Confidence regression coefficients

Factors with regression coefficients having confidence coefficients > 90% None ANOVA F-ratIo 13.65... 

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... 

Model and Parameter Sta stics (Model Diagnostic) Table 5-13 displays the variables selected for a model constructed to predict caustic. The table lists summary statistics for the regression model as weU as information about the estimated regression coefficients. Six variables in addition to an intercept are found to be significant at the 95% confidence level. 

The regression coefficient (r2) for this relationship was 0.8685. The significance of the calculated F(1/3) = 22.2 can be ascertained by consulting tables of F values. From such a table, it is found that the F(13)a0 025 distribution is 17.4. Because the F(13) of 22.2 is greater than 17.4, it can be assumed that the equation is significant at the 2.5% confidence level. This relationship showed that the kinetic rate increases as ELUMO increases. A set of aliphatic compounds was used for the correlation between ELUMO as molecular descriptor and activation energy as a predictive molecular descriptor. The dataset of... 

One particular challenge in the effective use of MLR is the selection of appropriate X-variables to use in the model. The stepwise and APC methods are some of the most common empirical methods for variable selection. Prior knowledge of process chemistry and dynamics, as well as the process analytical measurement technology itself, can be used to enable a priori selection of variables or to provide some degree of added confidence in variables that are selected empirically. If a priori selection is done, one must be careful to select variables that are not highly correlated with one other, or else the matrix inversion that is done to calculate the MLR regression coefficients (Equation 8.24) can become unstable, and introduce noise into the model. 

The calculated value of analysis of variance is F=1343.6 for the null hypothesis HqiP O. However, since the tabular value is F1 g 0 95=5.32 the null hypothesis is rejected and the alternative hypothesis accepted that the regression coefficient p, with 95% confidence level is statistically significant. 

Since the calculated value is F=57.52>FTAB=4.28 it can be with 95% confidence level asserted that the regression coefficient Pi is statistically greater than zero and that it should be kept in the linear regression. 

By comparing absolute values of regression coefficients with calculated intervals, we may assert with 95% confidence level that all regression coefficients except bi2 and b22 are statistically significant. The regression equation becomes ... 

By comparing absolute values of regression coefficients with errors in their estimates, it becomes evident that all regression coefficients are statistically significant with 0.95% confidence, except for bn and b22. A check of lack of fit of the obtained regression model proved that it is adequate with 95% confidence (FR[Pg.332]

By comparing absolute values of regression coefficients with their interval estimates, with 95% confidence, these regression coefficients are statistically significant b0, b3, b44 and bss. Following this the regression model (2.97) becomes ... 

All regression coefficients are significant with 95% confidence. The second-order regression model has the form ... 

A check of statistical significance of regression coefficients (Sect. 2.4.2), indicates that regression coefficients bn and b22 are statistically insignificant. The final form of the second-order regression model with 95% confidence may be given in the form ... 

The obtained regression model is adequate with 95% confidence. A check of significance of regression coefficients, in accord with Sect. 2.4.2, is completed thus ... 

A tabular value FT=1.85 is obtained for fAD=10 and fE=25(25-l)=600 from Table E. The regression model is with 95% confidence adequate because FRstatistical significance of regression coefficients, use formulas from Sect. 2.4.2. 

A check of significance of regression coefficients shows that with 95% confidence the following regression coefficients are statistically significant b0 hi b2 b3 b12 and b22. The second-order regression model has this form ... 

A regression coefficient is statistically significant if its absolute value is higher than the confidence interval. 

Thus, a regression coefficient is statistically significant with 95% confidence if its value comes from this interval. 

Thus, for a significance level (/-5%, limits of the confidence interval of a regression coefficient are given by expression ... 

With 95% confidence we may assert that all regression coefficients are statistically... 

All regression coefficients except bn and b22 are with 95 % confidence statistically significant since they are above the errors in their estimations (Ab). 

Only b0 and bx of all regression coefficients with 99% confidence, are statistically significant. Hence, the final form of the regression model is ... 

All regression coefficients are statistically significant with 95% confidence except for b34 b24 b234 b134 and b124. The regression model looks like this ... 

Statistically significant regression coefficients with 99% confidence ... 

All regression coefficients in Table 3.56 are statistically significant with 99% confidence. 

Confidence Interval and Hypothesis Tests for Regression Coefficients... 

After calculating calibration coefficients, it is worthwhile to examine the errors existing in b and establish confidence intervals. The standard error of each regression coefficient is computed according to... 

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