## Introduction

MIGA is a program for constructing 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)

*c*each of size

_{1},c_{2},...,c_{m}*n*with

*n*/2 +1's and

*n*/2 -1's. We now use each row

*i*of

*D*to construct a vector

*J*of length

_{i}

_{m}C_{2}+

_{m}C_{3}+

_{m}C_{4}. Define the first

_{m}C_{2}element as

*c*

_{i1}c

_{i2},

*c*

_{i1}c

_{i3},... the next

_{m}C_{3}as

*c*

_{i1}c

_{i2}c

_{i3},

*c*

_{i1}c

_{i2}c

_{i3}, ... and the last

_{m}C_{4}as

*c*

_{i1}c

_{i2}c

_{i3}c

_{i4},

*c*

_{i1}c

_{i2}c

_{i3}c

_{i5 }...where

*c*

_{ij}is the

*i*th entry of column

*c*. Let

_{j}*J=∑ J*.

_{i}Now, define *B*_{2}(*D*),
*B*_{3}(*D*) and *B*_{4}(*D*)
as
1/*n*^{2} x the sum of squares of of the first
_{m}C_{2}, the next _{m}C_{3}
and the last _{m}C_{4} elements of *J*
respectively (see Eq. 2 of Ingram & Tang, 2005). Therefore,
*B*_{2}(*D*) provides the extent of aliasing
between main
effects in *D*, *B*_{3}(*D*) provides
the extent
of aliasing between main effects and 2-factor interactions in *D*,
and
*B*_{4}(*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 *G*_{2}
aberration
than *D*' if

*B*_{2}(*D*)<B_{2}(*D'*) or*B*_{2}(*D*)=*B*_{2}(*D'*) and*B*_{3}(*D*)*<B*(_{3}*D'*) or*B*_{2}(*D*)=*B*_{2}(*D*') and*B*_{3}(*D*)*=B*(_{3}*D*') and*B*_{4}(*D*)*<B*(_{4}*D*').

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 G_{2} 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 7 as the the **number of
2-level
factors** and 32 as the **number of runs**:

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**, MIGA will start running and
in less than a second, the 2^{7-2} resolution IV
design will
pop up in the MIGA output window:

Note that the **START** button has been changed to the **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. A 7-factor design with the first three
strongly clear
factors 1-3, for example, should have the following 15 clear 2fi's: 12, 13, 14, 15, 16, 17, 23, 24, 25, 26, 27, 34,
35, 36 and 37 (Example 3).

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

- 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.
- V: In a resolution V design, all main effects are strongly clear.
- 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 3-9), 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 5-6).Table 1. Preset values of *m*'*m**n*=16*n*=32*n*=646 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 which is now used as a base design:

## Output

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

- Try number;
- The number of iterations;
*B*_{2}(*D*),*B*_{3}(*D*)*B*_{4}(*D*) values. MIGA automatically stops when all three values become 0.- The design plan and the associated random seed.
- The non-zero elements in
*J*if any. - Resolution of the design.
- A list of clear 2fi's if the design is of resolution IV.
- The time in seconds MIGA used to construct the above design.

## Examples

- A 2
^{5-1}resolution V design (http://designcomputing.net/gendex/miga/f16x5.html). - A 2
^{6-2}resolution IV design (http://designcomputing.net/gendex/miga/f16x6.html). - A 2
^{7-2}resolution IV design with 3 strongly clear main effects (http://designcomputing.net/gendex/miga/f32x7.html). - A 2
^{8-3}resolution IV design with 2 strongly clear main effects (http://designcomputing.net/gendex/miga/f32x8.html). - A 2
^{9-4}resolution IV design with 1 strongly clear main effect (http://designcomputing.net/gendex/miga/f32x9.html). - A 2
^{9-4}resolution IV design with 2 strongly clear main effects (http://designcomputing.net/gendex/miga/f32x9bis.html). - A 2
^{9-3}resolution IV design with 5 strongly clear main effects (http://designcomputing.net/gendex/miga/f64x9.html). - A 2
^{10-4}resolution IV design with 4 strongly clear main effects (http://designcomputing.net/gendex/miga/f64x10.html). - A 2
^{11-3}resolution IV design with 3 strongly clear main effects (http://designcomputing.net/gendex/miga/f64x11.html). - A 2-level design for 4 factors in 12 runs (http://designcomputing.net/gendex/miga/h12x4.html).
- A 2-level design for 6 factors in 24 runs (http://designcomputing.net/gendex/miga/h24x6.html).
- A 2-level for 15 factors in 16 runs (http://designcomputing.net/gendex/miga/h16x15.html).
- A 2-level design for 23 factors in 24 runs (http://designcomputing.net/gendex/miga/h24x23.html).
- A 2-level design with minimal aliasing for 6 factors in 12 runs (http://designcomputing.net/gendex/miga/h12x6bis.html).

**Notes**:

- Example 1-2: The first 4 factors of these designs (generated by
MIGA) make up a 2
^{4}factorial. - Example 3: The first 5 factors of these designs (generated by
MIGA) make up a 2
^{5}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*B*_{2}(*D*) and*B*_{3}(*D*) values of MIGA designs are 0. For*m*>12, although only*B*_{2}(*D*) values of the MIGA designs are 0, none of them has the 2fi's totally confounded. - Example 15: This is an an example of a design with minimal aliasing constructed by using option
**Minimal aliasing**(Cf. the design in the file h12x6.html and the one in Table 2 of Jones & Nachtsheim, 2011)

^{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.

Jones B. & Nachtsheim C.J. (2011) Efficient Designs With Minimal Aliasing. *Technometrics* **53**, 62-71.

Tang, B. & Deng, L.Y. (1999) Minimum G_{2}-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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