Optimal Incomplete Block Designs Research Articles

Correction to: Optimal Incomplete-Block Designs with Low Replication: A Unified Approach Using Graphs

Correction to: Optimal Incomplete-Block Designs with Low Replication: A Unified Approach Using Graphs

Optimal Incomplete-Block Designs with Low Replication: A Unified Approach Using Graphs

An incomplete-block design defines both a concurrence graph and a Levi graph. Properties of either graph can be used to compare designs with respect to D-optimality and with respect to A-optimality. In this paper, we show that optimality of the design implies strong conditions on connectivity properties of the graph, and use this to classify the optimal designs when the number of observational units is close to minimal.

Optimal Triangular Partially Balanced Incomplete Block Designs for Complete Diallel Cross

I t is well known that complete diallel cross system IV and triangular partially balanced incomplete block designs have one to one correspondence. In the present paper, we have identified some triangular partially balanced incomplete block designs which generates two types of optimal incomplete block designs for complete diallel cross system IV.(i)The first type is with minimal number of experimental units in proper and non-proper setting for estimating the general combining ability (gca) effects of lines (ii)the second types is with minimum number of experimental units in proper setting for estimating general combining ability (gca) ) effects of lines in presences of specific combining ability (sca) Keywords: Triangular partially balanced incomplete block design, complete diallel cross, General combining ability, of Specific combining ability, optimality DOI: 10.7176/JAAS/68-03 Publication date: September 30 th 2020

On Φp-optimality of Incomplete Block Designs: An Algorithm

Since statisticians are interested in optimal incomplete block designs, the existence of algorithms to construct them can be notable. This article presents an algorithm to search for Φp-optimal or nearly Φp-optimal incomplete block designs. It is a variant of exchange and interchange procedures of Jones and Eccleston. The algorithm starts by constructing a cyclic design as the initial design. Then it enters a modified exchange and interchange procedures. Since our criterion for optimality is Φp, the algorithm utilizes the eigenvalues of the information matrix of designs. In most of the cases, optimal or nearly optimal designs were found, which shows the good performance of the algorithm. In addition, a new virtual design for has been obtained. So, for situations in which optimal incomplete block designs are required, this algorithm can be used.

Formula omitted]-optimal designs when errors follow a conditional autonormal process

This paper deals with E -optimal incomplete block designs in blocks of size three when observations within each block are correlated according to a conditional autonormal process. The optimality problem is addressed within various subclasses of connected designs under generalized least squares estimation of treatment contrasts.

Cost constrained MV-efficient optimal incomplete block designs

Cost constrained MV-efficient optimal incomplete block designs

An algorithmic approach to constructing mixed-level orthogonal and near-orthogonal arrays

Due to run size constraints, near-orthogonal arrays (near-OAs) and supersaturated designs, a special case of near-OA, are considered good alternatives to OAs. This paper shows (i) a combinatorial relationship between a mixed-level array and a non-resolvable incomplete block design (IBD) with varying replications (and its dual, a resolvable IBD with varying block sizes); (ii) the relationship between the criterion E ( d 2 ) proposed by Lu and Sun [Lu, X., Sun, Y., 2001. Supersaturated designs with more than two levels. Chinese Ann. Math. B 22, 183–194] or E ( f N O D ) proposed by Fang et al. [Fang, K.T., Lin, D.K.J., Liu, M.Q., 2003b. Optimal mixed-level supersaturated design. Metrika 58, 279–291] used in the (near-) OA construction and the ( M , S ) -optimality criterion used in the IBD construction; (iii) the derivation of a tighter bound for E ( d 2 ) ; (iv) how to modify the IBD algorithm of Nguyen [Nguyen, N.-K., 1994. Construction of optimal incomplete block designs by computer. Technometrics 36, 300–307] to obtain efficient (near-) OA algorithms. Some new (near-) OAs are presented and some near-OAs are compared with arrays constructed by other authors. Examples showing the use of the constructed arrays are given.

Optimal Incomplete Block Designs

Optimal incomplete block designs are pursued through the E criterion of minimizing maximal variance. Methodology is developed for design choice based on graphs and an extensive catalog of downloadable designs is compiled. Along with the general methodology, a near complete solution for E-optimal block designs is provided for up to 15 treatments. E optimality is revealed to be a flexible criterion that, depending on the application, can offer many choices for good designs.

