CUT: Program for Multi-Dimensional Blocking Fractional Factorial and Response Surface Designs

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

Introduction

Using CUT

Let's assume all Gendex class files are in the directory c:\gendex and suppose you want to divide an unblocked Box-Behnken design (BBD) for 4 factors in 28 runs into two rows and two columns (Example 29). This design is in the file unblocked.txt in the working directory:

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

At the working directory, type the following command at the command prompt (case is important):

java -cp c:\gendex cut 

The CUT window will pop up. Enter unblocked.txt at the File text field and 2 as the number of blocking factors at 2-level. The CUT window will become:

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 screen will pop up:

There are four model options: (i) main effects; (ii) interaction; (iii) full and (iv) full (with the first m' <m main effects strongly clear). Choose full and click OK. An output window showing the blocked design will pop up and the CUT window will become:

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 CUT for a new design problem.

We now modify the definition of clear and strongly clear of an effect in WH p. 181 to suite our explanation of model options. A main effect or 2-factor interaction (2fi) is clear if it is orthogonal to other main effects, 2fi's and block effects. 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 16). Here, we use the hexadecimal system to represent the factor, i.e. 10 is represented by 'a' and 11 by 'b', etc.

The four model options are:

1. Main-effect model: only includes the main-effect terms.
2. Interaction model: includes all main-effect terms and 2-factor interaction (2fi) terms;
3. Full model: includes all main-effect terms, 2fi terms and squared terms (for 3-level factors).
4. Full model (with the first m'<m main effects strongly clear): choose this model when you want all main-effects clear and the first m' main effects strongly clear.

Option 4 have been used to construct designs in Table 1. Some designs in this table are in Examples 1-10. This table shows the block factor (BF) values of designs for m factors and n runs in b blocks. The values of b 's are in brackets. BF is a measure of orthogonality of a blocked design (Nguyen, 2001). BF equals 1 means the design is orthogonally blocked.

Note that when the unblocked design consists of runs from a mixture experiment, you will then have two options for model: (i) linear and (ii) quadratic.

Output

The output message containing the result of the best try will appear in a window and is also saved in the file cut.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. The objective function f . The session automatically stops if f becomes 0, i.e. when the design becomes orthogonally blocked. If either one of the last model option is specified, CUT 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 (or with m clear main effects and the first m' strongly clear main effects) is obtained.
5. |X'X|;
6. The standardized determinant |X'X|/n ;
7. BF value;
8. Factor levels for each block;
9. The X'X matrix;
10. (X'X)^-1;
11. The time in seconds CUT used to construct the above design;

Examples

1. Divide a 2³ factorial into two blocks (http://designcomputing.net/gendex/cut/c1.html).
2. Divide a 2³ factorial into four blocks (http://designcomputing.net/gendex/cut/c2.html).
3. Divide a 2⁴ factorial into four blocks (http://designcomputing.net/gendex/cut/c3.html).
4. Divide a 2⁵ factorial into eight blocks (http://designcomputing.net/gendex/cut/c4.html).
5. Divide a 2⁶ factorial into 16 blocks (http://designcomputing.net/gendex/cut/c5.html).
6. Divide a 2^6-1 fractional factorial into four blocks (http://designcomputing.net/gendex/cut/c6.html).
7. Divide a 2^6-1 fractional factorial into eight blocks (http://designcomputing.net/gendex/cut/c7.html).
8. Divide a 2^7-1 fractional factorial into 16 blocks (http://designcomputing.net/gendex/cut/c8.html).
9. Divide a 2^8-2 fractional factorial into eight blocks (http://designcomputing.net/gendex/cut/c9.html).
10. Divide a 2^8-2 fractional factorial into 16 blocks (http://designcomputing.net/gendex/cut/c10.html).
11. Divide a 2^7-2 fractional factorial into four blocks (http://designcomputing.net/gendex/cut/c11.html).
12. Divide a 2^8-3 fractional factorial into eight blocks (http://designcomputing.net/gendex/cut/c12.html).
13. Divide a 2^9-1 fractional factorial into eight blocks (http://designcomputing.net/gendex/cut/c13.html).
14. Divide a 2^9-1 fractional factorial into eight blocks (http://designcomputing.net/gendex/cut/c14.html).
15. Divide a 2^10-1 fractional factorial into eight blocks (http://designcomputing.net/gendex/cut/c15.html).
16. Divide a 2^11-1 fractional factorial into eight blocks (http://designcomputing.net/gendex/cut/c16.html).
17. Divide 18 runs (which includes a 2⁴ factorial) into three blocks (http://designcomputing.net/gendex/cut/c17.html).
18. Divide four replicates of a 2³ factorial into eight blocks(http://designcomputing.net/gendex/cut/c18.html).
19. Divide a 3³ factorial into three rows and three columns (http://designcomputing.net/gendex/cut/c19.html).
20. Divide a 3-factor RSD in Table 1 of Gilmour & Trinica (2003) into seven rows and four columns (http://designcomputing.net/gendex/cut/gt1.html).
21. Divide a 2-factor RSD in Table 2 of Gilmour & Trinica (2003) into four rows and four columns (http://designcomputing.net/gendex/cut/gt2.html).
22. Divide a 3-factor RSD in Table 3 of Gilmour & Trinica (2003) into four rows and six columns (http://designcomputing.net/gendex/cut/gt3.html).
23. Divide a 2⁴ factorial into four rows and four columns (http://designcomputing.net/gendex/cut/cg1.html).
24. Divide a two replications of a 2⁴ factorial into four rows and eight columns (http://designcomputing.net/gendex/cut/cg2.html).
25. Divide a 2⁵ factorial into four rows and eight columns (http://designcomputing.net/gendex/cut/cg3.html).
26. Divide the 3-factors BBD into three blocks (http://designcomputing.net/gendex/cut/b3.html).
27. Divide the 4-factors BBD into three blocks (http://designcomputing.net/gendex/cut/b4.html).
28. Divide the 4-factors BBD into two rows and two columns (http://designcomputing.net/gendex/cut/b4bis.html).
29. Divide the 5-factors BBD into two rows and two columns(http://designcomputing.net/gendex/cut/b5.html).
30. Divide the design D636 of Nguyen & Borkowski (2008) into two blocks (http://designcomputing.net/gendex/cut/b6.html).
31. Divide the design D736 of Nguyen & Borkowski (2008) design into two rows and two columns (http://designcomputing.net/gendex/cut/b7.html).
32. Divide the 4-factor augmented pair design of Morris (2000) into two blocks (http://designcomputing.net/gendex/cut/b8.html).
33. Divide 6 distinct binary blends into two blocks (http://designcomputing.net/gendex/cut/m1.html).
34. Divide 24 distinct 4-component blends into two two rows and two columns(http://designcomputing.net/gendex/cut/m2.html).
35. Divide 16 distinct binary blends into two two rows and two columns (http://designcomputing.net/gendex/cut/m3.html).
36. Divide 24 distinct 3-component blends into two two rows and two columns(http://designcomputing.net/gendex/cut/m4.html).

