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I am running experiments for a paper and I am looking for an interesting book / website to understand properly how ANOVA and ANCOVA work. I have a good math background so I don't necessarily need a vulgarized explanation.
I'd also like to know how to determine when to use ANOVA instead of ANCOVA.
The classics I think are Winer and Kirk, both cover essentially only ANOVA and ANCOVA. You can probably get used copys for cheap (e.g., I own a Winer second edition from 71 bought via AMAZON for less than 10$):
Winer - Statistical Principles In Experimental Design
Kirk - Experimental Design
A more contemporary book is the one by Maxwell & Delaney. Besides ANOVA and ANCOVA it covers other methods, e.g., multivariate and multilevel:
Maxwell & Delaney - Designing Experiments and Analyzing Data: A Model Comparison Perspective
Perhaps it is the best to go with this last one. It is pretty good.
So, in addition to this paper, Misunderstanding Analysis of Covariance, which enumerates common pitfalls when using ANCOVA, I would recommend starting with:
This is mostly R-oriented material, but I feel you might better catch the idea if you start playing a little bit with these models on toy examples or real datasets (and R is great for that).
As for a good book, I would recommend Design and Analysis of Experiments by Montgomery (now in its 7th ed.); ANCOVA is described in chapter 15. Plane Answers to Complex Questions by Christensen is an excellent book on the theory of linear model (ANCOVA in chapter 9); it assumes a good mathematical background. Any biostatistical textbook should cover both topics, but I like Biostatistical Analysis by Zar (ANCOVA in chapter 12), mainly because this was one of my first textbook.
And finally, H. Baayen's textbook is very complete, Practical Data Analysis for the Language Sciences with R. Although it focus on linguistic data, it includes a very comprehensive treatment of the Linear Model and mixed-effects models.
Applied Linear Statistical Models by Neter, Kutner, Wasserman, and Nachtscheim, has a very exhaustive (and exhausting!) treatment of ANOVA and ANCOVA.
It also covers power analysis, linear regression, multilinear regression, and introduces some MANOVA. It's a very long text, but does a very thorough job. I've linked you to the fourth edition. I doubt there's a huge difference over the fifth edition, and it's substantially cheaper.
Gelman has a good discussion paper on ANOVA Analysis of variance—why it is more important than ever
In my line of work, I've found this one to be quite useful: Statistical Methods for Psychology (Howell, 2009)
The R book does a good job on that. You can see that it dedicates one chapter to each one of those methods (11 and 12). If you are new to R, this is a great book to start with.
