sod: Program for Constructing Second-order Designs

Introduction

sod is a program for constructing 3-level second-order response surface designs (RSDs) for m factors in n runs Dmxn where each factor takes only levels -1, 0 and +1. The constructed designs include Box-Behnken designs (BBDs) of Box & Behnken (1960), Central Composite designs (CCDs) of Box & Wilson (1951), Small Composite Designs (SCDs) of Draper & Lin (1990), etc. sod can also augment a design with additional runs.

Let D be 3-level RSD for m factors in n runs with the same numbers of +1's and -1's. We have ∑xi=∑x3i=0 and ∑ x2i=∑ x4i =bi where bi is the number of ±1 of factor i. Let's impose the following conditions on D:

(i) ∑x2ixj=0 for i<j<k;
(ii) ∑x2ixjxk=0 for i<j<k;
(iii) ∑ xixj=0 (and x3ixj=0) for i<j;
(iv) ∑xixjxk=0 for i<j<k;
(v) ∑xixjxkxl=0 for i<j<k<l;
(vi) ∑x2ix2j-bibj/n=0 for i<j.

where the summations are taken over the n design points . It can be seen that these conditions are the conditions for D to be orthogonal (see Section 10.2 of John, 1971). A 3k full factorial, an orthogonal design will satisfy all six conditions. The CCDs and BBDs will satisfy the first five conditions (i)-(v), while the first three conditions (i)-(iii) will imply the orthogonal quadratic effects(OQE) property (see Section 1 of Nguyen & Lin, 2011).

Let A1 be the sum of squares of the sums in (i)-(iii), A2 be the sum of squares of the sums in (i)-(v) and A3 be the sum of squares of the sums in (i)-(vi). sod construct D by starting with a 3-level design and assign an equal number of ±1's to columns of Dmxn. sod then sequentially minimizes the A's by switching the positions of -1, 0 and +1 in each columns of D. The detailed account of the sod algorithm appears in Nguyen & Lin (2011). A similar approach has been used by Nguyen (1996a, 1996b) to construct supersaturated designs and near-orthogonal arrays.

Using sod

Let's assume all Gendex class files are in the directory c:\gendex and suppose you want to construct an RSD for five factors (three at 3-level and one at 2-level) in 26 runs (including two center runs). Also, you want each 3-level factor to have 8 zeros. At the working directory, type the following command at the command prompt (case is important and no space is allowed before/after the equal sign):

java -cp c:\gendex sod m2=3 m3=1 n=16 ncenters=2 nzeros=8

sod will start running and after try 1, the plan of the constructed design for this try will be displayed at the terminal window (as all A's have reached 0) and then sod stops:

sod 10.0: Constructing 3-level and Mixed-level Second-Order Designs
(c) 2018 Design Computing (http://designcomputing.net/)

Note: Best design for 4 factors in 18 runs.

Try	iter	A1	A2	rmax	rave	trace	det*
1	9	0	0	0	0	1.2	0.64

Design Plan for try 1 using seed 1533555013655:

1	1	1	-1
-1	-1	1	1
0	1	-1	-1
1	-1	1	1
0	1	-1	-1
1	1	-1	1
1	-1	-1	-1
0	-1	-1	1
0	-1	1	-1
-1	1	-1	1
-1	1	1	-1
0	-1	-1	1
0	-1	1	-1
0	1	1	1
-1	-1	-1	-1
0	1	1	1
0	0	0	0
0	0	0	0

X'X

18	8	0	0	0	0	0	0	0	0	0	0
8	0	0	0	0	0	0	0	0	0	0
8	0	0	0	0	0	0	0	0	0
16	0	0	0	0	0	0	0	0
16	0	0	0	0	0	0	0
16	0	0	0	0	0	0
8	0	0	0	0	0
8	0	0	0	0
8	0	0	0
16	0	0
16	0
16

inverse(X'X)

0.1	-0.1	0	0	0	0	0	0	0	0	0	-0
0.225	0	0	0	0	0	0	0	0	0	-0
0.125	0	0	0	0	0	0	0	0	-0
0.062	0	0	0	0	0	0	0	-0
0.062	0	0	0	0	0	0	-0
0.062	0	0	0	0	0	-0
0.125	0	0	0	0	-0
0.125	0	0	0	-0
0.125	0	0	-0
0.062	0	-0
0.062	-0
0.062

Note: sod used 0.171 seconds.
Note: sod.htm has been created

Note that the default random seed is the one obtained from the system clock and the default number of tries is 1000. You can change these default values by specifying the seed number and the number of tries, e.g.

