# Is it possible to perform a three-way ANOVA on non-normal data?

I am not very experienced in statistics so I'm not sure how to tackle this data. I have run an experiment where we recorded the proportion of time some beetles spent in 4 different habitats.

Response variable = proportion of time in a habitat (P1, P2, P3 and P4)
Explanetory variables = Species of beetle, size of beetle, and time of day (morning, midday and afternoon)

The idea was do an anova using the following equations.

P1 ~ species * size * time * speceis:size * species:time * size:time


then repeat for the other proportions.

Of course, for this you need normally distributed data and homogeneous variances, though I also know this can be an issue for proportions due to the bounds at 0 and 1

The problem I have is that the data is not homogeneous. If I have to use a non-parametric version, I would be happy but I can't think of how that would work. Does anyone have any advice on where to go from here. I've put my proportion histograms by species below.

• What follows "of course" is incorrect: ANOVA makes no assumptions about the (marginal) data distribution. It's good to have conditional distributions that aren't terribly skewed, and even that can be overcome with sufficient amounts of data because what matters are the distributions of the test statistics (sums of squares and so on). But, since you're analyzing proportions, you might consider approaches appropriate to them, such as GLMs.
– whuber
Commented Apr 24 at 20:09
• 1. With a single continuous proportion in (0,1) you might consider a beta regression as one possibility. 2. When your response is a set of 4 continuous proportions that add to 1, you might be better to consider some form of GLM-like model with a Dirichlet-distributed response (a single multivariate model). 3. On the other hand if exact 0's or 1's are possible (e.g. if one or more values among of the 4 sets of times might be exactly 0, a 0-inflated or 0-1 inflated model might be needed). Commented Apr 26 at 2:58
• Would GLMs not require normal data? Commented Apr 27 at 17:18
• The point (well, one point ...) is that they can assume some other (conditional) distribution than normal ... Commented Apr 30 at 21:11
• Look into Dirichlet regression. Commented Apr 30 at 21:29

Assuming the proportions $$P_1+P_2+P_3+P_4$$ sums to one, they are dependent and should be modeled together, not one by one. Such variables summing to one is called , peruse that tag! you will find useful advice. There are special models for compositional data that you can try, you can start with