Interpretation of the Precision Model in Dirichlet regression

Question

I use dirichlet regression to analyze the result of a membership matrix (compositional data). For each observation, I have membership probabilities to different groups and these probabilities sum to one. For sake of simplicity, let’s say that I have three groups. I use the DirichletReg package in R to estimate the model using the so-called second parameterization. If I got it right, when using this modeling strategy, we estimate the probability to be in one group against the probability to be in a baseline group. In this modeling strategy, we also have a model for the “precision” parameter.

My question is how can I correctly interpret this model for the “precision” parameter. I have the following output for the precision model. I want to be sure to correctly interpret these results.

PRECISION MODEL:
------------------------------------------------------------------
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)   1.3266     0.0259   51.15  < 2e-16 ***
Grammaryes   -0.1304     0.0467   -2.79   0.0053 ** 
gcse5eqyes   -0.2291     0.0380   -6.02  1.7e-09 ***
funempyes     0.0711     0.0490    1.45   0.1466    
------------------------------------------------------------------
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Here are my questions:

• If I understood things correctly, it means that we have some kind of “precision” for each observation. Is that right? How can I interpret this precision?
• Is it correct to say here that having Grammaryes (a dummy) decreases the precision of the memberships, or in other words, that observations with Grammaryes tend to be in between groups (for instance membership of 1/3 for each group)?
• Can I think of this precision as somewhat linked to the concept of entropy of the classification of each individual between the different groups?

For instance, if I have the following individuals.

     Group1           Group2         Group3
[1,]   0.996         0.00103         0.00255
[2,]   0.135         0.59898         0.26567

Observation 1 would have a high precision (all things being equal), because it is well classified (almost entirely in group1), whereas I can expect observation 2 to have a lower precision (because it is more in-between groups)? Is that correct?

Thank you all in advance for your answers!

Answer 1

Q1

The DirichletReg package has a great documentation and on their website you will find a link to a report

Maier, M. J. (2014). DirichletReg: Dirichlet Regression for Compositional Data in R. Research Report Series / Department of Statistics and Mathematics, 125. WU Vienna University of Economics and Business, Vienna. http://epub.wu.ac.at/4077/

which explains the concept.

The precision parameter refers to the shape of the Dirichlet distribution and is the sum of the parameters $\alpha$. The package allows to use an alternative parameterization in which this parameter is itself one of the estimated variables (dropping one of the $\alpha$).

The image below from wikipedia / wikicommons demonstrates the effect of the different $\alpha$ on the shape of the distribution. The shown distributions are for (clockwise, starting from the upper left corner) precision 4 (1.3, 1.3, 1.3), 9 (3,3,3), 21 (7,7,7), 19 (2,6,11), 28 (14, 9, 5), 14 (6,2,6).

Q2,Q3

Your model is not entirely clear but it seems that you predict the distribution as a function of the bacteria groups. So your individuals

        Group1           Group2         Group3
    [1,]   0.996         0.00103         0.00255
    [2,]   0.135         0.59898         0.26567

are predictions of the expectation values of dependent variables.

The model based on the intercept 1.3266, and coefficients for independent variables Grammaryes -0.1304, gcse5eqyes -0.2291, funempyes 0.0711 are for the precision, which is related to the error of the predicted expectation value.

The precision does not refer to one of the means being close to 1 or not. So you could have a low 'precision' in your first row and high 'precision' in the second row.