RAT: Program for Constructing Trend-free Fractional Factorial and Response Surface Designs

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

Introduction

RAT is a Gendex module for constructing trend-free fractional factorial designs (FFDs) and response surface designs (RSDs). RAT is the acronym for robust against trends. The following briefly describes the approach used by RAT. The algorithm which implements this approach is described in Nguyen (1998).

The uth row of the extended design matrix X for n runs with m₀ trend columns, m factors (input columns or variables) and p-1-m-m₀ derived variables is (w_u1,..., w_um0, 1, x_u1,..., x_um,...,x_u(p-1-m0)). Partition X as (W|X). The condition for columns of X to be trend-free is that these columns be orthogonal to the columns of W, i.e. W'X=0. The approach of RAT to construct a trend-free design is to find a suitable design and allocate the n runs of this design to the n time points such that the objective function f is minimized where f is the sum of squares of the elements of W'X.

In this note John (1990) is abbreviated as John and Atkinson & Donev (1996) is abbreviated as AD.

Using RAT

Let's assume all Gendex class files are in the directory c:\gendex and suppose you want to construct a trend-free 2⁴ factorial. This factorial is in the file 16runs.txt in the working directory:

-1     -1     -1     -1
 1     -1     -1     -1
-1      1     -1     -1
 1      1     -1     -1
-1     -1      1     -1
 1     -1      1     -1
-1      1      1     -1
 1      1      1     -1
-1     -1     -1      1
 1     -1     -1      1
-1      1     -1      1
 1      1     -1      1
-1     -1      1      1
 1     -1      1      1
-1      1      1      1
 1      1      1      1

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

java -cp c:\gendex rat

The RAT window will pop up. Enter 16runs.txt at the File text field, choose Full model in the Model options and Quadratic in the Trend options,  the RAT window will become: 

 Now click Start, the following window will pop up:

Choose 1 and click OK. Since you choose one run at each time point, there will be 16 time points. If you choose four runs at each time point, there will be four time points. Now, click OK, the OUTPUT window (not shown) will pop up and the RAT window will become:

Note:

1. The two trend options are linear and quadratic. If the trend is linear, there will be one trend column. This column is created by by scaling a column of numbers (1,..., 16)', i.e. by subtracting each number from their mean and then dividing the resulting number by the largest one. If the trend is quadratic, there will be two trend columns. The second trend column was created by scaling a column obtained by squaring each element of the first trend column. We say a term in the model is L+ when it is orthogonal to the linear trend and is Q+ when it is orthogonal to both linear and quadratic trends. Similarly, a design is L+ when all terms are orthogonal to the linear trend and is Q+ when all terms are orthogonal to both linear and quadratic trend.

2. The four model options are: (i) Main-effect model: only includes the main-effect terms; (ii) Interaction model: includes the main-effect terms and 2-factor interaction (2fi) terms; (iii) Full model: includes the main-effect terms, 2fi terms and squared terms (for 3-level factors). If this model is chosen and the number of parameters exceeds the number of runs, this option will automatically switch to the next one (full model with all main-effects clear). (iv) Full model (with all main-effects clear): choose this model when you want all main-effects clear of (orthogonal to) time trends. When this option is specified, RAT uses two objective functions g and f. Partition X as (X₁|X₂) where X₁ is an n x (1+m) matrix, then g is the sum of squares of the elements W'X₁. A design is selected if it has a smaller g than the previous design or the same g but smaller f (the sum of squares of the elements of W'X).

3. 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.

Output

The output message containing the result of the best try will appear in a window and is also saved in the file rat.htm in the working directory. This file can be read by a browser such as IE or Nestcape. 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 when f becomes 0, i.e. when the design becomes trend-free. If the last model is specified, RAT 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 all main effects clear of time trend(s) is obtained (Example 2).
5. |X'X|;
6. The standardized determinant |X'X|/n^p;
7. TF value. TF=(|X'X|/|W'W|/|X'X|)^1/(p-m0). When the design becomes trend-free, TF will become 1 (AD p. 334).
8. Factor levels;
9. X'X;
10. V;
11. The time in seconds RAT used to construct the above design.

Examples

1. A near-Q+ 2⁴ factorial (http://designcomputing.net/gendex/rat/r1.html).
2. A near-Q+ 2⁴ factorial (http://designcomputing.net/gendex/rat/r2.html).
3. A Q+ L₁₆(2⁵) (http://designcomputing.net/gendex/rat/r3.html).
4. An L+ L₁₂(2¹¹) (http://designcomputing.net/gendex/rat/r4.html).
5. A Q+ L₁₂(2⁵) (http://designcomputing.net/gendex/rat/r5.html).
6. An L+ 2-level design in 15 runs (http://designcomputing.net/gendex/rat/r6.html).
7. A near-Q+ 2-level design in 15 runs (http://designcomputing.net/gendex/rat/r7.html).
8. An L+ 2-level design in 15 runs with 3 runs at each time point (http://designcomputing.net/gendex/rat/r8.html).
9. A near Q+ 2-level design in 15 runs with 3 runs at each time point (http://designcomputing.net/gendex/rat/r9.html).
10. An L+ BBD design for 3 factors in 15 runs (http://designcomputing.net/gendex/rat/r10.html).
11. An L+ BBD design for 4 factors in 27 runs (http://designcomputing.net/gendex/rat/r11.html).
12. An L+ BBD design for 5 factors in 46 runs (http://designcomputing.net/gendex/rat/r12.html).

Notes:

• Example 1 and 2: These examples uses model options 3 and 4 respectively. Like plan DW in John p. 275, all main effects of the design in Example 2 are Q+ (all main effects are orthogonal to both linear and quadratic trends).
• Example 3: See the corresponding design in Plan 6 of John.
• Example 4: See the corresponding design in Plan 7 of John.
• Example 5: See the corresponding design in Plan 8 of John.
• Example 6: See the corresponding design in AD Table 3. The input design is two replicates of the 2³ factorial with one run (111) removed.
• Example 7: The constructed design slightly improves the corresponding design in AD Table 4.
• Example 8: The constructed design is trend free and slightly improves the corresponding design in AD Table 6.
• Example 9: See the corresponding design in AD Table 7.
• Example 10-12: The input designs for these examples are those in Box & Behnken (1960). These designs also appear in Examples 16-18 of the FRAC.

References

Box,G.E.P. & Behnken, D.W. (1960) Some new three level designs for the study of quantitative variables. Technometrics 2, 455-477.
Atkinson, A.C. & Donev, A.N. (1996) Experimental designs optimally balanced for trend. Technometrics 38, 333-341
John, P.W.M. (1990) Time trend and factorial experiments. Technometrics 32, 275-282.
Nguyen, N-K. (2004) Making experimental designs robust against trends. Submitted.

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