Complete diallel crosses plans through balanced incomplete block designs

The present investigation involves the methods of construction of complete diallel cross plans using balanced incomplete block (BIB) designs. Furthermore, the analysis of complete diallel crosses plans are carried out to estimate the general combining ability of the ith line (i=1, r 2, r …, r v) where the intra- block analysis of the adjusted sum of squares for GCA and the unadjusted block sum of squares are also obtained, thereafter the relationship between the estimates of BIB design and the estimates of the GCA effect of CDC plan has been established. Moreover, it has also been shown that the complete diallel crosses design obtained through two BIB designs satisfying v1=b1= 4 5 1+3=v2=b2, r r1=2 5 1+1=r2=k1=k2 and 5 1= 5 2 are universally optimum. These results are further supported by a suitable example of each. However, the need of this study is to show that the analysis of the CDC plan is reducible to the analysis of generating the BIB design.     

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.     

Incomplete block designs for complete diallel crosses and their analysis

A large number of incomplete block designs for Griffing's complete diallel cross-systems I, II and III, involving from five to 12 lines, are suggested, using two-associate triangular partially balanced incomplete block designs. Analysis of incomplete block designs for complete diallel cross-systems has been carried out assuming the most appropriate model for genetic yield, as advocated by Hinklemnann. This includes estimation of the general combining ability, specific combining ability and reciprocal cross- effects. An illustration of the design for each system is presented.     

Robust designs for diallel crosses against the missing of one block

Dey & Midha (1996) showed that some of the complete diallel crosses plans, obtained by using triangular partially balanced designs with two associate classes, are optimal. In this investigation, it is derived that these optimal designs for diallel crosses are robust also against the unavailability of one block.     

Optimal row-column designs for complete diallel crosses

Universally optimal row-column designs for complete, diallel crosses are investigated. Three series of designs that require just one replication of the crosses are provided. A series of designs having two replications of each cross is also provided.     

Robustness of complete diallel crosses plans to the unavailability of one block

The present investigation involved the estimation of the general combining ability of CDC plans subject to the unavailability of one block for Griffing's system IV. Further, it has been shown that CDC plans are fairly robust to the unavailability of one block.     

Search for suitable incomplete block designs for complete diallel cross systems

The suitability of incomplete block designs for each complete diallel cross system I, II, III and IV, under the general genetic model is examined, and a set of necessary conditions obtained. In this connection, modifications in available designs are suggested and illustrated. A table of suitable designs with higher efficiency for complete diallel cross systems is presented.     

Robustness of a complete diallel crosses plan with an unequal number of crosses to the unavailability of one block

The present investigation involved the estimation of the general combining ability of complete diallel crosses (CDC) plans with unequal numbers of crosses, subject to the unavailability of one block for Griffing's system IV . Further, it has been shown that CDC plans with unequal numbers of crosses are fairly robust to the unavailability of one block.     

A-optimal diallel crosses for test versus control comparisons

A-optimality of block designs for control versus test comparisons in diallel crosses is investigated. A sufficient condition for designs to be A-optimal is derived. Type S₀ designs are defined and A-optimal type S₀ designs are characterized. A lower bound to the A-efficiency of type S₀ designs is also given. Using the lower bound to A-efficiency, type S₀ designs are shown to yield efficient designs for test versus control comparisons.     

On the recovery of intra- and interblock information in n-ary designs

This note presents an extension of Q-method of analysis for binary designs given by Rao (1956) to n-ary balanced and partially balanced block designs. Here a linked n-ary block (LNB) design is defined as the dual of balanced n-ary (BN) design. Having a note on Yates’ (1939, 1940) method of P-analysis, we further extend the expressions for binary linked block (LB) designs given by Rao (1956) to linked n-ary block (LNB) designs which admit easy estimation of parameters for these type of all n-ary designs.     

Partial diallel crosses based on three associate class association schemes

Designs of partial diallel crosses obtained by including parents based on rectangular and cubic association schemes have been presented. In addition, a simplified method of their analysis by making use of latent roots and idempotent matrices has also been presented. The method has been illustrated with the help of numerical data.     

Balanced block designs

In the present paper an attempt has been made to characterize and unify the three different concepts of balancing in incomplete block designs, namely (i) variance balance, (ii) efficiency balance and (iii) pairwise balance. Simple characterizations of variance balance and efficiency balance have been given using the P matrix. A method of constructing efficiency balanced (EB) and variance balanced designs has also been presented.     

Orthogonal arrays and designs for partial triallel crosses

Abstract

Some block designs for Partial triallel crosses (PTC) are constructed in this paper using (a) orthogonal arrays (p², r, p, 2), where p is prime or a power of prime and (b) semi - balanced arrays (p (p???1)/2, p, p, 2), where p is a prime or power of an odd prime. Designs constructed are optimal in the sense of Kiefer (1975 Kiefer, J. 1975. Construction and optimality of generalized Youden designs. In A survey of statistical design and linear models, ed. J. N. Srivastava, 333–353. Amsterdam: North Hollond. [Google Scholar]). Some new designs for PTC experiments are also obtained.     

On the block structure of proper block designs

In this paper we consider proper block designs and derive an upper bound for the number of blocks which can have a fixed number of symbols common with a given block of the design. To arrive at the desired bound, a generalization of an integer programming theorem due to Bush (1976) is first obtained. The integer programming theorem is then used to derive the main result of this paper. The bound given here is then compared with a similar bound obtained by Kageyama and Tsuji (1977).     

The robustness of chain block designs and coat-of-mail designs

Chain block designs are relatively vulnerable to loss of information when missing values or outliers may occur. An alternative class of designs, coat-of-mail designs, are proposed and the relative robustness of the two types of design are compared.     

Interblock information in multidimensional block designs

In many experimental situations, d-way heterogeneity among experimental units may be controlled through use of multiple blocking criteria. In some cases it is reasonable to regard some or all of the block effects as random. Then the model is mixed and observations within blocks are correlated. Very general estimators of treatment effects and their dispersion matrix with recovery of interblock information are provided. They apply to designs with d > 1 blocking criteria that may be crossed, nested, or a combination thereof. These general results may be specialized to provide analyses of new classes of MBD's or used directly for numerical analyses of designs in the general class, perhaps through use as the basis for very general computer programs. Estimation of variance components is discussed, and an example is provided to illustrate adaptation of the general results.     

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.     

On nested block designs geometry

Nested block designs and block designs properties such as orthogonality, orthogonal block structure and general balance are examined using the concept of a commutative quadratic subspace and standard properties of orthogonal projectors. In this geometrical context conditions for existence of the best linear unbiased estimators of treatment contrasts are also discussed.