Klaus Hinkelmann

PARTIAL DIALLEL CROSSES IN INCOMPLETE BLOCKS

Various forms of diallel crosses play an important role in evaluating the breeding potential of genetic material in plant and animal breeding (e.g., Hinkelmann, 1975, inA Survey of Statistical Design and Linear Models, 243-269. Amsterdam: North Holland). In this paper we consider the use of partial diallel crosses in incomplete block designs as environment designs. We discuss in detail a particular method of constructing such combined mating-environment designs by relating both component designs to certain types of partially balanced incomplete block designs. This method is compared with a generally applicable method of embedding partial diallel crosses in existing incomplete block designs as environment designs.

Linear trend-free Box–Behnken designs

Abstract We give a general description of constructing Box–Behnken designs by combining ideas and concepts from incomplete block designs and 2k factorial designs. The structure is then exploited to construct linear trend-free designs. The individual steps in this process are described and illustrated by examples.

ESTIMATION OF HERITABILITY FROM EXPERIMENTS WITH INBRED AND RELATED INDIVIDUALS

SUMMARY Assuming an arbitrary amount of inbreeding and certain types of relationships among individuals in the parent population, it is shown how for nested and diallel mating designs the genetic and environmental variance components can be estimated from the mean squares in the usual analyses of variance. The systems of equations are given explicitly for the case of equal numbers of offspring per mating. A method for dealing with unequal numbers of offspring is outlined. In a previous paper (Hinkelmann [1969]) modifications of the usual procedures for estimating heritability from nested and diallel mating designs were given for the case that the dams are related. It was shown how this relationship induces a certain covariance structure for some random effects in the underlying linear model. However, just as the variance components can be expressed in terms of genetic and environmental variance components, these covariance components are simply functions of the genetic variance components and the coefficients of relationship. Since the coefficients of relationship can be evaluated from the pedigree, it is then easy to estimate the genetic and environmental variance components from the mean squares in the usual analysis of variance, and hence estimate heritability. In the present paper this method is extended to allow for an arbitrary amount of inbreeding and certain types of relationships in the parental population. The most general situation would be to allow for all types of relationships, i.e., among sires, among dams, and among sires and dams. This, however, may lead to inbreeding in the offspring generation which in turn leads to problems in evaluatiing covariances between relatives, an approach on which the results in this paper are based. Harris [1964] has developed formulae for the covariance between inbred relatives which involve additional genetic parameters not present in the usual covariance formulae for non-inbred relatives (e.g. Kempthorne [1957]). These parameters are not estimable from the usual mating designs. Also, Cockerham [1963] and Harris [19641 give special conditions under which the covariance between inbred

Estimation of heritability from experiments with related dams.

This paper considers modifications of the usual procedures for estimating heritability from nested or diallel mating designs assuming various types of relationships among the dams in the parent population. For the underlying linear models a covariance structure is introduced. It is shown how the resulting structural variance and covariance components can be expressed in terms of genetic and environmental variance components and certain types of average coefficients of relationship. This correspondence is used in evaluating the E(MS) in the analysis of variance table which then leads to estimators for h2.

Analysis of Partial Diallel Crosses in Incomplete Blocks

With a large number of lines in a diallel cross experiment, the number of crosses becomes unmanageable to be accommodated in homogeneous blocks. To overcome this problem, a sample of crosses, known as partial diallel cross (PDC) is often used. The selection of a PDC is based on the criterion of high efficiency for the estimation of general combining ability (gca) effects. Even with a moderately large number of crosses, the use of incomplete blocks is necessary to obtain homogeneous experimental units. The analysis of data from a general PDC grown in general incomplete block designs is being described. An iterative scheme is being developed for obtaining a generalized inverse of the information matrix used in estimating gca effects. Properties such as connectedness and efficiency of mating designs embedded in environment designs are being examined. The paper also examines the universal optimality of some designs in a class of designs. An illustration of the numerical procedure is also presented.

Measures of association between disease and genotype.

This paper is concerned with the statistical aspects of the phenomenon of disease occurring more frequently in individuals with some genotypes than in individuals with others. A correlation coefficient is defined to quantify association between disease and genotype. A distinction is made between the concepts of independence of allele and disease and independence of genotype and disease. This distinction is used to define two components of association which describe separate aspects of association of disease with genotype. One component is a measure of the association of disease with allele; the other a measure of the effect of allele interaction on association of disease and genotype. One aspect of the usefulness of the partition into components which is discussed is in expressing the recurrence risk of disease for a relative of an affected individual. A chi-squared analysis is provided to test hypotheses about the components of association and other hypotheses of genetic interest. This analysis is illustrated using a study done to determine the effect of the sex-linked dwarfing gene in male chickens on resistance to E. coli infection. This analysis shows a significant allele interaction effect on resistance to disease but no association of disease with alleles. In conclusion, some extensions and limitations of the proposed concepts and procedures are discussed.

The Efficiency of Blocking: How to Use MS(Blocks)/MS(Error) Correctly

Abstract Even though there are no valid tests of block effects in randomized complete block and Latin square experiments, it is noted that a commonly used measure of efficiency is monotonically related to the F ratios used inappropriately for testing the effectiveness of blocking. Because of this relationship one can give beginning students a useful interpretation of otherwise inappropriate F statistics without introducing concepts of relative efficiency.

Missing Values in Partial Diallel Cross Experiments

Formulae are presented for estimating one or two missing values in partial diallel crosses types I and II treating the complete diallel crosses as special cases with explicit formulae given for the complete diallel cross type II. The estimation procedures are illustrated by examples. The implication of estimating missing values for the analysis of variance is discussed. The case of several missing values is mentioned briefly.

A new series of EGD-PBIB designs☆

Abstract A method is proposed for constructing EGD-PBIB designs with t = Πi = 1n ti treatments and blocks of size ti provided that ti≠max(t1,t2,…,tn). The method of construction allows for a rather large number of possibilities for the remaining parameters of the design. Application to balanced factorial experiments is discussed.

Partial tetra-allel crosses.

SummaryThe construction of partical tetra-allel crosses (PTAC) is considered using BIB and PBIB designs with blocks of size four. It is shown how this can lead to certain types of balanced designs. An explicit procedure is given for constructing circulant PTACs. The analysis of PTACs — estimation of general effects of the lines involved and analysis of variance — is illustrated in terms of an example.ZusammenfassungIn dieser Arbeit wird die Konstruktion von unvollständigen Tetra-Allelen (PTAC) mit Hilfe von unvollständigen Blockversuchsplänen (BIB- und PBIB-Plänen) betrachtet. Es wird gezeigt, wie dies zu gewissen ausgewogenen Plänen führen kann. Eine Methode zur Konstruktion von zirkularen PTACs wird explizit angegeben. Die Analyse von PTACs, d.h. das Schätzen von allgemeinen Effekten der betrachteten Linien und die Varianzanalyse, wird an einem Beispiel erläutert.