Row-column designs with minimal units

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 proposed designs are treatment-connected, i.e., all paired comparisons of treatments in the designs are estimable in spite of the existence of row and column effects. The connectedness of the designs is justified from two perspectives: linear model and contrast estimability. Comparisons with other designs are studied in terms of A-, D-, E-efficiencies as well as design balance.     

Pseudo-Youden designs balanced for intersection

If the row-column intersections of a row-column design A form a balanced incomplete block design, then A is said to be balanced for intersection. This property was originally defined for triple arrays by McSorley et al. (2005a), Section 8, where an example was presented and questions of existence were raised and discussed. We give sufficient conditions for the class of balanced grids in order to be balanced for intersection, and prove that a family of binary pseudo-Youden designs has this property.     

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.     

Some partially variance balanced structurally incomplete row-column designs

Earlier results by the authors are used to provide the intrablock analysis for row-column designs that have observations at nodes of the row-column lattice, the design being structurally incomplete when some nodes are empty. Construction, properties, and intrablock analyses of some special b× b row-column designs with b empty nodes taken along the principal diagonal of the lattice are developed. The designs discussed have m > 1 associate classes and are said to be partially variance balanced. The special designs fall in two classes and are shown to be nearly optimal in a specified class of designs. A small catalog of designs constructed is provided and they should be useful when empty nodes do not represent wasted experimental units, perhaps because the row and column assignments of treatments are sequenced.     

Equineighboured balanced nested row-column designs

This paper presents equineighboured balanced nested row-column designs for v treatments arranged in b blocks each comprising pq units further grouped into p rows and q columns. These are two-dimensional designs with the property that all pairs of treatments are neighbours equally frequently at all levels in both rows and columns. Methods of construction are given both for designs based on Latin squares and those where pq≤v. Cyclic equineighboured designs are defined and tabulated for 5≤v≤10, p≤3, q≤5, where r=bpq/v is the number of replications of each treatment.     

Neighbor-Balanced Row-column Designs

In this article, row-column designs incorporating directional neighbor effects have been studied. A row-column design is said to be neighbor balanced if every treatment has all other treatments appearing as a neighbor a constant number of times. We considered here three different situations under row-column setup incorporating neighbor effects viz., row-column design with one-sided neighbor effect, two-sided neighbor effect, and four-sided neighbor effect. The information matrices for all the situations for estimating the direct and neighbor effects of treatments have been derived. Methods of constructing neighbor-balanced row-column designs have been developed and its characterization properties have been studied.     

A basic lemma and the analysis of block and Kronecker product designs

In this paper the analysis of the class of block designs whose C matrix can be expressed in terms of the Kronecker product of some elementary matrices is considered. The analysis utilizes a basic result concerning the spectral decomposition of the Kronecker product of symmetric matrices in terms of the spectral decomposition of the component matrices involved in the Kronecker product. The property (A) of Kurkjian and Zelen (1963) is generalised and the analysis of generalised property (A) designs is given. It is proved that a design is balanced factorially if and only if it is a generalised property (A) design. A method of analysis of Kronecker product block designs whose component designs are equi-replicate and proper is also suggested.     

THE ROBUSTNESS OF BINARY AND NON-BINARY BALANCED NESTED ROW-COLUMN DESIGNS UNDER THE UNAVAILABILITY OF BLOCKS: A COMPARISON

In cases where both exist, the balanced, binary nested row-column designs are known to be inferior to a class of balanced non-binary designs. However, if it is possible for blocks of observations to become unavailable after an experiment has commenced, a binary nested row-column design may possibly be better than a non-binary one. This paper investigates the robustness of binary and non-binary variance-balanced nested row-column designs to the unavailability of one or more blocks of observations. Robustness is measured through the C-matrices of the designs resulting from removing blocks, using optimality criteria such as A-, D-, E- and MV-optimality.     

Characterization of certain incomplete block designs

It is shown that certain inequalities known for partially balanced incomplete block (PBIB) designs remain valid for general incomplete block designs. Some conditions for attaining their bounds are also given. Furthermore, the various types of PBIB designs are characterized by relating blocks of designs with association schemes. The approach here is based on the spectral expansion of NN' for the incidence matrix N of an incomplete block design.     

On the construction of m-associate PBIB designs I

The m-associate triangular association scheme has been discussed, and several series of partially balanced incomplete block (PBIB) designs with m-associate triangular association scheme have been obtained in Section 1. In Section 2, an m-associate triangular-group divisible association scheme $\text{(T}{}_{\mathrm{q}}\text{—}\text{GD}\text{m}\text{, 1<q<m)}$ has been introduced and several series of PBIB designs with m-associate triangular group divisible association scheme, from m-associate triangular PBIB designs have been constructed. Some numerical values to the three associate triangular designs, and three associate triangular group divisible designs in the range b, v≦100; r, k≦10, with their average efficiencies are given, respectively, in Tables I and II, in Section 3, where as usual v denotes the number of treatments, b the number of blocks, r the number of replications of each treatment, and k the block size.     

On the analysis of circular balanced crossover designs

The paper considers Azaïs' (J. Roy. Statist. Soc. B, 49 (1987) 334–345) randomization procedure for circular balanced crossover designs. It is shown that this randomization does not justify the assumption of independent identically distributed errors when the estimates are corrected for carryover effects. This might lead to underestimation of the variance of treatment estimates. Similar to the results of Kunert (Biometrics, 43 (1987) 833–845) and Kunert and Utzig (J. Roy. Statist. Soc. B, 55 (1993) 919–927), we give constants, such that multiplication with this constant makes the usual estimate of variance conservative.     

