Statistical analysis regression coefficient

Because of the large difference in the behavior of the thin plywood and the gypsum board, the type of interior finish was the dominant factor in the statistical analysis of the total heat release data (Table III). Linear regression of the data sets for 5, 10, and 15 min resulted in squares of the correlation coefficients R = 0.88 to 0.91 with the type of interior finish as the sole variable. For the plywood, the average total heat release was 172, 292, and 425 MJ at 5, 10, and 15 min, respectively. For the gypsum board, the average total heat release was 25, 27, and 29 MJ at 5, 10, and 15 min, respectively. 

Statistics. Data were subjected to analysis of varlatice and regression analysis by using the general linear model procedure of the Statistical Analysis System (16). Correlation coefficients between growth parameters were determined with the same system. Equations were best fitted to the ata based on significance level of the terms of the equation and R values. 

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. 

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. 

Statistical data analysis has offered these regression coefficient values foryj fory2 ... 

Leaving out the z0 column, these coefficients multiplied by 2 immediately yield the effects in the statistical or variance analytical sense. This difference in the computation of the regression coefficients and the effects comes from the different models used in regression analysis and in analysis of variance. The first refers to the zero or medium level, the second refers to the lowest level. 

Figure 2.3. Linear regression analysis with Excel. Simple linear regression analysis is performed with Excel using Tools -> Data Analysis -> Regression. The output is reorganized to show regression statistics, ANOVA residual plot and line fit plot (standard error in coefficients and a listing of the residues are not shown here).

In contrast to the results returned by FINEST, the output is clearly labeled, and additional statistical data is provided. Regression data for the example shown in Figure 11-1 is shown in the three tables of Figure 11-14. Three tables are produced regression statistics, analysis of variance, and regression coefficients. (The coefficients table has been broken into two parts to fit the page.)... 

Figure 11-14. Data obtained by using Regression from the Analysis ToolPak (from top) Regression Statistics, Analysis of Variance, Regression Coefficients and Statistics.

Statistical methods. Certainly one of the most important considerations in QSAR is the statistical analysis of the correlation of the observed biological activity with structural parameters - either the extrathermodynamic (Hansch) or the indicator variables (Free-Wilson). The coefficients of the structural parameters that establish the correlation with the biological activity can be obtained by a regression analysis. Since the models are constructed in terms of multiple additive contributions the method of solution is also called multiple linear regression analysis. This method is based on three requirements (223) i) the independent variables (structural parameters) are fixed variates and the dependent variable (biological activity) is randomly produced, ii) the dependent variable is normally and independently distributed for any set of independent variables, and iii) the variance of the dependent variable must be the same for any set of independent variables. 

It is notable that such kinds of error sources are fairly treated using the concept of measurement uncertainty which makes no difference between random and systematic . When simulated samples with known analyte content can be prepared, the effect of the matrix is a matter of direct investigation in respect of its chemical composition as well as physical properties that influence the result and may be at different levels for analytical samples and a calibration standard. It has long since been suggested in examination of matrix effects [26, 27] that the influence of matrix factors be varied (at least) at two levels corresponding to their upper and lower limits in accordance with an appropriate experimental design. The results from such an experiment enable the main effects of the factors and also interaction effects to be estimated as coefficients in a polynomial regression model, with the variance of matrix-induced error found by statistical analysis. This variance is simply the (squared) standard uncertainty we seek for the matrix effects. 

The calculation of the dependency of the molar volume of the ternary system LiF-NaF-K2NbFy on composition was performed according to Eq. (5.17). The regression coefficients were calculated using the multiple linear regression analysis omitting the statistical non-important terms on the 0.99 confidence level. The molar volume of... 

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