# Why is compositional data not analyzed using anova?

Compositional data looks like this: $[p_1, ...., p_n]$ where $\sum p_i = 1$.

My question is, I know that we can analyze this using log-ratio analysis but ...

Why not just use anova? Take each of the $n$ components, and use them as factors. So each data row is turned into $n$ rows, with response $p_1, ...., p_n$, and the actual component is then a factor.

For example, if we have % of wage spent on milk, bread, and butter, then instead of treating [20%, 10%, 70 %] as a response, we can treat 20% as a "percent money spent on item", where "item" is the factor, here milk.

Then if the mean for milk is higher or lower, it tells us how much we spend on it compared to other items.

• Considering the case $n=2$ makes it clear something's wrong compared to the usual ANOVA situation: your data would be perfectly redundant.
– whuber
Mar 28 '17 at 17:01
• You have the following reference idescat.cat/sort/sort392/39.2.4.martin-etal.pdf were they show how to perform Manova analysis for compositional data. In essence, it is a Manova analysis using a well defined distance on the Simplex. Mar 31 '17 at 12:36

The problem is that the original proportional data exists in the simplex which is a constrained non-orthogonal geometric space. Standard anova models do not do well with such constrained data. For example, the resulting model would give non-zero probability mass to a datapoint where $\sum p_i > 1$ or where $p_i < 0$. This does not make any sense and can lead to some nasty and spurious conclusions if used.