FRAC: Program for Constructing Fractional Factorial and Response Surface Designs

1. Introduction
2. Using FRAC
3. Output
4. Examples
5. References

Introduction

FRAC is a Gendex module for constructing orthogonal and near-orthogonal 2-level fractional factorial designs (FFDs) and response surface designs (RSDs). FRAC can also augment hard-to-change (HTC) factors with additional easy-to-change (ETC) factors. Split-plot RSDs which satisfy the balanced equivalent estimation property expressed in equation (4) of Parker, et al. (2006)can be constructed by this facility (Examples 25-33).

For an orthogonal 2-level m-factor FFD in n runs, the A=X'X matrix takes the form nI_p and A^-1 takes the form (1/n)I_p where X is the expanded design matrix and p is the number of parameters.

For a second-order m-factor RSD in n runs, imposing the conditions

1. Σ x_i=0, Σx_ix_j=0, Σx_ix_j²=0, Σ x_i³=0, Σx_ix_j³=0, Σx_ix_jx_k²=0, Σx_ix_jx_k=0 and Σx_ix_jx_kx_l=0 for i≠j≠k≠l;
2. Σ x_i²=b;
3. Σx_i²x_j²=c;
4. Σ x_i⁴=c+d;

with the summations being taken over the n design points, the A matrix will take a special form (see John, 1971 Section 10.2). If c=b²/n, the design is said to be orthogonal.

The approach used by FRAC to construct the above designs is to assign an equal number of -1's and +1's to each factor in the design and minimize the objective function f by switching the positions of -1 and +1 in each 2-level factor and the positions of -1, 0 and +1 in each 3-level factor. Here, f is a function of the upper-diagonal elements of A and defined such that when the design is orthogonal, f becomes 0. The detailed account of this function will appear elsewhere. A similar approach has been used by Nguyen (1996a, 1996b) to construct supersaturated designs and near-orthogonal arrays.

In this note Wu & Hamada (2000) is abbreviated as WH.

Using FRAC

Let's assume all Gendex class files are in the directory c:\gendex and suppose you want to construct a 2^6-2 resolution VI design by augmenting a 2⁴ factorial with two additional 2-level factors. At the working directory, type the following command at the command prompt (case is important):

java -cp c:\gendex frac

The FRAC window will pop up. Enter 16runs.txt in the File field and 2 in the Number/Additional number of 2-level factors field. 16runs.txt is a file containing the 2⁴ factorial:

Note that the default random seed is the one obtained from the system clock and the default number of tries is 100. You can change these default values if you wish to. Now, click START, the following window will pop up:

There are three resolution choices: (i) III, (ii) V, (iii) IV (with the first m'<m strongly clear main effects). Since you pick the third choice, you will have to set m' (which ranges from 0 to m-1).

Choose 0 and click OK. The FRAC window will become:

and the output window showing the 2^6-2 resolution IV design will pop up. Note that the START button has been changed to the STOP one. If you close the pop-up window, the STOP button will become a RESET one. If you click this RESET button, the output will disappear and you can use FRAC for a new design problem.

We now modify the definition of clear and strongly clear of an effect in WH p. 156 to suite our explanation of the resolution options. A main effect or 2-factor interaction (2fi) is clear if it is orthogonal to other main effects and 2fi's. A main effect is strongly clear if it is clear and any 2fi's involving it is clear. An 11-factor design with the first three strongly clear factors, for example should have the following clear 2fi's: 12, 13, 14, 15, 16, 17, 18, 19, 1a, 1b, 23, 24, 25, 26, 27, 28, 29, 2a, 2b, 34, 35, 36, 37, 38, 39, 3a, 3b (Example 9). Here, we use the hexadecimal system to represent the factor, i.e. 10 is represented by 'a' and 11 by 'b', etc.

The three resolution options are:

1. III: In a resolution III design all main effects are pair-wise orthogonal. Resolution III design can be constructed more efficiently with the NOA module of the Gendex DOE toolkit.
2. V: In any resolution V design, all main effects are strongly clear.
3. IV (with the first m' main effects strongly clear): In a resolution IV design, all main effects are clear. However, the numbers of m' (<m), the first strongly clear main effects of two resolution IV designs of the same size might differ. To construct the resolution IV designs for 6=m=11, and n=16, 32 and 64 (Examples 2-10), you have to use Table 1. In this table, the preset values of m' are given for each value of m and n. The numbers in the brackets in this table refer to the numbers of clear 2fi's associated with each value of m'. Note that for m=9 and n=32, you have two choices of m' (Examples 4 and 5).
   

