Orthogonal main-effect plans in row–column designs for two-level factorial experiments

Row–column designs for two-level factorial experiments are constructed to estimate all the main effects. We give the interactions for row and column blockings. Based on these blockings, independent treatment combinations are proposed to establish the whole design so that practitioners can easily apply it to their experiments. Some examples are given for illustrations. The estimation of two-factor interactions in these designs is discussed.     

Confounded row–column designs

Confounded row–column designs for factorial experiments are considered and a simple method of construction using the classical method of confounding is described. Partially confounded designs are also studied and a method for generating Rao's (1946) designs is presented.     

Generalized confounded row–column designs

Generalized Confounded Row–Column (GCRC) designs for factorial experiments have been introduced and methods of constructing GCRC designs have been discussed. Fractionally replicated GCRC designs have also been constructed. The designs obtained ensure balancing with respect to estimable effects.     

A class of saturated row–column designs

A new class of row–column designs is proposed. These designs are saturated in terms of eliminating two-way heterogeneity with an additive model. The (m,s)-criterion is used to select optimal designs. It turns out that all (m,s)-optimal designs are binary. Square (m,s)-optimal designs are constructed and they are treatment-connected. Thus, all treatment contrasts are estimable regardless of the row and column effects.     

Note on Draper and Lin's article “capacity considerations for two-level fractional factorial designs”

In Table 4 of Draper and Lin (1990) the authors present eleven 2^{k − p}_R designs for which doubt is expressed as to whether they are saturated. This note removes doubt on nine of these designs indicating that they are indeed saturated. Some observations are made in terms of the resolution of a design for the remaining two designs among the 11. The conclusions are derived by properly interpreting an updated table of binary linear codes of Verhoeff (1987).     

Optimal row–column design for three treatments

The A-optimality problem is solved for three treatments in a row–column layout. Depending on the numbers of rows and columns, the requirements for optimality can be decidedly counterintuitive: replication numbers need not be as equal as possible, and trace of the information matrix need not be maximal. General rules for comparing 3×3$3\times 3$ information matrices for their A-behavior are also developed, and the A-optimality problem is also solved for three treatments in simple block designs.     

Variance-balanced row–column designs involving diallel crosses incorporating specific combining abilities for comparing test lines with a control line

In this paper, the problem of comparing t test lines with a control line under a row–column setup in complete diallel cross experiment is investigated when specific combining ability (sca) effect is included in the model. Three classes of Mating-Environmental Row–Column (MERC) designs have been obtained which are variance balanced for estimating the contrasts pertaining to general combining ability (gca) effects free from sca effects.     

A Mann–Whitney type effect measure of interaction for factorial designs

We propose a measure for interaction for factorial designs that is formulated in terms of a probability similar to the effect size of the Mann–Whitney test. It is shown how asymptotic confidence intervals can be obtained for the effect size and how a statistical test can be constructed. We further show how the test is related to the test proposed by Bhapkar and Gore [Sankhya A, 36:261–272 (1974)]. The results of a simulation study indicate that the test has good power properties and illustrate when the asymptotic approximations are adequate. The effect size is demonstrated on an example dataset.     

Some new augmented fractional Box–Behnken designs

The augmented Box–Behnken designs are used in the situations in which Box–Behnken designs (BBDs) could not estimate the response surface model due to the presence of third-order terms in the response surface models. These designs are too large for experimental use. Usually experimenters prefer small response surface designs in order to save time, cost, and resources; therefore, using combinations of fractional BBD points, factorial design points, axial design points, and complementary design points, we augment these designs and develop new third-order response surface designs known as augmented fractional BBDs (AFBBDs). These AFBBDs have less design points and are more efficient than augmented BBDs.