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.
Find the survivor function \(S(y)\), the hazard function \(h(y)\) and the cumulative hazard function \(H(y)\).
Show that the median survival time is \(e^{-\theta / \lambda}\).
Plot the hazard function for \(\lambda = 1\) and \(\lambda = 5\) with \(\theta = -5\), \(\theta = -2\) and \(\theta = 1/2\).
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 \(N(0, \sigma_\epsilon^2)\).
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 \[\begin{eqnarray*} \widehat{\mu} &=& \frac{\sum_{ij} y_{ij}}{na}, \\ \widehat{\sigma}_{\epsilon}^2 &=& \frac{\text{SSE}}{a(n-1)}, \\ \widehat{\sigma}_{\alpha}^2 &=& \frac{\text{SSA}/(a-1) - \widehat{\sigma}_{\epsilon}^2}{n}. \end{eqnarray*}\]
Derive the MLE estimate for \(\mu\), \(\sigma_\alpha^2\), and \(\sigma_{\epsilon}^2\). Hint: write down the log-likelihood and find the maximizer.
(Optional) Derive the REML estimate for \(\mu\), \(\sigma_\alpha^2\), and \(\sigma_{\epsilon}^2\).
For all three estimates, check that your results match those we obtained using R for the pulp example in class.
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 \[ \mathbb{E} (\boldsymbol{\gamma} \mid \mathbf{y}) = \boldsymbol{\Sigma} \mathbf{Z}^T (\mathbf{Z} \boldsymbol{\Sigma} \mathbf{Z}^T + \sigma^2 \mathbf{I})^{-1} (\mathbf{y} - \mathbf{X} \boldsymbol{\beta}). \]
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.