Optimal Row-column Designs Research Articles

Correction to: ‘Optimal row-column designs’

Correction to: ‘Optimal row-column designs’

Optimal row-column designs for CDC method (1)

In the present article, we are presenting row-column designs for Griffing’s complete diallel cross methods (1) for p parents by using a complete set of (p-1) mutually orthogonal Latin squares, when p is prime or a power of prime. The row-column designs for Griffing’s methods (1) are new and universally optimal in the sense of Kempthrone (1956) and Kiefer (1975). The row-column designs for methods (1) are orthogonally blocked designs. In an orthogonally blocked design no loss of efficiency on the comparisons of interest is incurred due to blocking. The analysis includes the analysis of variance (ANOVA), estimation of general combining ability (gca), specific combining ability (sca) and reciprocal combining ability (rca). Tables of universally optimal row-column designs have been provided.

Optimal row-column designs

Summary Row-column designs have been widely used in experiments involving double confounding. Among them, one that provides unconfounded estimation of all main effects and as many two-factor interactions as possible is preferred, and is called optimal. Most current work focuses on the construction of two-level row-column designs, while the corresponding optimality theory has been largely ignored. Moreover, most constructed designs contain at least one replicate of a full factorial design, which is not flexible as the number of factors increases. In this study, a theoretical framework is built up to evaluate the optimality of row-column designs with prime level. A method for constructing optimal row-column designs with prime level is proposed. Subsequently, optimal full factorial three-level row-column designs are constructed for any parameter combination. Optimal fractional factorial two-level and three-level row-column designs are also constructed for cost saving.

Optimal Block and Row–Column Designs for CDC Methods

A simple method of construction of three series of optimal block designs for complete diallel cross method (4) has been presented by using nested balanced incomplete block designs of Dey et al. (Calcutta Statist. Assoc. Bull.35, 161–167 1986). The proposed designs are different from Gupta and Choi (Communication in Statistics-Theory and Methods27(11), 2827–2835 1998) and Das et al. (Statistics and Probability Letters36, 427–436 1998)’s designs in parametric values. From the proposed designs we have also derived three series of optimal row–column designs for complete diallel cross method (4) and three series of A–optimal row–column designs for complete diallel cross method (2), respectively. By using the proposed method, we have also obtained some more designs of the above types from nested balanced incomplete block designs of Morgan et al. (Discrete Mathematics231, 351–389 2001). Tables of these optimal /A-optimal designs have been provided. Illustration of the layouts of the proposed designs along with their analysis has been provided.

Some Optimal Row-Column Designs

Abstract The optimality of two classes of row-column designs with two rows is established. Studies of row-column designs with two rows arise, for example, from designs of two-color microarray experiments. Each mRNA sample to be labeled with green or red fluorescent dyes and hybridized on a slide is a treatment, each slide defines a column, and the green and red dyes define the rows. We show the optimality of designs such that the treatments appear nearly equally often in the two rows, and the column designs are either balanced incomplete block designs or group-divisible designs with two groups and λ2 = λ1 + 1, two well known classes of optimal block designs.

Optimal Row–Column Designs in High-Throughput Screening Experiments

Saturated row–column designs are studied to eliminate row and column effects in primary high-throughput screening experiments. All paired comparisons of treatments in the designs recommended are estimable within each microplate in spite of the existence of row and column effects. The (M, S)-criterion is used to select optimal and eliminate inefficient designs. It turns out that all (M, S)-optimal designs are binary, that is, no treatments appear twice in any row or column. Optimal designs are not unique with respect to design isomorphism. A series of (M, S)-optimal designs is constructed and all paired comparisons of treatments in the constructed designs are estimable regardless of the two-way heterogeneity. An (M, S)-optimal design for 8 ×12 microplates is provided and optimal designs of other dimensions can be constructed systematically.

