Why Beta/Dirichlet Regression are not considered Generalized Linear Models?

Question

The premise is this quote from vignette of R package betareg¹.

Further-more, the model shares some properties (such as linear predictor, link function, dispersion parameter) with generalized linear models (GLMs; McCullagh and Nelder 1989), but it is not a special case of this framework (not even for fixed dispersion)

This answer also makes allusion to the fact:

[...] This is a type of regression model that is appropriate when the response variable is distributed as Beta. You can think of it as analogous to a generalized linear model. It's exactly what you are looking for [...] (emphasis mine)

Question title says it all: why Beta/Dirichlet Regression are not considered Generalized Linear Models (are they not)?

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As far as I know, the Generalized Linear Model defines models built on the expectation of their dependent variables conditional on the independent ones.

$f$ is the link function that maps the expectation, $g$ is probability distribution, $Y$ the outcomes and $X$ the predictiors, $\beta$ are linear parameters and $\sigma^2$ the variance.

$$f\left(\mathbb E\left(Y\mid X\right)\right) \sim g(\beta X, I\sigma^2)$$

Different GLMs impose (or relax) the relationship between the mean and the variance, but $g$ must be a probability distribution in the exponential family, a desirable property which should improve robustness of the estimation if I recall correctly. The Beta and Dirichlet distributions are part of the exponential family, though, so I'm out of ideas.

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[1] Cribari-Neto, F., & Zeileis, A. (2009). Beta regression in R.

Answer 1

Check the original reference:

Ferrari, S., & Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31(7), 799-815.

as the authors note, the parameters of re-parametrized beta distribution are correlated, so

Note that the parameters $\beta$ and $\phi$ are not orthogonal, in contrast to what is verified in the class of generalized linear regression models (McCullagh and Nelder, 1989).

So while the model looks like a GLM and quacks like a GLM, it does not perfectly fit the framework.

Answer 2

The answer by @probabilityislogic is on the right track.

The beta distribution is in the two parameter exponential family. The simple GLM models described by Nelder and Wedderburn (1972) do not include all of the distributions in the two parameter exponential family.

In terms of the article by N&W, the GLM applies to the density functions of the following type (this was later named exponential dispersion family in Jørgensen 1987):

$$\pi(z;\theta,\phi) = \exp \left[ \alpha(\phi) \lbrace z\theta - g(\theta) +h(z)\rbrace +\beta(\phi,z) \right]$$

with an additional link function $f()$ and linear model for the natural parameter $\theta = f(\mu) = f(X\beta)$.

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So we could rewrite the above distribution also:

$$\pi(z;\mu,\phi) = exp \left[z(f(\mu)\alpha(\phi)) +h(z)\alpha(\phi) - g(f(\mu))\alpha(\phi) +\beta(\phi,z) \right]$$

The two parameter exponential family is:

$$ f(z;\theta_1,\theta_2) = exp \left[T_1(z)\eta_1(\theta_1,\theta_2) + T_2(z)\eta_2(\theta_1,\theta_2) - g(\theta_1,\theta_2) +h(z) \right] $$

which looks similar but more general (also if one of the $\theta$ is constant).

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The difference is clear, and also putting the beta distribution in a form as a GLM is not possible.

However, I lack sufficient understanding to create a more intuitive and well informed answer (I have a feeling that there can be much deeper and more elegant relationships to a variety of fundamental principles). The GLM generalizes the distribution of the error by using a single variate exponential dispersion model in place of a least squares model and generalizes the linear relationship in the mean, by using a link function.

The best and most simple intuition seems to be the dispersion-$\alpha(\phi)$-term in the exponential, which gets multiplied with everything and thus the dispersion does not vary with $\theta$. Whereas several two parameter exponential families, and quasi-likelihood methods, allow the dispersion parameter to be a function of $\theta$ as well.

Answer 3

I don't think the beta distribution is part of the exponential dispersion family. To get this, you need to have a density

$$f (y;\theta,\tau)=\exp\left (\frac {y\theta - c (\theta)}{\tau} + d (y,\tau)\right)$$

for specified functions $c ()$ and $d () $. The mean is given as $ c'(\theta)$ and the variance is given as $\tau c''(\theta) $. The parameter $\theta $ is called the canonical parameter.

The beta distribution cannot be written this way - one way to see this is by noting there is no $y $ term in the log likelihood - it has $\log [y] $ and $\log [1-y] $ instead

$$f_{beta}(y;\mu,\phi)=\exp\left (\phi\mu\log\left[\frac {y}{1-y}\right] +\phi\log [1-y] - \log [B (\phi\mu,\phi (1-\mu)]-\log\left[\frac {y}{1-y}\right]\right) $$

Yet another way to see that beta is not exponential dispersion family is that it can be written as $y=\frac {x}{x+z} $ where $x $ and $z $ are independent and both follow gamma distributions with the same scale parameter (and gamma is exponential family).