Factorial points

Sketch a 3-factor non-central composite design for which the center of the star coincides with one of the factorial points. 

The lower left panel in Figure 13.2 shows the central composite design in the two factors X, and X2. The factor domain extends from -5 to +5 in each factor dimension. The coordinate axes in this panel are rotated 45° to correspond to the orientation of the axes in the panel above. Each black dot represents a distinctly different factor combination, or design point. The pattern of dots shows a central composite design centered at (Xj = 0, Xj = 0). The factorial points are located 2 units from the center. The star points are located 4 units from the center. The three concentric circles indicate that the center point has been replicated a total of four times. The experimental design matrix 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... 

The full factorial central composite design includes factorial points, star points, and center points. The corresponding model is the complete quadratic surface between the response and the factors, as given by Eq. 1 ... 

A five-level-five-factor CCRD was employed in this study, requiring 32 experiments (Cochran and Cox, 1992). The fractional factorial design consisted of 16 factorial points, 10 axial points (two axial points on the axis of each design variable at a distance of 2 from the design center), and 6 center points. The variables and their levels selected for the study of biodiesel synthesis were reaction time (4-20 h) temperature (25-65 °C) enzyme amount (10%-50% weight of canola oil, 0.1-0.5g) substrate molar ratio (2 1—5 1 methanol canola oil) and amount of added water (0-20%, by weight of canola oil). Table 9.5 shows the independent factors (X,), levels and experimental design coded and uncoded. Thirty-two runs were performed in a totally random order. 

Finally it is often useful to be able estimate the experimental error (as discussed in Section 2.2.2), and one method is to perform extra replicates (typically five) in the centre. Obviously other approaches to replication are possible, but it is usual to replicate in the centre and assume that the error is the same throughout the response surface. If there are any overriding reasons to assume that heteroscedasticity of errors has an important role, replication could be performed at the star or factorial points. However, much of experimental design is based on classical statistics where there is no real detailed information about error distributions over an experimental domain, or at least obtaining such information would be unnecessarily laborious. 

In most cases, it is best to use a full factorial design for the factorial points, but if the number of factors is large, it is legitimate to reduce this and use a partial factorial design. There are almost always 2k axial points. 

Rotatability implies that the confidence in the predictions depends only on the distance from die centre of the design. For a two factor design, this means that all experimental points hi a circle of a given radius will be predicted equally well. This has useful practical consequences, for example, if the two factors correspond to concentrations of acetone and methanol, we know that the further the concentrations are from the central point the lower is the confidence. Methods for visualising diis were described in Section 2.2.5. Rotatability does not depend on the number of replicates in the centre, but only on die value of a, which should equal j/Nj, where Nf is the number of factorial points, equal to 2k if a full factorial is used, for diis property. Note that the position of the axial points will differ if a fractional factorial is used for the cubic part of die design. 

Fig. 6. Distribution of experimental points in central composite designs factorial points, O centre point, x axial points...

Fig. 4 Central composite design for three factors. The factorial points are shaded, the axial points unshaded, and the center point(s) filled.

One important conclusion from this is, that it is always useful to include at least one experiment at the center point when a factorial or fractional factorial design is run. This will not cause any difficulties if tbe design is evaluated by hand-calculation A least squares fit of the linear coefficients, hj, and the interaction coefficients, bjj, are detennined as usual from the factorial points. The intercept, b, is computed as the average of all experiments. The standard error of the estimated coefficient will, however, be slightly different. Let N be the total number of experiments, and let Np be the number of factorial points. If is the experimental enor variance, the standard error of the intercept b will be aNN and for the linear coefficients and interaction coefficients the standard enor will be aNNp. 

A CCD for three factors is listed in Table 4, and includes an imbedded 2 factorial design with center points and three pairs of star points. For rotatability, a = 1.682, as there are F = 8 factorial points in the design. A three-block sequential strategy for a three-factor CCD is listed in Table 5, where ... 

The 11-run design and resulting data are listed in Table 9. The first four runs are the imbedded 2 factorial points, the next four runs are the axial or star points, and the last three runs are the center points. The factor levels are listed in the actual units ( uncoded form) and in coded units. The value of a was set to 1 to minimize the number of factor levels required, instead of conducting a rotatable design... 

We continue the surfactant mixture (mixed micelles) solubility example introduced in the first part of this chapter. If only 4 experiments are carried out at the factorial points and the model equation contains 3 coefficients plus the constant term, then the design is saturated. The model will fit the data exactly ... 

Therefore, the solubility in a medium containing 0.1 M bile salt (X[ = 0) and an equimolar ratio of lecithin (X2 = 0) was determined in duplicate (experiments 5 and 5 as described in chapter 4) at the same time as the measurements at the factorial points. (Note that experiments at the test points should be done, if possible, at the same time as the other experiments, all of them in a random order.) The experimentally measured solubilities were 11.70 and 11.04 mg mL , a mean value of 11.37. This is a difference of 1.21 mg mL" with respect to the model calculation of 10.16, which appears large in comparison with the differences observed previously. We therefore believe that the response surface may not be an inclined plane, but a curved surface. We have detected this curvature in the centre of the domain, and we therefore require a more complex mathematical model. This conclusion for the moment is entirely subjective as we have not yet considered any statistical tests. We will demonstrate later on (section II.B) how it is possible to test if this difference is significative and we will show that in such a case it may be attributed to the existance of squared terms in the model. For the moment we will limit ourselves to the conclusion that a more complex mathematical model is necessary. 

The columns X, Xi and X2 can be seen to be identical for the factorial experiments. Following the same reasoning as in chapter 3, we conclude that the estimator bo for the constant term, obtained from the factorial points, is biased by any quadratic effects that exist. On the other hand the estimate b o obtained only from the centre point experiments is unbiased. Whatever the polynomial model, the values at the centre of the domain are direct measurements of Pq. So the difference between the estimates, bo - b o, (which we can write as U+22 using the same notation as in chaptw 3) is a measure of the curvature P, + P22- The standard deviation o o is s/v 8 (as it is the mean of 8 data of the factorial design) and that of b o is s/ /2. (being the mean of 2 centre points). We define a function t as ... 

The model coefficients were then estimated by least squares linear regression (table 5.12). Compare the values with those obtained with the factorial points, in chapter 3, section II.B.4. We see that only the interaction terms are unmodified the axial data are not used in calculating these. 

Consider a 2 design with one observation at each of the factorial points (, +),... 

Furthermore, if the factorial points in the design are unreplicated, one may use the center points to construct an estimate of error with tIq — degrees of freedom. 

It means that the variance of response at origin is eqnal to the variance of response at a nnit distance from the origin. Considering nniform precision, for three-factor experimentation, eight (2 ) factorial points, six axial points (2x3) and six centre rnns, a total of 20 experimental rnns may be considered and the valne of a is = 1.682. 

A central composite design then has c = 2 full factorial points with — 1 and -I-1 levels, n, = 2k star points with levels —a and +a, and no = center points with levels equal to zero. Each factor is encountered at five levels (—a, —1,0, -1-1, +a) or at three (for a = 1, which is, however, unusual). The level a used in the star designs represents the distance from the center point and varies according to the number of factors (k) studied in the design. All points (except the center point), i.e., both the cube and the star points, are situated on a circle so that the design is rotatable. Rotatability is achieved when a =... 

---