\documentclass{report}[11pt]
\usepackage{Sweave}
\usepackage{amsmath}
\addtolength{\textwidth}{1in}
\addtolength{\oddsidemargin}{-.5in}
\setlength{\evensidemargin}{\oddsidemargin}
%\VignetteIndexEntry{The survival package}

\SweaveOpts{keep.source=TRUE, fig=FALSE}
% Ross Ihaka suggestions
\DefineVerbatimEnvironment{Sinput}{Verbatim} {xleftmargin=2em}
\DefineVerbatimEnvironment{Soutput}{Verbatim}{xleftmargin=2em}
\DefineVerbatimEnvironment{Scode}{Verbatim}{xleftmargin=2em}
\fvset{listparameters={\setlength{\topsep}{0pt}}}
\renewenvironment{Schunk}{\vspace{\topsep}}{\vspace{\topsep}}

% I had been putting figures in the figures/ directory, but the standard
%  R build script does not copy it and then R CMD check fails
\SweaveOpts{prefix.string=surv,width=6,height=4}

\newcommand{\myfig}[1]{\includegraphics[height=!, width=\textwidth]
                        {surv-#1.pdf}}

\newcommand{\bhat}{\hat \beta}        %define "bhat" to mean "beta hat"
\newcommand{\Mhat}{\widehat M}        %define "Mhat" to mean M-hat
\newcommand{\zbar}{\bar Z}
\newcommand{\lhat}{\hat \Lambda}
\newcommand{\Ybar}{\overline{Y}}
\newcommand{\Nbar}{\overline{N}}
\newcommand{\Vbar}{\overline{V}}
\newcommand{\yhat}{\hat y}
\newcommand{\code}[1]{\texttt{#1}}
\newcommand{\co}[1]{\texttt{#1}}
\newcommand{\twid}{\ensuremath{\sim}}
\newcommand{\Lhat}{\hat\Lambda}


\setkeys{Gin}{width=\textwidth}
<<echo=FALSE>>=
options(continue="  ", width=70)
options(SweaveHooks=list(fig=function() par(mar=c(4.1, 4.1, .3, 1.1))))
pdf.options(pointsize=10) #text in graph about the same as regular text
options(contrasts=c("contr.treatment", "contr.poly")) #ensure default
library("survival")
palette(c("#000000", "#D95F02", "#1B9E77", "#7570B3", "#E7298A", "#66A61E"))

# These functions are used in the document, but not discussed until the end
crisk <- function(what, horizontal = TRUE, ...) {
    nstate <- length(what)
    connect <- matrix(0, nstate, nstate,
                      dimnames=list(what, what))
    connect[1,-1] <- 1  # an arrow from state 1 to each of the others
    if (horizontal) statefig(c(1, nstate-1),  connect, ...)
    else statefig(matrix(c(1, nstate-1), ncol=1), connect, ...)
}

state3 <- function(what, horizontal=TRUE, ...) {
    if (length(what) != 3) stop("Should be 3 states")
    connect <- matrix(c(0,0,0, 1,0,0, 1,1,0), 3,3,
                      dimnames=list(what, what))
    if (horizontal) statefig(1:2, connect, ...)
    else statefig(matrix(1:2, ncol=1), connect, ...)
}

state4 <- function() {
    sname <- c("Entry", "CR", "Transplant", "Transplant")
    layout <- cbind(c(1/2, 3/4, 1/4, 3/4),
                    c(5/6, 1/2, 1/2, 1/6))
    connect <- matrix(0,4,4, dimnames=list(sname, sname))
    connect[1, 2:3] <- 1
    connect[2,4] <- 1
    statefig(layout, connect)
}

state5 <- function(what, ...) {
    sname <- c("Entry", "CR", "Tx", "Rel", "Death")
    connect <- matrix(0, 5, 5, dimnames=list(sname, sname))
    connect[1, -1] <- c(1,1,1, 1.4)
    connect[2, 3:5] <- c(1, 1.4, 1)
    connect[3, c(2,4,5)] <- 1
    connect[4, c(3,5)]  <- 1
    statefig(matrix(c(1,3,1)), connect, cex=.8,...)
}
@

\title{A package for survival analysis in R}
\author{Terry Therneau}
\begin{document}
\maketitle
\clearpage
\tableofcontents

\chapter{Introduction}
\section{History}
Work on the survival package began in 1985 in connection with the analysis
of medical research data, without any realization at the time that the
work would become a package.
Eventually, the software was placed on the Statlib repository hosted by
Carnegie Mellon University. 
Multiple version were released in this fashion but I don't have a list of
the dates --- version 2 was the first to
make use of the \code{print} method that was introduced in `New S' in 1988,
which places that release somewhere in 1989.
The library was eventually incorporated directly in S-Plus, and from there it
became a standard part of R.

