Study of Central Composite Design and Orthogonal Array Composite Design

Abstract

Response surface methodology (RSM) is an effective tool for exploring the relationships between the response and the input factors. Central composite design (CCD) and orthogonal array composite design (OACD) are useful second-order designs in response surface methodology. In this work, we consider the efficiencies of the two classes of composite designs for general case. Assuming the second-order polynomial model, the D-efficiency of CCDs and OACDs are studied for general value of $\alpha$ in star points. Moreover, the determination of $\alpha$ is also discussed from the perspective of space-filling criterion.

Appendix

Lemma 10.7

Let $a\ne 0$, $b\ne 0$,

$$\begin{aligned} \begin{pmatrix}c_0 &{} c\mathbf{1 }^{\prime }_k \\ c\mathbf 1 _k &{} a\mathbf{J} _k+b\mathbf{I} _k \end{pmatrix}=b^{k-1}(bc_0+k(ac_0-c^2)). \end{aligned}$$

Lemma 10.8

Let E and F be two $n\times n$ nonnegative definite matrices with partions

$$\begin{aligned} E=\begin{pmatrix}E_1 &{} \mathbf 0 \\ \mathbf 0 &{} E_2 \end{pmatrix}\ge 0,\ F=\begin{pmatrix}F_1 &{} F_3 \\ F^{\prime }_3 &{} F_2 \end{pmatrix}\ge 0, \end{aligned}$$

where $E_1$ and $F_1$ are $m\times m$ matrices. Then

$$\begin{aligned} |E+F|\ge |E_2|\cdot |E_1+F_1|. \end{aligned}$$

Proof of Theorem 10.2 Denote $X_0=(\mathbf 1 _{n_0},\mathbf 0 ,\mathbf 0 ,\mathbf 0 )$ and $X_i=(\mathbf 1 _{n_i},Q_i,L_i,B_i)$, where $Q_i$, $L_i$, $B_i$ respectively are the quadratic, linear and bilinear terms of $d_i$ in the second-order model, $i=1,2$.

$$\begin{aligned} X^{\prime }_1X_1=\begin{pmatrix}n_1 &{} n_1\mathbf 1 ^{\prime }_k &{} \mathbf 0 &{} \mathbf 0 \\ n_1\mathbf 1 _k &{} n_1J_k &{} \mathbf 0 &{} \mathbf 0 \\ \mathbf 0 &{} \mathbf 0 &{} n_1I_k &{} \mathbf 0 \\ \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} n_1I_q \end{pmatrix}=\begin{pmatrix}n_1J_{k+1} &{} \mathbf 0 &{} \mathbf 0 \\ \mathbf 0 &{} n_1I_k &{} \mathbf 0 \\ \mathbf 0 &{} \mathbf 0 &{} n_1I_q\end{pmatrix}, \end{aligned}$$

$$\begin{aligned} X^{\prime }_2X_2=\begin{pmatrix}n_2 &{} \frac{2}{3}n_2\alpha ^2\mathbf 1 ^{\prime }_k &{} \mathbf 0 &{} \mathbf 0 \\ \frac{2}{3}n_2\alpha ^2\mathbf 1 _k &{} \frac{4}{9}n_2\alpha ^4J_k+\frac{2}{9}n_2\alpha ^4I_k &{} \mathbf 0 &{} Q_2^{\prime }B_2 \\ \mathbf 0 &{} \mathbf 0 &{} \frac{2}{3}n_2\alpha ^2I_k &{} L_2^{\prime }B_2 \\ \mathbf 0 &{} B_2^{\prime }Q_2 &{} B_2^{\prime }L_2 &{} B_2^{\prime }B_2 \end{pmatrix}, \end{aligned}$$

let $Y=X^{\prime }_2X_2+X^{\prime }_0X_0$, then

$$\begin{aligned} Y=\begin{pmatrix}n_2+n_0 &{} \frac{2}{3}n_2\alpha ^2\mathbf 1 ^{\prime }_k &{} \mathbf 0 &{} \mathbf 0 \\ \frac{2}{3}n_2\alpha ^2\mathbf 1 _k &{} \frac{4}{9}n_2\alpha ^4J_k+\frac{2}{9}n_2\alpha ^4I_k &{} \mathbf 0 &{} Q_2^{\prime }B_2 \\ \mathbf 0 &{} \mathbf 0 &{} \frac{2}{3}n_2\alpha ^2I_k &{} L_2^{\prime }B_2 \\ \mathbf 0 &{} B_2^{\prime }Q_2 &{} B_2^{\prime }L_2 &{} B_2^{\prime }B_2 \end{pmatrix}, \end{aligned}$$

denote

$$\begin{aligned} B_{11}=\begin{pmatrix}n_2+n_0 &{} \frac{2}{3}n_2\alpha ^2\mathbf 1 ^{\prime }_k \\ \frac{2}{3}n_2\alpha ^2\mathbf 1 _k &{} \frac{4}{9}n_2\alpha ^4J_k+\frac{2}{9}n_2\alpha ^4I_k\end{pmatrix}, B_{13}=\begin{pmatrix}{} \mathbf 0 \\ Q_2^{\prime }B_2\end{pmatrix}, \end{aligned}$$

then

$$\begin{aligned} X^{\prime }X=X^{\prime }_1X_1+Y=\begin{pmatrix}B_{11}+n_1J_{k+1} &{} \mathbf 0 &{} B_{13} \\ \mathbf 0 &{} (\frac{2}{3}n_2\alpha ^2+n_1)I_k &{} L_2^{\prime }B_2 \\ B_{13}^{\prime } &{} B_2^{\prime }L_2 &{} n_1I_q+B_2^{\prime }B_2\end{pmatrix}, \end{aligned}$$

