Comparison of Dirichlet and Multivariate Beta Regression

#1

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 .

π‘€βˆΌπ·π‘–π‘Ÿπ‘–π‘β„Žπ‘™π‘’π‘‘([1,π›½π‘Žβˆ—π‘‘π‘,π›½π‘βˆ—π‘‘π‘])M∼Dirichlet([1,Ξ²aβˆ—tb,Ξ²bβˆ—tb])

or

𝑀1βˆΌπ΅π‘’π‘‘π‘Ž(1,π›½π‘Žβˆ—π‘‘π‘)M1∼Beta(1,Ξ²aβˆ—tb)

𝑀2βˆΌπ΅π‘’π‘‘π‘Ž(1,π›½π‘βˆ—π‘‘π‘)M2∼Beta(1,Ξ²bβˆ—tb)

𝑀3βˆΌπ΅π‘’π‘‘π‘Ž(1,π›½π‘βˆ—π‘‘π‘)M3∼Beta(1,Ξ²cβˆ—tb)

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 ?

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#2

Can you share your model?

I can’t tell much, but you realise that the prior of a Dirichlet is tricky for regression? You can reparametrise

~ dirichlet(precision * softmax([rates_of_abundance]))

rates of abundance should have N - 1 degrees of freedom, and can be the result of your linear equation (in that case the thing that have N - 1 degrees of freedom is the N intercept terms).

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