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Let $\textbf{X} = (X_1, \ldots, X_n)$ be a vector of responses, where $X_i = (p_1, \ldots, p_k)$ is itself a vector of probabilities.

What method does one use to analyze such data? I want the logic/steps/ideas/concepts behind the method outlined in a short summary, so that I know what's going on before I study it in depth.

To be more specific: I am particularly interested in how one would go about developing the analogue of a linear model: we have some covariates for each response, and we may be interested in whether the mean of the probabilities is influenced by some linear function of the covariates, and then we want to find these parameters, test them, do confidence intervals, and predictions.

What's the "general" idea here? One way could be to just straight-up assume that $X_i$ follows a Dirichlet distribution, but I am more interested in the standard "log-ratio analysis" approach, and what the underlying plan is.

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  • $\begingroup$ Hi Margin. Welcome to CV! Can you say whether your outcome variables (your $X_{i}$s, right?) are probabilities based on count data? Or are they continuous between 0 and 1 probability measures? You can edit your question using the "edit" at its lower left to clarify and improve. $\endgroup$ – Alexis Mar 12 '17 at 21:39
  • $\begingroup$ Does $\sum_{j=1}^k p_j =1$ for given $X_i$? Also can we assume that you are having "standard" univariate predictor variables? Or these are compositional data themselves? (By the way, usually the response variables are denoted as $Y$ and $X$ is reserved for the explanatory variables.) $\endgroup$ – usεr11852 Mar 13 '17 at 0:21
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The standard idea is to transform (using a log-ratio transform) the data from the simplex to an unconstrained space where standard linear models can be applied. The Isometric Log-ratio transform is particularly powerful in that it transforms the compositional data to real space where just about any standard linear model can be applied, used, and interpreted.

Note that a multivariate normal model in either Additive Log-ratio space or Isometric Log-ratio space corresponds to a logistic-normal model applied to the data in the simplex.

The Logistic-normal model has a number of advantages over the Dirichlet models. The LN models allow for complex covariance of the variables $(X_1,\dots,X_n)$ beyond the covariance introduced by the sum constraint.

For some reading material I would recommendModeling and Analysis of Compositional Data by Pawlowsky-Glahn, Egozcue, and Tolosana-Delgado. It is a really wonderful book. For a very applied book Analyzing Compositional Data in R by van den Boogaart and Tolosana-Delgado.

Both books have entire sections on linear models for compositional data. You can also seem my answer to this question.

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