Construction Of Optimal Minimum Replicate Designs

A binary block design of size D(t,b,k,r) has ttreatments, set out in b blocks, each block having kunits, and each treatment occurring once or not at all in each block and r=bk/t the number of replicates. The focus in every experimental situation is to search for a design that provides maximum information but utilizes minimum experimental material. A design which can guarantee high precision in estimating parameters with small probabilities of committing Type I and Type II errors is considered useful to the experimenter. Therefore, the aim of this research work was to construct a new class of optimal designs with minimum replication. The specific objectives were to; (i) develop an algorithm for the optimal binary block designs, (ii) assess the binary designs constructed using a new optimality criterion, Minimum Variance MV={∑_(i=1)^(t-1)▒(e_i-e ̅ )^2 } where e_iis the i^th eigenvalue of the information matrix D(t,b,kr)and e ̅ the mean of the eigenvalues, (iii) classify the obtained designs as partially balanced incomplete block design PBIBD with massociate classes, with m=1,2,. . .. The best designs being cases withm=1 or 2, namely PBIBD/1 and PBIBD/2, respectively.rnClasses of optimal designs were constructed using the developed algorithm for the following parameter combinations, namely, 2 ≤r≤t, 6≤t≤15, and 2 ≤k≤t/2. The information matrix was obtained for each design D(t,b,kr)and the one with minimum MV was selected. Designs with equal eigenvalues were classified as the PBIBD/1, those with only two distinct eigenvalues as PBIBD/2.rnThe following were the major findings from this study:rn The algorithm developed had been able to search for the optimal design amongst a class of designs of same sizeD(t,b,kr).rn The MV criterion used in design assessment outperformed the well known A-, D-, or the E-Optimality criteria and it was found to be less cumbersome in computation,rn A new class of optimal designs were constructed and results obtained for parameters combinations 2 ≤r≤t, 6≤t≤15, and 2 ≤k≤t/2.rnThe study concluded that the algorithm for constructing small size replicates optimal experiments was effective. The study also confirmed that increasing the number of replications or block sizes would lead to improved efficiency but higher cost for the experiment. Thus, the research has opened up efficient designs for pilot survey and for easier computation with less cost. For an experimenter to obtain optimal design for the replication and unit constraint problem, construction of designs and their complementary designs should be considered. Optimal minimum replicate design is recommended for the construction of a better design for a particular anti-symmetric design with an improved frequency distribution of concurrence matrix.

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