I suspect that 
one of the primary reasons for the package's success is that all of the
functions have been written to solve real analysis questions that arose from
real data sets; theoretical issues were explored when necessary but they 
have never played a leading role.  
As a statistician in a major medical center, the central focus of my department
is to advance medicine; statistics is a tool to that end.
This also highlights one of the deficiencies of the package: if a particular
analysis question has not yet arisen in one of my studies then the survival 
package will also have nothing to say on the topic. 
Luckily, there are many other R packages that build on or extend
the survival package, and anyone working in the field (the author included) 
can expect to use more packages than just this one.
I certainly never foresaw that the library would become as popular 
as it has.

This vignette is an introduction to version 3.x of the survival package. 
We can think of versions 1.x as the S-Plus era and 2.1 -- 2.44 as maturation of
the package in R.
Version 3 had 4 major goals.
\begin{itemize} 
  \item Make multi-state curves and models as easy to use as an ordinary
    Kaplan-Meier and Cox model.
  \item Deeper support for absolute risk estimates.
  \item Consistent use of robust variance estimates.
  \item Clean up various naming inconsistencies that have arisen over time.
\end{itemize}

With over 600 dependent packages in 2019, not counting Bioconductor, other
guiding lights of the change are
\begin{itemize}
  \item We can't do everything (so don't try).
  \item Allow other packages to build on this one.  That means clear 
    documentation of all of the results that are produced, the use of simple
    S3 objects that are easy to manipulate, and setting up many 
    of the routines as a pair.  For example, \code{concordance} and 
    \code{concordancefit}; the former is the user front end and the latter does 
    the actual work.  Other package authors might want to access the lower level
    interface, while accepting the penalty of fewer error checks. 
  \item Don't mess it up!
\end{itemize}

This meant preserving the current argument names as much as possible.
Appendix \ref{sect:changes} summarizes changes that were made which are not
backwards compatible.

The two other major changes are to collapse many of vignettes into this single
large one, and the parallel creation of an actual book. 
Documentation is an ongoing process, and there are still things the package
can do which are not well described. That said,  
we've recognized that the package needs more than a vignette.  
With the book's (eventual) appearance this vignette can also
be more brief, essentially leaving out a lot of the theory.

Version 3 will not appear all at once, however; it will take some time to get 
all of the documentation sorted out in the way that we like.

\section{Survival data}
The survival package is concerned with time-to-event analysis.
Such outcomes arise very often in the analysis of medical data:
time from chemotherapy to tumor recurrence, the durability of a joint
replacement, recurrent lung infections in subjects with cystic fibrosis,
the appearance of hypertension, hyperlipidemia and other comorbidities 
of age, and of course death itself, from which the overall label of
``survival'' analysis derives.  
A key principle of all such studies is that ``it takes time to observe
time'', which in turn leads to two of the primary challenges.
\begin{enumerate}
  \item Incomplete information.  At the time of an analysis, not everyone
    will have yet had the event.  This is a form of partial information
    known as \emph{censoring}: if a particular subject was enrolled in a
    study 2 years ago, and has not yet had an event at the time of
    analysis, we only know that their time to event is $>2$ years.
  \item Dated results.  In order to report 5 year survival, say, from a 
    treatment, patients need to be enrolled and then followed for 5+ years.
    By the time recruitment and follow-up is finished, analysis done, 
    the report finally published the treatment in question might be 8
    years old and considered to be out of date.  This leads to a tension
    between early reporting and long term outcomes.
\end{enumerate}

\begin{figure}
<<states, fig=TRUE, echo=FALSE>>=
oldpar <- par(mar=c(.1, .1, .1, .1), mfrow=c(2,2))
sname1 <- c("Alive", "Dead")
cmat1 <- matrix(c(0,0,1,0), nrow=2, 
                dimnames=list(sname1, sname1))
statefig(c(1,1), cmat1)

sname2 <- c("0", "1", "2", "...")
cmat2 <- matrix(0, 4,4, dimnames= list(sname2, sname2))
cmat2[1,2] <- cmat2[2,3] <- cmat2[3,4] <- 1
statefig(c(1,1,1,1), cmat2, bcol=c(1,1,1,0))

sname3 <- c("Entry", "Transplant", "Withdrawal", "Death")
cmat3 <- matrix(0, 4,4, dimnames=list(sname3, sname3))
cmat3[1, -1] <- 1
statefig(c(1,3), cmat3)

sname4 <- c("Health", "Illness", "Death")
cmat4 <- matrix(0, 3, 3, dimnames = list(sname4, sname4))
cmat4[1,2] <- cmat4[2,1] <- cmat4[-3, 3] <- 1
statefig(c(1,2), cmat4, offset=.03)

par(oldpar)
@
\caption{Four multiple event models.}
\label{fig:multi}
\end{figure}

