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Add gamma regression #25

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24 changes: 24 additions & 0 deletions gen_linear_regression/gamma.qmd
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---
title: Gamma model
---

## Description

Linear regression with gamma distributed residuals.

## Definition

For continuous outcome $y$ lower bounded at zero and predictors $x$, the model is:

$$
\begin{align}
y_i \sim \text{gamma}(s, \alpha/exp(\eta)) \\
\eta_i = \alpha + \beta \cdot x_i
\end{align}
$$
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@aloctavodia aloctavodia Sep 13, 2024 •
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The Gamma distribution can be a tricky one given the different parameterizations.
In PyMC/Bambi/PreliZ we have (alpha, beta) and (mu, sigma) see here for details https://preliz.readthedocs.io/en/latest/examples/gallery/gamma.html

Usually, we use the mu, sigma parametrization like this

\begin{align}
y_i \sim \text{gamma}(\text{mu}=\mu_i, \text{sigma}=\frac{\mu_i}{\alpha^{0.5}}) \
\mu_i = \exp(\eta_i)\
\eta_i = \beta_0 + \beta_1 \cdot x_i
\end{align}

In brms, I think we have (mu, alpha) and Stan use (alpha, beta) [the same as the PyMC's default]

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Good point, there are clear typos here too. However the issue of different parameterizations is something we will run into in various places, also when recommending priors (e.g. exponential prior). I will open an issue about handling different parametrisations in general


## Parameters needing priors

- $\alpha$ (intercept)
- $\beta$ (predictor weights)
- $s$ shape parameter