Analysis of (co)variance is a core element of statistics particularly in the behavioural sciences and this book aims to provide a clinical guide to the topic bridging the divide between accessible research method books which all too often merely dip their toes into the water and the beefy specialist tome with its reliance on unappealing (to non-mathematicians) mathematical formulae. This book specifically describes the derivation of analysis of (co)variance [an(c)ova] tables used to interpret designs involving random, and fixed, factors. Indeed a more appropriate title for this book might have been something like ‘Mixed Models in Analysis of Variance’. Although covariates are mentioned, a bigger issue strikes me as the different formulations for random and fixed sources of variance rather than whether a source of variation is a factor or a covariate.
A website (http://www.soton.ac.uk/~cpd/anovas/datasets/) is briefly mentioned in the opening pages which features notation used in the book, a useful flow chart for choosing an appropriate design and worked examples of an(c)ovas. This website is useful in giving an idea of the issues, style and notation of the book.
The book is divided into three sections. The first 60 pages illustrate general concepts in constructing an(c)ovas. These concepts include the summing of variance components to form mean squares with a simple rule for constructing these given in a short paragraph in the section from pages 35 to 37. Variance components are important in understanding which source of variation to use as the error sum of squares in forming the F ratio. An(c)ova tables are presented in numeric form for worked case studies. Other issues include framing hypotheses of interest, pooling error terms and illustrating jargon such as cross-factoring, nesting, blocking and, of course, covariate and factor, random and fixed. The reader is also introduced to the notation used to describe each design and its variance components. It is important that the reader understands this terminology and notation as they are used extensively to describe the designs in the subsequent main section.
The main section of 176 pages focuses on particular types of an(c)ova featuring up to three factors. There are 28 designs listed in the index and summarized at the back of the book! These are grouped into seven general design types: one factor, nested, factorial, randomized-block, split-plot, repeated measures and unreplicated factorial with a brief introduction to each. The presentation of each design includes its model notation, the general hypothesis the design is testing, description of the factors in the design and, perhaps most helpfully, the format of the data and descriptions of example datasets which have this design. Each design is compared with other designs mentioned in the book (giving the subsection they are mentioned in) and assumptions underlying the use of each design are discussed. Each design is then further subdivided depending on whether its factors are random or fixed. General forms of an(c)ova tables are given which comprise variance components, which are summed to yield mean squares, described using the algebraic notation of section 1. The emphasis is on the theory behind the use of the model consequently there are no numerical illustrations of any of the designs in this section. It is also, importantly, noted when a design precludes the testing of certain interactions.
The final ‘Further Topics’ section of 52 pages includes brief mentions of balanced designs, contrasts, design comparisons, effect sizes and reporting of methods and results. There is a subsection detailing frequently asked questions on an(c)ovas, a glossary of terms and a comprehensive bibliography.
The three sections are not mutually exclusive, e.g. certain types of an(c)ova are mentioned in section 1 and re-introduced in section 2. One might also argue that since section 3 deals with issues which are of general interest to an(c)ovas they could also have been dealt with in section 1.
This is an authoritatively written book aimed at people who already have a good grasp of analysis of (co)variance using fixed factor an(c)ova, who are not afraid of algebraic notation and who wish to understand the background to the comprehensive range of study designs described which incorporate covariates and random factors.
