On properties of efficiency-balanced designs

We consider a class of efficiency-balanced block designs which are design patterns for the analytical simplification of statistical analysis. This paper consists of eight sections which investigate various problems (such as, bounds on parameters, characterizations, existence and nonexistence, and dual designs) for efficiency-balanced block designs.     

A new special class of peb designs

A new class of partially efficiency-balanced designs is introduced from a practical point of view. This new design includes all equireplicated incomplete block designs available in literature as special cases. The fundamental properties of the design are clarified with relation to other block designs.     

Remark on Construction of Efficiency-Balanced Designs

In this note, all the efficiency-balanced block designs constructed by utilizing two methods of Dey & Singh (1980) are completely presented within a practical range of parameters.     

Some methods for constructing efficiency-balanced block designs

Three methods are presented for constructing connected efficiency-balanced block designs from other block designs with the same properties. The resulting designs differ from the original ones in the number of blocks and/or in the number of experimental units and their arrangement, while the number of treatments remains unaltered. Some remarks on the proposed methods of construction refer also to variance-balanced block designs.     

The efficiency and optimality characterization of certain six-Treatment incomplete-Block designs emanating from some quasi-Semi-Latin squares

Incomplete-block designs were constructed from three types of three-factor block-structured combinatorial designs which are not necessarily semi-Latin squares but whose treatments are ab initio arranged as in the semi-Latin square. The basic consideration for the construction involves the exploitation of the different block structures of the experimental units of the Quasi-semi-Latin Squares (QSLS) for the same treatments. All the emanating designs are found to be connected, only the designs in nine blocks are both variance- and efficiency-balanced and offer gain in efficiency but lead to about 46 percent loss of information.     

A note on generalized binary proper efficiency-balanced block designs

Some methods of construction of generalized binary proper efficiency-balanced block (GB-EB) designs with two different replications are derived and the solution of the unknown designs given by Das and Kageyama (1991) are obtained.     

Estimation of a response surface from-a general incomplete block design

If an incomplete block design is used to compare v treatments and later it is decided to fit a second order response surface using the fact that the v treatments are v combinations of levels of m different factors, how will one proceed to obtain this? This paper demonstrates how the usual analysis of an incomplete block design with or without recovery of interblock information can be augmented to yield estimates of response surface coefficients.     

Designs for two-colour microarray experiments

Summary.  Designs for two-colour microarray experiments can be viewed as block designs with two treatments per block. Explicit formulae for the A- and D-criteria are given for the case that the number of blocks is equal to the number of treatments. These show that the A- and D-optimality criteria conflict badly if there are 10 or more treatments. A similar analysis shows that designs with one or two extra blocks perform very much better, but again there is a conflict between the two optimality criteria for moderately large numbers of treatments. It is shown that this problem can be avoided by slightly increasing the number of blocks. The two colours that are used in each block effectively turn the block design into a row–column design. There is no need to use a design in which every treatment has each colour equally often: rather, an efficient row–column design should be used. For odd replication, it is recommended that the row–column design should be based on a bipartite graph, and it is proved that the optimal such design corresponds to an optimal block design for half the number of treatments. Efficient row–column designs are given for replications 3–6. It is shown how to adapt them for experiments in which some treatments have replication only 2.     

A characterization of a universally optimal design within a class of block designs

It is shown that within the class of connected binary designs with arbitrary block sizes and arbitrary replications only a symmetic balanced incomplete block design produces a completely symmetric information matrix for the treatment effects whenever the number of blocks is equal to the number of treatments and the number of experimental units is an integer multiple of the number of treatments. Such a design is known to be universally optimal.     

Resolvable row–column designs

This paper discusses resolvable row-column designs for v treatments arranged in b blocks each comprising pq units further grouped into p rows and q columns. A resolvable row-column design has v =pqs treatments set out in r groups of s blocks. Each rectangular block has p rows and q columns.     

Neighbor-Balanced Bipartite Block Designs

This article deals with the neighbor-balanced block design setting when there are two disjoint sets of treatments, one set consisting of test treatments and the other of control treatments. The interest here is to estimate the contrasts pertaining to test treatments vs. control treatments (with respect to direct and neighbors) with as high precision as possible. Some series of neighbor-balanced block designs for comparing a set of test treatments to a set of control treatments have been developed. The designs obtained are totally balanced in the sense that all the contrasts among test treatments for direct and neighbor effects are estimated with same variance and all the contrasts pertaining to test vs. control for direct and neighbor effects are estimated with the same variance.     

Upper Bounds on the True Coverage of Bootstrap Percentile Type Confidence Intervals

A simple derivation of expected mean squares is given for the randomized (complete) block design, showing that “experimental error,” the error term for testing treatments, is comprised of three sources of variability: block by treatment interaction, within block plot-to-plot variability, and within experimental plot sampling variation. The approach could readily be extended to incorporate measurement error as a fourth component of experimental error.     