Optimal partially balanced nested row-column designs

A partially balanced nested row-column design, referred to as PBNRC, is defined as an arrangement of v treatments in b p × q blocks for which, with the convention that p ⩽ q, the information matrix for the estimation of treatment parameters is equal to that of the column component design which is itself a partially balanced incomplete block design. In this paper, previously known optimal incomplete block designs, and row-column and nested row-column designs are utilized to develop some methods of constructing optimal PBNRC designs. In particular, it is shown that an optimal group divisible PBNRC design for v = mn ⩾ kn treatments in p × q blocks can be constructed whenever a balanced incomplete block design for m treatments in blocks of size k each and a group divisible PBNRC design for kn treatments in p × q blocks exist. A simple sufficient condition is given under which a group divisible PBNRC is Ψ f -better for all f > 0 than the corresponding balanced nested row-column designs having binary blocks. It is also shown that the construction techniques developed particularly for group divisible designs can be generalized to obtain PBNRC designs based on rectangular association schemes.

Optimal incomplete block designs for general dependence structures

The paper introduces strongly equineighboured designs, which are relevant designs for the usual case of dependent observations which have a centrosymmetric dispersion matrix. The designs are shown to be weakly universally optimal BIB designs under ordinary least-squares for any dependence structure. They are also shown to be universally optimal binary designs under generalized least-squares for any dependence structure, and optimal over all designs for some dependence structures. Some other designs are considered for the cases when the strongly equineighboured designs are not optimal. The results unify previous work on this topic.

COMPUTER‐AIDED CONSTRUCTION OF SMALL (M, S)‐OPTIMAL INCOMPLETE BLOCK DESIGNS

SummaryA new exchange algorithm for the construction of (M, S)‐optimal incomplete block designs (IBDS) is developed. This exchange algorithm is used to construct 973 (M, S)‐optimal IBDs (v, k, b) for v= 4,…,12 (varieties) with arbitrary v, k (block size) and b (number of blocks). The efficiencies of the “best” (M, S)‐optimal IBDs constructed by this algorithm are compared with the efficiencies of the corresponding nearly balanced incomplete block designs (NBIBDs) of Cheng(1979), Cheng & Wu (1981) and Mitchell & John(1976).

The Construction of Optimal Imcomplete Block Designs When Observations Within a Block are Correlated

SummaryAn algorithmic method is described for the construction of optimal incomplete block designs when a known correlation structure is assumed for observations from plots in the same block. The method is applicable to a wide class of designs and correlation structures. Some examples are given to illustrate the procedure.

Optimal Incomplete Block Designs for Comparing Treatments with a Control

The problem of finding optimal incomplete blocks designs for comparing $p$ test treatments with a control is studied. B.I.B. designs are found to be $D$-optimal. $A$- and $E$-optimal designs are also obtained. For a large class of functions $\phi$, conditions for a design to be $\phi$-optimal are found. Most of the optimal designs are certain types of B.T.I.B. designs, introduced by Bechhofer and Tamhane (1981), which are binary in test treatments.

Incomplete Block Designs for Comparing Treatments Wth a Control: General Theory

In this paper we develop a theory of optimal incomplete block designs for comparing several treatments with a control. This class of designs is appropriate for comparing simultaneously p ≥ 2 test treatments with a control treatment (the so-called multiple comparisons with a control (MCC) problem) when the observations are taken in incomplete blocks of common size k < p + 1. For this problem we propose a new general class of incomplete block designs that are balanced with respect to (wrt) test treatments. We shall use the abbreviation BTIB to refer to such designs. We study their structure and give some methods of construction. A procedure for making exact joint confidence statements for this multiple comparisons problem is described. By using a new concept of admissibility of designs, it is shown how “inferior” designs can be eliminated from consideration, and attention limited to a small class of BTIB designs that can be constructed from so-called generator designs in the minimal complete class of such d...

Optimal Incomplete Block Designs

Summary The class of regular graph (RG) designs, conjectured to be optimal among all incomplete block designs, is defined. The results of a systematic search of the class of RG designs are given for all cases in which v ≤ 12, r ≤ 10 and v ≤ b. In nearly every case, the optimum design is determined for each of the three optimality criteria: A, D and E. For v &amp;gt; b, we present the duals of the optimal designs for v &amp;lt; b. Many of the designs are new and these are tabulated in a back-up report.