Survival data is often represented as
a pair $(t_i, \delta_i)$ where $t$ is the time until endpoint or last
follow-up, and $\delta$ is a 0/1 variable with 0= ``subject was censored at
$t$'' and 1 =``subject had an event at $t$'',
or in R code as \code{Surv(time, status)}.  
The status variable can be logical, e.g., \code{vtype=='death'} where 
\code{vtype} is a variable in the data set.  
An alternate view is to think of time to event data as a multi-state process
as is shown in figure \ref{fig:multi}. 
The upper left panel is simple survival with two states of alive and dead,
``classic'' survival analysis.
The other three panels show repeated events of the same type (upper right)

competing risks for subjects on a liver transplant waiting list(lower left) 
and the illness-death model (lower right).
In this approach interest normally centers on the transition rates or hazards
(arrows) from state to state (box to box).
For simple survival the two multistate/hazard and the time-to-event viewpoints 
are equivalent, and we will move
freely between them, i.e., use whichever viewpoint is handy at the moment.
When there more than one transition the rate approach is particularly useful.

The figure also displays a 2 by 2 division of survival data sets, one that
will be used to organize other subsections of this document.
\begin{center}
  \begin{tabular}{l|cc}
   & One event & Multiple events \\
   & per subject& per subject \\ \hline
 One event type & 1 & 2 \\
 Multiple event types & 3 & 4
\end{tabular}
\end{center}

\section{Overview}
   The summary below is purposefully very terse.  If you are familiar
with survival analysis {\em and} with other R modeling functions it will
provide a good summary.  Otherwise, just skim the section to get an overview
of the type of computations available from this package, and move on to
section 3 for a fuller description.

\begin{description}
\item[Surv()] A {\em packaging} function; like I() it doesn't
    transform its argument.  This is used for the left hand side of all the
    formulas.
\begin{itemize}
\item    \code{Surv(time, status)}  --  right censored data
\item    \code{Surv(time,  endpoint=='death')} -- right censored data, where the
  status variable is a character or factor
\item    \code{Surv(t1, t2, status)} -- counting process data
\item    \code{Surv(t1, ind, type='left')} -- left censoring
\item    \code{Surv(time, fstat} -- multiple state data, fstat is a factor
\end{itemize}
\item[aareg] Aalen's additive regression model.
  \begin{itemize}
    \item The \code{timereg} package is a much more comprehensive implementation
      of the Aalen model, so this document will say little about \code{aareg}
  \end{itemize}
\item[coxph()] Cox's proportional hazards model.
\begin{itemize}
\item    \code{coxph(Surv(time, status) {\twid}x, data=aml)}     --
                        standard Cox model
\item    \code{coxph(Surv(t1, t2, stat) {\twid} (age + surgery)* transplant)} --
                           time dependent covariates.
\item    \code{y <- Surv(t1, t2, stat)} \\
         \code{coxph(y {\twid} strata(inst) * sex + age + treat)} --  Stratified model, with a
                            separate baseline per institution, and institution
                            specific effects for sex.
\item    \code{coxph(y {\twid} offset(x1) + x2)} --     force in a known term,
            without estimating a coefficient for it.
\end{itemize}
\item[cox.zph] Computes a test of proportional hazards for the fitted
Cox model.
\begin{itemize}
\item    \code{zfit <- cox.zph(coxfit); plot(zfit)}
\end{itemize}
\item[pyears] Person-years analysis
\item[survdiff] One and k-sample versions of the Fleming-Harrington $G^\rho$
 family.
    Includes the logrank and Gehan-Wilcoxon as special cases.
\begin{itemize}
\item    \code{survdiff(Surv(time, status) {\twid} sex + treat)} --  Compare
the 4 sub-groups formed by sex and treatment combinations.
\item    \code{survdiff(Surv(time, status) {\twid} offset(pred))} -  One-sample test
\end{itemize}
\item[survexp] Predicted survival for an age and sex matched cohort of
    subjects, given a baseline matrix of known hazard
    rates for the population.  Most often these are US mortality tables,
    but we have also used local tables for stroke rates.
\begin{itemize}
\item    \code{survexp(entry.dt, birth.dt, sex)} --  Defaults to
                 US white, average cohort survival
\item    \code{pred <- survexp(entry, birth, sex, futime, type='individual')}  Data to enter into
            a one sample test for comparing the given group to a
            known population.
\end{itemize}
\item[survfit] Fit a survival curve.
\begin{itemize}
\item    \code{survfit(Surv(time, status))} --        Simple Kaplan-Meier
\item    \code{survfit(Surv(time, status) {\twid} rx + sex)} --  Four groups
\item    \code{fit <- coxph(Surv(time, stat) {\twid} rx + sex)} \\
         \code{survfit(fit, list(rx=1, sex=2))} --    Predict curv
\end{itemize}
\item[survreg]  Parametric survival models.
\begin{itemize}
\item \code{survreg(Surv(time, stat) {\twid} x, dist='loglogistic')} - Fit a
        log-logistic distribution.
\end{itemize}
\item[Data set creation]
  \begin{itemize}
    \item \code{survSplit} break a survival data set into disjoint portions
      of time
    \item \code{tmerge} create survival data sets with time-dependent covariates
      and/or multiple events
    \item \code{survcheck} sanity checks for survival data sets
  \end{itemize}
\end{description}

