The notion of general balance due to Nelder is discussed in relation to the eigenvectors of an information matrix, combinatorial balance and the simple combinability of information from uncorrelated sources in an experiment.

/pdf/balance-in-designed-experiments-with-orthogonal-block-20u8mv8011.pdf

Balance in Designed Experiments with Orthogonal Block Structure

The earliest important papers on planned experiments in Biometrika appeared within a year of each other in 1917 and 1918. Roughly speaking, one was concerned with design optimality and industrial experiments, the other with agricultural trials and blocking. We find this approximate division of the subject into two parts helpful in describing its development and growth. As a result of the hostility between Fisher and Karl Pearson, much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of Biometrika. Despite this, we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and Bayesian versions of design optimality.

One hundred years of the design of experiments on and off the pages of Biometrika

An algorithm is proposed for deriving optimal connected block designs. The method employed is to improve a given design by interchanging treatments between blocks, the treatment replications being kept fixed. Examples illustrating the use of the algorithm are given and its performance is discussed.

An Algorithm for Deriving Optimal Block Designs

https://rss.onlinelibrary.wiley.com/doi/pdf/10.2307/2982800

The Non-Orthogonal Design of Experiments

This text is essentially an application of principal component analysis (PCA) and partial least squares (PLS) to multivariate (X0X matrix full rank) and megavariate (X0X not full rank) problems in the physical sciences. The software used is SIMCA-P and MODDE from Umetrics. Also included is a CD with all of the data  les used as examples in the text. Chapter 0 is a Preface that includes no mention of the software/version used. I guess because SIMCA-P and MODDE are mentioned as registered trademarks on the cover page, the reader should know this fact. However, I had SIMCA-P, version 8 on my computer, but could not read any of the data  les until I upgraded to version 9. Part I comprises four introductory chapters on the concepts and principles of projections along with discussions and derivations of PCA and PLS. Graphs are used extensively throughout the discussion, and overall this is an excellent introduction to PCA and PLS. In addition, examples are used to explain the output/terminology from SIMCA-P. Part II, Chapters 5–8, covers the following applications: multivariate characterization (mixture of qualitative and quantitative variables, with the goal of quantifying discrete changes in the qualitative variables), multivariate calibration, multivariate process modeling and monitoring, multivariate classi cation, and discriminant analysis). Part III, Chapters 9–11, discusses preprocessing of the data with respect to transformation, scaling, and signal correction and compression. Part IV, Chapters 12–15, discusses multivariate statistical process control charts for batch and continuous data, multivariate time series, and design of experiments. Part V, Chapters 16–20, covers QSAR modeling, combinatorial chemistry, bioinformatics, and cheminformatics. Finally, Part VI, Chapters 21– 23, discusses nonlinear PLS modeling, analysis of preference (sensory) data, and validation. I liked a number of features of this book, including (1) illustration of the concepts using real data, (2) clear and straightforward discussion of the different application types, (3) thorough discussion/explanation of the SIMCA-P and MODDE output, and (4) the broad range of application topics. What I disliked about the book is that after Part I, the text becomes mainly an application guide for SIMCA-P and MODDE. There is no discussion on alternative methods of analysis or, for that matter, when PCA and PLS might not be appropriate. The basic message is that when one has a multivariate problem, do either a PCA analysis or a PLS analysis, or both. The chapters became very predictable—a few pages of discussion of the application, presentation of one or more datasets, and analysis using SIMCA-P.

Linear Regression Analysis

SUMMARY The least squares solution for a block design is considered in some detail for both the intrablock and the interblock cases. It leads to the identification of certain contrasts, here called basic, that are estimated independently and with efficiency factors that are readily determined. Basic contrasts indicate possible partitions of the treatment sum of squares and aid in the computation of the appropriate analysis of variance. Also they help to clarify the relationship between three methods of computing the analysis for the general case of block designs, namely those of (i) Tocher, (ii) Kuiper and Corsten and (iii) Wilkinson.

T. Caliński

Papers

The basic contrasts of an experimental design with special reference to the analysis of data