ADJUSTED ORTHOGONAL NESTED ROW-COLUMN DESIGNS

Adjusted orthogonality in nested row-column designs is defined and a sufficient condition established for its existence. It is shown that the properties of an adjusted orthogonal nested row-column design are directly related to those of its separate row and column component designs. A method for constructing efficient adjusted orthogonal designs involving a single replicate of every treatment in each of two blocks is given.     

Comparing more than one treatment to more than one control in incomplete blocks

The class of balanced treatment incomplete block designs is generalized to allow for comparison of v₁ test treatments and v₂ control treatments. The generalized class is equivalent to the class of balanced bipartite block designs considered by Jaggi, Gupta, and Parsad. Some results on design construction and A-optimality are given for small values of v₁ and v₂. Algorithms are developed for computing simultaneous confidence bounds for all test treatment versus control contrasts.     

Balanced and partially balanced incomplete block designs with autocorrelation errors

The paper aims to find variance balanced and variance partially balanced incomplete block designs when observations within blocks are autocorrelated and we call them BIBAC and PBIBAC designs. Orthogonal arrays of type I and type II when used as BIBAC designs have smaller average variance of elementary contrasts of treatment effects compared to the corresponding Balanced Incomplete Block (BIB) designs with homoscedastic, uncorrelated errors. The relative efficiency of BIB designs compared to BIBAC designs depends on the block size k and the autocorrelation ρ and is independent of the number of treatments. Further this relative efficiency increases with increasing k. Partially balanced incomplete block designs with autocorrelated errors are introduced using partially balanced incomplete block designs and orthogonal arrays of type I and type II.     

Improved algorithms for closed step-down procedures to test heterogeneity of all subsets of means

A common problem in analysis of variance is testing for heterogeneity of different subsets of the full set of k population means. A step-down procedure tests a given subset of p means only after rejecting homogeneity for all sets that contain it. The Peritz and Gabriel closed procedure rejects homogeneity for the subset if every partition of the k means that includes the subset includes some rejected set. The Begun and Gabriel closure algorithm reduces computations, but the number of tests still increases exponentially with respect to the number of complementary means, m=k−p. We propose a new algorithm that tests only the m−1 pairs of adjacent ordered complementary sample means. Our algorithm may be used with analyses of variance test statistics in balanced and unbalanced designs, and with Studentized ranges except in extremely unbalanced designs. Seaman, Levin, and Serlin proposed a more powerful closure criterion that cannot exploit the Begun and Gabriel algorithm. We propose a new algorithm in this case as well.     

On optimality properties of simple block designs in the approximate design theory

Optimality properties of approximate block designs are studied under variations of (1) the class of competing designs, (2) the optimality criterion, (3) the parametric function of interest, and (4) the statistical model. The designs which are optimal turn out to be the product of their treatment and block marginals, and uniform designs when the support is specified in advance. Optimality here means uniform, universal, and simultaneous j_p-optimality. The classical balanced incomplete block designs are embedded into this approach, and shown to be simultaneously j_p-optimal for a maximal system of identifiable parameters. A geometric account of universal optimality is given which applies beyond the context of block designs.     

Bayes A-optimal designs for comparing test treatments with a control

The problem of comparing v test treatments simultaneously with a control treatment when k, v ⩾ 3 is considered. Following the work of Majumdar (1992), we use exact design theory to derive Bayes A-optimal block designs and optimal Г-minimax designs for a more general prior assumption for the one-way elimination of heterogeneity model. Examples of robust optimal designs, highly efficient designs, and the comparisons of the approximate optimal designs that are derived by our methods and by some other existing rounding-off schemes when using Owen's procedure are also provided.     

Weighted A-optimality for fractional 2m factorial designs of resolution V

A weighted A-optimality (WA-optimality) criterion is discussed for selecting a fractional 2^m factorial design of resolution V. A WA-optimality criterion having one weight may be considered for designs. It is shown that designs derived from orthogonal arrays are WA-optimal for any weight. From a WA-optimal design, a procedure for finding WA-optimal designs for various weights is given. WA-optimal balanced designs are presented for 4 ⩽ m ⩽ 7 and for the values of n assemblies in certain ranges. It is pointed out that designs for m = 7 and for n = 41, 42 given in Chopra and Srivastava (1973a) or in the corrected paper by Chopra et al. (1986), are not A-optimal.     

Some New Residual Treatment Effects Designs for Comparing Test Treatments with a Control

Pigeon & Raghavarao (1987) introduced control balanced residual treatment effects designs for the situation where one treatment is a control or standard and is to be compared with the v test treatments, and they have also given methods of construction of control balanced residual treatment effects designs and have investigated their efficiencies. In this paper we have developed some new families of control balanced residual treatment effects designs, which are Schur-optimal.     

MULTI-FACTOR BALANCED BLOCK DESIGNS WITH COMPLETE ADJUSTED ORTHOGONALITY FOR ALL PAIRS OF TREATMENT FACTORS

A new series of multi-factor balanced block designs is introduced. Each of these designs has the following properties: (i) each of its k– 1 treatment factors is disposed in a cyclic or multi-cyclic balanced incomplete block design with parameters (v,b,r,k,Λ) = (a(k-l) + 1,a²(k-1) +a, ak, k, k) (a > 1); (ii) the incidence of any one of the treatment factors on any other is balanced; and (iii) after adjustment for blocks only, the relationship between any two of the treatment factors is that of adjusted orthogonality. The treatment factors are thus orthogonal to one another in the within-blocks stratum of the analysis of variance. The designs provide a benchmark with which other designs may be compared.