1. 1 2006 International Conference on Design of Experiments and Its Applications July 9-13, 2006, Tianjin, P.R. China Sung Hyun Park, Hyuk Joo Kim and Jae-Il Cho Optimal central composite designs for fitting second order response surface regression models

2. 2 Introduction Contents 1 Orthogonality, rotatability and slope rotatability 2 The alphabetic design optimality 3 Optimal CCDs when the true model is of third order 4 Concluding remarks 5

3. 3 1. Introduction  The central composite design (CCD) is a design widely used for estimating second order response surfaces. It is perhaps the most popular class of second order designs.  Let denote the explanatory variables being considered. Much of the motivation of the CCD evolves from its use in sequential experimentation. It involves the use of a two-level factorial or fraction (resolution Ⅴ ) combined with the following 2k axial points:

4. 4 1. Introduction  As a result, the design involves, say, F=2 k factorial points (or F=2 k-p fractional factorial points), 2k axial points, and n 0 center points. The CCDs were first introduced by Box and Wilson(1951). -α-α 0 … 0 α 0 … 0 0 -α-α… 0 0 α… 0 … 00 …-α-α 00 …α

5. 5 2. Orthogonality, rotatability and slope rotatability Let us consider the model represented by where x iu is the value of the variable x i at the uth experimental point, and ε u 's are uncorrelated random errors with mean zero and variance σ 2. This is the second order response surface model.

6. 6 2. Orthogonality, rotatability and slope rotatability 2.1. Orthogonality In this subsection, we consider the model with the pure quadratic terms corrected for their means, that is, where and. In regard to orthogonality, this model is often used for the sake of simplicity in calculation.

7. 7 2. Orthogonality, rotatability and slope rotatability Let denote the least squares estimators of respectively. In the CCD, all the covariances between the estimated regression coefficients except are zero. But if the matrix is a diagonal matrix, then also becomes zero. This property is called orthogonality. It is well-known (See Myers (1976, p.134) and Khuri and Cornell (1996, p.122).) that the condition for a CCD to be an orthogonal design is that

8. 8 2. Orthogonality, rotatability and slope rotatability

9. 9 2.2. Rotatability It is important for a second order design to possess a reasonably stable distribution of throu- ghout the experimental design region. Here is the estimated response at the point. A rotatable design is one for which has the same value at any two locations that have the same distance from the design center. In other words, is constant on spheres. The rotatability property was first introduced by Box and Hunter(1957).

10. 10 2. Orthogonality, rotatability and slope rotatability It is well-known that the condition for a CCD to be rotatable is that This means that the value of α for a rotatable CCD does not depend on the number of center points.

11. 11 2. Orthogonality, rotatability and slope rotatability Table 2.2 gives the values of α for rotatable CCDs for various k. Note that for k=5 and 6, a CCD is also suggested in which a fractional factorial is used instead of a complete factorial. Also tabulated are F and T, where T=2k+1. The designs considered in the table contain a single center point. This by no means implies that one would always use only one center point.

12. 12 2. Orthogonality, rotatability and slope rotatability 2.3. Slope rotatability Suppose that estimation of the first derivative of η is of interest (η is the expected value of the response variable y). For the second order model, The variance of this derivative is a function of the point x at which the derivative is estimated and also a function of the design.

13. 13 2. Orthogonality, rotatability and slope rotatability Hader and Park (1978) proposed an analog of the Box-Hunter rotatability criterion, which requires that the variance of be constant on circles (k=2), spheres (k=3), or hyperspheres (k≥4) centered at the design origin. Estimates of the derivative over axial directions would then be equally reliable for all points equidistant from the design origin. They referred to this property as slope rotatability, and showed that the condition for a CCD to be a slope-rotatable design is as follows:

14. 14 2. Orthogonality, rotatability and slope rotatability

15. 15 2. Orthogonality, rotatability and slope rotatability

16. 16 2. Orthogonality, rotatability and slope rotatability Table 2.3 gives slope-rotatable values of α for 2≤k≤6. For k=5 and 6, CCDs involving fractional factorials are also considered.

17. 17 3. The alphabetic design optimality 3.1. D-optimality The best known and most often used criterion is D- optimality. D-optimality is based on the notion that the experimental design should be chosen so as to achieve certain properties in the matrix. Here is the following matrix:

18. 18 3. The alphabetic design optimality Suppose the maximum, arithmetic mean, and geometric mean of the eigenvalues of are indicated by and. It turns out that an important norm on the moment matrix is the determinant; that is, where p is the number of parameters in the model.

19. 19 3. The alphabetic design optimality Under the assumption of independent normal errors with constant variance, the determinant of is inversely proportional to the square of the volume of the confidence region for the regression coefficients. The volume of the confidence region is relevant because it reflects how well the set of coefficients are estimated. A D-optimal design is one in which is maximized; that is, where max ξ implies that the maximum is taken over all design ξ’s.

20. 20 3. The alphabetic design optimality 3.3. E-optimality The criterion E, evaluation of the smallest eigenvalue, also gains in understanding by a passage to variances. It is the same as minimizing the largest eigenvalue of the dispersion matrix; that is, where i=1,2,…,p In terms of variance, it is a minimax approach. Thus the E-optimal design is defined as

21. 21 3. The alphabetic design optimality 3.4. Application to the CCD For fitting the two factor second order model, we can consider the following CCD. It consists of (i) a 2 2 factorial, at levels ±1, (ii) a one-factor-at-a-time array and (iii) n 0 center points. That is, the matrix X is given by

22. 22 3. The alphabetic design optimality Then the matrix is given by where N is the number of experimental points, F is the number of factorial points, a=F+2α 2 and b=F+2α 4 For the two factor CCD, for example, the value of D is where n 0 is the number of center points.

