Author | Anne Penfold Street, Deborah Street |
---|---|
Language | English |
Genre | Mathematics |
Publisher | Oxford University Press |
Publication date | 1987 |
Combinatorics of Experimental Design is a textbook on the design of experiments, a subject that connects applications in statistics to the theory of combinatorial mathematics. It was written by mathematician Anne Penfold Street and her daughter, statistician Deborah Street, and published in 1987 by the Oxford University Press under their Clarendon Press imprint.
The book has 15 chapters. Its introductory chapter covers the history and applications of experimental designs, it has five chapters on balanced incomplete block designs and their existence, and three on Latin squares and mutually orthogonal Latin squares. Other chapters cover resolvable block designs, finite geometry, symmetric and asymmetric factorial designs, and partially balanced incomplete block designs. [1] [2]
After this standard material, the remaining two chapters cover less-standard material. The penultimate chapter covers miscellaneous types of designs including circular block designs, incomplete Latin squares, and serially balanced sequences. The final chapter describes specialized designs for agricultural applications. [1] [2] The coverage of the topics in the book includes examples, clearly written proofs, [3] historical references, [2] and exercises for students. [4]
Although intended as an advanced undergraduate textbook, this book can also be used as a graduate text, or as a reference for researchers. Its main prerequisites are some knowledge of linear algebra and linear models, but some topics touch on abstract algebra and number theory as well. [1] [2] [4]
Although disappointed by the omission of some topics, reviewer D. V. Chopra writes that the book "succeeds remarkably well" in connecting the separate worlds of combinatorics and statistics. [2] And Marshall Hall, reviewing the book, called it "very readable" and "very satisfying". [3]
Other books on the combinatorics of experimental design include Statistical Design and Analysis of Experiments (John, 1971), Constructions and Combinatorial Problems in Design of Experiments (Rao, 1971), Design Theory (Beth, Jungnickel, and Lenz, 1985), and Combinatorial Theory and Statistical Design (Constantine, 1987). Compared to these, Combinatorics of Experimental Design makes the combinatorial aspects of the subjects more accessible to statisticians, and its last two chapters contain material not covered by the other books. [1] However, it omits several other topics that were included in Rao's more comprehensive text. [4]
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.
The design of experiments, also known as experiment design or experimental design, is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associated with experiments in which the design introduces conditions that directly affect the variation, but may also refer to the design of quasi-experiments, in which natural conditions that influence the variation are selected for observation.
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.
Raj Chandra Bose (or Basu) (19 June 1901 – 31 October 1987) was an Indian American mathematician and statistician best known for his work in design theory, finite geometry and the theory of error-correcting codes in which the class of BCH codes is partly named after him. He also invented the notions of partial geometry, association scheme, and strongly regular graph and started a systematic study of difference sets to construct symmetric block designs. He was notable for his work along with S. S. Shrikhande and E. T. Parker in their disproof of the famous conjecture made by Leonhard Euler dated 1782 that for no n do there exist two mutually orthogonal Latin squares of order 4n + 2.
In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as blocks, chosen such that frequency of the elements satisfies certain conditions making the collection of blocks exhibit symmetry (balance). Block designs have applications in many areas, including experimental design, finite geometry, physical chemistry, software testing, cryptography, and algebraic geometry.
In the design of experiments, optimal experimental designs are a class of experimental designs that are optimal with respect to some statistical criterion. The creation of this field of statistics has been credited to Danish statistician Kirstine Smith.
In the statistical theory of the design of experiments, blocking is the arranging of experimental units that are similar to one another in groups (blocks) based on one or more variables. These variables are chosen carefully to minimize the affect of their variability on the observed outcomes. There are different ways that blocking can be implemented, resulting in different confounding effects. However, the different methods share the same purpose: to control variability introduced by specific factors that could influence the outcome of an experiment. The roots of blocking originated from the statistician, Ronald Fisher, following his development of ANOVA.
The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. In algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and the theory of error-correcting codes. In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups.
Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of balance and/or symmetry. These concepts are not made precise so that a wide range of objects can be thought of as being under the same umbrella. At times this might involve the numerical sizes of set intersections as in block designs, while at other times it could involve the spatial arrangement of entries in an array as in sudoku grids.
Fisher's inequality is a necessary condition for the existence of a balanced incomplete block design, that is, a system of subsets that satisfy certain prescribed conditions in combinatorial mathematics. Outlined by Ronald Fisher, a population geneticist and statistician, who was concerned with the design of experiments such as studying the differences among several different varieties of plants, under each of a number of different growing conditions, called blocks.
Algebraic statistics is the use of algebra to advance statistics. Algebra has been useful for experimental design, parameter estimation, and hypothesis testing.
In mathematics, an orthogonal array is a "table" (array) whose entries come from a fixed finite set of symbols, arranged in such a way that there is an integer t so that for every selection of t columns of the table, all ordered t-tuples of the symbols, formed by taking the entries in each row restricted to these columns, appear the same number of times. The number t is called the strength of the orthogonal array. Here are two examples:
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Anne Penfold Street (1932–2016) was one of Australia's leading mathematicians, specialising in combinatorics. She was the third woman to become a mathematics professor in Australia, following Hanna Neumann and Cheryl Praeger. She was the author of several textbooks, and her work on sum-free sets became a standard reference for its subject matter. She helped found several important organizations in combinatorics, developed a researcher network, and supported young students with interest in mathematics.
Combinatorics of Finite Geometries is an undergraduate mathematics textbook on finite geometry by Lynn Batten. It was published by Cambridge University Press in 1986 with a second edition in 1997 (ISBN 0-521-59014-0).
Introduction to the Theory of Error-Correcting Codes is a textbook on error-correcting codes, by Vera Pless. It was published in 1982 by John Wiley & Sons, with a second edition in 1989 and a third in 1998. The Basic Library List Committee of the Mathematical Association of America has rated the book as essential for inclusion in undergraduate mathematics libraries.
Deborah Street is an Australian statistician known for her research in the design of experiments. She is a professor at the University of Technology Sydney, where she is a core member of the Centre for Health Economics Research and Evaluation (CHERE).
Independence Theory in Combinatorics: An Introductory Account with Applications to Graphs and Transversals is an undergraduate-level mathematics textbook on the theory of matroids. It was written by Victor Bryant and Hazel Perfect, and published in 1980 by Chapman & Hall.