On the study of balanced incomplete block designs with repeated blocks and other incomplete block designs

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Nonparametric analysis of unbalanced paired-comparison or ranked data

Suppose we have t objects C[subscript]1,...,C[subscript]t, and that objects C[subscript]i and C[subscript]j are judged pairwise in n[subscript]ij independent comparisons, for i,j = 1,...,t; i ≠ j. In the simplest of such \u27paired-comparison\u27 experiments, all pairs of objects are compared an equal number of times (i.e., all n[subscript]ij = n); much of the paired-comparison literature pertains to the design and analysis of such \u27completely balanced\u27 experiments. Yet it is often inconvenient or impractical to carry out such a design: some pairs of objects might be compared more often than others, and some pairs might not be compared at all. Most of the available methods for analysis of unbalanced paired-comparison data are parametric, in the sense that a (paired-comparison) linear model generates, for each pair of objects, the \u27preference probability\u27 [pi][subscript]ij with which C[subscript]i is preferred to C[subscript]j. The few existing nonparameteric approaches are critically examined. David (1987) proposes a simple method of scoring objects from unbalanced paired-comparison data that takes into account differences in the strength of the competition encountered by each object as well as possible differences in the number of comparisons on each pair of objects. Statistical properties of the proposed scores are developed for the general unstructured case and for special cases of partial balance, such as when objects are arranged in a group divisible design. The asymptotic distribution of these scores leads to several approximate tests of hypotheses, including a test for equality of the objects. Through some numerical examples this proposed method will be compared with the few other nonparametric method designed for unbalanced data. The approach is then extended to unbalanced ranked data. It is shown that the previous nonparametric rank approaches fail to account adequately for the aspects of unbalanced data of concern in this dissertation. Numerical examples of unbalanced ranked data illustrate the comparison between the proposed method and the existing rank methods;Reference. David, H. A. (1987). Ranking from unbalanced paired-comparison data. Biometrika 74, 2, 432-6

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

Constructions of tt-designs from weighing matrices and walk-regular graphs

We provide a method to construct $t$-designs from weighing matrices and walk-regular graphs. One instance of our method can produce a $3$-design from any (symmetric or skew-symmetric) conference matrix, thereby providing a partial answer to a question of Gunderson and Semeraro JCTB 2017. We explore variations of our method on some matrices that satisfy certain combinatorial restrictions. In particular, we show that there exist various infinite families of partially balanced incomplete block designs with block size four on the binary Hamming schemes and the $3$-class association schemes attached to symmetric designs, and regular pairwise balanced designs with block sizes three and four.Comment: 31 page