# SPLIT: Program for Constructing Second-order Split-Plot Designs

## Introduction

Several industrial experiments are set up in second-order split-plot designs (SPDs). These experiments have two types of factors: whole-plot (WP) factors and sub-plot (SP) factors. WP factors, also called hard-to-change factors are factors whose levels are hard or expensive to change. SP factors, also called easy-to-change factors are factors whose levels are easy or less expensive to change. In a split-plot experiment, the WP factors which are often fixed are confounded with blocks. SPDs which appear most useful are SPDs which possesses the equivalent-estimation property. For SPDs with this property, ordinary least-squares estimates of the model are equivalent to the generalized least-squares ones.

SPLIT is a program for constructing D-efficient equivalent-estimation SPDs (EE-SPDs) by interchanging the levels of the SP factors within each WP. The produced EE-SPDs are not only highly D-efficient but also attractive from the experimenters' point of view as their information matrices are close to the structure possessed by the ones of popular designs such as the Box-Behnkens designs or BBDs (Box & Behnkens, 1960), central-composite designs or CCDs (Box & Wilson, 1951) and BBD- and CCD-based SPDs (Parker et al, 2006, 2007). In addition, they could improve all EE-SPDs with non-integer levels reported in Jones Goos (2012). The detailed account of the SPLIT algorithm appears in Nguyen & Pham (2015).

## Using SPLIT

Let's assume all Gendex class files are in the directory c:\gendex and suppose you want to construct an EE-SPD for one WP factor and two SP factors in five WPs of size three. Create a file called d25.in containing a starting (or input) design with the first column representing the WP factor and the second and third columns representing the SP factors:

```1    1    1
1    0    0
1   -1   -1

1    1    1
1    0    0
1   -1   -1

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

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

0    1    1
0   -1   -1
0    0    0```

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

`java -cp c:\gendex SPLIT`

The SPLIT GUI will pop up. Enter name of the file containing the starting design, number of WP factors and the number of WPs and choose the design criterion (i) orthogonal or (ii) equivalent-estimation (default):

Now click START, SPLIT will start running and in less than a second, the plan of the obtained EE-SPD will pops up in the SPLIT output window. This is the most D-efficient EE-SPD among the 1,000 tries:

Note that the START button has changed to RESET button. If you click this RESET button, the output in the SPLIT output window will disappear and you can use SPLIT for a new design problem. Also 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.

The method of construction of the starting design when there are more than one WP factors and for different number of WPs is described in Nguyen & Pham (2015).

## Output

The result of the best try is displayed in the SPLIT output window and is also saved in the file SPLIT.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 f value. The fact that f=0 indicates that the information matrix M=X'V-1X (where X is the expanded design matrix and V is the variance-covariance matrix of the model) has a particular orthogonal structure. For SPDs with this structure, the mS SP main effects and mC2 interactions can be estimated orthogonally.
4. trace(C'C). The fact that trace(C'C)=0 indicates that the constructed SPD is an EE-SPD.
5. det*=|M|1/p/n where n is the number of runs and p is the number of parameters in the model;
6. det=|M|;
7. The design plan and the associated random seed;
8. The time in seconds SPLIT used to construct the above design.

## Examples

1. An EE-SPD for 1 WP factors and 2 SP factors in 4 WPs of size 4 (http://designcomputing.net/gendex/split/pkv1.html).
2. An EE-SPD for 1 WP factors and 3 SP factors in 4 WPs of size 12 (http://designcomputing.net/gendex/split/pkv2.html).
3. An EE-SPD for 1 WP factors and 2 SP factors in 3 WPs of size 5 (http://designcomputing.net/gendex/split/pkv3.html).
4. An EE-SPD for 2 WP factors and 2 SP factors in 9 WPs of size 5 (http://designcomputing.net/gendex/split/pkv4.html).
5. An EE-SPD for 1 WP factors and 3 SP factors in 3 WPs of size 9 (http://designcomputing.net/gendex/split/pkv5.html).
6. An EE-SPD for 1 WP factors and 2 SP factors in 5 WPs of size 3 (http://designcomputing.net/gendex/split/d25.html).
7. An EE-SPD for 2 WP factors and 2 SP factors in 12 WPs of size 4 (http://designcomputing.net/gendex/split/ceramic.html).
8. An EE-SPD for 1 WP factors and 3 SP factors in 6 WPs of size 6 (http://designcomputing.net/gendex/split/d48.html.html).
9. An EE-SPD for 2 WP factors and 2 SP factors in 10 WPs of size 3 (http://designcomputing.net/gendex/split/d94.html).
10. An EE-SPD for 2 WP factors and 3 SP factors in 8 WPs of size 6 (http://designcomputing.net/gendex/split/d109.html).

Notes:

• Examples 1-5: The EE-SPDs in these examples correspond to those in Tables 1-4 and 9 of Parker, et al. (2006). The starting designs for these examples are in the files: pkv1.in, pkv2.in, pkv3.in, pkv4.in and pkv5.in.
• Examples 6-10: The EE-SPDs in these examples correspond to those in Tables 2-6 of Nguyen & Pham (2015). The EE-SPDs in examples 6, 8-10 also correspond to the ones for scenarios 25, 48, 94 and 109 of Jones & Goos (2012). The EE-SPD in examples 7 also corresponds to the one for an experiment on the strength of ceramic pipe of Vining et al. (2006). The starting designs for these examples are in the files: d25.in, d48.in, d94.in, d109.in and ceramic.in.

## References

Box, G.E.P. & Behnken, D.W. (1960) Some new three level designs for the study of quantitative variables. Technometrics 2, 455-477.
Box, G.E.P. & Wilson, K.B. (1951) On the experimental attainment of optimum conditions. J. Roy. Statist. Soc. Ser. B 13, 1-45.
Jones, B.& Goos, P. (2012) An algorithm for finding D-efficient equivalent-estimation second-order split-plot designs, Jounal of Quality Technology, 44, 363-374.
Nguyen, N-K & Pham, T-D (2015) Searching for D-efficient Equivalent-Estimation Second-Order Split-Plot Designs. Journal of Quality Technology, 47, 54-65.
Parker, P. A., Kowalski, S. M. & Vining, G. G. (2006) Classes of split-plot response surface designs for equivalent-estimation, Quality and Reliability Engineering International 22, 291-305.
Parker, P. A., Kowalski, S. M. & Vining, G. G. (2007) Construction of balanced equivalent-estimation second-order split-plot designs. Technometrics 49, 56-65.
Vining, G. G., Kowalski, S. M. & Montgomery, D. C. (2005) Response surface designs within a split-plot structure. Journal of Quality Technology 37, 115-129.