---
title: "Biostat 200C Homework 4"
subtitle: Due ~~May 29~~ May 31 @ 11:59PM
output: 
  html_document:
    toc: true
    toc_depth: 4
---

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

## Q1. Log-logistic survival model

The **log-logistic distribution** with the probability density function
$$
f(y) = \frac{e^\theta \lambda y^{\lambda - 1}}{(1 + e^{\theta} y^{\lambda})^2}
$$
is sometimes used for modelling survivial times.

1. Find the survivor function $S(y)$, the hazard function $h(y)$ and the cumulative hazard function $H(y)$.

2. Show that the median survival time is $e^{-\theta / \lambda}$. 

3. Plot the hazard function for $\lambda = 1$ and $\lambda = 5$ with $\theta = -5$, $\theta = -2$ and $\theta = 1/2$. 

## Q2. Balanced one-way ANOVA random effects model

Consider the balanced one-way ANOVA random effects model with $a$ levels and $n$ observations in each level
$$
y_{ij} = \mu + \alpha_i + \epsilon_{ij}, \quad i=1,\ldots,a, \quad j=1,\ldots,n.
$$
where $\alpha_i$ are iid from $N(0,\sigma_\alpha^2)$, $\epsilon_{ij}$ are iid from $N(0, \sigma_\epsilon^2)$. 

1. Derive the ANOVA estimate for $\mu$, $\sigma_\alpha^2$, and $\sigma_{\epsilon}^2$. Specifically show that
\begin{eqnarray*}
  \mathbb{E}(\bar y_{\cdot \cdot}) &=& \mathbb{E} \left( \frac{\sum_{ij} y_{ij}}{na} \right) = \mu \\
  \mathbb{E} (\text{SSE}) &=& \mathbb{E} \left[ \sum_{i=1}^a \sum_{j=1}^n (y_{ij} - \bar{y}_{i \cdot})^2 \right] = a(n-1) \sigma_{\epsilon}^2 \\
  \mathbb{E} (\text{SSA}) &=& \mathbb{E} \left[ \sum_{i=1}^a \sum_{j=1}^n (\bar{y}_{i \cdot} - \bar{y}_{\cdot \cdot})^2 \right] = (a-1)(n \sigma_{\alpha}^2 + \sigma_{\epsilon}^2),
\end{eqnarray*}
which can be solved to obtain ANOVA estimate
\begin{eqnarray*}
\widehat{\mu} &=& \frac{\sum_{ij} y_{ij}}{na}, \\
\widehat{\sigma}_{\epsilon}^2 &=& \frac{\text{SSE}}{a(n-1)}, \\
\widehat{\sigma}_{\alpha}^2 &=& \frac{\text{SSA}/(a-1) - \widehat{\sigma}_{\epsilon}^2}{n}.
\end{eqnarray*}

2. Derive the MLE estimate for $\mu$, $\sigma_\alpha^2$, and $\sigma_{\epsilon}^2$. Hint: write down the log-likelihood and find the maximizer.

3. (**Optional**) Derive the REML estimate for $\mu$, $\sigma_\alpha^2$, and $\sigma_{\epsilon}^2$. 

4. For all three estimates, check that your results match those we obtained using R for the `pulp` example in class.

## Q3. Estimation of random effects

1. Assume the conditional distribution
$$
\mathbf{y} \mid \boldsymbol{\gamma} \sim N(\mathbf{X} \boldsymbol{\beta} + \mathbf{Z} \boldsymbol{\gamma}, \sigma^2 \mathbf{I}_n)
$$
and the prior distribution
$$
\boldsymbol{\gamma} \sim N(\mathbf{0}_q, \boldsymbol{\Sigma}).
$$
Then by the Bayes theorem, the posterior distribution is
\begin{eqnarray*}
f(\boldsymbol{\gamma} \mid \mathbf{y}) &=& \frac{f(\mathbf{y} \mid \boldsymbol{\gamma}) \times f(\boldsymbol{\gamma})}{f(\mathbf{y})}, \end{eqnarray*}
where $f$ denotes corresponding density. Show that the posterior distribution is a multivariate normal with mean
$$
\mathbb{E} (\boldsymbol{\gamma} \mid \mathbf{y}) = \boldsymbol{\Sigma} \mathbf{Z}^T (\mathbf{Z} \boldsymbol{\Sigma} \mathbf{Z}^T + \sigma^2 \mathbf{I})^{-1} (\mathbf{y} - \mathbf{X} \boldsymbol{\beta}).
$$

2. For the balanced one-way ANOVA random effects model, show that the posterior mean of random effects is always a constant (less than 1) multiplying the corresponding fixed effects estimate.

## Q4. ELMR Exercise 11.1 (p251)