proportions.Rmd

# Proportions

## Beta Regression

This is for a response variable that is a proportion, $y_i \in (0, 1)$,
$$
y_i \sim \dbeta(\alpha_i, \beta_i)
$$
and
$$
\begin{aligned}[t]
\mu_i &= g^{-1}(\vec{x}_i' \vec{\gamma}) \\
\alpha_i &= \mu_i \phi \\
\beta_i &= (1 - \mu_i) \phi
\end{aligned}
$$
Additionally, the $\phi$ parameter could also be modeled.

In Stan:

-   `beta(y | alpha, beta)` with positive prior successes plus one,
    $\alpha > 0$, and negative prior failures plus one, $\beta > 0$.
    Then $\E(Y) = \alpha / (\alpha + \beta)$ and
    $\Var(Y) = \alpha\beta / ((\alpha + \beta)^2 (\alpha + \beta + 1))$.

**rstanarm** function `r rdoc("rstasnarm", "stan_betareg")`

See:

-   @FerrariCribari-Neto2004a, @Cribari-NetoZeileis2010a, and
    @GruenKosmidisZeileis2012a on beta regression.

-   **rstanarm** documentation [Modeling Rates/Proportions using beta
    regression with rstanarm](https://cran.r-project.org/web/packages/rstanarm/vignettes/betareg.html)