alpha: Program for Constructing α-designs

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

Introduction

Patterson & Williams (1976a) introduced a new class of resolvable incomplete block design called α-designs. Since their introduction, α-designs have become popular among designers of experiments due to two main reasons:

• α-designs are available for many (r,k,s) combinations where r is the number of replicates, k is the block size and s is the number of blocks per replicate (the number of treatments v=ks). A square lattice design, for example does not exist for the (4,6,6) combination as there is no Graeco-Latin square of side six. Similarly, a rectangular lattice design does not exist for the (4,5,6) combination. However, efficient α-designs do exist for these combinations.
• the computer revolution has brought the PC to the desk of most designers of experimenters. As such, the flexibility of the design has succeeded computational simplicity as their criterion in design selection.

The construction of an α-design of size (r,k,s) begins with an r×k array α with elements in set of residues mod s. Each row of α is used to generate s-1 further rows by cyclic substitution. Denote the intermediate rs×k array by α*. Now add s to the second column of α*, 2s to the third column of α* and so on. The rows of the resulting array are blocks of the α-design. Each set of rows generated from the same row of α forms a complete replication. An α-design with pairs of treatments appearing in blocks either 0 or 1 times is referred to as α(0,1)-design and either 0, 1 or 2 times is referred to as α(0,1,2)-design. Chapter 4 of John & Williams (1995) gives an excellent summary of resolvable incomplete block designs including α-designs. In this note this book is abbreviated as JW.

alpha is a Gendex module for constructing optimal or near-optimal α-design. alpha handles up to 10,000 treatments. Designs constructed by alpha can be used as column components of resolvable row-column designs (see Nguyen & Williams 1993). alpha uses a 2-stage optimization process. Each stage of the optimization process uses an algorithm similar to the cyclic-coordinate exchange algorithm described in Nguyen (2002) . The algorithm uses extensively some theoretical results described in Patterson & Williams (1976b).

Using alpha

Let's assume all Gendex class files are in the directory c:\gendex and suppose you want to construct an α-design of size (r,k,s)=(2,4,5). At the working directory, issue the following command at the Command Prompt (case is important and no space is allowed before and after the equal sign):

java -cp c:\gendex alpha r=2 k=4 s=5

alpha will start running and after try 1, the plan of the constructed design for this try will be displayed at the terminal window (as the ratio E/U reaches 1) and alpha stops:

alpha 10.0: Program for constructing alpha designs
(c) 2018 Design Computing (http://designcomputing.net/)

Note: Best Alpha design for v=20, r=2, k=4 and s=5

try	iter	E	E/U	Concurrences
1	0	0.677	1	0(130) 1(60) 

Alpha array (r x k) for try 1 using seed 1533482021362:

0	0	0	0	
2	4	3	1	

Plan (blocks are columns):

7	14	13	16	5	
12	19	18	1	15	
17	9	8	6	0	
2	4	3	11	10	

6	15	7	14	16	
18	12	11	3	2	
4	1	19	17	9	
10	8	0	5	13	

Note: alpha used 0.137 seconds.
Note: alpha.htm has been created.

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 by specifying the seed number and the number of tries, e.g.

java -cp c:\gendex ibd alpha r=2 k=4 s=5 seed=1234 tries=1000

The result of the best try is displayed at the terminal window and is also saved in the file alpha.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. E, the efficiency factor of this design;
4. The ratio E/U where U is the upper bound of an IBD. U=min(U_J,U_WP) for resolvable IBDs. U_WP is the bound of Williams & Patterson (1977) good for any resolvable IBDs with v≥b. U_J is the bound of Jarrett (1989) good for any regular graph design (RGD). An RGD is an IBD with concurrences differing by at most 1. The program automatically stops if this ratio reaches 1.
5. The distribution of the concurrences of this design;
6. The design plan with blocks as columns and the associated random seed. If you want to print the blocks as rows, please specify the option -r on the command line.
7. The time in seconds alpha used to construct this design;

Examples

1. An α-design of size (r,k,s)=(3,4,3) (http://designcomputing.net/gendex/alpha/a1.html).
2. An α-design of size (r,k,s)=(4,4,6) (http://designcomputing.net/gendex/alpha/a2.html).
3. An α-design of size (r,k,s)=(2,3,4) (http://designcomputing.net/gendex/alpha/a3.html).
4. An α-design of size (r,k,s)=(2,9,6) (http://designcomputing.net/gendex/alpha/a4.html).
5. An α-design of size (r,k,s)=(3,5,8) (http://designcomputing.net/gendex/alpha/a5.html).
6. An α-design of size (r,k,s)=(3,6,4) (http://designcomputing.net/gendex/alpha/a6.html).
7. An α-design of size (r,k,s)=(2,30,33) (http://designcomputing.net/gendex/alpha/a7.html).

Notes:

• Example 1: See JW Example 4.2.
• Example 2. See JW Example 4.3.
• Example 3: See the example in JW Section 4.7.
• Example 4: See JW Example 4.4.
• Example 5: See JW Example 4.5.
• Example 6: See JW Example 4.7.

References

Jarrett, R.G. (1989). A review of bounds for the efficiency factor of block designs. Austral. J. Statist. 31, 118-129.
John, J.A. & Williams E.R. (1995). Cyclic designs and computer-generated designs. New York: Chapman & Hall.
Nguyen, N-K (2002) A modified cyclic-coordinate exchange algorithm as illustrated by the construction of minimum-point second-order designs. Advances in Statistics, Combinatorics and Related Areas. Edited by C. Gulati, Y-X Lin, S. Mishra, J. Rayner. World Scientific Publishing Co. Pty. Ltd., 205-210.
Nguyen, N-K. and Williams, E.R. (1993). An algorithm for constructing optimal resolvable row-column designs. Austral. J. Statist. 35, 363-370.
Patterson, H.D. & Williams, E.R. (1976a). A new class of resolvable incomplete block designs. Biometrika 63, 83-92.
Patterson, H.D. & Williams, E.R. (1976b) Some theoretical results on general block designs. In Proceedings of the 5th British Combinatorial Conference. Congressus Numeratium XV, 489-496, Utilitas Mathematica, Winnipeg.
Williams & Patterson (1977) Upper bound for efficiency factors in block designs. Austral. J. Statist. 19, 194-201.

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