A Note on Block Designs with Nested Rows and Columns

Block designs with nested rows and columns have been introduced by Singh and Dey (1979). This note gives some results regarding the total relative loss of information and patterns of efficiency balanced block designs with nested rows and columns in variable replications of treatments.     

On the Construction of Balanced and Partially Balanced Ternary Design

A method of constructing balanced and partially balanced ternary designs from balanced and partially balanced incomplete block designs, respectively, and two methods of constructing partially balanced ternary designs from association schemes are obtained. Two new and efficient balanced ternary designs having K < V and R ≦ 20 are obtained by the first method.     

Dual Designs and Their Application in Genetical Experiments

The dual of incomplete block designs has been studied with th́eir applications in genetical experiments. Partial diallel crosses (PDC) of type I have been constructed using balanced incomplete block (BIB) designs, partially balanced incomplete block (PBIB) designs and their dual designs. Simplified analysis of PDC has been presented using the dual property of these designs. List of optimal PDC having simple analysis has been given.     

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.     

A Simple Approach to Analyse Non-Orthogonal Data

The present paper deals with an alternative and simple procedure to analyse the non-orthogonal data. The procedure is general in nature but has some advantages for the non-orthogonal data due to some missing observations. The procedure is applied to (i) two way classification with unequal number of observations per cell; (ii) randomized block designs with some missing observations and (iii) balanced incomplete block designs and also illustrated with the help of numerical examples.     

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.     

Fractional Plans to Estimate Main Effects and Two Factor Interactions Inclusive of a Specific Factor

Balanced incomplete block designs are used to construct non‐geometric 2ⁿ fractional factorial plans to estimate all n main effects and n – 1, 2 factor interactions with a specific factor included in each interaction. When the balanced incomplete block design is of Family (A), the resulting fractional factorial plan has the same number of runs as a fold‐over Hadamard matrix giving same variances for the estimates; however, some new designs are shown to be non‐isomorphic to the fold‐over Hadamard matrix plans.     

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.     

Analysis of data from incomplete block designs

This paper presents an organized solution to the problem of computing inter- and intrablock analyses of incomplete block designs, based on the modified maximum likelihood principle proposed by Patterson and Thompson (1971, Biometrika 58, 545-554). The calculations are set out to be easily programmed on a microcomputer. The method is attractive because it is simple, yet sufficiently general to handle a wide class of designs, including partially balanced incomplete block designs, designs with unequal block sizes, designs with missing values, and generally unbalanced split-plot experiments.     

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.     

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.     

Small Sample Properties of Rank Tests for Incomplete Unbalanced Designs

A rank test is presented for analysis of incomplete unbalanced designs, i.e. for designs that may have been originally planned to be either balanced or unbalanced and where some observations may be missing at random. This test is a modification of the procedure of Benard and van El-teren (1953) based on a generalization of block weights proposed by Prentice (1979). It is compared with the tests of Haux, Schumacher, and Weckesser (1984) and Rai (1987). For incomplete or unbalanced designs with more than two treatments the quadratic forms proposed by these authors are proven to be invalid for small sample sizes, except for special cases. A necessary condition is given for test statistics to be valid also for small samples.     

New Constructions of BIB Designs

A method of construction of certain balanced incomplete block (BIB) designs is defined from which we get new series of BIB designs.     

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.     

A Simple Construction Procedure for Eesolvable Incomplete Block Designs for any Number of Treatments

A simple, straightforward procedure, which requires no special tables or generators, is presented for constructing resolvable incomplete block designs for v=pk, v=p²k, …, treatments, for k≥p, in incomplete blocks of size k. Also, it is shown, how to obtain incomplete block designs for any v in blocks of size k and k+1. The procedure allows construction of balanced incomplete block designs for p = k a prime number. For p = n not a prime number, incomplete block designs can be obtained by the procedure, but are not balanced. However, for p_s being the smallest prime factor of n, p_s + 1 for v = n², p_s²+ p_s + 1 for v = n³, …, arrangements can be obtained for which the occurrence of any treatment pair in the blocks is either zero or one. This is called a zero-one concurrence design. Procedures are described for obtaining additional zero-one concurrence arrangements. It is shown that the efficiency of these designs is maximum. Both intra-block and inter-block analyses are described.     

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.     

A necessary condition for balance of a block design

It is shown that, barring orthogonal designs, a necessary condition for a block design, BD(v, b, N), to be J-balanced is that the rows of its incidence matrix, N, are linearly independent. This strengthens the condition b v known in the literature as FISHER'S inequality. V-balanced block designs are also discussed.     

FACTORS INFLUENCING THE EFFICIENCY OF INCOMPLETE BLOCK DESIGNS IN SENSORY EVALUATION EXPERIMENTS

The efficiencies of incomplete block designs were investigated by comparing two hundred and twenty eight analyses from eleven trials using hedonic scales with corresponding randomized complete block analyses. Of the ten explanatory factors examined, only the panelist, the product type, the number of samples per session and the average score of the data had an effect on the efficiency of incomplete block designs. The effect of product type was attributed to influences of produce consumed outside the trial, and the effect of the data mean reflected decreased conscientiousness with products the panelists disliked. With three and four samples per session, incomplete block designs were 31 % and 2 % more efficient, respectively, than randomized complete block designs. When five or more samples were tested, the incomplete block designs were markedly less efficient. The practical implications of all these effects on experimental design are discussed.     

Analysis of Line × Tester Progenies Compared in Orthogonally Supplemented Efficiency Balanced Incomplete Block Design

An analysis of experimental design is suggested for comparing progenies of lineXtester mating system. The line × tester mating system is an expanded top-cross mating system, used frequently in practice. This system is of interest for breeders dealing with estimation and testing of general as well as specific combining abilities. In the suggested design, treatments, i.e. hybrids obtained in crosses, and testers are studied in orthogonally supplemented incomplete block design. Analysis of variance has been presented, estimators of expected values for progenies and for testers have been given, testing of hypotheses concerning contrasts of interest for progenies, lines and testers has been discussed. Estimators of effects of general and specific combining abilities as well as tests permitting for testing the significance of various genetic parameters, have also been given.     

A Note on the Analysis of a p × 22 Design

TYAGI has given a method of constructing a balanced factorial experiment of the p × 2² type. A complete analysis of such a design with and without recovery of interblock information, using the eigenyalues and eigenvectors of the C-matrix of the design, is given in this paper. KHATRY & SHAH'S method of producing uniformly better estimators is employed to combine the inter and intra block estimates.