sequential.Rnw

<<echo=FALSE>>=
library(survival, quietly=TRUE)
opts_chunk$set(comment=NA, tidy=FALSE, highlight=FALSE, echo=FALSE,
               fig.with=7, fig.height=5.5, fig.path="figures/",
               out.width="\\textwidth", out.height="!", device="pdf",
               cache=FALSE, background="#ffffff",
               warning=FALSE, error=FALSE, prompt=TRUE)
options(contrasts= c("contr.treatment", "contr.poly"),
        show.signif.starts = FALSE, continue=" ", width=60)
par(mar=c(4.1, 4.1, 1.1, 1.1))
@ 

\section{Sequential Events}
\begin{frame}
  {\Large Sequential Events}
\end{frame}

\begin{frame}
  \begin{itemize}
    \item One of the first applications, widely used.
    \item Data sets in the survival package (book by Therneau and Grambsch)
      \begin{itemize}
        \item Sequential events
          \begin{itemize}
            \item Recurrent bladder cancer
            \item Repeated infections in children with chronic granulomatous
             disease
           \item rhDNase for the treatment of cystic fibrosis
           \item Failure of kidney catheters
          \end{itemize}
        \item Parallel events
          \begin{itemize}
            \item Left and right eyes in diabetic retinopathy
            \item Multiple liver sequelae in a UDCA trial
          \end{itemize}
      \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{Parallel events}
  \begin{itemize}
    \item Uncommon
    \item Decisions
      \begin{itemize}
        \item Multiple strata?
          \begin{itemize}
            \item Diabetes: no
            \item UDCA in PBC: yes
          \end{itemize}
          \item strata by covariate interactions
      \end{itemize}
    \item Data setup: stacked 
    \item Analysis: robust variance
  \end{itemize}
\end{frame}

\begin{frame}{Stacked data sets}
  \begin{itemize}
    \item Diabetic retinopathy
     \begin{itemize}
       \item 2n observations
       \item Data set for the right eye, status of 0/1
       \item Data set for the left eye
     \end{itemize}
   \item Parallel failures after UCDA
     \begin{itemize}
       \item 7 endpoints
       \item 7n observations
     \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}[fragile]{Diabetic retinopathy}
  \begin{itemize}
    \item Two eyes per subject, one randomized to laser coagulation
  \end{itemize}
<<diabetic>>=
juvenile <- 1*(diabetic$age < 20)
coxph(Surv(time, status) ~ trt + juvenile + cluster(id), diabetic)
@   
\end{frame}

\begin{frame}{Sequential events}
  \begin{itemize}
    \item Single stratum or multiple strata?
      \begin{itemize}
        \item Does the baseline risk reset to a new level after each event?
        \item CGD data set: no
        \item Repeat cardiac events: maybe
      \end{itemize}
    \item strata by covariate iteractions?
    \item time scale: age, time since enrollment, time since last event, \ldots
  \end{itemize}
\end{frame}

\begin{frame}{Models}
  \begin{itemize}
    \item Andersen-Gill model
      \begin{itemize}
        \item single stratum
        \item an event is an event is an event
      \end{itemize}
    \item Prentice-Williams-Petersen
      \begin{itemize}
        \item new stratum for each event
        \item time normally resets to zero
        \item dangerous!
      \end{itemize}
    \item Wei, Lin, and Weissfeld
      \begin{itemize}
        \item pretend that we have parallel event data
        \item never do this
      \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}[fragile]{cgd}
<<cgd1>>= 
cgd[1:13, c('id', 'treat', 'age', 'tstart', 'tstop', 'status')]
coxph(Surv(tstart, tstop, status) ~ treat + age + 
      steroids + cluster(id), cgd)
@
\end{frame}

\begin{frame}{Hidden covariates}
  \begin{itemize}
    \item Assume an important covariate $Z$ is not in the model
    \item Single event model
      \begin{itemize}
        \item $\beta$ biased towards zero
        \item amount is proportional to $se(\gamma Z)$
      \end{itemize}
    \item Multiple event model
      \begin{itemize}
        \item stratify by number of prior events, or
        \item add number prior events as a covariate
        \item $\beta$ is severely biased,  and can actually change sign
      \end{itemize}
    \item A random effect per subject can help
  \end{itemize}
\end{frame}

\begin{frame}{AG simplicity}
  \begin{itemize}
    \item For many studies, the coefficient(s) from an AG model often have
      the same interpretation as an ordinary Cox model
    \item higher rate $\leftrightarrow$ shorter time to next event
    \item Cumulative hazard = E(number of events so far)
    \item Survival curve = Pr(no events at all) is more complex, 
      but often not of interest
  \end{itemize}
\end{frame}