# Looking for mathematical account of ANOVA

I am looking for a freely available, mathematical description of the standard analysis of variance (several factors, one dependent variable). It should be self-contained and be readable by a person without any knowledge of statistics but a good background in mathematics and probability theory.

Almost all texts I find are either step-by-step instructions on example data-sets, heuristic outlines written for scientists from applied areas or they use a lot of statistical lingo to explain the concept (with only crippled math).

Ideally, the text would include a derivation of the F-distribution (why does it emerge in this setting?) and the F-test as well as the relation to regression.

• ANOVA viewed through the lens of linear regression theory is, to me, the most appealing and unified method of learning and understanding the topic. A mathematician of most any stripe will feel quite comfortable thinking in terms of linear algebra and, in particular, projections. The $F$-test can then be understood both in terms of ratios of sums of squares and as a monotone transformation of a likelihood-ratio test, which give two complementary viewpoints. For one such text, Seber & Lee (2003), Linear Regression Analysis, 2nd edition, Wiley is a choice I can recommend, though it's not free. – cardinal Nov 3 '11 at 20:30
• @cardinal Another good one is Plane Answers to Complex Questions, by Christensen (ISBN 978-0387953618). – chl Nov 3 '11 at 22:52
• @chl: I have heard of this book, but am not familiar with it. Judging by the table of contents, though, it seems to be a good option. I might even consider adding it to my library. Thanks for mentioning it. – cardinal Nov 3 '11 at 23:16

Everything you ask for is beautifully accomplished by Jack Kiefer in his classic Introduction to Statistical Inference (Springer-Verlag 1987). ANOVA is introduced as a special case of the General Linear Model (i.e., regression) in chapter 5, then taken up again at the end of chapter 8 as an application of normal-theory tests. Chapter 8 begins with a statement of the distributions associated with the Normal--t, F, and chi-squared along with their noncentral versions--and shows from first principles how they arise in each setting. General background on estimation is developed in chapters 1-4.

Because this is not a math text, no effort is made to derive explicit formulations of the PDFs of these distributions: the focus is on specifying certain optimal properties and additional criteria (such as invariance, unbiasedness, or linearity) and then deriving, from first principles, the tests that satisfy them and critically evaluating the characteristics of those tests. The math is reasonably rigorous and all statistical terminology is developed ab initio and clearly defined in mathematical terms.

I'm not sure what you mean by "freely available," but this book remains in print, is not expensive, and used copies can be had for very little.

• thanks for the suggestion, I will check it out. I was hopen for an online-text, though :-) – thias Nov 4 '11 at 11:18

This paper of Terry Speed may interest you.

Special invited paper: What is an Analysis of Variance?
T. P. Speed Annals of Statistics Vol 15 No 3 (1987)

Available from project Euclid with commentary from some big names including Tukey (who says he's not equipped to comment adequately on the mathematical niceties and careful craftsmanship of the paper.)

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&page=toc&handle=euclid.aos/1176350470