Tsagris 2018 - Dirichlet Regression with Zeros - Dirichlet Regression

Summary

Dirichlet regression in mean-precision form: $\mathbf{Y}\sim \text{Dir}(\varphi ,{\mathbf{a}}^{\ast })$ with $\sum a_i^{\ast }=1$ and the multinomial-logit link tying ${\mathbf{a}}^{\ast }$ to covariates. It needs $d(p+1)+1$ parameters instead of the classical $D(p+1)$, and $\varphi$ may also depend on covariates through a log link.

Overview

The Dirichlet is a natural distribution on the simplex $S^d=\{(y_1,\dots ,y_D):y_i\ge 0,\sum y_i=1\}$, $d=D-1$. Earlier Dirichlet regressions: Gueorguieva et al. (2008), Hijazi and Jernigan (2009). Aitchison (2003) instead applies log-ratios and multivariate linear regression.

Main Content

Dirichlet in mean-precision form

$\mathbf{Y}\sim \text{Dir}(\varphi ,{\mathbf{a}}^{\ast })$ has density

$$f(\mathbf{y})=\frac{\Gamma \left({\sum }_{i=1}^D\varphi a_i^{\ast }\right)}{{\prod }_{i=1}^D\Gamma (\varphi a_i^{\ast })}\prod _{i=1}^D{y}_i^{\varphi a_i^{\ast }-1},\sum _i{a}_i^{\ast }=1.$$

${\mathbf{a}}^{\ast }=E(\mathbf{Y})$ is the mean and $\varphi >0$ the precision (the usual Dirichlet parameters are ${\alpha }_i=\varphi a_i^{\ast }$).

Multinomial-logit link (Eq. 1)

With predictors $\mathbf{x}=(1,X_1,\dots ,X_p{)}^{\top }$ and coefficient vectors ${\mathbf{\beta }}_i=({\beta }_{0i},\dots ,{\beta }_{pi}{)}^{\top }$, $i=2,\dots ,D$:

$$a_1^{\ast }=\frac{1}{1+{\sum }_{k=2}^D{e}^{{\mathbf{x}}^{\top }{\mathbf{\beta }}_k}},a_i^{\ast }=\frac{e^{{\mathbf{x}}^{\top }{\mathbf{\beta }}_i}}{1+{\sum }_{k=2}^D{e}^{{\mathbf{x}}^{\top }{\mathbf{\beta }}_k}},i=2,\dots ,D.$$

Component 1 is the reference. $\log (a_i^{\ast }/a_1^{\ast })={\mathbf{x}}^{\top }{\mathbf{\beta }}_i$ so interpretation matches Aitchison’s ALR regression.

Log-likelihood (Eq. 2)

With $n$ observations $y_{ij}$ (component $i$, observation $j$) and common $\varphi$:

$$\ell =n\log \Gamma (\varphi )-\sum _{j=1}^n\sum _{i=1}^D\log \Gamma (\varphi a_{ij}^{\ast })+\sum _{j=1}^n\sum _{i=1}^D(\varphi a_{ij}^{\ast }-1)\log y_{ij}.$$

Parameter count: $d(p+1)+1$.

Covariate-dependent precision (Eqs. 3-4)

${\varphi }_j^{\ast }=e^{{\mathbf{x}}_j^{\top }\mathbf{\gamma }}$, $\mathbf{\gamma }=({\gamma }_0,\dots ,{\gamma }_p)$, and the log-likelihood becomes

$$\ell =\sum _{j=1}^n\log \Gamma ({\varphi }_j^{\ast })-\sum _{j=1}^n\sum _{i=1}^D\log \Gamma ({\varphi }_j^{\ast }{a}_{ij}^{\ast })+\sum _{j=1}^n\sum _{i=1}^D({\varphi }_j^{\ast }{a}_{ij}^{\ast }-1)\log y_{ij}$$

with $p$ additional parameters. Tested against the constant-precision model by a likelihood-ratio test.

Examples

Interpreting a coefficient

In the foraminifera fit (4 components, covariate $\log$ depth), ${\hat{\beta }}_{\text{slope}}$ for Obesa is 0.117: relative to the reference Triloba, Obesa’s share increases with depth by factor $e^{0.117}\approx 1.12$ per unit of log depth. See Tsagris 2018 - Dirichlet Regression with Zeros - Examples and Simulation Results.

Connections

Aitchison’s alternative: Greenacre 2023 - Logratios ALR CLR ILR and Log-Contrasts; PyMC form: Greenacre 2023 - Compositional Outcome Models in PyMC. The Dirichlet is not in the linear exponential family (Gourieroux et al. 1984), so consistency of estimates requires correct specification.

See Also

Tsagris 2018 - Dirichlet Regression with Zeros - Overview