23. 23 3. The alphabetic design optimality Figure 3.1 shows a plot of D versus for α the indicated values of n 0 for a CCD in k=2 factors.

24. 24 3. The alphabetic design optimality In CCDs, the determinant of moment matrix has a tendency of increase as α increases. That is, a larger value of α is recommendable for D-optimal sense. But in a practical experiment, the region of interest is usually restricted and the conditions of experiment cannot be set for a large α. So it is necessary for the experimenter to choose as large as possible within the controllable region of interest. On the other hand, for the two factor CCD, the value of A is

25. 25 3. The alphabetic design optimality Figure 3.2 shows plots of A versus for the indicated values of n 0 for CCDs in k=2 factors. Table 3.1 shows the results of optimal α values for two factor CCDs.

26. 26 3. The alphabetic design optimality

27. 27 4. Optimal CCDs when the true model is of third order Suppose that we fit the second order response surface model, but the true model is of third order. For this case, what value of α should be used in the CCD? We can generally formulate the problem by supposing that the experimenter fits a model of order d 1 in a region R of the explanatory variables. However, the true model is a polynomial of order d 2, where d 2 >d 1. Then, a reasonable design criterion is the minimization of

28. 28 4. Optimal CCDs when the true model is of third order The multiple integral in Eq. (3) actually represents the average of the expected squared deviations of the true response from the estimated response over the region R. Writing the integral

29. 29 4. Optimal CCDs when the true model is of third order The first quantity in Eq. (4) is the variance of, integrated or, rather averaged over the region R, whereas the second quantity is the square of the bias, similarly averaged. Thus M is naturally divided as follows: M=V+B where V is the average variance of, and B is the average squared bias of. In this section, as a reasonable choice of design we will consider the design which minimizes B. Such a design is called the all-bias design.

30. 30 4. Optimal CCDs when the true model is of third order It is assumed here that the experimenter desires to fit a quadratic response surface in a cuboidal region R but that the true function is best described by a cubic polynomial. The actual measured variables have been transformed to which are scaled so that the region of interest R is a unit cube. Also the assumption on the design is made that its center of gravity is at the origin (0,0,…,0) of the cube.

31. 31 4. Optimal CCDs when the true model is of third order The equation of the fitted model is where The true relationship is written as where contains the cubic contribution to the actual model. The vector contains the coefficients corresponding to terms in ; terms such as are included.

32. 32 4. Optimal CCDs when the true model is of third order The matrix X 1 is given by In this case the matrix X 2 is

33. 33 4. Optimal CCDs when the true model is of third order Let us now write where. One can write the bias term as where the a 2 vector is merely. (See Myers (1976, p.213))

34. 34 4. Optimal CCDs when the true model is of third order The first term in the square brackets in Eq. (5) contains only the region moment matrices and thus is independent of the design. The bias term can be no smaller than the positive semidefinite quadratic form. So the experimenter has to use designs which minimize the positive semidefinite quadratic form

35. 35 4. Optimal CCDs when the true model is of third order Now we will find out the value of which makes the optimal design in the CCDs. But, let's assume that a 2 is a vector of ones. That is, a 2 is (1,1,1,1) for the two factor CCDs when d 1 =2 and d 2 =3. And, if we assume that the region of interest -1≤x i ≤1 is where i=1,2, …,k then we can obtain region moment matrices(μ 11 and μ 12 ).

36. 36 4. Optimal CCDs when the true model is of third order For example, let's consider the second order CCD which minimizes the squared bias from the third order terms for k=2. The design consists of four factorial points, four axial points at a distance α from the origin, and two center points. Then we obtain the following design moment matrices and region moment matrices.

37. 37 4. Optimal CCDs when the true model is of third order

38. 38 4. Optimal CCDs when the true model is of third order So is obtained as The value of which minimizes is found to be A very interesting fact is that f( α) has nothing to do with the number of center points. Table 4.1 gives the appropriate values of α for second order CCD which minimize the squared bias from the third order terms for k factors.

39. 39 4. Optimal CCDs when the true model is of third order

40. 40 5. Concluding remarks In this paper, we found out values of α which optimize CCDs for fitting second order response surface models under several criteria. Table 5.1 gives the value of α in Tables 2.1, 2.2 and 4.1.

41. 41 5. Concluding remarks From Table 5.1, we can find that the values of tend to increase in the following order : Minimum bias<Orthogonality<Rotatability <Slope rotatability<Alphabetic optimality

42. 42 5. Concluding remarks Note that the optimal value of α under the minimum bias and rotatability criteria does not depend on the number of center points. Also, an interesting fact is that the optimal value of α under the minimum bias criterion is very similar to that under the orthogonality criterion with one center point. In conclusion, we will consider reasonable choice of CCD for fitting the second order model according to the following cases: 1. when the true model is of second order (d 2 =2) 2. when the true model is of third order (d 2 =3)

43. 43 5. Concluding remarks Table 5.2 shows values of α recommended for the CCD considering the order d 2.

44. 44 Thank you