Design matrix evaluation for response surface quadratic model
Design matrix evaluation implies that there are no aliases for the quadratic model. In general, a minimum of degrees of freedom 3 and 4 has been recommended for lack-of-fit and pure error, respectively. Therefore, degrees of freedom obtained in this study ensured a valid lack of fit test (Table 5).
Table 5 Degrees of freedom for evaluation
Model 14
Residuals 15
Lack of Fit 10
Pure Error 5
The standard error (SE) used to measure the precision of the estimate of the coefficient. The smaller standard error implies the more accurate the estimate. The variables of A, B, C and D have a standard errors = 0.16. The interceptions of AB, AC, AD, BC, BD and CD have slightly high standard errors = 0.2, while A2, B2, C2 and D2 have standard errors = 0.15. An approximate 95 % confidence interval for the coefficient is given by the estimate plus and minus 2 times the standard error. For example, with 95 % confidence can be said that the value of the regression coefficient A is between 6.49 and 5.85 (6.17 ± 2 × 0.16).
The quadratic model coefficients for the CCD are shown in Table 6. This results suggested that the variables coefficients and their interactions are estimated adequately without multicollinearity. The low Ri-squared for independent variables and their interactions imply that the model is a good fit. In general, power should be approximately 80 % for detecting an effect [60]. In this study, there are more than 99 % chance of detecting a main effect while it is twice the background sigma.
Table 6 The Quadratic model coefficients for the CCD
Term StdErr** VIF Ri-Squared Power at 5 % Power at 5 % Power at 5 %
SN = 0.5 SN = 1 SN = 2
A 0.16 1 0 20.90 % 63.00 % 99.50 %
B 0.16 1 0 20.90 % 63.00 % 99.50 %
C 0.16 1 0 20.90 % 63.00 % 99.50 %
D 0.16 1 0 20.90 % 63.00 % 99.50 %
AB 0.2 1 0 15.50 % 46.50 % 96.20 %
AC 0.2 1 0 15.50 % 46.50 % 96.20 %
AD 0.2 1 0 15.50 % 46.50 % 96.20 %
BC 0.2 1 0 15.50 % 46.50 % 96.20 %
BD 0.2 1 0 15.50 % 46.50 % 96.20 %
CD 0.2 1 0 15.50 % 46.50 % 96.20 %
A2 0.15 1.05 0.0476 68.70 % 99.80 % 99.90 %
B2 0.15 1.05 0.0476 68.70 % 99.980 % 99.90 %
C2 0.15 1.05 0.0476 68.70 % 99.80 % 99.90 %
D2 0.15 1.05 0.0476 68.70 % 99.80 % 99.90 %
**Basis Std. Dev. = 1.0