Stanisław Mejza

Incomplete split plot designs

We propose an incomplete split plot design where levels of one factor (say A) are applied to the wholeplots and levels of the other (say B) to subplots, and where the number of subplots in each wholeplot may be less than the number of levels of factor B. The t levels of factor A are arranged in a completely randomized design. The s levels of factor B are arranged in a connected and proper incomplete block design within each level of factor A, by considering the wholeplots as blocks.

On some properties of the dual of a totally balanced block design

Abstract In this paper it is shown that the dual of a totally balanced block design with t = b, is also a totally balanced block design. It is shown that Fishers inequality b≧t for BIB designs, holds also for a totally balanced block design.

A comparison between joint regression analysis and the AMMI model: a case study with barley

Joint regression analysis (JRA) and additive main effects and multiplicative interaction (AMMI) models are compared in order to (i) access the ability of describing a genotype by environment interaction effects and (ii) evaluate the agreement between the winners of mega-environments obtained from the AMMI analysis and the genotypes in the upper contour of the JRA. An iterative algorithm is used to obtain the environmental indexes for JRA, and standard multiple comparison procedures are adapted for genotype comparison and selection. This study includes three data sets from a spring barley (Hordeum vulgare L.) breeding programme carried out between 2004 and 2006 in Czech Republic. The results from both techniques are integrated in order to advise plant breeders, farmers and agronomists for better genotype selection and prediction for new years and/or new environments.

On the Notion of Efficiency of a Block Design

A general definition of an orthogonal block design and, subsequently, of the efficiency of a block design are given. The efficiency of a block design is first considered for an individual estimable contrast of treatment parameters, then as a mean efficiency for all estimable contrasts. It appears that the two most common definitions of efficiency, one relating the precision of a block design to that of an equireplicate orthogonal block design with the same total number of plots, the other relating the precision to that of an orthogonal block design with the same numbers of treatment replications, are particular cases of the hither introduced general definition of efficiency.

A note on the inequality b≥t for block designs

Abstract A generalization of Fishers inequality b≥t for block designs, and some related results, are presented.

An incomplete split-block design generated by GDPBIBD(2)s

This paper deals with two-factorial experiments laid out in incomplete split-block designs. We propose an incomplete split-block design that is obtained as the Kronecker product of two designs, one for row treatments and the second one for column treatments. For such a design we investigate statistical properties such as general balance and efficiency balance. Especially, we investigate the above properties when the design is the Kronecker product of two GDPBIBD(2)s. The so-obtained design is termed incomplete split-block design generated by GDPBIBD(2)s.

Optimality and constructions of incomplete split-block designs

A sufficient condition for an incomplete split-block design to be universally optimal is given. Optimal properties are examined under two linear models, i.e., with interaction effects and without these effects. Furthermore, some methods of constructing universally optimal incomplete split-block designs are presented.

Incomplete Split-Plot Designs Generatd By GDPBIBD(2)

The paper deals with split-plot types of experiment in which some kind of incompleteness can be accepted. In particular, the considered designs can be incomplete with regard to the wboleplot treatments or with regard to the subplot treatments. In such a case the incomplete treatments are arranged in a Gtoup Divisible Partially Balanced Incomplete Block Design with Two Associate Classes (GDPBIBD{2)). Hence, the resulting desian is called incomplete split-plot design generated by GDPBIBD(2). AMS Subject Classification: 62K10, 62K15.

Analyzing genotype-by-environment interaction using curvilinear regression

In the context of multi-environment trials, where a series of experiments is conducted across different environmental conditions, the analysis of the structure of genotype-by-environment interaction is an important topic. This paper presents a generalization of the joint regression analysis for the cases where the response (e.g. yield) is not linear across environments and can be written as a second (or higher) order polynomial or another non-linear function. After identifying the common form regression function for all genotypes, we propose a selection procedure based on the adaptation of two tests: (i) a test for parallelism of regression curves; and (ii) a test of coincidence for those regressions. When the hypothesis of parallelism is rejected, subgroups of genotypes where the responses are parallel (or coincident) should be identified. The use of the Scheffe multiple comparison method for regression coefficients in second-order polynomials allows to group the genotypes in two types of groups: one with upward-facing concavity (i.e. potential yield growth), and the other with downward-facing concavity (i.e. the yield approaches saturation). Theoretical results for genotype comparison and genotype selection are illustrated with an example of yield from a non-orthogonal series of experiments with winter rye (Secalecereale L.). We have deleted 10 % of that data at random to show that our meteorology is fully applicable to incomplete data sets, often observed in multi-environment trials.

Incomplete Split-Plot Designs Supplemented by a Single Control

The article deals with the constructing methods for experiments carried out in an incomplete split-plot design supplemented by an additional treatment, called a single control. The control treatment has been treated usually as one specific factor level while not necessarily. The control cannot be connected with treatment combinations in an experiment. This distinguishes this article from others in the area considered. The proposed supplementation of whole incomplete split-plot designs leads to the designs with generally accepted methodological requirements, especially randomization. Moreover, we propose a few methods for constructing considered types of the designs with desirable statistical properties such as general balance and efficiency balance of the design with respect to treatment contrasts.