\section{Mathematical Notation}
We start with some mathematical background and notation, simply because it
will be used later.
A key part of the computations is the notion of a \emph{risk set}.  
That is, in time to event analysis a given subject will only be under 
observation for a specified time.
Say for instance that we are interested in the patient experience after a 
certain treatment, then a patient recruited on March 10 1990 and followed
until an analysis date of June 2000 will have 10 years of potential follow-up,
but someone who recieved the treatment in 1995 will only have 5 years
at the analysis date.
Let $Y_i(t), \,i=1,\ldots,n$ be the indicator that subject $i$ is at 
risk and under observation at time t.
Let $N_i(t)$ be the step function for the ith subject, which counts
the number of ``events'' for that subject up to time t.  
There might me things that can happen multiple times such as rehospitalization,
or something that only happens once such as death.
The total number of events that have occurred up to time $t$ will be
$\Nbar(t) =\sum N_i(t)$, and the number of subjects at risk at time $t$ will
be $\Ybar(t) = \sum Y_i(t)$.
Time-dependent covariates for a subject are the vector $X_i(t)$.
It will also be useful to define $d(t)$ as the number of deaths that occur
exactly at time $t$.


\chapter{Survival curves}
\section{One event type, one event per subject}
\index{function!survfit}%
\index{survival curves!one event}
The most common depiction of survival data is the Kaplan-Meier curve,
which is a product of survival probabilities:
\begin{equation}
    \hat S_{KM}(t) = \prod_{s < t} \frac{\Ybar(ts) - d(s)}{\Ybar(s)}\,.
\end{equation}
Graphically, the Kaplan-Meier survival curve appears as a step function with
a drop at each death.  Censoring times are often marked on the plot as
``$+$'' symbols.
KM curves are created with the \code{survfit} function.
The left-hand side of the formula will be a Surv object and the right hand
side contains one or more categorical variables that will divide the 
observations into groups.  For a single curve use $\sim 1$
as the right hand side.

<<survfit1>>=
fit1 <- survfit(Surv(futime, fustat) ~ resid.ds, data=ovarian)
print(fit1, rmean= 730)

summary(fit1, times= (0:4)*182.5, scale=365)
@ 

The default printout is very brief, only one line per curve, showing the
number of observations, number of events, median survival, and
optionally the restricted mean survival time (RMST) in each of
the groups.  In the above case we used the value at 2.5 years = 913 days
as the upper threshold for the RMST, the value of 453 for females 
represents an average survival for 453 of the next 913 days after enrollment
in the study.
The summary function gives a more complete description of the curve,
in this case we chose to show the values every 6 months for the first two years.
In this case the number of events (\code{n.event}) column is the number of
deaths in the interval between two time points, all other columns reflect
the value at the chosen time point.

Arguments for the survfit function include the usual
\code{data}, \code{weights}, \code{subset} and \code{na.action} 
arguments common to modeling formulas.
A further set of arguments have to do with standard errors and confidence
intervals, defaults are shown in parenthesis.


\begin{itemize}
  \item se.fit (TRUE): compute a standard error of the estimates.  
    In a few rare circumstances omitting the standard error  can save 
    computation time.
  \item conf.int (.95): the level of confidence interval, or FALSE if
    intervals are not desired.
  \item conf.type ('log'): transformation to be used in computing the
    confidence intervals. 
  \item conf.lower ('usual'): optional modification of the lower interval.
\end{itemize}

\index{survfit!confidence intervals}
For the default \code{conf.type} the confidence intervals are computed as
$
   \exp[\log(p) \pm 1.96 {\rm se}(\log(p))]
$
rather than the direct formula of $p \pm 1.96 {\rm(se)}(p)$, where 
$p = S(t)$ is the survival probability.
Many authors have investigated the behavior of transformed intervals, and a 
general conclusion is that the direct intervals do not behave well, particularly
near 0 and 1, while all the others are acceptable. 
Which of the choices of log, log-log, or logit is ``best'' depends on the 
details of any particular simulation study,
all are available as options in the function.
(The default corresponds to the most recent paper the author had read, at
the time the default was chosen; a current meta review might give a slight edge
to the log-log option.)

