R-squared for Dirichlet regression (`r2`)

Hi! First of all, thank you for creating and maintaining this package!

I have come across an unexpected behaviour when applying r2() to Dirichlet regression fitted with the DirichletReg package. In short, the Dirichlet regression extends the beta regression to C categories: bounded responses (0, 1) across more than 2 categories. This regression comes in two parametrizations: common (a separate model is fitted to each of the C categories) vs alternative (a separate model is fitted to C-1 categories, and precision is modelled separately). Each model can use a different set of explanatory variables, separated by pipes |.

r2() appears to return Nagelkerke's R2, but the value is very high for models with the alternative parametrization. For instance, a value close to 0.9, when the squared correlation between fitted and observed values is no higher than 0.75 for any category. The value for a model with the common parametrization is more sensible (i.e. in line with the correlations between fitted and observed values). I suspect this has to do with how a null model is declared, based on comparisons with MuMIn::rsquaredLR(). A reproducible example is available in an issue over at the DirichletReg package.

https://github.com/maiermarco/DirichletReg/issues/12

I am not an expert, so perhaps the r2 values actually make sense. The analysis of proportions across categories is quite interesting, and given a recent review (https://besjournals.onlinelibrary.wiley.com/doi/full/10.1111/2041-210X.13234) its popularity might increase in ecology and evolution. If performance can work with such models it would be a very useful extension!

Thank you!

Start with the r2() handling for DirichletReg models and review the reproducible example in DirichletReg issue 12. Compare the common and alternative parametrizations with MuMIn::rsquaredLR(), focusing on how the null model is declared. Done means determining whether the reported values are correct and, if needed, supporting Dirichlet regression with validated R-squared results.