---
title: "Biostat 200C Homework 3"
subtitle: Due May 5 @ 11:59PM
output: 
  html_document:
    toc: true
    toc_depth: 4
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, cache = FALSE)
```

To submit homework, please upload both Rmd and html files to Bruinlearn by the deadline.

## Q1. Concavity of Poisson regression log-likelihood 

Let $Y_1,\ldots,Y_n$ be independent random variables with $Y_i \sim \text{Poisson}(\mu_i)$ and $\log \mu_i = \mathbf{x}_i^T \boldsymbol{\beta}$, $i = 1,\ldots,n$.

### Q1.1

Write down the log-likelihood function.

### Q1.2

Derive the gradient vector and Hessian matrix of the log-likelhood function with respect to the regression coefficients $\boldsymbol{\beta}$. 

### Q1.3

Show that the log-likelihood function of the log-linear model is a concave function in regression coefficients $\boldsymbol{\beta}$. (Hint: show that the negative Hessian is a positive semidefinite matrix.)

### Q1.4

Show that for the fitted values $\widehat{\mu}_i$ from maximum likelihood estimates
$$
\sum_i \widehat{\mu}_i = \sum_i y_i.
$$
Therefore the deviance reduces to
$$
D = 2 \sum_i y_i \log \frac{y_i}{\widehat{\mu}_i}.
$$

## Q2. Show negative binomial distribution mean and variance 

Recall the probability mass function of negative binomial distribution is 
$$
\mathbb{P}(Y = y) = \binom{y + r - 1}{r - 1} (1 - p)^r p^y, \quad y = 0, 1, \ldots
$$
Show $\mathbb{E}Y = \mu = rp / (1 - p)$ and $\operatorname{Var} Y = r p / (1 - p)^2$.

## Q3. ELMR Chapter 5 Exercise 5 (page 100)

## Q4. Uniform association 

For the uniform association when all two-way interactions are included, i.e., 
$$
\log \mathbb{E}Y_{ijk} = \log p_{ijk} = \log n + \log p_i + \log p_j + \log p_k + \log p_{ij} + \log p_{ik} + \log p_{jk}.
$$

Proof the odds ratio (or log of odds ratio) across all stratum $k$ 
$$
\log \frac{\mathbb{E}Y_{11k}\mathbb{E}Y_{22k}}{\mathbb{E}Y_{12k}\mathbb{E}Y_{21k}}
$$

is a constant, i.e., the estimated effect of the interaction term "i:j" in the uniform association model