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.     

A Simple Procedure for Constructing Experiment Designs with Incomplete Blocks of Sizes 2 and 3

In order to maximize control of heterogeneity within complete blocks, an experimenter could use incomplete blocks of size k = 2 or 3. In certain situations, incomplete blocks of this nature would eliminate the need for such spatial types of analyses as nearest neighbor. The intrablock efficiency factors for such designs are relatively low. However, with recovery of interblock information, FEDERER and SPEED (1987) have presented measures of design efficiency factors which demonstrate that efficiency factors approach unity for certain ratios of the intrablock and interblock variance components. Hence with recovery of interblock information, even incomplete block designs with k = 2 or 3 have relatively high efficiency factors. The reduction in the intrablock error variance over the complete block error variance in many situations will provide designs with high efficiency. A simple procedure for constructing incomplete blocks of sizes 2 and 3 is presented. It is shown how to obtain additional zero-one association confounding arrangements when v = 4 t, t an integer, and for v = pk, k ≤ p. It is indicated how to do the statistical analysis for these designs.     

Designs Suited for Varietal Trials with Multiple Basals

DAS (1960) gave a method of construction of confounded balanced asymmetrical factorial designs of the type v × 2² by using BIB designs. In the present paper a method has been given for construction of balanced asymmetrical factorial designs of the type (v–t) × 2² by using truncated balanced incomplete block designs obtainable by omitting t treatments. Likewise, partially balanced asymmetrical factorial designs can also be obtained by omitting any particular treatment alongwith its first or second associate treatments from the v treatments of a PBIB design. We can get a large number of new designs not available in literature through this technique. These designs are well suited for varietal trials with multiple basals.     

Application of Partially Balanced Block Designs to Factorial Experiments

In this paper we present partially balanced block designs with F_p and C_p association schemes. These designs play a great role in the experimental theory since they can be applied to factorial experiments. A particular stress is put on the formulation of association schemes _p and C_p, presentation of exemplary constructions of these designs and their analysis of variance. The presented considerations are abundantly illustrated with examples.     

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     

Incomplete Split-Block Designs

The paper deals with modelling of experiments carried out in incomplete split-block designs. Moreover, the characterization of the design with respect to a general balance property is given. Also, some statistical properties of the design are examined. The problems considered are illustrated with examples.     

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.     

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.     

Construction of Resolvable Designs with Nested Treatment Structure

The construction of resolvable incomplete block and row-column designs is investigated when the treatments have a nested structure. Some theoretical results are derived for lattice designs. Efficient designs for unequal-sized treatment groups are obtained by defining a multiple objective function and carrying out a computer search using an interchange algorithm.     

A New Class of Matrices with Application in Experimental Designs

A very general class of balanced matrices is defined in this paper. These matrices are generalisations of the balanced orthogonal designs of Rao (1970) and generalised balanced matrices of Dey and Midha (1976). Some methods of construction of these balanced matrices are discussed. An application of these matrices in experimental designs is also given.     

Model Building and Analysis for Block Designs with Nested Rows and Columns

The paper deals with two main problems: it gives a unified approach to model building for observations obtained in a block design with nested rows and columns and it provides the analysis of variance for such data. Also, some statistical properties of the design are examined.     

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.     

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.