   Table 1. Preset values of m'

   m	n=16	n=32	n=64
   6	0 (0)	-	-
   7	0 (0)	3 (15)	-
   8	0 (0)	2 (13)	-
   9	-	1 (8) 2 (15)	5 (30)
   10	-	0 (0)	4 (30)
   11	-	0 (0)	3 (27)

   
   

Output

The output message containing the result of the best try will appear in a window and is also saved in the file frac.htm in the working directory. This file can be read by a browser such as IE or Firefox. Information for this try includes:

1. Try number;
2. The random seed used;
3. The number of iterations;
4. f. The program automatically stops when f=0. If the resolution IV is specified, FRAC will use two objective functions f and g. f will be followed by the value of g in brackets. g=0 indicates that a design with m clear main effects and the first m' strongly clear main effects is obtained (Examples 2-11). For an RSD, f will be followed by an (R) if the constructed RSD is slope-rotatable (see Park, 1987 for the discussion on slope-rotatability).
5. The number of clear 2fi's if resolution IV is specified;
6. |X'X|;
7. The standardized determinant |X'X|/n^p;
8. Trace V of the constructed design where V=(X'X)^-1;
9. Factor levels;
10. X'X;
11. V;
12. A list of clear 2fi's if resolution IV is specified;
13. The number of columns which from the base design;
14. The time in seconds FRAC used to construct the above design;

Note: Items 6-8 and 11 are not available for designs when |X'X|=0.

Examples

1. A 2^5-1 resolution V design (http://designcomputing.net/gendex/frac/f1.html).
2. A 2^6-2 resolution IV design (http://designcomputing.net/gendex/frac/f2.html).
3. A 2^7-2 resolution IV design with 3 strongly clear main effects (http://designcomputing.net/gendex/frac/f3.html).
4. A 2^8-3 resolution IV design with 2 strongly clear main effects (http://designcomputing.net/gendex/frac/f4.html).
5. A 2^9-4 resolution IV design with 1 strongly clear main effect (http://designcomputing.net/gendex/frac/f5.html).
6. A 2^9-4 resolution IV design with 2 strongly clear main effects (http://designcomputing.net/gendex/frac/f6.html).
7. A 2^9-3 resolution IV design with 5 strongly clear main effects (http://designcomputing.net/gendex/frac/f7.html).
8. A 2^10-4 resolution IV design with 4 strongly clear main effects (http://designcomputing.net/gendex/frac/f8.html).
9. A 2^11-3 resolution IV design with 3 strongly clear main effects (http://designcomputing.net/gendex/frac/f9.html).
10. A 2-level near-resolution V design for 4 factors in 12 runs (http://designcomputing.net/gendex/frac/n1.html).
11. A 2-level resolution IV design for 6 factors in 24 runs (http://designcomputing.net/gendex/frac/n2.html).
12. A mixed-level resolution V design for 3 factors in 12 runs (http://designcomputing.net/gendex/frac/m1.html).
13. A mixed-level resolution V design for 4 factors in 24 runs (http://designcomputing.net/gendex/frac/m2.html).
14. A mixed-level resolution IV design for 5 factors in 24 runs (http://designcomputing.net/gendex/frac/m3.html).
15. An Box-Behnken type for 3 factors in 15 runs (http://designcomputing.net/gendex/frac/b3.html).
16. An Box-Behnken type for 4 factors in 27 runs (http://designcomputing.net/gendex/frac/b4.html).
17. An Box-Behnken type for 5 factors in 46 runs (http://designcomputing.net/gendex/frac/b5.html).
18. An Box-Behnken type for 6 factors in 54 runs (http://designcomputing.net/gendex/frac/b6.html).
19. An Box-Behnken type for 7 factors in 62 runs (http://designcomputing.net/gendex/frac/b7.html).
20. A Draper-Lin design for 3 factors in 10 runs (http://designcomputing.net/gendex/frac/c3.html).
21. A Draper-Lin design for 4 factors in 16 runs (http://designcomputing.net/gendex/frac/c4.html).
22. A Draper-Lin design for 5 factors in 21 runs (http://designcomputing.net/gendex/frac/c5.html).
23. A Draper-Lin design for 6 factors in 28 runs (http://designcomputing.net/gendex/frac/c6.html).
24. A Draper-Lin design for 7 factors in 36 runs (http://designcomputing.net/gendex/frac/c7.html).
25. A split-plot RSD with 1 HTC factor and 2 ETC factors in 4 blocks of 4 runs each (http://designcomputing.net/gendex/frac/s1.html).
26. A split-plot RSD with 1 HTC factor and 3 ETC factors in 4 blocks of 12 runs each (http://designcomputing.net/gendex/frac/s2.html).
27. A split-plot RSD with 1 HTC factor and 2 ETC factors in 3 blocks of 5 runs each (http://designcomputing.net/gendex/frac/s3.html).
28. A split-plot RSD with 2 HTC factors and 2 ETC factors in 9 blocks of 5 runs each (http://designcomputing.net/gendex/frac/s4.html).
29. A split-plot RSD with 2 HTC factors and 2 ETC factors in 9 blocks of 4 runs each (http://designcomputing.net/gendex/frac/s5.html).
30. A split-plot RSD with 1 HTC factor and 4 ETC factors in 3 blocks of 12 runs each (http://designcomputing.net/gendex/frac/s6.html).
31. A split-plot RSD with 1 HTC factor and 4 ETC factors in 3 blocks of 12 runs each (http://designcomputing.net/gendex/frac/s7.html).
32. A split-plot RSD with 1 HTC factor and 4 ETC factors in 3 unequal-sized blocks. (http://designcomputing.net/gendex/frac/s8.html).
33. A split-plot RSD with 1 HTC factor and 3 ETC factors in 4 unequal-sized blocks. (http://designcomputing.net/gendex/frac/s9.html).
34. A split-plot RSD with 2 HTC factors and 2 ETC factors in 6 blocks of 4 runs each (http://designcomputing.net/gendex/frac/s10.html).