The \code{conf.lower} option is mostly used for graphs.  If a study has a 
long string of censored observations, it is intuitive that the precision
of the estimated survival must be decreasing due to a smaller sample size,
but the formal standard error will not change until the next death.
This option widens the confidence interval between death times, proportional
to the number at risk, giving a visual clue of the decrease in $n$.
There is only a small (and decreasing) population of users who make use of this.

\index{function!plot.survfit}
The most common use of survival curves is to plot them, as shown below.
<<survfit2, fig=TRUE>>=
plot(fit1, col=1:2, xscale=365.25, lwd=2, mark.time=TRUE,
     xlab="Years since study entry", ylab="Survival")
legend(750, .9, c("No residual disease", "Residual disease"), 
       col=1:2, lwd=2, bty='n')
@ 

Curves will appear in the plot in the same order as they are listed by 
\code{print}; this is a quick way to remind ourselves of which subset maps
to each color or linetype in the graph.
Curves can also be labeled using the \code{pch} option to place marks on 
the curves.
The location of the marks is controlled by the \code{mark.time} option
which has a default value of FALSE (no marks).  A vector of numeric values
specifies the location of the marks, optionally a value of 
\code{mark.time=TRUE} will cause a 
mark to appear at each censoring time; this can result in far too many marks
if $n$ is large, however.
By default confidence intervals are included on the plot of there is a single
curve, and omitted if there is more than one curve.  

Other options:
\begin{itemize}
  \item xaxs('r')  It has been traditional to have survival curves touch 
    the left axis
    (I will not speculate as to why).
    This can be accomplished using \code{xaxs='S'}, which was the default
    before survival 3.x.  The current default is the standard R style,
    which leaves space between the curve and the axis.
  \item The follow-up time in the data set is in days.  This is very common in
    survival data, since it is often generated by subtracting two dates.  
    The xscale argument has been used to convert to years.
    Equivalently one could have used \code{Surv(futime/365.25, status)} in the
    original call to convert all output to years.
    The use of \code{scale} in print and summary and \code{xscale} in plot
    is a historical mistake.
  \item Subjects who were not followed to death are \emph{censored} at the time
    of last contact.  These appear as + marks on the curve.  
    Use the \code{mark.time} option to suppress or change the symbol.
  \item By default pointwise 95\% confidence curves will be shown if the plot
    contains a single curve; they are by default not shown if the plot
    contains 2 or more groups. 
  \item Confidence intervals are normally created as part of the \code{survfit}
    call.  However, they can be omitted at that point, and added later by
    the plot routine.
  \item There are many more options, see \code{help('plot.survfit')}.
\end{itemize}

The result of a \code{survfit} call can be subscripted.  This is useful
when one wants to plot only a subset of the curves.
Here is an example using a larger data set collected on a set of
patients with advanced lung cancer \cite{Loprinzi94}, which better
shows the impact of the Eastern Cooperative Oncolgy Group (ECOG) score.
This is a simple measure of patient mobility:
\begin{itemize}
  \item 0: Fully active, able to carry on all pre-disease performance 
    without restriction
  \item 1:Restricted in physically strenuous activity but ambulatory and 
    able to carry out work of a light or sedentary nature, e.g.,
    light house work, office work
  \item 2: Ambulatory and capable of all selfcare but unable to carry out 
    any work activities. Up and about more than 50\% of waking hours
  \item 3:  Capable of only limited selfcare, confined to bed or chair 
    more than 50\% of waking hours 
  \item 4: Completely disabled. Cannot carry on any selfcare. 
    Totally confined to bed or chair
\end{itemize}

\index{survfit!subscript}
<<survfit3, fig=TRUE>>=
fit2 <- survfit(Surv(time, status) ~  sex + ph.ecog, data=lung)
fit2
plot(fit2[1:3], lty=1:3, lwd=2, xscale=365.25, fun='event',
       xlab="Years after enrollment", ylab="Survival")
legend(550, .6, paste("Performance Score", 0:2, sep=' ='), 
       lty=1:3, lwd=2, bty='n')
text(400, .95, "Males", cex=2)
@ 
The argument \code{fun='event'} has caused the death rate $D = 1-S$ to be
plotted.  
The choice between the two forms is mostly personal, but some areas
such as cancer trial always plot survival (downhill) and other such
as cardiology prefer the event rate (uphill).

