CIBD: Program for Constructing Cyclic Incomplete Block Designs

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

Introduction

Cyclic incomplete block designs (IBDs) are IBDs generated by the cyclic development of one or more suitably chosen initial blocks. Cyclic IBDs accounts for a large number of balanced IBDs or BIBDs in Fisher & Yates (1963) and Rao (1961). They also provide efficient alternatives to many partially balanced IBDs, or PBIBDs, catalogued in Clatworthy (1973). Chapter 3 of John & Williams (1995) gives an excellent summary of cyclic designs. In this note this book is abbreviated as JW. When the number of replications r is equal to or is a multiple of the block size k, cyclic IBDs render automatic elimination of heterogeneity in two directions (see Section 5.7 of John & Williams 1995). Being an important class of IBD, cyclic IBDs have been extensively catalogued by John et al. (1972).

CIBD is a Gendex module for generating optimal or near-optimal cyclic IBDs. CIBD handles up to 10,000 treatments. Designs constructed by CIBD can be used as column components of row-column designs (see Nguyen 1997). CIBD 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 detailed account of the CIBD algorithm will be given elsewhere.

Using CIBD

Let's assume all Gendex class files are in the directory c:\gendex and suppose you want to construct a cyclic IBD of size (v,k,r)=(6,2,4). At the working directory, type the following command at the Command Prompt (case is important):

java -cp c:\gendex cibd

The CIBD window will pop up. Enter the the number of (v,k,r)=(6,2,4), the IBD window will become:

Now click START, the following window will pop up:

You encounter this question when the number of replications r is equal to or is a multiple of the block size k. The order of blocks generated by the same initial block and the order of treatments within blocks are always randomized. Answering Yes to this question if you do not want to randomize the treatments within blocks (you want to construct a Youden square type design). Click Yes, the CIBD window will become:

and the output window showing the constructed CIBD of size (v,k,r)=(6,2,4) will pop up. Note that the START button has been changed to the STOP one. If you close the pop-up window, the STOP button will become a RESET one. If you click this RESET button, the output will disappear and you can use CIBD 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 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 cibd.htm in the working directory. This file can be read by a browser such as IE or Netscape. Information for this try includes:

  1. try number;
  2. the random seed used;
  3. the number of iterations;
  4. E, the efficiency factor of this design;
  5. the ratio E/U where U is the upper bound of an IBD. U=UJ where UJ 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.
  6. the distribution of the concurrences of this design;
  7. the design plan;
  8. the time in seconds CIBD
    used to construct this design;

An additional output of a CIBD session is the file Form.htm. The following is the file form.htm of the above CIBD session:

Block Plot  Treat
1     1     1
1     2     5
2     1     4
2     2     2
3     1     5
3     2     3
4     1     2
4     2     0
5     1     3
5     2     1
6     1     0
6     2     4
7     1     4
7     2     5
8     1     3
8     2     4
9     1     5
9     2     0
10    1     0
10    2     1
11    1     2
11    2     3
12    1     1
12    2     2

Examples

  1. A cyclic IBD of size (v,k,r)=(7,3,15) (http://designcomputing.net/gendex/cibd/c1.html).
  2. A cyclic IBD of size (v,k,r)=(8,4,35) (http://designcomputing.net/gendex/cibd/c2.html).
  3. A cyclic IBD of size (v,k,r)=(11,4,4) (http://designcomputing.net/gendex/cibd/c3.html).
  4. A cyclic IBD of size (v,k,r)=(8,4,6) (http://designcomputing.net/gendex/cibd/c4.html).
  5. A cyclic IBD of size (v,k,r)=(12,4,6) (http://designcomputing.net/gendex/cibd/c5.html).
  6. A cyclic IBD of size (v,k,r)=(12,4,8) (http://designcomputing.net/gendex/cibd/c6.html).
  7. A cyclic IBD of size (v,k,r)=(15,4,4) (http://designcomputing.net/gendex/cibd/c7.html).
  8. A cyclic IBD of size (v,k,r)=(91,10,10) (http://designcomputing.net/gendex/cibd/c8.html).
  9. A cyclic IBD of size (v,k,r)=(133,12,12) (http://designcomputing.net/gendex/cibd/c9.html).
  10. A cyclic IBD of size (v,k,r)=(183,14,14) (http://designcomputing.net/gendex/cibd/c10.html).
  11. A cyclic IBD of size (v,k,r)=(43,7,14) (http://designcomputing.net/gendex/cibd/c11.html).
  12. A cyclic IBD of size (v,k,r)=(61,5,15) (http://designcomputing.net/gendex/cibd/c12.html).

Notes:

References

Clatworthy, W.H. (1973) Tables of two-associates- class partially balanced designs. Applied. Math. Ser. 63. National Bureau of Standards, Washington.
Fisher, R.A. & Yates, F. (1963) Statistical tables for biological, agricultural and medical research. London: Oliver & Boyd.
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.
John, J.A., Wolock, F.W. & David, H.A. (1972) Cyclic Designs. Applied. Math. Ser. 62. National Bureau of Standards, Washington.
Nguyen, N-K (2001) 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. Pte. 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.
Rao, C.R. (1961) A study of BIB design with replication 11 to 15. Sankhya A 23, 117-127.

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