Small Youden Rectangles, Near Youden Rectangles, and Their Connections to Other Row-Column Designs

In this paper we study Youden rectangles of small orders. We have enumerated all Youden rectangles for all small parameter values, excluding the almost square cases, in a large scale computer search. For small parameter values where no Youden rectangles exist, we also enumerate rectangles where the number of symbols common to two columns is always one of two possible values. We refer to these objects as \emph{near Youden rectangles}. For all our designs we calculate the size of the autotopism group and investigate to which degree a certain transformation can yield other row-column designs, namely double arrays, triple arrays and sesqui arrays. Finally we also investigate certain Latin rectangles with three possible pairwise intersection sizes for the columns and demonstrate that these can give rise to triple and sesqui arrays which cannot be obtained from Youden rectangles, using the transformation mentioned above.Comment: 33 pages, 21 Table

Balanced semi-Latin rectangles : properties, existence and constructions for block size two

There exists a set of designs which form a subclass of semi-Latin rectangles. These designs, besides being semi-Latin rectangles, exhibit an additional property of balance; where no two distinct pairs of symbols (treatments) differ in their concurrences, that is, each pair of distinct treatments concur a constant number of times in the design. Such a design exists for a limited set of parameter combinations. We designate it a balanced semi-Latin rectangle (BSLR) and give some properties, and necessary conditions for its existence. Furthermore, algorithms for constructing the design for experimental situations where there are two treatments in each row-column intersection (block) are also given.Publisher PDFPeer reviewe

Relations among partitions

Combinatorialists often consider a balanced incomplete-block design to consist of a set of points, a set of blocks, and an incidence relation between them which satisfies certain conditions. To a statistician, such a design is a set of experimental units with two partitions, one into blocks and the other into treatments: it is the relation between these two partitions which gives the design its properties. The most common binary relations between partitions that occur in statistics are refinement, orthogonality and balance. When there are more than two partitions, the binary relations may not suffice to give all the properties of the system. I shall survey work in this area, including designs such as double Youden rectangles.PostprintPeer reviewe

Computer construction of experimental plans

Experimental plans identify the treatment allocated to each unit and they are necessary for the supervision of most comparative experiments. Few computer programs have been written for constructing experimental plans but many for analysing data arising from designed experiments. In this thesis the construction of experimental plans is reviewed so as to determine requirements for a computer program. One program, DSIGNX, is described. Four main steps in the construction are identified: declaration, formation of the unrandomized plan (the design), randomization and output. The formation of the design is given most attention. The designs considered are those found to be important in agricultural experimentation and a basic objective is set that the 'proposed' program should construct most designs presented in standard texts (e.g. Cochran and Cox (1957)) together with important designs which have been developed recently. Topics discussed include block designs, factorial designs, orthogonal Latin squares and designs for experiments with non-independent observations. Some topics are discussed in extra detail; these include forming standard designs and selecting defining contrasts in symmetric factorial experiments, general procedures for orthogonal Latin squares and constructing serially balanced designs. Emphasis is placed on design generators, especially the design key and generalized cyclic generators, because of their versatility. These generators are shown to provide solutions to most balanced and partially balanced incomplete block designs and to provide efficient block designs and row and column designs. They are seen to be of fundamental importance in constructing factorial designs. Other versatile generators are described but no attempt is made to include all construction techniques. Methods for deriving one design from another or for combining two or more designs are shown to extend the usefulness of the generators. Optimal design procedures and the evaluation of designs are briefly discussed. Methods of randomization are described including automatic procedures based on defined block structures and some forms of restricted randomization for the levels of specified factors. Many procedures presented in the thesis have been included in a computer program DSIGNX. The facilities provided by the program and the language are described and illustrated by practical examples. Finally, the structure of the program and its method of working are described and simplified versions of the principal algorithms presented

Some 6 x 11 Youden squares and double Youden rectangles

5-cyclic Youden squares of size 6x11 are enumerated and classified into 70 species. In a certain standardised form, there are 249 such Youden squares. Of the 61 752 ordered pairs of these, just 1352 can be used to construct 5-cyclic 6 x 11 double Youden rectangles (DYRs); this gives a 'success rate' of 2.2%. The DYRs fall into 175 species, of which 12 exhibit some symmetry. Hitherto, just a single 5-cyclic 6 x 11 DYR has appeared in print. The results are introduced via the analogous results for 4 x 7 Youden squares and DYRs, and provide encouragement in the search for 2p x (4p - 1)DYRs with p > 3; no such DYR with p > 4 has yet been found; the paper is motivated by the need to explore possible ways of finding some

A holistic evaluation concept for long-term structural health monitoring

[no abstract