\paragraph{Mean and median}
For the Kaplan-Meier estimate,
the estimated mean survival is undefined if the last observation is censored.
One solution, used here, is to redefine the estimate to be zero beyond the
last observation.  This gives an estimated mean that is biased towards zero,
but there are no compelling alternatives that do better.  With this
definition, the mean is estimated as
$$   \hat \mu = \int_0^T \hat S(t) dt
$$
where $\hat S$ is the Kaplan-Meier estimate
and $T$ is the maximum observed follow-up time in the study.
The variance of the mean is
$$
 {\rm var}(\hat\mu) = \int_0^T \left ( \int_t^T \hat S(u) du \right )^2
                         \frac{d \Nbar(t)}{\Ybar(t) (\Ybar(t) - \Nbar(t))}
$$
where $\bar N = \sum N_i$ is the total counting process and $\bar Y = \sum Y_i$
is the number at risk.

The sample median is defined as the first time at which $\hat S(t) \le .5$.  Upper
and lower confidence intervals for the median are defined in terms of
the confidence intervals for $S$: the upper confidence interval is the first
time at which the upper confidence interval for $\hat S$ is $\le .5$.  This
corresponds to drawing a horizontal line at 0.5 on the graph of the survival
curve, and using intersections of this line with the curve and its upper
and lower confidence bands.
  In the very rare circumstance that the survival curve has a horizontal
portion at exactly 0.5 (e.g., an even number of subjects and no censoring
before the median) then the average time of that horizonal segment is used.
This agrees with
usual definition of the median for even $n$ in uncensored data.

\section{Repeated events}
\index{survival curves!repeated events}
This is the case of a single event type, with the possibility of multiple
events per subject.
Repeated events are quite common in industrial reliability
data.
As an example, consider a data set on the replacement times of
diesel engine valve seats.
The simple data set \code{valveSeats} contains an engine identifier, time, and
a status of 1 for a replacement and 0 for the end of the inspection interval
for that engine; the data is sorted by time within engine.
To accommodate multiple events for an engine we need to rewrite the data in
terms of time intervals.
For instance, engine 392 had repairs on days 258 and 328 and
a total observation time of 377 days, and will be represented as three
intervals of (0, 258), (258, 328) and (328, 377) thus:
<<echo=FALSE>>=
data.frame(id=rep(392,3), time1=c(0, 258, 328), time2=c(258, 328, 377),
           status=c(1,1,0))
@ 
Intervals of length 0 are illegal for \code{Surv} objects.  
There are 3 engines that
had 2 valves repaired on the same day, which will create such an interval.
To work around this move the first repair back
in time by a tiny amount.  

<<survival4>>=
vdata <- with(valveSeat, data.frame(id=id, time2=time, status=status))
first <- !duplicated(vdata$id)
vdata$time1 <- ifelse(first, 0,  c(0, vdata$time[-nrow(vdata)]))
double <- which(vdata$time1 == vdata$time2)
vdata$time1[double] <- vdata$time1[double] -.01
vdata$time2[double-1] <- vdata$time1[double]
vdata[1:7, c("id", "time1", "time2", "status")]
survcheck(Surv(time1, time2, status) ~ 1, id=id, data=vdata)
@ 

Creation of (start time, end time) intervals is a common data manipulation 
task when there are multiple events per subject. 
A later chapter will discuss the \code{tmerge} function, which is very often
useful for this task. 
The \code{survcheck} function can be used as check for some of more common
errors that arise in creation; 
it also will be covered in more detail in a later section.
(The output will be also be less cryptic for later cases, where the states 
have been labeled.) 
In the above data, the engines could only participate in 2 kinds of transitions:
from an unnamed initial state to a repair, (s0) $\rightarrow$ 1, or from one
repair to another one, 1 $\rightarrow$ 1, or reach end of follow-up.
The second table printed by \code{survcheck} tells us that 17 engines had 0
transitions to state 1, i.e., no valve repairs before the end of observation
for that engine, 9 had 1 repair, etc.
Perhaps the most important message is that there were no 
warnings about suspicious data.

We can now compute the survival estimate.  When there are multiple observations
per subject the \code{id} statement is necessary.
(It is a good idea any time there \emph{could} be multiples, even if there are none,
as it lets the underlying routines check for doubles.)

