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

## Introduction

RAT is a Gendex module for constructing trend-free fractional factorial designs (FFDs) and response surface designs (RSDs). The 2015 version of RAT is modified to construct trend-free definitive screening designs (DSDs) with augmented 2-level factors. 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 (2014).

The u th 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 (wu1,..., wum0, 1, xu1,..., xum,...,xu(p-1-m0)). Partition X as (W|X). The condition for columns of X to be trend-free is that these columns are 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 in the file 16runs.txt is in the working directory. At the working directory, type the following command at the Command Prompt (case is important):

`java -cp c:\gendex RAT`

The RAT GUI will pop up. Enter 16runs.txt at the File text field and click Linear in the Trend option, the RAT window will become: Now click Start, the following window will pop up: Choose Interaction

The following window will then pop up: Choose 1 and since you choose one run at each time point, there will be 16 time points. Now, click OK, RAT will start running and after the 3rd try, the plan of the constructed design for this try pops up in the RAT output window (as the TF value reaches 1) and RAT then stops: Note:

1. The are two trend options are linear and quadratic. If the trend option is linear, there will be one trend column. This column is created 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 option is quadratic, there will be two trend columns. The second trend column is 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 trends.

2. The four model options are: (i) Linear: only includes the main-effect terms; (ii) Interaction: includes the main-effect terms and the 2-factor interaction (2fi) terms; (iii) Quadratic: includes the squared terms (for 3-level factors), the main-effect terms and the 2fi terms; (iv) Pure-quadratic: includes the squared terms (for 3-level factors) and the main-effect terms.

3. For models other than the linear model, if you want all main effects to be clear of (orthogonal to) the time trends then click Yes for the Clear main effects option. When this option is specified, RAT uses two objective functions g and f. Partition X as (X1|X2) where X1 is a matrix associated with the main effects and 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).

4. 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.

## Output

The result of the best try will appear in the RAT output 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 Google Chrome. Information for this try includes:

1. Try number;
2. The number of iterations;
3. The objective function f. The session automatically stops when f becomes 0, i.e. when the design becomes trend-free. If the Clear primary effects option is chosen, RAT will use two objective functions f and g. f will be followed by the value of g. g=0 indicates that a design with all primary effects clear of time trend(s) is obtained (e.g. Example 2).
4. The standardized determinant|X'X|1/p/n;
5. The 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).
6. Factor levels and the associated random seed;
7. X'X;
8. (X'X)-1
9. The time in seconds RAT used to construct the above design.

## Examples

1. An L+ 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 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/b3.html).
11. An L+ BBD design for 4 factors in 27 runs (http://designcomputing.net/gendex/rat/b4.html).
12. An L+ BBD design for 5 factors in 46 runs (http://designcomputing.net/gendex/rat/b5.html).
13. A near-L+ DSD for 5 3-level factors in 16 runs (http://designcomputing.net/gendex/rat/d1.html).
14. A near-L+ DSD for 4 3-level and 2 2-level factors in 14 runs (http://designcomputing.net/gendex/rat/d2.html).
15. A near-L+ DSD for 4 3-level and 3 2-level factors in 18 runs (http://designcomputing.net/gendex/rat/d3.html).
16. A near-L+ DSD for 4 3-level and 4 2-level factors in 18 runs (http://designcomputing.net/gendex/rat/d4.html).

Notes:

• Example 1 and 2: 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 15runs.txt has 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 http://designcomputing.net/bbd/.
• Example 13-16: The DSDs with augmented 2-levels factors in these examples are from Jones & Nachtsheim (2013). The squared terms in these examples are L+.

## References

Atkinson, A.C. & Donev, A.N. (1996) Experimental designs optimally balanced for trend. Technometrics 38, 333-341
Box,G.E.P. & Behnken, D.W. (1960) Some new three level designs for the study of quantitative variables. Technometrics 2, 455-477.
John, P.W.M. (1990) Time trend and factorial experiments. Technometrics 32, 275-282.
Jones, B., & Nachtsheim, C. J. (2013). Definitive Screening Designs with Added Two-Level Categorical Factors. Journal of Quality Technology, 45, 121-129.
Nguyen, N-K. (2014) Making Experimental Designs Robust Against Time Trend Statistics & Applications, 11, 79-86.