Robust row-column designs for complete diallel cross experiments

This paper investigates the robustness of variance-balanced row-column designs for complete diallel cross experiments for estimating the comparisons among the general combining ability parameters against the loss of observations. A necessary and sufficient condition of robustness as per connectedness criterion is obtained. The robustness of optimal row-column designs of Gupta and Choi (1998) has been investigated for the loss of any m(≥1) observations in a column and for the loss of any two observations in the design. The study of robustness has also been conducted as per A-efficiency criterion.

Optimality of a class of efficiency-balanced designs

A class of binary efficiency-balanced (called GB-EB) designs are shown to be D- and E f -optimal among all block designs with given replication numbers. These designs have very high A- and D-efficiencies among designs with given v, b and k. Under certain conditions, it is shown that the GB-EB designs can be used to obtain optimal row-column designs.

Optimal row-column designs for complete diallel crosses

Universally optimal row-column designs for complete, diallel crosses are investigated. Three series of designs that require just one replication of the crosses are provided. A series of designs having two replications of each cross is also provided.

A note on universally optimal row-column designs with empty nodes

Identification of universally optimal row-column designs is investigated. This paper shows that Kunert's (1993) examples of universally optimal generalized non-binary designs are not special cases. One can construct an universally optimal generalized non-binary design by use of a binary one.

Optimal Row-Column Designs for Multiple Comparisons With a Control: A Complete Catalog

The problem of identification and construction of optimal designs for comparing several test treatments with a control treatment is studied. Two-way heterogeneity in the experimental units is considered under the usual additive model. A general class of designs—named balanced treatment row-column designs—appears to be natural candidates for the A-optimality criterion. Catalogs of A-optimal balanced treatment row-column designs have been prepared and tabulated for a practical range of parameter values. The use of the catalogs is explained with examples.

Optimal Row-Column Designs for Multiple Comparisons with a Control: A Complete Catalog

The problem of identification and construction of optimal designs for comparing several test treatments with a control treatment is studied. Two-way heterogeneity in the experimental units is considered under the usual additive model. A general class of designs—named balanced treatment row-column designs—appears to be natural candidates for the A-optimality criterion. Catalogs of A-optimal balanced treatment row-column designs have been prepared and tabulated for a practical range of parameter values. The use of the catalogs is explained with examples.

On the determination and construction of optimal row-column designs having unequal row and column sizes

In this paper we consider experimental situations requiring usage of a row-column design where v treatments are to be applied to experimental units arranged in b1 rows and b2 columns where row i has size k1i, i=1,..., b1 and column j has size k2j, j=1,..., b2. Conditions analogous to those given in Kunert (1983, Ann. Statis., 11, 247–257) and Cheng (1978, Ann. Statist., 6, 1262–1272) are given which can often be used to establish the optimality of a given row-column design from the optimality of an associated block design. In addition, sufficient conditions are derived which guarantee the existence of an optimal row-column design which can be constructed by appropriately arranging treatments within blocks of an optimal block design.

Some Universally Optimal Row-Column Designs with Empty Nodes

Some Universally Optimal Row-Column Designs with Empty Nodes

Some universally optimal row-column designs with empty nodes

General forms of the reduced coefficient matrix for estimation of treatment effects and the intrablock analysis of variance of row-column designs with n experimental units and v treatments are obtained from earlier work by the authors. These results are used to identify desirable properties for row-column designs with empty nodes. A need for such designs is apparent when the blocking criteria are implemented in sequence and empty nodes do not represent wasted experimental units. The construction and properties of three special classes of row-column designs with some empty nodes are discussed and examples given. In particular, it is shown that, if a row-column design belongs to one of these classes, then it is universally optimal for specified design parameters, where universally optimal designs are designs that maximize a generalized optimality criterion as defined by Kiefer.

Optimal designs for comparisons between two sets of treatments

Suppose v treatments are to be compared in b blocks of size k each. Also suppose that the treatments are divided into 2 sets of u and w= v− u treatments. A-optimal designs are obtained for estimating all the differences of two treatments, one from each set. Optimal row-column designs are also obtained. Some new optimal designs for comparing several treatments with a single control are obtained as special cases.