# Departures from normality in factorial design

This paper (http://psycnet.apa.org/journals/med/6/4/147/) states that departures from normality can be tolerated for one-way ANOVA.

"The results give strong support for the robustness of the ANOVA under application of non-normally distributed data."

I have a factorial design and some of my data is not normal distributed. Although it is only a minor fraction of the total data set (28 out of 340 samples) I wonder if it is legitimate to proceed with a factorial design ANOVA.

My dependant variable is the relative absorption in an IR range (defined by DRIFT analysis) and my independent variables are timepoint, treatment, exposition, depth. Per sampling condition (example: timepoint=0, treatment=x, exposition=north, depth= 0-5cm) I have 3 replicates.

I have 4 treatment, 3 timepoints, 2 exposition and 2 depth. As some replicates are missing, my design is unbalanced and I used type III Anova (from car package) in R . I assumed a linear model with interactions. (linear Model = Absorption ~ Treatment * Timepoint * Depth * Exposition)

• Are you sure you really need all those interactions? Treatment * Timepoint * Depth * Exposition is 6 2-way interactions, 4 3-way interactions & the 4-way interaction. W/ multiple levels in some factors, that's a lot of df consumed. You will have very little power for those tests unless the effects are massive. Is every one of those interactions theoretically supported & important? – gung Mar 4 '15 at 1:12
• What do you mean that "some of [your] data is not normal distributed"? Do you mean the distributions of some of your cells are not normal? You only have nj<=3 in each cell. How did you conclude that some are not normal? What is "relative absorption"? Is it a ratio of something? – gung Mar 4 '15 at 1:15
• On what basis do you conclude non-normality for some subsets? [Wow, I just read the abstract for that paper. That seems a rather odd way to run a simulation. How can you conclude anything about its behavior on say exponential data if you throw away 90% of perfectly valid exponential samples?]... on Type III SS, you may find Venables' Exegeses on linear models interesting – Glen_b Mar 4 '15 at 7:53
• @ Gung: The type 3 Anova gave me two significant interactions. So no I probably don't need all the interactions and could just assume the interactions Treatment:Timepoint and Treatment:Depth. I just assumed that there could be interactions because theoretically that would be possible. (Or let's say I don't have any prove that there are no interactions possible.) When I do the type 2 Anova the main effects are significant in a similar magnitude. Can I therefore conclude that the interactions have only a minor effect on the main effects? – CuriousIndeed Mar 4 '15 at 18:12
• @ Gung: I meant that 28 out of 340 sampling conditions (each with 3 replicates) are not normal distributed. I tested each of these 340 samples with the Shapiro-Wilk test separately. Should I test them in any other way? In DRIFT analysis I measured the absorption of 12 different peaks. Each of these peaks had been delineated from the crude curve by a tangent line.This procedere yields an area for each peak. The areas of all the peaks has been summed and the rel. Absorption for one peak is defined as one Peak / Sum of all peaks. – CuriousIndeed Mar 4 '15 at 18:20