Notes:

• Example 1: Obtained by adding a 2-level factor to a 2⁴ factorial.
• Example 2: Obtained by adding two 2-level factors to a 2⁴ factorial. See the corresponding design in WH Table 4.A2. All main effects are clear.
• Example 3: Obtained by adding two 2-level factor to a 2⁵ factorial. See the corresponding design in WH Table 4.A3. Both designs have 3 strongly clear main effects and 15 clear 2fi's.
• Example 4: Obtained by adding a 2-level factor to the design in Example 3. See the corresponding design in WH Table 4.A3. Both designs have 2 strongly clear main effects and 13 clear 2fi's.
• Example 5: Obtained by adding a 2-level factor to the design in Example 4. See the corresponding design in WH Table 4.A3. Both designs have 1 strongly clear main effects and 8 clear 2fi's.
• Example 6: Obtained by adding a 2-level factor to the design in Example 4. See the corresponding design in WH Table 4.A3. Both designs have 2 strongly clear main effects and 15 clear 2fi's. There is a small price to pay for maximizing the number of strongly clear main effects. While this design has 21 pairs of non-orthogonal 2fi's, the design in Example 5 (also called minimum aberration design) has only 18 pairs of non-orthogonal 2fi's. See WH Section 4.5 for the minimum aberration and related criteria in design selection.
• Example 7: Obtained by adding a 2-level factor to a 2^8-2 resolution V design . See the corresponding design in WH Table 4.A5. Both designs have 5 strongly clear main effects and 30 clear 2fi's.
• Example 8: Obtained by adding a 2-level factor to the design in Example 7. See the corresponding design in WH Table 4.A5. The FRAC design has 4 strongly clear main effects and 30 clear 2fi's. The WH design has 2 strongly clear main effects and 33 clear 2fi's.
• Example 9: Obtained by adding a 2-level factor to the design in Example 7. See the corresponding design in WH Table 4.A5. The FRAC design has 3 strongly clear main effects and 27 clear 2fi's. The WH design has 1 strongly clear main effects and 34 clear 2fi's.
• Examples 10-14: These designs correspond to designs IF0412, IF0624, MF0312, MF0424 and MF0524 respectively in Haaland (1989).
• Examples 15-19: These designs correspond to those for 3-7 factors in Box & Behnken (1960) and Nguyen & Borkowski (2008). Note that the one for 6 factors improves the corresponding Box-Behnken design in terms of optimality as well as rotatability (see Nguyen & Borkowski, 2008).
• Examples 20-24: These designs correspond to those for 3-7 factors in Draper & Lin (1990). Note that the one for 7 factor improves the corresponding Draper-Lin design in terms of D-optimality.
• Examples 25-34: These split-plot RSDs were constructed by augmenting hard-to-change (HTC) factors with additional easy-to-change (ETC) factors. It can be verified that the constructed designs in these examples satisfy the the balanced equivalent estimation property expressed in equation (4) of Parker, et al. (2006). Designs in Examples 25-30 correspond to those in Table I-IV and VII-IX in Parker et al. (2006). Designs in Examples 31-32 correspond to those in Table 8 and 10 in Parker et al. (2006). Note that the HTC factors of the design in Example 33 are of mixed-level.

References

Box, G.E.P. & Behnken, D.W. (1960) Some new three level designs for the study of quantitative variables. Technometrics 2, 455-477.
Draper, N.R. & D.K.J. Lin (1990) Small response surface designs. Technometrics 2, 187-194.
Haaland, P. (1989) Experimental design in biotechnology, New York: Marcel Dekker.
John, P.M.W. (1971) Statistical design and analysis of experiments, New York: McMillan.
Morris, M.D. (2000) A class of three-level experimental design for response surface modelling. Technometrics 2, 111-121.
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.
Park, S. H. (1987) A class of multifactor designs for estimating the slope of response surfaces. Technometrics 29, 449-453.
Parker, A.P., Kowalski, S.M. & Vining, G.G. (2006) Classes of split-plot response surface designs for equivalent estimation. Quality & Reliability Engineering International 22, 291-305.
Parker, A.P., Kowalski, S.M. & Vining, G.G. (2007) Unbalanced and minimal point equivalent estimation second-order split-plot designs. J. of Quality Technology 39, 376-388.
Wu, C.F.J & M. Hamada (2000) Experiments: Planning, Analysis and Parameter Design Optimization. New York: John Wiley & Sons, Inc.

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