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DESIGN OF EXPERIMENTS

Efficient Block Designs for Dependent Observations—A Computer-Aided Search

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Pages 1187-1223 | Received 20 Jan 2006, Accepted 27 Jul 2006, Published online: 27 Apr 2007
 

Abstract

In this article, the exchange and interchange algorithm of Zergaw (Citation1989) and Martin and Eccleston (Citation1992) have been modified and used for searching efficient block designs for making all possible pairwise treatment comparisons when observations are dependent. The lower bounds to the A- and D-efficiencies of the designs in a given class of the designs have been obtained for correlated observation structure and the procedure of computing lower bounds to A- and D-efficiencies has been incorporated in the algorithm. The algorithm has been translated into a computer program using Microsoft Visual C++. Using this program, a search for efficient designs for making all possible pairwise treatment comparisons has been made for v ≤ 10, b ≤ 33, k ≤ 10 such that bk ≤ 100 and v > k. The block designs considered are usual block designs (rectangular block designs) and circular block designs. Nearest neighbor (NN), autoregressive of order 1 (AR(1)) correlation structures are studied. The ranges of correlation coefficients for different correlation structures investigated are |ρ|≤0.50 for NN correlation structure in rectangular blocks, |ρ|≤0.45 for NN correlation structure in circular blocks, and |ρ|≤0.95 for AR(1) correlation structure. For these ranges, the matrix of correlation coefficients among observations within a block is positive definite. Robustness aspects of designs that are efficient for a given value of correlation have been investigated against other values of correlation coefficients. Robustness aspects of designs that are efficient for independent observations have also been studied for experimental situations with dependent observations.

Mathematics Subject Classification:

Acknowledgments

The financial support received by the first author from CSIR/JRF is duly acknowledged to carry out this work. The authors gratefully acknowledge the help received from the referee in terms of valuable suggestions that strengthened the contents of the article.

Notes

AE: A-efficiency; DE: D-efficiency; CV-A: percent coefficient of variation of A-efficiencies over different correlation values; CV-D: percent coefficient of variation of D-efficiencies over different correlation values.

∗denotes the replication differ by 2.

∗∗denotes the replication differ by 3.

∗∗∗denotes the replications differ by 4.

∗denotes the replication differ by 2.

∗∗denotes the replication differ by 3.

∗∗∗denotes the replications differ by 4.

∗∗∗∗denotes the replications differ by 5.

∗denotes the replication differ by 2.

∗∗denotes the replication differ by 3.

∗∗∗denotes the replications differ by 4.

Note: For block size 3, the efficiencies for circular NN correlation structure remain the same for all values of ρ.

#ρNNL, ρNNU are lower limit and upper limit of ρ for NN correlation structures, respectively. #ρARL, ρARU are same for AR(1) correlation structures. #ρCNNL, ρCNNU are lower limit and upper limit of ρ for circular NN correlation structures, respectively. Note: For block size 3, the efficiencies for circular NN correlation structure remain the same for all values of ρ.

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