Center point, replicated experiments

For the ten-experiment design with six replicates at the center point. 

The parameter estimates with this design will be more precise than they would be with either a five-experiment design (see Equation 11.8) or, with the exception of the estimate of PJ, a ten-experiment design with six replicates at the center point (see Equation 11.11). We will use as our design here, the allocation represented in Equation 11.12. 

Inspection of the coded experimental design matrix shows that the first four experiments belong to the two-level two-factor factorial part of the design, the next four experiments are the extreme points of the star design, and the last four experiments are replicates of the center point. The corresponding matrix for the six-parameter model of Equation 12.54 is... 

In Figure 13.9, instead of carrying out four replicate experiments at the center point (as in Figure 13.2), the four replicates are carried out such that one experiment is moved to each of the existing four factorial points. The experimental design matrix is... 

In this design there are only six distinctly different factor combinations. Thus, there are no degrees of freedom for lack of fit when fitting a two-factor FSOP model with six parameters. There are three degrees of freedom for purely experimental uncertainty because of the four replicate experiments at the center point. 

The complete design is seen in the score space with replicate center points clearly visible. Note that the interpretation of scores plots is not always as straightforward as in this example. The experimental design is not seen if the experiment is not well designed or if the problem is high dimensional. The level of impEcidy modeled components (e.g., component O also has an effect on the relative position of the samples in score space. For this example, the effect of C on the relative placement of the samples in score space is small. 

Finally, the problem was resolved by irradiating standards and mixtures of standards in a factorial experiment. The experiment design was a full factorial experiment with three variables, mercury, selenium, and ytterbium, at two levels with replication and with a center point added to test higher order effects. The pertinent information on treatments and levels of variables are shown in Table VII. 

The experiment consisted of 32 design points, six design points replicated in the design center and 16 half-replica design points of type 25 1. The design of experiments matrix with the outcomes of experiments is shown in Table 2.140. The second-order regression model has this form ... 

Estimate ofthe experimental error Replication of the center point experiment gives an independent estimate, s of the experimental error variance, o, which can be used to asses the significance of the model. It can be used to evaluate the lack of fit by comparison with the residual mean square, as weU as to assess the significance of the individual terms in the model. (2) Check of curvature If a linear or a second-order interaction model is adequate, the constant term ho will correspond to the expected response, y(0), at the center point. If the difference y(0) - should be significantly greater than the standard deviation of the experimental error as determined by the r-statistic... 

If a significant curvature is found, run the axial experiments and fit a quadratic model. Use the independent estimate of the experimental error from the replicated center point experiments to check the fit of model. 

From the replicated experiments at the center point, we can obtain an estimate, with (iVo -1) degrees of freedom, of the experimental error... 

Sometimes it is seen that the residual mean square / (N - p) is compared to an estimate of the pure error from the replicated center point experiments. This is not quite correct, but will reveal a highly significant lack-of-fit. 

A normal probability plot of the residuals is shown in Fig. 12.4. There are evidently some abnormally high errors associated with the replicated center point experiments. Maybe they were run on a different occasion. The remaining residuals... 

Besides the design experiments, frequently additional experiments are performed (7). For example, to estimate the experimental error, the center point or one or several design experiment(s) can be replicated. To evaluate the prediction performance, additional points, different from the experimental design points, for example, the predicted optimum, can be measured. 

Central Composite Designs. CCDs are the most often used response surface designs (1,7,17).These designs are constructed by combining a two-level full factorial design (2 experiments), a star design (2/experiments), and a center point, which is often replicated a number of times. Thus, to examine / factors, at least = 2 + 2/ + 1 experiments are required. For more... 

A.2.3. Example of an Applied Response Surface Design. In the optimization phase of the development of a CE method for the chiral enantio-separation of a nonsteroidal anti-inflammatory drug (28), a circumscribed CCD was performed. The applied symmetrical response surface design is as shown in Table 2.14, with a = 2 f = 1.68. The center point (experiment 15 in Table 2.14) was replicated five times (experiments 15-19). 

At this point, the required experiments can be defined. For this purpose, the levels (e.g., -a, -1,0, -f1, H-a) in the theoretical experimental design (e.g., Tables 2.8, 2.14, and 2.9) are replaced by the real factor levels (e.g., Tables 2.2-2.4, respectively).This results in the experimental conditions for each experiment. The dummy factor columns in PB designs can be ignored at this point. Often a number of replicated experiments at nominal or center point conditions are added to the setup (see above). 

SE)e can be estimated in different ways, that is, from the variance of replicated experiments, for instance, at the nominal or center point level, from a priori declared negligible effects or from a posteriori defined negligible effects (4, 5,7,16, 24,31,74,105,106,109-114). 

Exercise 6.2. Using the estimate = 2.33 obtained from the replicate experiments at the center point of Fig. 6.1, calculate the standard errors of the average of all seven response values and of the effects determined in the preceding exercise. Compare your results with the errors determined for the coefficients of the fitted model (Eq. (6.3)). 

In a Box-Behnken design, the experimental points lie on a hypersphere equidistant from the center point as exemplified for a three-factor design in Figure 4.13 and Table 4.11. In contrast to the central composite design, the factor levels have only to be adjusted at three levels. In addition, if two replications are again performed in the center of the three-factor design, the total number of experiments is 15 compared to 17 with the central composite design. 

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