Notes:

• Example 2: See the corresponding design in WH Table 3.A. Each block of the blocked design forms a fold-over pair.
• Example 3-10: The main-effects of the designs in these examples are clear. Some designs also have the first few main-effects strongly clear. Unlike the corresponding designs in WH Table 3.A and 4B, these designs do not completely confound any 2fi with blocks. These designs are useful in general settings where the experimenters are not prepared to sacrifice any 2fi.
• Example 11-16: The main-effects of the designs in these examples are clear. They also have 3, 2, 2, 5, 3 and 2 strongly clear main-effects respectively. The number of clear effects of the designs in Examples11-14 are the same as the corresponding designs in WH Table 4B.
• Example 17: See Nguyen (2001) Example 2. The unblocked design for this example consists of the 2⁴ factorial and a fold-over pair (1) and abcd . The main effects are orthogonal to block effects as the user specify full model (with all main-effects clear). See an alternative solution in Cook & Nachtsheim (1989) Section 3.2.
• Example 20-22: See the corresponding designs in Tables 1-3 of Gilmour & Trinica (2003).
• Examples 23-25: See the corresponding designs in Examples 3.1, 3.2 and 4.1 of Choi & Gupta (2008).
• Examples 26-32: With the exception of the design in Example 31, all designs are orthogonally blocked. Note that the 4-factor BBD in two orthogonal blocks is not available in Box & Behnken (1960). 
• Example 33: See Cornell (1990), p. 438. It is conventional to add a centroid blend to each block. In this example the centroid blend is (0.3333, 0.3333, 0.3333). To analyze this design and the constructed design in the next four examples, reduce the number of block terms by one (See Cornell, 1990 or Draper et al ., 1993).
• Example 34: See Cornell (1990), p. 439.
• Example 35: See Table 2 of Draper et al ., 1993.
• Example 36: See Table 2 of Nguyen (2001). The unblocked design consists of 24 distinct 3-component blends (0.00, 0.05, 0.25, 0.70), etc.

References

Box, G.E.P. & Behnken, D.W. (1960) Some new three-level designs for the study of qualitative variables. Technometrics 2, 455-475.
Choi, K.C. & S. Gupta (2008) Confounded row-column designs. J. of Statistical Planning & Inference 138, 196-202.
Cornell, J.A. (1990) Experiments with mixture designs, models and the analysis of mixture data. 2nd ed. New York: John Wiley & Sons,Inc.
Cook, R.D. & Nachtsheim, C.J. (1989) Computer-aided blocking of factorial and response surface designs. Technometrics 31, 339-346.
Draper, N.R., Prescott, P., Lewis, S.M., Dean, A.M., John, P.W.M & Tuck, M.G. (1993) Mixture designs for four components in orthogonal blocks. Technometrics 35, 268-276.
Gilmour, S.G. & L.A. Trinica (2003) Row-column response surface designs. J. of Quality Technology 2, 184-193.
Morris, M.D. (2000) A class of 3-level experimental designs for response surface modelling. Technometrics 42, 111-121.
Nam-Ky Nguyen (2001) Cutting experimental designs into blocks. Austral. & New Zealand J. of Statistics 43, 367-374.
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. 
Wu, C.F.J & M. Hamada (2000) Experiments: Planning, Analysis and Parameter Design Optimization. New York: John Wiley & Sons, Inc.

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