On the efficiency of some non-orthogonal split-plot×split-block designs with control treatments

Many split-plot×split-block (SPSB) type experiments used in agriculture, biochemistry or plant protection are designed to study new crop plant cultivars or chemical agents. In these experiments it is usually very important to compare test treatments with the so-called control treatments. It happens yet that experimental material is limited and it does not allow using a complete (orthogonal) SPSB design. In the paper we propose a non-orthogonal SPSB design for consideration. Two cases of the design are presented here, i.e. when its incompleteness is connected with a crossed treatment structure only or with a nested treatment structure only. It is assumed the factors' levels connected with the incompleteness of the design are split into two groups: a set of test treatments and a set of control treatments. The method of constructions involves applying augmented block designs for some factors' levels. In a modelling data obtained from such experiments the structure of experimental material and appropriate randomization scheme of the different kinds of units before they enter the experiment are taken into account. With respect to the analysis of the obtained randomization model the approach typical to the multistratum experiments with orthogonal block structure is adapted. The proposed statistical analysis of linear model obtained includes estimation of parameters, testing general and particular hypotheses defined by the (basic) treatment contrasts with special reference to the notion of general balance.     

Optimal Circular Block Designs for Neighbouring Competition Effects

Competition or interference occurs when the responses to treatments in experimental units are affected by the treatments in neighbouring units. This may contribute to variability in experimental results and lead to substantial losses in efficiency. The study of a competing situation needs designs in which the competing units appear in a predetermined pattern. This paper deals with optimality aspects of circular block designs for studying the competition among treatments applied to neighbouring experimental units. The model considered is a four-way classified model consisting of direct effect of the treatment applied to a particular plot, the effect of those treatments applied to the immediate left and right neighbouring units and the block effect. Conditions have been obtained for the block design to be universally optimal for estimating direct and neighbour effects. Some classes of balanced and strongly balanced complete block designs have been identified to be universally optimal for the estimation of direct, left and right neighbour effects and a list of universally optimal designs for v<20 and r<100 has been prepared.     

SOME SERIES OF EFFICIENCY BALANCED DESIGNS

Two series of efficiency balanced designs with v*+ 1 treatments have been constructed using balanced incomplete block designs having v* treatments.     

Bayes a-opttmal and efficient block designs for comparing test treatments with a standard treatment

A sufficient condition for the Bayes A-optimality of block designs when comparing a standard treatment with v test treatments is given by Majumdar. (In:Optimal Design and Analysis of Experiments, Y. Dodge, V. V. Fedorov and H. P. Wynn (Eds.), 15-27, North-Holland, 1988). The priors that he considers depend on a constant α ε [0, ∞), with α - 0 corresponding to no prior information at all. The given sufficient condition, consequently, also depends on a. Large families of optimal and highly efficient designs are only known for the case α - 0. We will show how some of the results for α - 0 can be extended to obtain large families of optimal and highly efficient designs for arbitrary values of α. In addition, these results are useful when considering design robustness against an improper choice of α.     

Sweeping,alignment and the analysis of unbalanced data

The terms sweeping and alignment refer to the same process. Sweeping/alignment is used by data analysts as a technique for describing the effects of a model factor (e.g., treatments in a randomized block design) after the effects of nuisance parameters (e.g., blocks) have been removed from the data. In this paper sweeping/alignment is used as the basis for developing tests of factors in unbalanced experimental design models. Formulas are presented for treatment effects in randomized block designs with missing observations, and for interaction and main effects in unbalanced two-way factorial designs with empty cells.     

Efficiency balanced designs through reinforcement

In this investigation, general efficiency balanced (GEB) and efficiency balanced (EB) designs with (v + t) treatments, using (i) balanced incomplete block (BIB), (ii) symmetrical BIB, (iii) f -resolvable BIB, (iv) group divisible (GD) and (v) resolvable GD designs have been constructed with smaller number of replications and block sizes.     

Fourth exact moment results for mrbp tests with two or three treatments

MRBP tests were proposed by Mielke and Iyer (1982) to analyze multivariate data for the randomized block design, based on permutation procedures. They obtained the first three exact moments of the MRBP test statistic to approximate its permutation distribution. Tracy and Khan (1991) derived its fourth exact moment, to obtain a better approximating distribution, when there are four or more treatments. In this paper we obtain the fourth exact moment when the number of treatments is less than four.     

RCBDs with a control

The randomized complete block designs, RCBDs, are among the most popular of block designs for comparing a set of experimental treatments. The question of this design's effectiveness when one of the treatments is a control is examined here. Optimality ranges are established for the RBCD in terms of the strength of interest in control comparisons. It is found that if the control treatment is of secondary interest, the RCBD, when not best, is typically near best. This is not so when comparisons with the control are of greater interest than those among the other treatments.