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\title{Competing Risks\footnote{See last slide for copyright information.}}
\subtitle{STA312 Spring 2019}
\date{} % To suppress date

\begin{document}

\begin{frame}
  \titlepage
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\begin{frame}
\frametitle{Background Reading}
%\framesubtitle{} 

\begin{itemize}
    \item Chapter 9 (Especially 9.2.4) in \emph{Applied Survival Analysis Using R} by Dirk Moore
\end{itemize}

\end{frame}

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\begin{frame}
\frametitle{Competing Risks: Common in practice} \pause
%\framesubtitle{} 
\begin{itemize}
    \item Death from cancer, death from other cause, censored. \pause
    \item For kidney patients: death, transplant, censored. \pause
    \item For Ph.D. students: Graduate, withdraw, disappear, censored. \pause
    \item Marriage: Divorce, separation, widowed, censored. 
\end{itemize}
\end{frame}

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\begin{frame}
\frametitle{Three Approaches}
\framesubtitle{There are probably more}  \pause
\begin{itemize}
    \item Analyze the data for each outcome in turn, treating the others as censored. \pause
    \item Variation on the first method, with stratification and frailty. \pause
    \item ``Sub-distribution" approach.
\end{itemize}
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\begin{frame}
\frametitle{One-at-a-time} \pause
%\framesubtitle{} 
Analyze the data for each outcome in turn, treating the others as censored. \pause
\begin{itemize}
    \item Straightforward and easy, but \pause
    \item Is the censoring mechanism independent of the failure process? \pause
    \item If $x_k$ affects outcome 2 differently from outcome 3, it's hard to test.
\end{itemize}
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\begin{frame}
\frametitle{Variation on one-at-a-time}
\framesubtitle{Therneau and Grambsch, pp.~175-177}  \pause
%{\footnotesize
\begin{itemize}
    \item Create multiple lines of data for each participant\pause, one for each outcome \pause (except censored). \pause
    \item For all the outcomes that did \emph{not} happen, the outcome is recorded as censored. \pause 
    \item So for example, \pause
                \begin{itemize}
                    \item If there are 3 outcomes in addition to censoring, \pause each case contributes 3 lines. \pause
                    \item At most one line has $\delta=1$. \pause The others have $\delta=0$; they are censored. \pause
                    \item ``Endpoint" is a variable with different values for the 3 lines. \pause
                    \item Stratify on endpoint. \pause
                    \item Also, tie the lines together with a random effect for \texttt{id}. \pause
                    \item This is meant to take care of lack of independence. \pause
                    \item Different regression coefficients for the strata (outcomes) are possible.
                \end{itemize}
\end{itemize}
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\begin{frame}[fragile]
\frametitle{Example}
\framesubtitle{Of the last method described} 
%{\footnotesize % or scriptsize
\begin{verbatim}
> age1 = age * (endpoint=='death')
> age2 = age * (endpoint='transp')
> coxme(Surv(time, status) ~ hemoglobin + age1 + age2
                           + strata(endpoint) + (1|id) )
\end{verbatim} \pause \vspace{5mm}
%} % End size
Objection? \pause
\begin{itemize}
    \item The random effect is affecting all the outcomes in the same way. \pause
    \item Maybe there should be a different ``frailty" for each outcome. \pause
    \item Does it matter?
\end{itemize}
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\begin{frame}
\frametitle{Sub-distribution method}
\framesubtitle{Fine and Grey (1999)}  \pause
\begin{itemize}
    \item The ``sub-distribution" function $F_k(t)$ is like a cdf, but only applies to outcome (cause of death) $k$. \pause
    \item Instead of approaching one as $t \rightarrow \infty$, it approaches a limiting probability that the person will die of cause $k$. \pause
    \item Corresponding to the sub-distribution function is the sub-distribution hazard \pause
\end{itemize}
\begin{displaymath}
    h_k(t) = \lim_{\Delta \rightarrow 0}\frac{P(t < T_k < t + \Delta \, | \, E)}{\Delta},
\end{displaymath} \pause
where the event $E = \{ T_k>t\} \cup \{ T_{k^\prime} \leq t, k^\prime \neq k \}$. \pause
\vspace{3mm}

The meaning of this notation is \pause  ``\ldots the risk set includes not only those
currently alive and at risk for the $k$th event type, but also those who died earlier of
other causes." (Our text, p.~129) \pause Fine and Grey call this ``unconventional."
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\begin{frame}
\frametitle{Sub-distributions and sub-hazard functions} \pause
%\framesubtitle{} 
\begin{itemize}
    \item The sub-hazard is related to the sub-distribution function by \pause
\begin{displaymath}
    h_k(t) = -\frac{d}{dt} \log(1-F_k(t))
\end{displaymath} \pause
    \item $ h_k(t) =  h_{0,k}(t) \,  e^{\mathbf{x}^\top \boldsymbol{\beta}_k}$\pause: Baseline hazard times regression function. \pause
    \item In theory, there is a separate set of regression coefficients for each outcome. \pause
    \item In software, each set is estimated in a different run. \pause
    \item The method extends to time-varying covariates, in both theory and software.
\end{itemize}
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\begin{frame}
\frametitle{Copyright Information}

This slide show was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistics, 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:
\href{http://www.utstat.toronto.edu/~brunner/oldclass/312s19} {\footnotesize \texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/312s19}}

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