java -cp c:\gendex sod m2=3 m3=1 n=16 ncenters=2 nzeros=8 seed=1234 tries=10000

To construct the BBD-type designs for 3-7 factors (Examples 4-8) you have to use the preset values nf (factorial runs from a 3m factorial), nc (center runs) and 0's (number of 0-level for each 3-level factor) in Table 1:

Table 1. Preset values of nf, and n0 and 0's
m n nf nc 0's
3 15 12 3 4
4 27 24 3 12
5 46 40 6 24
6 54 48 6
24
7 62 56 6 32

To construct the CCDs of Box & Wilson (1951) and the SCDs of Draper & Lin (1990) in examples 9-13, you have to specify a file which include the na axial runs and the number of 2-level factorial runs using the preset values na and nf in Table 2.Here is an example:
java -cp c:\gendex sod in=a3.txt n=4
Table 2. Preset values of nf
m n nf na
3 10 4 6
4 16 8 8
5 22 12 10
6 28 16 12
7 38 24 14

Output

The result of the best try is displayed at the terminal window and is also saved in the file sod.htm in the working directory. This file can be read by a browser such as IE, or Google Chrome. Information for this try includes:

1. Try number;
2. The number of iterations;
3. The value of A's. If A2=0 and ∑x2ix2j=const., A3 will be followed by an (R) indicating that the constructed design is slope-rotatable (see Park, 1987). The program automatically stops when A3 reaches 0 or when the constructed design is slope-rotatable.
4. det*=|X'X|1/p/n where n is the number of runs and p is the number of parameters in the model.
5. det=|X'X| where X is the expanded design matrix;
6. The design plan and the associated random seed;
7. The non-zero elements in J.
8. The number of runs of the base design if any;
9. The time in seconds sod used to construct the above design.

Example

1. An RSD for 4 factors in 27 runs (http://designcomputing.net/gendex/sod/s3.html).
2. An mixed-level RSD for 5 factors in 36 runs (http://designcomputing.net/gendex/sod/s4.html).
3. An BBD-type design for 3 factors in 15 runs (http://designcomputing.net/gendex/sod/b3.html).
4. An BBD-type design for 4 factors in 27 runs (http://designcomputing.net/gendex/sod/b4.html).
5. An BBD-type design for 5 factors in 46 runs (http://designcomputing.net/gendex/sod/b5.html).
6. A SCD for 3 factors in 10 runs (http://designcomputing.net/gendex/sod/c3.html).
7. A SCD for 4 factors in 16 runs (http://designcomputing.net/gendex/sod/c4.html).
8. A SCD for 5 factors in 21 runs (http://designcomputing.net/gendex/sod/c5.html).
9. A SCD for 6 factors in 28 runs (http://designcomputing.net/gendex/sod/c6.html).
10. A SCD for 7 factors in 38 runs (http://designcomputing.net/gendex/sod/c7.html).

Notes:

• Examples 2-5: These designs correspond to those for 3-5 factors in Box & Behnken (1960) and Nguyen & Borkowski (2008).
• Examples 6-10: These designs with OQE property correspond to those for 3-7 factors in Draper & Lin (1990). Note that the one for 7 factor improves the corresponding SCD in terms of D-optimality.

References

Box, G.E.P. & Behnken, D.W. (1960) Some new three level designs for the study of quantitative variables. Technometrics 2, 455-477.
Box, G.E.P. & Wilson, K.B. (1951) On the experimental attainment of optimum conditions. J. Roy. Statist. Soc. Ser. B 13, 1-45.
Draper, N.R. & D.K.J. Lin (1990) Small response surface designs. Technometrics 2, 187-194.
John, P.M.W. (1971) Statistical design and analysis of experiments, New York: McMillan.
Nguyen, N-K. (1996a) An algorithmic approach to constructing supersaturated designs. Technometrics 38, 205-209.
Nguyen, N-K. (1996b) A note on the construction of near-orthogonal arrays with mixed levels and economic run size.Technometrics 38, 279-283.
Nguyen, N-K & J.J. Borkowski (2008) New 3-level response surface designs constructed from incomplete block designs. J. of Statistical Planning & Inference 138, 294-305.
Nguyen, N-K & D.K.J. Lin (2011) A note on small composite designs for sequential experimentation, Journal of Statistics Theory and Practice 5 109-117.
Park, S. H. (1987) A class of multifactor designs for estimating the slope of response surfaces. Technometrics 29, 449-453.