# 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

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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 '11 at 21:02
thanks. I am reading it. Best. Pegah –  Pegah Jan 23 '11 at 21:15
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/…. –  caracal Jan 25 '11 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"

There probably is no "simple" way to interpret a 3-way (or higher) ANOVA, but here's how I do it. I think of it like a contingency table (or "contingency cube" in 3-way case). You have a factor for the row, a factor for the column, and a factor for the "depth". As you move around the cells in the table, the mean changes and we can observe patterns in how the mean varies as we move around the table in a "structured way".

If the factors are "independent" in their association with the response, then this indicates that the mean value only depends on which row, column, and depth we are in. For instance, if you change the row (but leave the column and depth fixed) then the effect of the column is unchanged, and the effect of the depth is unchanged.

If all factors are present, then it matters which particular cell the observation is in (i.e. each cell has a unique mean under the model). For instance, if you change the row (but leave the column and depth fixed) then the effect of the column always changes, and the effect of the depth always changes.

For the "in-between" interpretation where 2-way interactions are present but 3-way interaction are not, it is a bit more tricky to interpret. This goes along the lines of if you change the row (but leave the column and depth fixed) then the effect column-by-row always changes, and the effect of the depth-by-row always changes, but the effect of the depth-by-column remains the same.

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.

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