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{\Large \textbf{Sample Questions: Proportional Hazards Regression Part One}}%\footnote{}
\vspace{1 mm}

STA312 Fall 2023. Copyright information is at the end of the last page.
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\begin{enumerate} 

\item For a proportional hazards regression model with hazard function $h_i(t|\boldsymbol{\beta}) =  h_0(t) \,  e^{\beta_0 +\mathbf{x}_i^\top \boldsymbol{\beta}}$, the partial likelihood is 
\begin{displaymath}
    \mbox{PL}(\boldsymbol{\beta}) = \displaystyle \prod_{i=1}^D 
    \left( \frac{h_{(i)}(t_{(i)}|\boldsymbol{\beta}) }
     { \displaystyle  \sum_{j \in R_{(i)}} h_j(t_{(i)}|\boldsymbol{\beta}) }\right) 
=
    \displaystyle \prod_{i=1}^D 
    \left( \frac{\displaystyle  e^{\mathbf{x}_{(i)}^\top \boldsymbol{\beta}} }
     {\displaystyle  \sum_{j \in R_{(i)}}  e^{\mathbf{x}_j^\top \boldsymbol{\beta}}}     
     \right).
\end{displaymath}
    \begin{enumerate}
        \item What happened to $\beta_0$? \vspace{50mm}
        \item Write the log partial likelihood for a model with \emph{one explanatory variable}. Differentiate and set to zero.
    \end{enumerate}

\newpage

\item Let $ h(t) =  h_0(t) \,  e^{\mathbf{x}^\top \boldsymbol{\beta}}$. Show $S(t) =  S_0(t)^{\exp\{\mathbf{x}_i^\top \boldsymbol{\beta} \}}$.

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

\item Adult volunteers who were unemployed were randomly assigned to either Job Training Program $A$, Job Training Program $B$, or a wait list control group. The response variable was time until employment. It might be censored because the person was still unemployed at the end of the study, or if they left the study for other reasons. Age is a covariate.

    \begin{enumerate} 
        \item \label{haz} Assuming a proportional hazards regression model, write the hazard function, denoting the length of time until employment by $t$. Denote age by $a$. 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}

$h(t) = $

\vspace{3mm}

        \item In the table below, make columns showing how your dummy variables are defined. In the last column, write the hazard function $h(t)$ for the appropriate vector of explanatory variable values $\mathbf{x}$, using the notation of your model from Question~\ref{haz} above. If \emph{symbols} for your dummy variables appear in the last column, the answer is wrong. \vspace{4mm}

\hspace{3.8in} $h(t)$
\begin{center}
\renewcommand{\arraystretch}{2.5}
\begin{tabular}{|l|c|c|}  \hline
Wait List   & \hspace{30mm} & \hspace{70mm} \\ \hline
Program $A$ &     &  \\ \hline
Program $B$ &     &  \\ \hline
\end{tabular}
\renewcommand{\arraystretch}{1.0}
\end{center} \vspace{10mm}

        \item In the notation of your model, what is the risk of employment at time $t$ for a 25-year-old participant on the wait list? \vspace{15mm}

        \item For a 60-year-old participant in Program $A$, the chances of finding a job at any time period are \underline{\hspace{20mm}} times as great as the chances for a 60-year-old on the wait list. Answer in terms of the Greek letters from your model. \vspace{10mm}

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

        \item For a 47-year-old participant in Program $A$, the hazard of finding a job at time $t$ is \underline{\hspace{20mm}} times as great as the hazard for a 47-year-old in Program $B$. Answer in terms of the Greek letters from your model. \vspace{25mm}


        \item You want to know whether, controlling for age, training program ($A$, $B$ or neither) has any effect on time until employment. What is the null hypothesis? Answer in terms of the Greek letters from your model. \vspace{25mm}
        
        \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 hazard function for the restricted model in the space below. \vspace{8mm}
                    
$h(t) =$
 \vspace{8mm}
                    \item Suppose you decided on a Wald test. Write $H_0: \mathbf{L}\boldsymbol{\beta} = \mathbf{0}$ in terms of specific matrices. \vspace{30mm}
                \end{enumerate}
    \end{enumerate} % end of job training question.




\end{enumerate} % End of all the questions

~

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This document 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/312f23} {\small\texttt{http://www.utstat.toronto.edu/brunner/oldclass/312f23}}
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