Minimal-point second-order designs

According to some previous works that related to how to find the small composite designs, and the key point of these works is to replace the first-order design of central composite design with another small and proper design. Here we are interesting in finding the minimal-point second-order designs based on the different first-order designs over a k-ball with radiusby the D-optimal criterion.

Example:

1. Minimal-point composite design based on 2^k factorial or 2^k fractional factorial design with resolution V

two factors、three factors、five factors

2. Minimal-point composite design based on 2^k fractional factorial design with resolution III^*

four factors、six factors

3. Minimal-point composite design with Plackett and Burman Design

three factors、five factors、seven factors、eight factors

4. Comparison

Now, the P_eff values for all minimal-point designs, that we found here, are in Table 2. To compare CCDs with other minimal-point designs, we use the relative efficiency, i.e. P_eff(Min) / P_eff(CCD).

The ratios are shown in the last column of Table 2.

k	parameter	Peff(CCD)	Peff(V)	Peff(III^*)	Peff(P-B)	Peff(Min) / Peff(CCD)
2	6	0.6285	0.5733	ND	ND	0.9122 V
3	10	0.7116	0.6048	ND	0.6680	0.8499 V 0.9387 P-B
4	15	0.7673	ND	0.7115	ND	0.9273 III^*
5	21	0.8002	0.7669	ND	0.7580	0.9581 V 0.9473 P-B
6	28	0.8384	ND	0.7810	ND	0.9313 III^*
7	36	0.8547	ND	ND	0.6886	0.8057 P-B
8	45	0.8787	ND	ND	0.6278	0.7145 P-B

Table 2. The point efficiencies of CCDs and our minimal-point designs for k = 2, …, 8.

Excluding the center points, the number of the experiment design points for CCDs and our minimal-point designs are shown in Table 3.

Factor	2	3	4	5	6	7	8
CCD	8	14	24	26	44	78	80
Minimal-point design	6	10	15	21	28	36	45

Table 3. Excluding the center points, total supports in central composite designs and minimal-point designs.

Hartley (1959) propose the small composite design which contains 2k fractional factorial design with resolution III^*, 2k star points and some replicated center points. Here we compare our minimal-point designs with the Hartley designs in Table 4 by D-efficiency.

k	parameter	Peff(Hartley)	Peff(Min)	Peff(Hartley) / Peff(Min)
2	6	0.5714	0.5733	0.9967 V
3	10	0.5908	0.6048	0.9769 V
4	15	0.6503	0.7115	0.9140 III^*
5	21	ND	0.7669	ND
6	28	0.6684	0.7810	0.8558 III^*

Table 4. The relative efficiencies between Hartley designs and our minimal-point designs.

We also compare our minimal-point designs based on P-B designs with the small composite designs of Draper and Lin (1990).

k	parameter	Peff(P-B)	Peff(Min)	Peff(P-B) / Peff(Min)
3	10	0.5908	0.6680	0.8844
5	21	0.5899	0.7580	0.7782
7	28	0.5067	0.6886	0.7358
8	45	0.4832	0.6278	0.7697

Table 5. The relative efficiencies between small composite designs of Draper and Lin (1990) and our minimal-point designs based on P-B designs.

Here we compare our minimal-point designs with the designs of Lucas (1974); Notz (1982); Mitchell and Bayne (1976); Box and Draper (1974); Rechtschaffner (1967) and Katsaounis (1999) by point efficiency. The results are shown in Table 6, and the point efficiencies of Lucas (1974), Notz (1982), Mitchell and Bayne (1976), Box and Draper (1974), Rechtschaffner (1967) and Katsaounis (1999) are previously published in Katsaounis (1999). From this table, our minimal-point designs according to D-efficiency are better than other minimal-point designs.

k	Lucas	Notz	Mitchell and Bayne	Box and Draper	Rechtschaffine	Katsaounis		D-optimal minimal-point design
	(1974)	(1982)	(1976)	(1974)	(1967)	(1999)
						Pattern 1	Pattern 2
3	0.152 (0.251 , 0.228)	0.400 (0.661 , 0.599)	0.410 (0.678 , 0.614)	0.423 (0.699 , 0.633)	0.400 (0.661 , 0.599)	0.400 (0.661 , 0.599)	0.41 (0.678 , 0.614)	0.605 V 0.668 P-B
4	0.096 (0.135)	0.392 (0.551)	0.425 (0.597)	0.423 (0.594)	0.392 (0.551)	0.393 (0.552)	0.425 (0.597)	0.712 III^*
5	0.066 (0.086 , 0.087)	0.459 (0.598 , 0.606)	0.456 (0.595 , 0.602)	0.374 (0.488 , 0.493)	0.450 (0.587 , 0.594)	0.459 (0.598 , 0.606)	0.459 (0.598 , 0.606)	0.767 V 0.758 P-B
6	0.048 (0.064)	0.446 (0.571)	ND	0.317 (0.406)	0.428 (0.548)	0.446 (0.571)	0.460 (0.589)	0.781 III^*
7	0.036 (0.052)	ND	ND	0.227 (0.329)	0.383 (0.556)	0.448 (0.650)	0.451 (0.655)	0.689 P-B
8	0.028 (0.045)	ND	ND	0.193 (0.307)	0.336 (0.535)	0.434 (0.691)	0.446 (0.710)	0.628 P-B

Note: Parentheses indicate the relative efficiency between our designs and previous designs.

Table 6. The comparison of P_eff for selected minimal-point design.