<<survival5, fig=TRUE>>=
vfit <- survfit(Surv(time1, time2, status) ~1, data=vdata, id=id)
plot(vfit, cumhaz=TRUE, xlab="Days", ylab="Cumulative hazard")
@ 

By default, the \code{survfit} routine computes both the survival and
the Nelson cumulative hazard estimate
$$
  \hat\Lambda(t) = \sum_{i-1}^n \int_0^t \frac{dN_i(s)}{\Ybar (s)}
$$
Like the KM, the Nelson estimate is a step function, it starts at zero and has a
step of size $d(t)/\Ybar(t)$ at each death.
To plot the cumulative hazard the \code{cumhaz} argument of
\code{survfit} is used.
\index{cumulative hazard function}%
\index{mean cumulative function|see cumulative hazard function}  
In multi-event data, the cumulative hazard is an estimate of the expected
\emph{number} of events for a unit that has been observed for the 
given amount of time, whereas the survival $S$ estimates the probability that
a unit has had 0 repairs.
The cumulative hazard is the more natural quantity to plot in such 
studies; in reliability analysis it is also known as the 
\emph{mean cumulative function}.  

The estimate is also important in multi-state models.
An example is the occurene of repeated infections in children with
chronic granultomous disease, as found in the \code{cgd} data set.
<<cgd1d, fig=TRUE>>=
cgdsurv <- survfit(Surv(tstart, tstop, status) ~ treat, cgd, id=id)
plot(cgdsurv, cumhaz=TRUE, col=1:2, conf.times=c(100, 200, 300, 400),
     xlab="Days since randomization", ylab="Cumulative hazard")
@ 

\section{Competing risks}
The case of multiple event types, but only one event per subject is commonly
known as competing risks. 
We do not need the (time1, time2) data form for this case, since each subject
has only a single outcome, but we do need a way to identify different outcomes.
In the prior sections, \code{status} was either a logical or 0/1 numeric variable
that represents censoring (0 or FALSE) or an event (1 or TRUE),
and the result of \code{survfit} was a single survival curve for each group.
For competing risks data 
\code{status} will be a factor;
the first level of the factor is used to code censoring while
the remaining ones are possible outcomes.


\subsection{Simple example}
Here is a simple competing risks example where the three endpoints are
labeled as a, b and c.
<<simple1>>=
crdata <- data.frame(time= c(1:8, 6:8),
                     endpoint=factor(c(1,1,2,0,1,1,3,0,2,3,0),
                                     labels=c("censor", "a", "b", "c")),
                     istate=rep("entry", 11),
                     id= LETTERS[1:11])
tfit  <- survfit(Surv(time, endpoint) ~ 1, data=crdata, id=id, istate=istate)
dim(tfit)
summary(tfit)
@
The resulting object \code{tfit} contains an estimate of $P$(state),
the probability of being in each state at each time $t$.
$P$ is a matrix with one row for each time and one column for
each of the four states a--c and ``still in the starting state''.
By definition each row of $P$ sums to 1. 
We will also use the notation $p(t)$ where $p$ is a vector with one element
per state and $p_j(t)$ is the fraction in state $j$ at time $t$.
The plot below shows all 4 curves.
(Since they sum to 1 one of the 4 curves is redundant, often the entry state
is omitted since it is the least interesting.)  
In the \code{plot.survfit} function there is the argument \code{noplot="(s0)"}
which indicates that curves for state (s0) will not be plotted.  
If we had not specified \code{istate} in the call to \code{survfit}, 
the default label for the initial state would have been ``s0'' and t
he solid curve would not have been plotted.

<<fig=TRUE>>=
plot(tfit, col=1:4, lty=1:4, lwd=2, ylab="Probability in state")
@ 

The resulting \code{survfms} object appears as a matrix and can be
subscripted as such, with a column for each state and rows for each
group,
each unique combination of values on the right hand side of the formula
is a group or strata.
This makes it simple to display a subset of the curves using plot
or lines commands.
The entry state in the above fit, for instance, can be displayed with
\code{plot(tfit[,1])}.

<<>>=
dim(tfit)
tfit$states
@

The curves are computed using the Aalen-Johansen estimator. 
This is an important concept, and so we work it out below.

1. The starting point is the column vector
      $p(0) = (1, 0, 0, 0)$, everyone starts in the first state.
    
2. At time 1, the first event time, form the 4 by 4 transition matrix $T_1$
\begin{align*}
 T(1)  &= \left( \begin{array}{cccc}
             10/11 & 1/11 & 0/11 & 0/11 \\
              0    & 1    &  0   & 0    \\
              0    & 0    &  1   & 0    \\
              0    & 0    &  0   & 1 \end{array} \right )
 p(1) &= p(0)T_1
 \end{align*}

The first row of $T(1)$ describes the disposition of everyone who is 
in state 1 and under observation at time 1: 10/11 stay in state 1 and 
1 subject transitions to state a.  
There is no one in the other 3 states, so rows 2--4 are technically
undefined; use a default ``stay in the same state'' row which has 1 on
the diagonal.
(Since no one ever leaves states a, b, or c, the bottom three rows of $T$ will 
continue to have this form.)

