Statistics regression coefficient

The model building step deals with the development of mathematical models to relate the optimized set of descriptors with the target property. Two statistical measures indicate the quality of a model, the regression coefficient, r, or its square, r, and the standard deviation, a (see Chapter 9). 

Quantitative stmcture—activity relationships have been estabUshed using the Hansch multiparameter approach (14). For rat antigoiter activities (AG), the following (eq. 1) was found, where, as in statistical regression equations, n = number of compounds, r = regression coefficient, and s = standard deviation... 

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. 

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... 

Isotherm plots of TOC data for only Bottom Ash Solid waste and isotherm equations for the different solid phases are shown in Fig. 12, and the isotherm parameters determined from statistical regression analyses with their coefficients are given in Table 4. 

In reference 88, response surfaces from optimization were used to obtain an initial idea about the method robustness and about the interval of the factors to be examined in a later robustness test. In the latter, regression analysis was applied and a full quadratic model was fitted to the data for each response. The method was considered robust concerning its quantitative aspect, since no statistically significant coefficients occurred. However, for qualitative responses, e.g., resolution, significant factors were found and the results were further used to calculate system suitability values. In reference 89, first a second-order polynomial model was fitted to the data and validated. Then response surfaces were drawn for... 

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. 

Designed Experiments Produce More Precise Models. In the context of linear regression, this is demonstrated by examining the statistical uncertainties of the regression coefficients. Equation 2.1 is the regression model where the response for the th sample (r ) of an instrument is shown as a linear function of the sample concentration (c.) with measurement error... 

Statistical Prediction Errors (Model and Sample Diag Jostic) Uncertainties in the concentrations can be estimated because the predicted concentrations are regression coefficients from a linear regression (see Equations 5.7-5.10). These are referred to as statistical prediction errors to distinguish them from simple concentration residuals (c — c). Tlie statistical prediction errors are calculated for one prediction sample as... 

Model statistics include R, adjusted R and root mean squared error. Parameter statistics are the estimated regression coefficients and associated statistics. 

A number of points should be noted concerning the statistics displayed In the table. First, If the researcher wishes to rank the variables In order of their Importance within the equation, absolute values of the beta values are the appropriate Indicators of rank (7, p. 284). Second, the t-values of the regression coefficients give us estimates of the statistical significance of the Independent variables used. Third, the R-square, or coefficient of determination. Is an estimate of the percent of variation In the dependent variable (the functional property) explained by the corresponding regression equation. 

Most suppliers of log P calculating software (see van de Waterbeemd, 1996) present a statistical evaluation of their program s performance. This usually is based on regression of calculated versus measured values and an examination of the regression coefficient and standard deviation. 

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. 

After obtaining regression coefficient values, both their statistical significance and lack of fit of the obtained regression model are checked. 

A check of statistical significance must be done for the calculated regression coefficients and a check of lack of fit for the regression model. Both checks are a subject of statistical analysis that will be elaborated in more detail in the next chapter. The check of the obtained regression model has shown that it is inadequate, so that we have to reduce variation intervals of factors and increase the number of design-point replications. 

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