler and laboratory abrasion ratings obtained on LAT 100 testing equipment. 

It has already been shown how a correlation between road test ratings and laboratory abrasion can be obtained over a range of energies and speeds (Table 26.6). Usually, a good correlation, i.e., a high correlation coefficient and a regression coefficient near 1 are obtained only over a limited range. 

As an example for MLR, we consider the data from Table 4.2 (Section 4.1) where only variables x, and x2 were in relation to the y-variable but not x3. Nevertheless, a regression model using all x-variables is fitted and the result is presented in Figure 4.13. The statistical tests for the single regression coefficients clearly show that variable x3 can be omitted from the model. 

FIGURE 4.24 PLS as a multiple linear regression method for prediction of a property y from variables xi,..., xm, applying regression coefficients b1,...,bm (mean-centered data). From a calibration set, the PLS model is created and applied to the calibration data and to test data. 

An advantage of LR in comparison to LDA is the fact that statistical inference in the form of tests and confidence intervals for the regression parameters can be derived (compare Section 4.3). It is thus possible to test whether the /th regression coefficient bj = 0. If the hypothesis can be rejected, the jth regressor variable xj... 

Since the stability data from individual lots are pooled, these data are examined for validity by the F test. The mean square of the regression coefficient (slope) is divided by the mean square of the deviation within lots, and similarly, the adjusted mean (y intercept) is divided by the common mean square to give the respective F ratios. The latter values then are compared with the critical 5% F values. When the calculated F values are smaller than the critical F values, the data may be combined and the pooled data analyzed. 

If plotting cannot be used in the selection of a proper geometric representation (and the corresponding equation form) of the Influence of each variable factor being studied, then accepted theory and past research results may aid the researcher in selection of variable and equation forms to be used. Within limits, the resuts of t-tests of regression coefficients may be used to select the most appropriate variables and variable transformations for use in the final... 

Testing the component and mixture melting points was done by the ASTM-D-127 method. Vaseline penetration testing was done by ASTM-D-937, and that of ozocerite, paraffin and the mixture by ASTM-D-1321 method. Calculations by Eq. (3.29) offered these values of regression coefficients ... 

The test of the regression coefficients is based on the /-test according to STUDENT ... 

The parameter c was found to be a linear function of Reynolds number with regression coefficients between 0.98 and 1.00. The shear rate constant, k, was within 10% of the values found by Donnelly (15) and Rieth (16) for a double-helical ribbon impeller. Furthermore, the Power Law could be used to describe corn stover suspension viscosity with correlation coefficients above 0.99 for all four concentrations tested. Finally, the yield stress predicted by direct data extrapolation and by the Herschel-Bulkley model was similar for each concentration of com stover. 

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