How does one Interpret a Dirichlet regression output? I have been exploring the Dirichlet regression in R to analyze my ecological dataset and in the summary output is have several coefficient estimates. How do I interpret this output?


In his paper, Marco Maier (the author of the package) explains the parametrization and interpretation of the coefficients. There are two implemented parametrizations available, the "common" (the default) and the "alternative" one. Using the "common" parametrization, each $\alpha_j$ for each outcome is modelled separately by a (possibly different) linear predictor. The link function is the log link. In this parametrization, the mean of each variable (i.e. the proportion) is $\operatorname{E}(y_j) = \alpha_j/\alpha_0$, where $\alpha_0 = \sum_{j=1}^{J}\alpha_j$ and $J$ is the number of variables ($3$ in your case).
For your variable $\texttt{SFA}$, the fitted model is:
$$
\log(\hat{\alpha}_{1})=5.22 + 2.30\times\texttt{dayD18} -0.40\times\texttt{dayD25} + 0.0012\times\texttt{Pconc} - 0.0051\times\texttt{dayD18}\times\texttt{Pconc} - 0.00033\times\texttt{dayD25}\times\texttt{Pconc}
$$
The coefficients are interpreted in the "usual" way as in any regression model. For example, the equation for the trajectory for the reference category of $\texttt{day}$ is $\log(\hat{\alpha}_{1})=5.22 + 0.0012\times\texttt{Pconc}$. For the category $\texttt{dayD18}$ it is $\log(\hat{\alpha}_{1})=(5.22 + 2.30) + (0.0012 - 0.0051)\times\texttt{Pconc} = 7.52 + 0.00073\times\texttt{Pconc}$ and for the category $\texttt{dayD25}$ it is $\log(\hat{\alpha}_{1})=(5.22 - 0.40) + (0.0012 - 0.00033)\times\texttt{Pconc} = 4.80 + 0.00091\times\texttt{Pconc}$. Exponentiating the resulting number will give the predicted $\alpha$ for any specific combination of covariate values.
Using the function predict, you can get the estimated proportions for a specific set of covariate values. Personally, I would use predict to create plots of the predicted proportion across reasonable values of $\texttt{Pconc}$ for each variable. Specifically, there would be three lines for each outcome, one for each level/category of $\texttt{day}$.
