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.     

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.     

Robustness of complete diallel crossing plans against exchange of one cross

The robustness aspects of block designs for complete diallel crossing plans against the exchange of one cross using connectedness and efficiency criteria have been investigated. The exchanged cross may have either no line in common or one line in common with the original cross. It has been found that randomized complete block (RCB) designs for complete diallel crosses and binary balanced block designs for complete diallel crosses are robust against the exchange of one cross in one observation. The RCB designs for diallel crosses have been shown to be robust against the exchange of one cross with another cross in all the blocks. The non-binary balanced block designs obtainable from Family 5 of Das et al. (1998) have also been found to be robust against the exchange of one cross.     

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.     

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.     

Optimal block designs for triallel cross experiments

Optimal block designs for a certain type of triallel cross experiments are investigated. Nested balanced block designs are introduced and it is shown how these designs give rise to optimal designs for triallel crosses. Several .series of nested balanced block designs, leading to optimal designs for triallel crosses are reported.     

Application of PBIB designs in CDC Method IV

In this paper we propose the use of some partially balanced incomplete block designs for blocking in complete diallel cross Method IV (Griffing, 1956) to deal with the situation when it is not desirable for all crosses to be accommodated in the block of a traditional randomized block design. A method is also proposed to analyse the MatingEnvironment designs for estimating the general combining ability effect of lines.     

Star-Type Systematic Sampling Schemes for Finite Populations

Two new sampling schemes namely, Star-Type Systematic (STS) sampling without replacement and Modified Star-Type Systematic (MSTS) sampling without replacement for estimation of finite population means are introduced. The relative performances of the proposed star-type systematic sample means along with those of the simple random and systematic sample means are assessed for a hypothetical population with a linear trend and also for certain natural populations. Furthermore, the usefulness of the proposed sampling schemes in quality control and for constructing partial diallel crosses in mating designs are briefly break discussed.     

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.     

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.     

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.     

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.     

Robustness of a Class of Partial Diallel Cross Designs to the Unavailability of a Complete Block of Observations

Complete and partial diallel cross designs are examined as to their construction and robustness against the loss of a block of observations. A simple generalized inverse is found for the information matrix of the line effects, which allows evaluation of expressions for the variances of the line-effect differences with and without the missing block. A-efficiencies, based on average variances of the elementary contrasts of the line-effects, suggest that these designs are fairly robust. The loss of efficiency is generally less than 10%, but it is shown that specific comparisons might suffer a loss of efficiency of as much as 40%.     

Partial diallel grosses based on generalized right angj lar association scheme

Some new plans for partial diallel crosses (PDC) based on Generalized Right Angular (GRA) association scheme excluding parents as well as including parents have been presented. Unlike existing plans based on GRA, these plans have wider existence and provide sufficient size of PDC to have more precise conclusions. Simplified method of analysis for these plans has also been presented.     

Use of n-ary block designs in diallel crosses evaluation

To fill the gap between theory and practice of modern statistical designs, this paper presents the use of n-ary block designs in the evaluation of balanced incomplete block (BIB) designs for all the four Griffing's (1956) complete dialled crosses (CDC) systems, the construction of which is proposed here by adopting a suitable BIB design and associating its treatments with the crosses under a CDC system.     

Invariance and factorial models

Two factors having the same set of levels are said to be homologous. This paper aims to extend the domain of factorial models to designs that include homologous factors. In doing so, it is necessary first to identify the characteristic property of those vector spaces that constitute the standard factorial models. We argue here that essentially every interesting statistical model specified by a vector space is necessarily a representation of some algebraic category. Logical consistency of the sort associated with the standard marginality conditions is guaranteed by category representations, but not by group representations. Marginality is thus interpreted as invariance under selection of factor levels ( I -representations), and invariance under replication of levels ( S -representations). For designs in which each factor occurs once, the representations of the product category coincide with the standard factorial models. For designs that include homologous factors, the set of S -representations is a subset of the I -representations. It is shown that symmetry and quasi-symmetry are representations in both senses, but that not all representations include the constant functions (intercept). The beginnings of an extended algebra for constructing general I -representations is described and illustrated by a diallel cross design.     

Minimax A-, c-, and I-optimal regression designs for models with heteroscedastic errors

It is well known that it is difficult to construct minimax optimal designs. Furthermore, since in practice we never know the true error variance, it is important to allow small deviations and construct robust optimal designs. We investigate a class of minimax optimal regression designs for models with heteroscedastic errors that are robust against possible misspecification of the error variance. Commonly used A-, c-, and I-optimality criteria are included in this class of minimax optimal designs. Several theoretical results are obtained, including a necessary condition and a reflection symmetry for these minimax optimal designs. In this article, we focus mainly on linear models and assume that an approximate error variance function is available. However, we also briefly discuss how the methodology works for nonlinear models. We then propose an effective algorithm to solve challenging nonconvex optimization problems to find minimax designs on discrete design spaces. Examples are given to illustrate minimax optimal designs and their properties.     

Optimal repeated measurements designs for comparing test treatments with a control

A-optimal and mv optimal repeated measurments designs for comparing serveral test treatments with a control are considered. the models considered are basically of two types: without preperides and the cirular model. It is shown known that some known strongly balanced uniform repeated measurements designs can be modified to obtain optimal designs for this problem. Some other methods of finding optimal designs are also given.     

Robustness properties of minimally‐supported Bayesian D‐optimal designs for heteroscedastic models

Bayesian D‐optimal designs supported on a fixed number of points were found by Dette & Wong (1998) for estimating parameters in a polynomial model when the error variance depends exponentially on the explanatory variable. The present authors provide optimal designs under a broader class of error variance structures and investigate the robustness properties of these designs to model and prior distribution assumptions. A comparison of the performance of the optimal designs relative to the popular uniform designs is also given. The authors' results suggest that Bayesian D‐optimal designs suported on a fixed number of points are more likely to be globaly optimal among all designs if the prior distribution is symmetric and is concentrated around its mean.     

Variance-balanced row–column designs involving diallel crosses incorporating specific combining abilities for comparing test lines with a control line

In this paper, the problem of comparing t test lines with a control line under a row–column setup in complete diallel cross experiment is investigated when specific combining ability (sca) effect is included in the model. Three classes of Mating-Environmental Row–Column (MERC) designs have been obtained which are variance balanced for estimating the contrasts pertaining to general combining ability (gca) effects free from sca effects.