I am conducting some analysis on my data I found a strange behavior and would greatly appreciate some guidance or suggestions.

I am trying to investigate the effect of a categorical variable ( **cl** ) to three percentages that sum 1 ( **M** ). Naturally, I conducted a *dirichlet regression* on my dataset and a multivariate *beta regression* , but when compared using *loo* the *beta regression* presented a significantly better fit the data than the *dirichlet* .

πβΌπ·ππππβπππ‘([1,π½πβπ‘π,π½πβπ‘π])MβΌDirichlet([1,Ξ²aβtb,Ξ²bβtb])

or

π1βΌπ΅ππ‘π(1,π½πβπ‘π)M1βΌBeta(1,Ξ²aβtb)

π2βΌπ΅ππ‘π(1,π½πβπ‘π)M2βΌBeta(1,Ξ²bβtb)

π3βΌπ΅ππ‘π(1,π½πβπ‘π)M3βΌBeta(1,Ξ²cβtb)

Strangely, the predicted variables sum varies between 50% to 150% which is nonsense. However, the fitted variables sum varies 95% to 105% that is an acceptable error.

Is it fair to compare the models? or due to the natural constraints of a *Dirichlet* model it yields worst fit than a *multivariate beta regression* ?