(10.13)

from Lemma 10.8, we get

$$\begin{aligned} |X^{\prime }X|=|X^{\prime }_1X_1+Y|\ge |n_1I_q|\cdot \begin{vmatrix} B_{11}+n_1J_{k+1}&\mathbf 0 \\ \mathbf 0&(\frac{2}{3}n_2\alpha ^2+n_1)I_k\end{vmatrix}=n_1^q\left( \frac{2}{3}n_2\alpha ^2+n_1\right) ^k|B_{11}+n_1J_{k+1}|, \end{aligned}$$

from Lemma 10.7, we have

$$\begin{aligned} |B_{11}+n_1J_{k+1}|=\left( \frac{2}{9}n_2\alpha ^2\right) ^k\left[ (1+2k\alpha ^2)n_0+n_2+n_1\left( 1+2k\alpha ^2+\frac{9kn_0}{2n_2\alpha ^2}+\frac{9k}{2\alpha ^2}-6k\right) \right] , \end{aligned}$$

therefore

$$\begin{aligned} |X^{\prime }X|\ge n^q_1\left[ \frac{(4n_2\alpha ^2+6n_1)\alpha ^2n_2}{27}\right] ^k\left[ (1+2k\alpha ^2)n_0+n_2+n_1\left( 1+2k\alpha ^2+\frac{9kn_0}{2n_2\alpha ^2}+\frac{9k}{2\alpha ^2}-6k\right) \right] , \end{aligned}$$

(10.14)

then we can obtain Theorem 10.2.

Proof of Theorem 10.3 When $s=L$, from Eq. (10.13) and Fischer inequality, we have

$$\begin{aligned} |X^{\prime }_{(L)}X_{(L)}|\le |B_{11}+n_1J_{k+1}|\cdot |n_1I_q+B_2^{\prime }B_2|, \end{aligned}$$

because all of the diagonal elements of $B_2^{\prime }B_2$ are $\frac{4}{9}n_2\alpha ^4$, we have

$$\begin{aligned} |n_1I_q+B_2^{\prime }B_2|\le \left( n_1+\frac{4}{9}n_2\alpha ^4\right) ^q, \end{aligned}$$

(10.15)

so

$$\begin{aligned} |X^{\prime }_{(L)}X_{(L)}|\le |B_{11}+n_1J_{k+1}|\cdot \left( n_1+\frac{4}{9}n_2\alpha ^4\right) ^q, \end{aligned}$$

then using Eq. (10.4) and (10.14), we obtain the lower bound of $D_L$-efficiency. Moreover, from Fischer inequality, we have

$$\begin{aligned} |X^{\prime }X|\le |X^{\prime }_{(L)}X_{(L)}|\cdot |X^{\prime }_LX_L|, \end{aligned}$$

so

$$\begin{aligned} \frac{|X^{\prime }X|}{|X^{\prime }_{(L)}X_{(L)}|}\le |X^{\prime }_LX_L|=\left| \left( \frac{2}{3}n_2\alpha ^2+n_1\right) I_k\right| =\left( \frac{2}{3}n_2\alpha ^2+n_1\right) ^k, \end{aligned}$$

then get the upper bound of $D_L$-efficiency, if the linear terms of $d_2$ are orthogonal to the bilinear terms of $d_2$, then

$$\begin{aligned} \frac{|X^{\prime }X|}{|X^{\prime }_{(L)}X_{(L)}|}=|X^{\prime }_LX_L|, \end{aligned}$$

and the upper bound of $D_L$-efficiency is achieved.

When $s=B$, from Eq. (10.13),

$$\begin{aligned} |X^{\prime }_{(B)}X_{(B)}|=\begin{vmatrix} B_{11}+n_1J_{k+1}&\mathbf 0 \\ \mathbf 0&(\frac{2}{3}n_2\alpha ^2+n_1)I_k\end{vmatrix}=|B_{11}+n_1J_{k+1}|\cdot \left( \frac{2}{3}n_2\alpha ^2+n_1\right) ^k, \end{aligned}$$

then follows from Eqs. (10.4) and (10.14), we get the lower bound of $D_B$-efficiency.

When $s=Q$, from Eq. (10.13) and Fischer inequality,

$$\begin{aligned} \begin{aligned} |X^{\prime }_{(Q)}X_{(Q)}|&=\begin{vmatrix} N&\mathbf 0&\mathbf 0 \\ \mathbf 0&(\frac{2}{3}n_2\alpha ^2+n_1)I_k&L_2^{\prime }B_2 \\ \mathbf 0&B_2^{\prime }L_2&n_1I_q+B_2^{\prime }B_2\end{vmatrix}\le N\left( \frac{2}{3}n_2\alpha ^2+n_1\right) ^k|n_1I_q+B_2^{\prime }B_2|\\&\le N\left( \frac{2}{3}n_2\alpha ^2+n_1\right) ^k \left( n_1+\frac{4}{9}n_2\alpha ^4\right) ^q, \end{aligned} \end{aligned}$$

then follows from Eq. (10.4), Theorem 10.2 and Eq. (10.15), we get the lower bound of $D_Q$-efficiency.