MIGA: Program for Constructing Minimum G-Aberration Designs

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

Introduction

MIGA constructs minimum G-aberration designs. The minimum G-aberration criterion proposed by Tang & Deng (1999) is a generalized version of the popular minimum aberration criterion of Fries & Hunter (1980). Designs constructed by MIGA include both regular and non-regular 2-level fractional factorial designs.

Let D_mxn be a design with m columns (factors) c₁,c₂,...,c_m each of size n with n/2 +1's and n/2 -1's. We now use each row i of D to construct a vector J_i of length _mC₂+_mC₃+_mC₄. Define the first _mC₂ element as c_i1c_i2, c_i1c_i3,... the next _mC₃ as c_i1c_i2c_i3, c_i1c_i2c_i3, ... and the last _mC₄ as c_i1c_i2c_i3c_i4, c_i1c_i2c_i3c_i5 ...where c_ij is the ith entry of column c_j. Let J=ΣJ_i.

Now, define B₂(D), B₃(D) and B₄(D) as 1/n² x the sum of squares of of the first _mC₂, the next _mC₃ and the last _mC₄ elements of J respectively (see eq. 2 of Ingram & Tang, 2005). Therefore, B₂(D) provides the extent of aliasing between main effects in D, B₃(D) provides the extent of aliasing between main effects and 2-factor interactions in D, and B₄(D) provides the extent of aliasing between pairs of 2-factor interactions in D (see Ingram & Tang, 2005)

A design D is said to have less G₂ aberration than D' if

1. B₂(D)<B₂(D') or
2. B₂(D)=B₂(D') and B₃(D)<B₃(D') or
3. B₂(D)=B₂(D') and B₃(D)=B₃(D') and B₄(D)<B₄(D').

If B₂(D)=B₂(D') and B₃(D)=B₃(D') and B₄(D)=B₄(D'), additional criteria will be used to select the design.

MIGA starts with a random design D with m columns, each having n/2 +1's and n/2 -1's. It then continues improving D with repect to the minimum G₂ aberration criterion by exchanging the positions of +1 and -1 in each column of D. Details of the MIGA algorithm will be reported elsewhere.

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

Using MIGA

Let's assume all Gendex class files are in the directory c:\gendex and suppose you want to construct a 2^7-2 resolution VI design. At the working directory, type the following command at the command prompt (case is important):

java -cp c:\gendex miga

The MIGA window will pop up. Enter 32 in the Number of runs field and 7 in the Number of 2-level factors field:

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 if you wish to. Now, click START, the following window will pop up:

Choose 3 (see Table 1 below) and click OK. The MIGA window will become:

and the output window showing the 2^7-2 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 MIGA 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 27 clear 2fi's: AB, AC, AD, AE, AF, AG, AH, A,I AJ, AK, BC, BD, BE, BF, BG, BH, BI, BJ, BK, CD, CE, CF, CG, CH, CI, CJ, CK (Example 9).

A constructed design will be classified into the following resolution type:

1. III: In a resolution III design, 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 a 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)

   
   

MIGA can also augment additional columns to existing design. To construct a 2^8-2 resolution VI design, you can add an additional column to the 2^7-2 design constructed previously as a base design.

In the following example, MIGA adds 5 additional columns to a 2-level design for 12 factors in 24 runs in the file h24x12.txt. As soon as you enter the file name in the Base design source, the number of runs and factors will appear in in the Number of runs and Number of 2-level factors fields. Now, change the number of 2-level factors in the latter from 12 to 17 and click Start, you will see the following result in the MIGA output screen:

Output

The output message containing the result of the best try will appear in a pop-up window when the program stops or when you stop the program by clicking the STOP button. It is also saved in the file miga.htm in the working directory. This file can be read by a browser such as IE, Firefox or Google Chrome. Information for this try includes:

1. Try number;
2. The random seed used;
3. The number of iterations;
4. B₂(D), B₃(D) and B₄(D) values. The program automatically stops when all three values become 0. When B_i(D) is not 0 (i=2, 3, 4), the maximum value of J (in terms of absolute value) used in the calculation of B_i(D) and the frequency of this value will be displayed.
5. The non-zero elements in J if any.
6. Resolution of the design.
7. A list of clear 2fi's if the design is of resolution IV.
8. The time in seconds MIGA used to construct the above design.

Examples

1. A 2^5-1 resolution V design (http://designcomputing.net/gendex/miga/f16x5.html).
2. A 2^6-2 resolution IV design (http://designcomputing.net/gendex/miga/f16x6.html).
3. A 2^7-2 resolution IV design with 3 strongly clear main effects (http://designcomputing.net/gendex/miga/f32x7.html).
4. A 2^8-3 resolution IV design with 2 strongly clear main effects (http://designcomputing.net/gendex/miga/f32x8.html).
5. A 2^9-4 resolution IV design with 1 strongly clear main effect (http://designcomputing.net/gendex/miga/f32x9.html).
6. A 2^9-4 resolution IV design with 2 strongly clear main effects (http://designcomputing.net/gendex/miga/f32x9bis.html).
7. A 2^9-3 resolution IV design with 5 strongly clear main effects (http://designcomputing.net/gendex/miga/f64x9.html).
8. A 2^10-4 resolution IV design with 4 strongly clear main effects (http://designcomputing.net/gendex/miga/f64x10.html).
9. A 2^11-3 resolution IV design with 3 strongly clear main effects (http://designcomputing.net/gendex/miga/f64x11.html).
10. A 2-level design for 4 factors in 12 runs (http://designcomputing.net/gendex/miga/h12x4.html).
11. A 2-level design for 6 factors in 24 runs (http://designcomputing.net/gendex/miga/h24x6.html).
12. A 2-level for 15 factors in 16 runs (http://designcomputing.net/gendex/miga/h16x15.html).
13. A 2-level design for 23 factors in 24 runs (http://designcomputing.net/gendex/miga/h24x23.html).

Notes:

• Example 1-2: The first 4 factors of these designs (generated by MIGA) make up a 2⁴ factorial.
• Example 3: The first 5 factors of these designs (generated by MIGA) make up 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 the J vector of this design has 7 non-zero elements or 21 pairs of non-orthogonal 2fi's, the one of the design in Example 5 (also called minimum aberration design) has only 6 non-zero elements or 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 MIGA 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 MIGA 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 and 11: These designs corresponds to designs IF0412 and IF0624 in Haaland (1989).
• Example 12: This design is constructed sequentially. The first 4, 5,...14 columns of this design form designs for m=4, 5,...15 and correspond to those in Table 2 of Tang & Deng (1999).
• Example 13: This design is constructed sequentially. The first 4, 5,...22 columns of this design form designs for m=4, 5,...22 and correspond to those in Tables 1 and 2 of Ingram & Tang (2005). Note that for m≤12, both B₂(D) and B₃(D) values of MIGA designs are 0. For m>12, although only B₂(D) values of the MIGA designs are 0, none of them has the 2fi's totally confounded.

References

Fries, A. & Hunter, W. G. (1980) Minimum Aberration 2^k-p designs. Technometrics 22, 601-608.
Ingram, D. & Tang, B. (2005) Minimum G Aberration design construction and design tables for 24 runs. J. of Quality Technology 37, 101-114.
Haaland, P. (1989) Experimental design in biotechnology, New York: Marcel Dekker.
Tang, B. & Deng, L.Y. (1999) Minimum G₂-aberration for nonregular fractional factorial designs. The Annals of Statistics 27, 1914-1926.
Wu, C.F.J & M. Hamada (2000) Experiments: Planning, Analysis and Parameter Design Optimization. New York: John Wiley & Sons, Inc.

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