# How to interpret a three-way ANOVA?

I have used a three-way ANOVA to analyze the the effect of genes, transcription factors, and different conditions on the gene expression. Now I have 9 elements, SSa, SSb, SSc, SSab, SSbc, SSac, SSabc, SSe. How can I interpret these values? I appreciate if you introduce me some resources, or share your own experience with interpreting 3-way ANOVA. Best

• I think you will get an answer soon, but maybe you could add some background information, i.e. what are you trying to show (or equivalently, what is/are your null hypothes{i|e}s), what is the sample size, etc. In the meantime, I think @caracal's brilliant response to this question What is the NULL hypothesis for interaction in a two-way ANOVA? might be helpful.
– chl
Jan 23, 2011 at 21:02
• I suggest Maxwell & Delaney, 2004, Designing Experiments and Analyzing Data as a thorough introduction to ANOVA. Chapter 8 discusses the 3-factor case, especially the different types of interaction: books.google.de/…. Jan 25, 2011 at 14:23

Just a (not so) brief note on the question as posed, before I give a way to intepret, you must be careful about which sums of squares you have calculated, for they relate to different kinds of null hypothesis. For example, the term $SSa$ may be for just the effect $a$ assuming everything else is in the model, or it may be any effect including $a$, which is $a+ab+ac+abc$, or some other null hypothesis which refers to effect $a$. The standard literature refers to four different Types of sums of squares, corresponding to different ways of analysing the data (e.g. total contribution, sequential contribution, marginal contribution). You should investigate whatever software you are using to see what sums of squares it gives you. If you are using R it is most likely sequential which means basically "in the order specified when you wrote the model"
So like I said, probably not "simple", but this is how I would interpret interactions in a 3-way ANOVA. And finally, the $SSe$ term is simply the variation which could not be explained by the full 3-way model (residual or error variance). The sums of squares are devices to make testing a "null hypothesis" more straight forward. But in multi-way ANOVA, there are many null hypothesis one can test, and so no "automatic" procedure know what the user will want in their particular case, so they give a default. It is important to understand what the default sums of squares can be used to test, and then if any of those tests are the one you want to do.