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{\Large \textbf{Sample Questions: Weibull Regression}}%\footnote{}
\vspace{1 mm}

STA312 Spring 2019. Copyright information is at the end of the last page.
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\begin{enumerate} 

\item Let the failure time 
{\Large $t_i = \exp\{\beta_0 + \beta_1x_{i,1} + \ldots + \beta_{p-1}x_{i,p-1} \} \cdot \epsilon_i^\sigma$. }
Show that if $x_{i,k}$ is increased by $c$ units, $t_i$ is multiplied by {\Large $e^{c\beta_k}$}.

\pagebreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item {\Large 
      Let $t_i = e^{\mu_i} \, \epsilon_i^\sigma$, where $-\infty < \mu_i < \infty$ and $\sigma>0$. The idea is that $\mu_i = \beta_0 + \beta_1x_{i,1} + \ldots + \beta_{p-1}x_{i,p-1}$.
    \begin{enumerate}
        \item Derive the density of $t_i$.

\pagebreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

        \item Re-parameterizing by $\lambda_i = e^{-\mu_i}$ and $\alpha = 1/\sigma$, verify that $t_i$ has a Weibull distribution.

\pagebreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

        \item The hazard function of a Weibull is $h(t) = \alpha\lambda^\alpha t^{\alpha-1}$. Going back to the $(\mu,\sigma)$ parameterization of the Weibull and substituting $\mu_i = \mathbf{x}_i^\top \boldsymbol{\beta}$, write the hazard function $h(t)$. \vspace{120mm}
%        \item If $x_{i,k}$ is increased by one unit, the hazard function is multiplied by \underline{\hspace{10mm}}. % This one is no good because of sigma.

\pagebreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \item The expected value of a Weibull is $\Gamma(1+\frac{1}{\alpha})/\lambda$. Write this in the $(\mu,\sigma)$ parameterization, letting $\mu_i = \mathbf{x}_i^\top \boldsymbol{\beta}$.  \vspace{120mm}
    \item If $x_{i,k}$ is increased by one unit, the expected failure time is multiplied by \underline{\hspace{10mm}}.
    
\pagebreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \item The median of a Weibull random variable is $\frac{[\log(2)]^{1/\alpha}}{\lambda}$. Write this in the $(\mu,\sigma)$ parameterization, letting $\mu_i = \mathbf{x}_i^\top \boldsymbol{\beta}$.  \vspace{120mm}
    \item If $x_{i,k}$ is increased by one unit, the median failure time is multiplied by \underline{\hspace{10mm}}.
    \end{enumerate}
      } % End size

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\item For a particular form of cancer, the standard treatment is a combination of chemotherapy and radiation therapy. Both chemotherapy and radiation have serious side effects. Some patients may be so weakened by the treatment that they die from other things (such as infections) that are apparently unrelated to the cancer.

Volunteer patents who were considering no treatment at all were randomly assigned to one of three experimental conditions. They received either Chemotherapy only, Radiation only, or Both treatments. The response variable is survival time, which in some cases will be right-censored. Age is an important predictor of survival, and is used as a covariate. 
    \begin{enumerate} 
        \item \label{wreg} Write the (multiplicative) Weibull regression equation, denoting the length of time between diagnosis and death (call it survival time) for patient $i$ by $t_i$. Denote age by $x_i$. There should be \emph{no interactions} in the model, in case you know what that is. You do not need to say how your dummy variables are defined. You will do that in the next part. Complete the equation below.
        
\vspace{3mm}

$t_i = $

\vspace{3mm}

        \item In the table below, make columns showing how your dummy variables are defined. In the last column, write the expected survival time, using the notation of your model from Question~\ref{wreg} above. If \emph{symbols} for your dummy variables appear in the last column, the answer is wrong. \vspace{4mm}

\hspace{3.3in} Expected Survival Time
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\begin{tabular}{|l|c|c|}  \hline
Chemotherapy & \hspace{30mm} & \hspace{70mm} \\ \hline
Radiation   &     &  \\ \hline
Both        &     &  \\ \hline
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        \item In the notation of your model, what is the expected survival time for a 25-year-old patient receiving both radiation and chemotherapy? \vspace{12mm}

        \item For a 60-year-old patient receiving radiation only, the expected survival time is \underline{\hspace{20mm}} times as great as the expected survival time for a 60-year-old receiving both radiation and chemotherapy. Answer in terms of the Greek letters from your model. \vspace{10mm}

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        \item For a 47-year-old patient receiving radiation only, the expected survival time is \underline{\hspace{20mm}} times as great as the expected survival time for a 47-year-old receiving chemotherapy only. Answer in terms of the Greek letters from your model. \vspace{10mm}


        \item You want to know whether, controlling for age, experimental treatment (Chemotherapy, Radiation, or Both) has any effect on average survival time. What is the null hypothesis? Answer in terms of the Greek letters from your model. \vspace{10mm}
        
        \item That last question could be answered with either a large-sample likelihood ratio test, or a Wald test.
                \begin{enumerate}
                    \item Suppose you decided on a likelihood ratio test. Write the multiplicative Weibull regression equation for the restricted model. \vspace{8mm}
                    
$t_i =$
 \vspace{8mm}
                    \item Suppose you decided on a Wald test. Write the $\mathbf{L}$ matrix for $H_0: \mathbf{L}\boldsymbol{\theta} = \mathbf{0}$. \vspace{30mm}
                \end{enumerate}

        \item You want to know whether it is better for patients to get both radiation and chemotherapy, or just radiation. What is the null hypothesis? Answer in terms of the Greek letters from your model. \vspace{12mm}
        
        \item You want to know whether it is better for patients to get both radiation and chemotherapy, or just chemotherapy. What is the null hypothesis? Answer in terms of the Greek letters from your model. \vspace{12mm}

        \item You want to know whether it is better for patients to get just radiation or just chemotherapy. What is the null hypothesis? Answer in terms of the Greek letters from your model. \vspace{25mm}
    \end{enumerate}


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\item Show that the multiplicative Weibull regression model has proportional hazards. Consider two patients with different vectors of explanatory variable values.

\end{enumerate} % End of all the questions

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\noindent
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This assignment was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Mathematical and Computational Sciences, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website: 
     
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\href{http://www.utstat.toronto.edu/~brunner/oldclass/312s19} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/312s19}}
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