I am well-acquainted to basic topics of experimental designs such as anova, ancova, crd, rbd, lsd, factorial design, robust design etc (as per the book of Dr. D. C. Montgomery). Now as per my recent ongoing research curriculum, I need to learn more mathematically oriented combinatorial design. The topics I need to learn are finite geometry, steiner systems, t-designs, resolvable designs, pairwise balanced design, directed design, pbd, tuscan squares, vector space design, packing, partial geometry, sequences with zero autocorrelation, D-optimal design etc and some other topics based on abstract and linear algebra. I have started with the book by W. D. Wallis but not all topics (such as equidistant permutation array, skolem sequences and many other topics aforementioned) are covered in this book. Also Wikipedia gives a list of references but I am not sure which one to pick beside the book I am using.

Would you kindly like to recommend books on combinatorial design that cover these topics? All suggestions and recommendations are valuable and appreciated.


2 Answers 2


There is a book on the topic which actually have its own page on wikipedia!. Start with that one. amazon link

Also Introduction to Combinatorial Designs (Discrete Mathematics and Its Applications) 2nd Edición by W.D. Wallis, with empasis on the maths, but ending with statistical applications.

  • 1
    $\begingroup$ +1 for the recommendation. I will definitely use it for study. $\endgroup$
    – vbm
    Aug 27, 2022 at 19:13

Disclaimer: These are some of the materials that I came across, some of which I used routinely and others as references. While the resources don't cover all the things OP mentioned, these are worth to study.

$\bullet$ Analysis and Design of Experiments, H. B. Mann, Dover, $1949.$

Author does an exceptional job in keeping the chapters short but detailed enough to not create any void in derivations. The book is not voluminous; though sometimes beating the bush to go to the main results but this is definitely not quaint: a sharp exposition. The highlights are the chapters on latin squares, Galois fields and Orthogonal Latin squares. More or less self-contained.

$\bullet$ Mathematics of Design and Analysis of Experiments, M. C. Chakrabarti, Asia Publishing House, $1962.$

These are collection of lectures by the author. Not recommended for first reading. The materials are terse. I liked his treatment of confounded arrangements in split-plot designs, application of Galois fields and finite geometry in constructing the confounded designs, hypercubes of strength $d.$ But again, I reiterate it is not for first reading.

$\bullet$ Constructions and Combinatorial Problems in Design of Experiments, Damaraju Raghavarao, Wiley, $1971.$

The author indeed has largely been successful in consolidating some of the important topics: construction of orthogonal arrays, products of orthogonal arrays, embeddings, MacNeish-Mann Theorem, duals of incomplete block designs, partial geometries. Essentially a formal treatment of the field: a good read.

$\bullet$ Optimal Design: An Introduction to the Theory for Parameter Estimation, S.D. Silvey, Chapman and Hall, $1980.$

My personal favorite. Silvey did an amazing job in writing such a beautiful treatise. If one knows nothing and wants to have an intuitive idea about information matrices, design criteria, $\mathcal D$ optimality, design algorithms, then without an iota of doubt, do read this classic.

$\bullet$ Optimal Design of Experiments, Friedrich Pukelsheim, SIAM, $2006.$

This is a magnum opus in true sense for it delves deep, I mean way deep, in covering optimality theory. It thoroughly discusses information matrices, optimality criteria, the equivalence theorems, $\cal D,~A, ~E, ~T$ optimality, Bayes designs, invariant designs, Loewner and Kiefer optimality, rotatibility.

$\bullet$ Design and Analysis of Experiments, M.K.Sharma, B. V. S. Sisodia, PHI Learning Pvt. Ltd., $2012 . $

A beginner's book with a copious writing style and a tolerable pace develops the finite field theory, finite geometries, difference sets and their applications in constructing the requisite designs. Exercises are easy to moderate.

$\bullet$ Robust Response Surfaces, Regression, and Positive Data Analyses, R. N. Das, CRC Press, $2014.$

The monograph provides a conspicuous elucidation of weak rotatibility, slope rotatibility, weak slope rotatibility, $\mathcal D$ optimal slope rotatibility, and also brief discussion of correlated structures. Although the main attention and emphasis are on robust response surfaces. It is replete with good applications.

  • 2
    $\begingroup$ +1 for the recommendations. I will definitely study the recommended books. $\endgroup$
    – vbm
    Aug 27, 2022 at 17:20

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