Constructions of efficiency-balanced designs

We present a number of methods of constructing efficiency-balanced binary block designs which are design patterns for simplification of statistical analysis. Furthermore, a method of construction of an efficiency-balanced block design with v+1 treatments from one with v treatments is generally characterized.     

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.     

On choosing between fixed and random block effects in some no-interaction models

The purpose of this article is to strengthen the understanding of the relationship between a fixed-blocks and random-blocks analysis in models that do not include interactions between treatments and blocks. Treating the block effects as random has been recommended in the literature for balanced incomplete block designs (BIBD) because it results in smaller variances of treatment contrasts. This reduction in variance is large if the block-to-block variation relative to the total variation is small. However, this analysis is also more complicated because it results in a subjective interpretation of results if the block variance component is non-positive. The probability of a non-positive variance component is large precisely in those situations where a random-blocks analysis is useful – that is, when the block-to-block variation, relative to the total variation, is small. In contrast, the analysis in which the block effects are fixed is computationally simpler and less subjective. The loss in power for some BIBD with a fixed effects analysis is trivial. In such cases, we recommend treating the block effects as fixed. For response surface experiments designed in blocks, however, an opposite recommendation is made. When block effects are fixed, the variance of the estimated response surface is not uniquely estimated, and in practice this variance is obtained by ignoring the block effect. It is argued that a more reasonable approach is to treat the block effects to be random than to ignore it.     

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.     

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

A 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).     

Construction of Balanced Incomplete Block Designs Using Cyclic Shifts

Balanced incomplete block designs (BIBDs) play important role in design of experiments, especially in field experiments. These designs ensure that treatments are compared with equal precision. Several methods are available in the literature to construct BIBDs but in this article some infinite series of these designs are presented by method of cyclic shifts. This method expresses some standard properties of a design just through examining the sets of shifts rather than studying the whole design.     

The Use of Balanced Incomplete Block Designs in Designing Randomized Response Surveys

This paper investigates the block total response method proposed by Raghavarao and Federer for providing accurate estimates of the base rates of sensitive characteristics during surveys. It determines the best balanced incomplete block design to use to estimate the base rates for three, four, five and six sensitive attributes respectively, given a maximum total number of 13 questions. The estimates obtained from this method have smaller variance than estimates obtained using the similar, but more popular, unmatched count technique.     

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.     

An algorithm for constructing optimal resolvable incomplete block designs

This paper describes an efficient algorithm for the construction of optimal or near-optimal resolvable incomplete block designs (IBDs) for any number of treatments v < 100. The performance of this algorithm is evaluated against known lattice designs and the 414 or-designs of Patterson & Williams [36]. For the designs under study, it appears that our algorithm is about equally effective as the simulated annealing algorithm of Venables & Eccleston [42]. An example of the use of our algorithm to construct the row (or column) components of resolvable row-column designs is given.     

PARTIALLY EQUINEIGHBOURED DESIGNS

This paper presents further results on a class of designs called equineighboured designs, ED. These designs are intended for field and related experiments, especially whenever there is evidence that observations in the same block are correlated. An ED has the property that every unordered pair of treatments occurs as nearest neighbours equally frequently at each level. Ipinyomi (1986) has defined and shown that ED are balanced designs when neighbouring observations are correlated. He has also presented ED as a continuation of the development of optimal block designs. An ED would often require many times the number of experimental materials needed for the construction of an ordinary balanced incomplete block, BIB, design for the same number of treatments and block sizes. Thus for a relatively large number of treatments and block sizes the required minimum number of blocks may be excessively large for practical use of ED. In this paper we shall define and examine partially equineighboured designs with n concurrences, PED (n), as alternatives where ED are practically unachievable. Particular attention will be given to designs with smaller numbers of blocks and for which only as little balance as possible may be lost.     

A-Optimal and MV-Optimal Incomplete Block Designs for Comparing Successive Treatments

Consider an incomplete block experiment in which observations are taken from t treatments using an incomplete block design with b blocks of size k < t. Suppose the interest is in estimating the differences of effects of successive treatments. This may occur, for example, if the treatments are different dosages or concentrations of a compound. This article presents A-optimal and MV-optimal incomplete block designs for estimating the the differences of successive treatment effects. Tables of optimal designs are given for k < t ≤ 5 with b ≤ 40.     

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.     

Notes on testing carry-over effects in continuous data under an incomplete block crossover design

We consider hypothesis testing and estimation of carry-over effects in continuous data under an incomplete block crossover design when comparing two experimental treatments with a placebo. We develop procedures for testing differential carry-over effects based on the weighted-least-squares (WLS) method. We apply Monte Carlo simulations to evaluate the performance of these test procedures in a variety of situations. We use the data regarding the forced expiratory volume in one second (FEV₁) readings taken from a double-blind crossover trial comparing two different doses of formoterol with a placebo to illustrate the use of test procedures proposed here.     

A-Optimal Block Designs with Additional Singly Replicated Treatments

Block designs to which have been added a number of singly-replicated treatments, known as secondary treatments, are particularly useful for experiments where only small amounts of material are available for some treatments, for example new plant varieties. The designs are of particular use in the microarray situation. Such designs are known as 'augmented designs'. This paper obtains the properties of these designs and shows that, with an equal number of secondary treatments in each block, the A-optimal design is obtained by using the A-optimal design for the original block design. It develops formulae for the variance of treatment comparisons, for both the primary and the secondary treatments. A number of examples are used to illustrate the results.     

Regular E-Optimal Spring Balance Weighing Designs with Correlated Errors

The problems linked with an E-optimal spring balance weighing design with correlated errors are discussed. The topic is focus on the determining the maximal eigenvalue of the inverse of the information matrix of estimators. The constructing method of the E-optimal design, based on the incidence matrices of balanced incomplete block designs, is presented.     

Single replicate orthogonal block designs for circulant partial diallel crosses

Some incomplete block designs for partial diallel crosses have been given in the literature. These designs are obtained by regarding the number of crosses as treatments, and consequently require several replications of each cross. The need for resorting to a partial diallel cross itself implies that it is desired to have fewer crosses. A method for constructing single replicate incomplete block designs for circulant partial diallel crosses is provided in this paper. The designs are orthogonal, and thus they retain full efficiency for estimation of the contrasts of interest.     

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.     

The Parshvanath yantram yields a constant-sum partially balanced incomplete block design

Yantrams have been used to generate mixture designs in the interior of a simplex. In this note, we show a connection between Parshvanath yantram and a particular partially balanced incomplete block design. This block design is rather special and somewhat unexpected due to the feature that sum of the treatment symbols in any block is constant.     

Measures of non orthogonality in block designs

Orthogonality is an important concept in block design. Necessary and sufficient condition for a connected block design to be orthogonal is well known. However, when a design is not orthogonal, it is not clear how much it deviates from orthogonality. In this paper, an attempt has been made to first define the measures of or indices to non orthogonality in block design and then to characterize designs possessing minimum non orthogonality indices. It is shown that a Balanced Incomplete Block Design (BIBD) and a Balanced Block Design (BBD), if exist, possess this property.