
# 13 Binomial Models

## Prerequisites

library("rstan")
library("rstanarm")
library("tidyverse")
library("recipes")
library("bayz")

## 13.1 Introduction

Binomial models are used to an outcome that is a bounded integer, $y_i \in 0, 1, 2, \dots, n .$ The outcome is distributed Binomial, \begin{aligned}[t] y_i \sim \dBinom \left(n_i, \pi \right) \end{aligned}

A binary outcome is a common special case, $y_i \in \{0, 1\},$ and \begin{aligned}[t] y_i &\sim \dBinom \left(1, \pi \right) & \text{for all i} \\ \end{aligned}

Depending on the link function, these are logit and probit models that appear in the literature.

## 13.3 References

For general references on binomial models see Stan Development Team (2016 Sec. 8.5), McElreath (2016 Ch 10), A. Gelman and Hill (2007) [Ch. 5; Sec 6.4-6.5], Fox (2016 Ch. 14), and A. Gelman, Carlin, et al. (2013 Ch. 16).

1. Since the cumulative distribution function of a distribution maps reals to $$(0, 1)$$, any CDF can be used as a link function.

2. Beck, Katz, and Tucker (1998) show that the cloglog link function can be derived from a grouped duration model with binary response variables.

3. Example from Zelig-logit.