How do you interpret parameters from logratio analysis of compositional data?

Question

In compositional data analysis as studied by John Aitchison, the analogue of simple linear regression is $$Y_i = \alpha\oplus\beta\odot X_i \oplus \epsilon_i$$ Here, $Y_i = [y_1,...,y_n]$ is the response, $X_i \in \mathbb{R}$ is a covariate, $\alpha, \beta = [\alpha_1,...,\alpha_n], [\beta_1,...,\beta_n]$ are the parameters, and $\epsilon = [\epsilon_1,...,\epsilon_n]$ is a random component.

Here, $\oplus$ and $\odot$ mean pertubation and powering.

Question: What is the interpretation of the parameters in this model? If I estimate $\alpha$ and $\beta$ using log-ratio analysis, how do I interpret their meaning and communicate that to laymen?

Answer 1

I believe that this is really a terrific question although somewhat vague. Here is my attempt at an answer:

In practice it depends on which space you want to interpret the data in. (So what do I mean by this?)

Perturbation and Powering reduce to standard Addition and Multiplication upon Log-ratio transformation. Thus lets say that you have transformed the data using the ALR transform $(X_i^{alr} = X_i/X_D)$. Then your simple linear regression can be interpreted as you would normally interpret a linear model but now your variables are log-ratios.

Now a very interesting question is What is perturbation and powering when you are not thinking in terms of log-ratios. The best answer I have found to this question comes from this paper. In it the authors describe how perturbation can be seen as the updating of prior information to posterior information in a Bayesian setting. They also describe how distance in the Aitchison geometry is a measure of something called evidence-information which is slightly different than Shannon Information. They also discuss powering in the paper but I find it easiest to view powering as repeated perturbation (that makes the most intuitive sense to me).

Overall, I think the first option (describing the linear models in the transformed space) is by far the easiest way to communicate it to laymen. More specifically, choose your log-ratio transform of interest and find log-ratios that have meaning, then describe a standard linear model acting upon those log-ratios.