Dirichlet regression - understand missing slope parameter

Modeling
1

I see that in brms the dirichlet regression is formulated so:

parameters {
  vector[Kc_muy2] b_muy2;  // population-level effects
  real Intercept_muy2;  // temporary intercept for centered predictors
  vector[Kc_muy3] b_muy3;  // population-level effects
  real Intercept_muy3;  // temporary intercept for centered predictors
  real<lower=0> phi;  // precision parameter
}
model {
  // initialize linear predictor term
  vector[N] muy2 = Intercept_muy2 + Xc_muy2 * b_muy2;
  // initialize linear predictor term
  vector[N] muy3 = Intercept_muy3 + Xc_muy3 * b_muy3;
  // linear predictor matrix
  vector[ncat] mu[N];
  for (n in 1:N) {
    mu[n] = [0, muy2[n], muy3[n]]';
  }
  // priors including all constants
  target += student_t_lpdf(Intercept_muy2 | 3, 0, 10);
  target += student_t_lpdf(Intercept_muy3 | 3, 0, 10);
  target += gamma_lpdf(phi | 0.01, 0.01);
  // likelihood including all constants
  if (!prior_only) {
    for (n in 1:N) {
      target += dirichlet_logit_lpdf(Y[n] | mu[n], phi);
    }
  }
}

I guess both intercept and slope are 0 with no uncertainty.

  • Can we have information about the distribution of the first component?
  • Can we use the other components to estimate uncertainty of the first?

This because I would like to know the intercept of which components overlap (to a certain degree)

  • Last question, if we constrain with a sum-to-zero (intercept and slope) would it be a fundamentally different model, with different interpretation of the parameters?

Thanks a lot!

2

From muy2, muy3 imaginary muy1 has been already subtracted.

muy2 = (Intercept_muy2_star - Intercept_muy1_star) + (Xc_muy2_star - Xc_muy1_star) * b_muy2

with (Intercept_muy2_star - Intercept_muy1_star) = Intercept_muy2;
and (Xc_muy2_star - Xc_muy1_star) = Xc_muy2.

and muy3 analog and muy1 = 0

I think so, we have 2 equations and 2 unknowns.

The estimated parameters are different in sum-to-zero case, because
muy1 = - Intercept_muy2 - Intercept_muy3 - Xc_muy3 * b_muy3 - Xc_muy2 * b_muy2

It’s a matter of the hyper-priors of the estimated parameters.

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3

Here Xc is the design matrix, and b the coefficients.

Don’t we have 5 equations and 6 unknowns (the star parameters)?

  • (Intercept_muy2 + Xc_muy2 * b_muy2) + (Intercept_muy3 + Xc_muy3 * b_muy3) = 1

  • (Intercept_muy2_star - Intercept_muy1_star) = Intercept_muy2

  • (b_muy2_star - b_muy1_star) = b_muy2

  • (Intercept_muy3_star - Intercept_muy1_star) = Intercept_muy3

  • (b_muy3_star - b_muy1_star) = b_muy3

(In a sense I would expect that the uncertainty of the four initial parameters would be reduced because “spread” among 6.)

Interesting comment, could you elaborate?

4

We are comparing muy2 with muy1. (muy3 analog) see dummy coding in:
https://stats.idre.ucla.edu/r/library/r-library-contrast-coding-systems-for-categorical-variables/#reverse
and deviation coding refering to the sum-to-zero constraint.

We are estimating something like:
(Intercept_muy2_star - Intercept_muy1_star) = Intercept_muy2 ~ student_t(3, 0, 10)

If the hyper-prior is informative, then different coding-schemes results in different posterior distributions.

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