Youden Square with Split Units
In this chapter we present the most important problems connected with the design of experiments using Youden squares with split units. In fact we consider two types of designs. The first is connected with different arrangements of subplot treatments on th
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Abstract
In this chapter we present the most important problems connected with the design of experiments using Youden squares with split units. In fact we consider two types of designs. The first is connected with different arrangements of subplot treatments on the units of Youden squares. The second is connected with the design of experiments when one or more treatments arranged in Youden squares are control or standard treatments. We characterize some of these designs with respect to general balance property and with respect to design efficiency factors.
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Introduction
In performing experiments we quite often use a row–column design in order to eliminate real or potential orthogonal disposed heterogeneity of experimental material. In this case the Latin square is the appropriate design. This design possesses many desirable and optimal statistical properties. In the Latin square every treatment occurs once in each row and once in each column. It means that this design uses many experimental units. We can reduce the number of experimental units by using a design in which every treatment occurs once in each row (and not in each column) or vice versa. Then the Youden square is the proper design, with many
S.F. Mejza () Department of Mathematical and Statistical Methods, Poznan University of Life Sciences, Wojska Polskiego 28, PL-60-637 Pozna´n, Poland e-mail: [email protected] S. Kuriki Department of Mathematical Sciences, Graduate School of Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan e-mail: [email protected] J. Lita da Silva et al. (eds.), Advances in Regression, Survival Analysis, Extreme Values, Markov Processes and Other Statistical Applications, Studies in Theoretical and Applied Statistics, DOI 10.1007/978-3-642-34904-1 1, © Springer-Verlag Berlin Heidelberg 2013
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S.F. Mejza and S. Kuriki
desirable statistical properties (see, e.g., [3]). In the Youden square the treatments occur in completely randomized blocks with respect to rows (columns), while with respect to columns (rows) they occur in a balanced incomplete block design (BIBD). In the experiments considered here the units of a Youden square are subdivided into the same number of subunits. The structure of the experimental material is described formally below. Let us assume that the experimental material is divided into k0 superblocks. Each superblock constitutes a row–column design with k1 rows and k2 columns. On each unit of the row–column design that is treated as a whole plot, the levels of a factor A .A1 ; A2 ; ; Aa / are arranged. These levels will be called whole-plot treatments. Additionally, each whole plot is divided into k3 small plots called subplots; on each subplot the levels of the second factor B .B1 ; B2 ; ; Bb / are arranged. These levels are called subplot treatments. In this chapter we will examine the statistical properties of a design in which each superblock has a Youden square structure with q rows and a columns. It is assumed that a subdesign o
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