This paper deals with two-factor experiments with split units. The whole plot treatments occur in a repeated Latin square, modified Latin square or Youden square, while subplot treatments occur in a block design within the whole plots. The statistical properties of the considered designs are examined. Special attention is paid to the case where one of the treatments is an individual control or an individual standard treatment. In addition, we give a brief overview of work on the design of experiments using the considered designs, as well as possible arrangements of controls in the experiments.
We consider a new method of constructing non-orthogonal (incomplete) split-split-plot designs (SSPDs) for three (A, B, C) factor experiments. The final design is generated by some resolvable incomplete block design (for the factor A) and by square lattice designs for factors B and C using a modified Kronecker product of those designs (incidence matrices). Statistical properties of the constructed designs are investigated under a randomized-derived linear model. This model is strictly connected with a four-step randomization of units (blocks, whole plots, subplots, sub-subplots inside each block). The final SSPD has orthogonal block structure (OBS) and satisfies the general balance (GB) property. The statistical analysis of experiments performed in the SSPD is based on the analysis of variance often used for multistratum experiments. We characterize the SSPD with respect to the stratum efficiency factors for the basic estimable treatment contrasts. The structures of the vectors defining treatment contrasts are also given.
This paper deals with the problems of selection in the early stages of a breeding program. During the improvement process, it is not possible to use an experimental design that satisfies the requirement of replicating all the treatments, because of the large number of genotypes involved, the small amount of seed and the low availability of resources. Hence unreplicated designs are used. To control the real or potential heterogeneity of experimental units, control (check) plots are arranged in the trial. There are many methods of using the information resulting from check plots. All of the usually applied adjusting methods for unreplicated experiments are appropriate for some specific structure of soil fertility. Their disadvantage is the fact that, before and also after the experiment, we usually do not know what a kind of soil structure is present in the experiment. Hence we cannot say which of the existing methods is appropriate for a given experimental situation. The method of inference presented below avoids this disadvantage. It is always appropriate, because of the fact that a trend of soil variability is identified and estimated. In the paper the main tool used to explore this information will be based on a response surface methodology. To begin with we will try to identify a response surface characterizing the experimental environments. We assume that observed yield (or another trait) results directly from two components, one of them due to soil fertility and the other due to the genotype effect. This means that difference between observed yield and forecast can be treated as the estimate of a genotype effect. The obtained response surface will then be used to adjust the observations for genotypes. Finally, the data so adjusted are used for inferences concerning the next stage of the breeding program. The theoretical considerations are illustrated with an example involving yields of spring barley.
We construct an incomplete split-block design (ISBD) by the semi- Kronecker product of two affine α-resolvable designs for row and column treatments. We characterize such ISBDs with respect to the general balance property and we give the stratum efficiency factors for the ISBDs.
We consider an incomplete split-plot design (ISPD) with two factors generated by the semi-Kronecker product of two α-resolvable designs. We use an α-resolvable design for the whole plot treatments and an affine α-resolvable design for the subplot treatments. We characterize the ISPDs with respect to the general balance property, and we give the stratum efficiency factors for the ISPDs.