Observation overviews

_quarto.yml

@@ -13,10 +13,12 @@ website:
             contents:
               - section: "Unbounded"
                 contents:
+                  - gen_linear_regression/continuous_unbounded_overview.md
                   - gen_linear_regression/normal.qmd
                   - gen_linear_regression/student_t.qmd
               - section: "Bounded"
                 contents:
+                  - gen_linear_regression/continuous_bounded_overview.md
                   - gen_linear_regression/beta.qmd
           - section: "Discrete outcome"
             contents:
@@ -25,6 +27,7 @@ website:
                 - gen_linear_regression/bernoulli_logit.qmd
               - section: "Count"
                 contents:
+                  - gen_linear_regression/discrete_count_overview.qmd
                   - gen_linear_regression/poisson.qmd
                   - gen_linear_regression/negative_binomial.qmd
                   - gen_linear_regression/binomial.qmd
@@ -54,7 +57,7 @@ website:
         href: tools.qmd
       - text: "Contributors"
         href: contributors.md
-        
+
 format:
   html:
     theme: cosmo

contributors.md

@@ -5,17 +5,17 @@
 	<tbody>
 		<tr>
             <td align="center">
-                <a href="https://github.com/aloctavodia">
-                    <img src="https://avatars.githubusercontent.com/u/1338958?v=4" width="100;" alt="aloctavodia"/>
+                <a href="https://github.com/n-kall">
+                    <img src="https://avatars.githubusercontent.com/u/33577035?v=4" width="100;" alt="n-kall"/>
                     <br />
-                    <sub><b>Osvaldo A Martin</b></sub>
+                    <sub><b>Noa Kallioinen</b></sub>
                 </a>
             </td>
             <td align="center">
-                <a href="https://github.com/n-kall">
-                    <img src="https://avatars.githubusercontent.com/u/33577035?v=4" width="100;" alt="n-kall"/>
+                <a href="https://github.com/aloctavodia">
+                    <img src="https://avatars.githubusercontent.com/u/1338958?v=4" width="100;" alt="aloctavodia"/>
                     <br />
-                    <sub><b>Noa Kallioinen</b></sub>
+                    <sub><b>Osvaldo A Martin</b></sub>
                 </a>
             </td>
             <td align="center">

gen_linear_regression/beta.qmd

@@ -4,27 +4,35 @@ title: Beta model
 
 ## Description
 
-Beta regression is used for outcomes on the [0, 1] interval. It is a
-distributional regression, not a generalized linear model.
+Beta regression is used for outcomes on the [0, 1] interval.
+
 
 ## Definition
 
 For continuous outcome $y$ bounded [0, 1] and predictors $x$, the model is:
 
 $$
 \begin{align}
-y_i \sim \text{beta}(\text{logit}^{-1}(\mu_i), \log{\phi_i}) \\
-\mu_i = \alpha_{\mu} + \beta_{\mu} \cdot x \\
-\phi_i = \alpha_{\phi} + \beta_{\phi} \cdot x \\
+y_i \sim \text{beta}(\text{logit}^{-1}(\mu_i), \kappa) \\
+\mu_i = \alpha + \beta \cdot x_i \\
 \end{align}
 $$
 
+The [logit](https://en.wikipedia.org/wiki/Logit) function is the quantile function associated with the standard logistic distribution. Its inverse maps $\mu_i$ from the real line into a value within the interval (0, 1).
+
+
 ## Parameters needing priors
 
-- $\alpha_{\mu}$ (intercept for $\mu$)
-- $\alpha_{\phi}$ (intercept for $\phi$)
-- $\beta_{\mu}$ (predictor weights for $\mu$)
-- $\beta_{\mu}$ (predictor weights for $\phi$)
+- $\alpha$ (intercept)
+- $\beta$ (predictor weights)
+- $\kappa$ (concentration parameter)
+
+The larger the value of $\kappa$, the more concentrated the distribution is around the mean. This parameter is also named the precision parameter or the sample size parameter. Greek letters $\nu$ and $\phi$ are also commonly used to refer to this parameter.
+
+
+## Prior for $\kappa$
+
+For weakly informative prior we can use a gamma distribution with low probability mass on very small values of $\kappa$, such priors disfavor u-shaped beta distributions which are generally less common in real-world data. For instance, we could choose $\text{gamma}(4, 0.1)$ as discussed by [Solomon Kurz](https://solomonkurz.netlify.app/blog/2023-06-25-causal-inference-with-beta-regression/), which place most of the prior mass on the double-digit range with long right tail allowing for greater concentrations if needed.
 
 
 ## See also

gen_linear_regression/continuous_bounded_overview.qmd

@@ -0,0 +1,9 @@
+---
+title: Continuous bounded outcome
+---
+
+For lower-bounded observations, there are the [gamma model](gamma.qmd),
+[lognormal](lognormal.qmd)
+
+For both upper- and lower-bounded observations, the [beta model] may
+be appropriate, particularly if the bounds are 0 and 1 (or can be transformed to this scale).

gen_linear_regression/continuous_unbounded_overview.qmd

@@ -0,0 +1,7 @@
+---
+title: Continuous unbounded outcome
+---
+
+When observations are unbounded and continuous, the most common model
+is the [normal model](normal.qmd). When the observations may include
+outliers, the [Student-t model](student_t.qmd) may be appropriate.

gen_linear_regression/discrete_count_overview.qmd

@@ -0,0 +1,16 @@
+---
+title: Discrete count outcome
+---
+
+Counts can be considered in different ways. If the count represents
+simple counting of observations, the [Poisson model](poisson.qmd) may
+be an appropriate starting point. However, if there is
+over-dispersion, the [negative binomial](negative_binomial.qmd) may
+better capture the data.
+
+If the counts are of the number of successes in repeated binary
+trials, then the [binomial model](binomial.qmd) is appropriate.
+
+In cases where there is an over-representation of zeroes in the
+data, this can be modelled with zero-inflated versions of the
+distributions.