On the E-Optimality of Block Designs with Unequal Block Sizes

In this paper an inequality for the smallest positive eigenvalues of the C-matrix of a block design are derived. This inequality is a generalization of a result by Constantine (1982) to the case of unequal block sizes. On the basis of the above result a certain E-optimality criterium of block designs is given. Furthermore coefficient e_d has been introduced which permits to assess how close the block design is from the optimal one.     

Analysis of Partially Balanced Incomplete Block Designs with Standards

The paper presents a detailed analysis of variance of PBIB designs supplemented by one or more standard treatments. Comparisons were carried out of treatment effects within the group of standard and test treatments and between the two groups. The analysis is supplemented by an example.     

On C-Property in Block Designs

Block designs having a property described in Caliński (1971) and Saha (1976), referred to herein as C-property, are considered. A necessary and sufficient condition for a block design to have C-property is derived, and some methods of construction of such block designs are presented.     

Searching for Optimal Block Designs when Block Effects are Random

An algorithm that searches for optimal block designs when block effects are random is described. Some situations in which such an algorithm will prove useful are considered. An illustrative example and some discussion are given.     

Construction of Partially Balanced Incomplete Block Designs with Nested Rows and Columns

A number of methods of construction of partially balanced incomplete block designs with nested rows and columns are developed and new balanced incomplete block designs with nested rows and columns are obtained as a by-product.     

Some Bounds on Parameters of Efficiency-Balanced Block Designs

Bounds on the eigenvalue of the C -matrix with respect to R and the number of blocks for efficiency-balanced block designs are given. AMS 1970 Subject Classification: 62K10     

Optimality of Some Class of Incomplete Block Designs

A class of incomplete block designs called C-design, was considered by Caliński (1971), Saha (1976), Ceranka (1983) and Ceranka , Kozłowska (1983, 1984). In this paper we extend the theory of block designs having the C-property. We consider optimality of C-designs with respect to any criterion of a described form. Das and Kageyama (1991) considered a class of E-optimal proper efficiency balanced designs (strictly speaking, Das and Kageyama considered E_R-optimality of some class of block designs). Hence we consider E_R, A_R, D_R optimality of C-designs.     

On the PBB Designs for Multiresponse Experiments

We present the idea of using multiresponse incomplete block designs when not all responses can be observed in all experimental units. For a special class of such designs, in which partial designs are PBB designs, a method for estimating natural treatment contrast is given. We also consider the problem of testing the hypotheses concerning the natural and any estimable treatment contrasts. For testing this hypothesis the Wald statistics, being asymptotically chi-square distributed, is proposed.     

On Analysis of Some Augmented Block Designs

Augmented designs are frequently used in practice. In this paper, a simplified procedure of analysis of some types of augmented designs is discussed using the concept of efficiency-balance and partially efficiency-balance.     

Construction of Equi-replicated Balanced Block Designs with Block Sizes Exceeding the Number of Treatments

In this note it is shown that the block design with incidence matrix Ñ = [NN… N], where N = c_1hN_h + c_oh (11′–N_h). c_oh and c_1h are any non-negative integers and N_h,h = 1, 2,…,p, are incidence matrices of balanced incomplete block designs with the same number of treatments t, is a balanced block design with the block sizes exceeding the number of treatments. In derivation the matrix M₀, introduced by CALIński (1971) is utilized.     

A Type of Orthogonal Contrasts for Unbalanced Data

A simple method of making a type of orthogonal contrasts among treatments for unbalanced two-way classification data has been described. The method begins with the direct use of absorbing equations, derived from normal equations. Then it is shown that the single components of the treatment sum of squares, adjusted for blocks, are independently distributed.     

On the Efficiency Factor for a Contrast of Treatment Parameters

In this paper a formula for computing the efficiency factor of a block design for any estimable contrast of treatment parameters is given. The formula is expressible in terms of the efficiency factors of some basic contrasts.     

Analysis of Triangular Diallel Table for Experiments in Block Designs

In this paper the analysis of genotypes obtained in diallel crossing system, including one set of F₁′s and parents, arranged in triangular table is given. This analysis is presented for data obtained from the experiments laid out in block designs. The analysis of variance, estimators of general and specific combining abilities as well as statistics for testing of hypotheses concerning those parameters are given.     

Selection Problems in Balanced Block Designs

If we have independent random variables the solution of BECHHOFER's selection problem in special cases is easily applied because we can use tables of a standardized multivariate normal distribution. The aim of this note is to show how such tables are also applicable in a case of correlated random variables like the estimates of treatment effects in BIB and BCB. Formulae for a choice of minimal sample sizes are also given.     

Analysis of Diallel Table for Experiments in Block Designs

In this paper the analysis of genotypes obtained in diallel crossing system, including parents, one set of F₁s and reciprocal F₁'s, is given. This analysis is presented for data obtained from the experiments laid out in block designs. The analysis of variance, estimator of reciprocal effects, estimators of general and specific combining abilities as well as statistics for testing of hypotheses concerning those parameters are given.     

An Annealing Algorithm for Searching Optimal Block Designs

This paper is concerned with a computer algorithm for searching optimal block designs. The algorithm uses a technique called simulated annealing. Exchange and interchange steps are defined in a way similar to JONES and ECCLESTON (1980). Two examples are used to show the performance of the algorithm.     

Characterisation of Certain Split-Block Designs with a Control

The paper deals with an incomplete split-block design in which one or two factors are split into two parts, the first one containing test treatments and the second one — a control treatment. The aim of the experiment carried out in such design is to compare the average of the test treatment effects with the control treatment effect and the test treatment effects with the control treatment effect individually. All those comparisons can be expressed by contrasts, elementary or basic ones. The aim of the paper is to characterise particular cases of the incomplete split-block designs with respect to general balance and to efficiency factors of the design with respect to the contrasts. In the paper we restrict our attention to the designs in which each block has two rows or/and two columns only. These designs are suitable for certain agricultural experiments.     

Use of Wishart Prior and Simple Extensions for Sparse Precision Matrix Estimation

A conjugate Wishart prior is used to present a simple and rapid procedure for computing the analytic posterior (mode and uncertainty) of the precision matrix elements of a Gaussian distribution. An interpretation of covariance estimates in terms of eigenvalues is presented, along with a simple decision-rule step to improve the performance of the estimation of sparse precision matrices and associated graphs. In this, elements of the estimated precision matrix that are zero or near zero can be detected and shrunk to zero. Simulated data sets are used to compare posterior estimation with decision-rule with two other Wishart-based approaches and with graphical lasso. Furthermore, an empirical Bayes procedure is used to select prior hyperparameters in high dimensional cases with extension to sparsity.