3. At time 2 the first row will be (9/10, 0, 1/10, 0), and
$p(2) = p(1)T(2) = p(0) T(1) T(2)$.

Continue this until the last event time.  
At a time point with only censoring, such as time 4, $T$ would be the identity
matrix.  

It is straightforward to show that when there are only two states of
alive -> dead, then $p_1(t)$ replicates 
the Kaplan-Meier computation.  
For competing risks data such as the simple example above, $p(t)$ replicates
the cumulative incidence (CI) estimator.
That is, both the KM and CI are both special cases of the Aalen-Johansen.  
The AJ is more general, however; a given subject can have multiple 
transitions from state to state, including transitions to a state that was 
visited earlier.


\subsection{Monoclonal gammopathy}
\label{mgusplot}
The \code{mgus2} data set contains information of 1384 subjects who were
who were found to have a particular pattern on a laboratory test
(monoclonal gammopathy of undetermined significance or MGUS).
The genesis of the study was a suspicion that such a result might indicate a
predisposition to plasma cell malignancies such a multiple myeloma;
subjects were followed forward to assess whether an excess did occur.
The mean age at diagnosis is 63 years, so death from other causes will be
an important competing risk.
Here are a few observations of the data set, one of which experienced
progression to a plasma cell malignancy.
<<mgus1>>=
mgus2[55:59, -(4:7)]
@ 

To generate competing risk curves create a new (etime, event) pair.
Since each subject has at most 1 transition, we do not need a multi-line
(time1, time2) dataset.

<<mgus2, fig=TRUE>>=
event <- with(mgus2, ifelse(pstat==1, 1, 2*death))
event <- factor(event, 0:2, c("censored", "progression", "death"))
etime <- with(mgus2, ifelse(pstat==1, ptime, futime))
crfit <- survfit(Surv(etime, event) ~ sex, mgus2)
crfit

plot(crfit, col=1:2,  noplot="",
     lty=c(3,3,2,2,1,1), lwd=2, xscale=12,
     xlab="Years post diagnosis", ylab="P(state)")
legend(240, .65, c("Female, death", "Male, death", "malignancy", "(s0)"),
       lty=c(1,1,2,3), col=c(1,2,1,1), bty='n', lwd=2)
@ 

There are 3 curves for females, one for each of the three states, and
3 for males. 
The three curves sum to 1 at any given time (everyone has to be somewhere),
and the default action for \code{plot.survfit} is to leave out the
``still in original state'' curve (s0) since it is usually the least
interesting, but in this case we have shown all 3.
We will return to this example when exploring models.

A common mistake with competing risks is to use the Kaplan-Meier
separately on each
event type while treating other event types as censored.
The next plot is an example of this for the PCM endpoint.
<<mgus3, fig=TRUE>>=
pcmbad <- survfit(Surv(etime, pstat) ~ sex, data=mgus2)
plot(pcmbad[2], mark.time=FALSE, lwd=2, fun="event", conf.int=FALSE, xscale=12,
     xlab="Years post diagnosis", ylab="Fraction with PCM")
lines(crfit[2,2], lty=2, lwd=2, mark.time=FALSE, conf.int=FALSE)
legend(0, .25, c("Males, PCM, incorrect curve", "Males, PCM, competing risk"),
       col=1, lwd=2, lty=c(1,2), bty='n')
@ 

There are two problems with the \code{pcmbad} fit.  
The first is that it attempts to estimate the expected occurrence of
plasma cell malignancy (PCM)
if all other causes of death were to be disallowed.
In this hypothetical world it is indeed true that many more subjects would
progress to PCM (the incorrect curve is higher), but it is also 
not a world that any of us will ever inhabit.
This author views the result in much the same light as  a discussion of
survival after the zombie apocalypse.
The second problem is that the computation for this
hypothetical case is only correct if all of the competing endpoints
are independent, a situation which is almost never true.
We thus have an unreliable estimate of an uninteresting quantity.
The competing risks curve, on the other hand,
estimates the fraction of MGUS subjects who \emph{will experience} 
PCM, a quantity sometimes known as the lifetime risk,
and one which is actually observable.

The last example chose to plot only a subset of the curves, something that is
often desirable in competing risks problems to avoid a
``tangle of yarn'' plot that simply has too many elements.
This is done by subscripting the \code{survfit} object.
For subscripting, multi-state curves behave as a matrix
with the outcomes as the second subscript. 
The columns are in order of the levels of \code{event},
i.e., as displayed by our earlier call to \code{table(event)}.   
The first subscript indexes the groups formed by the right hand side of
the model formula, and will be in the same order as simple survival curves.
Thus \code{mfit2[2,2]} corresponds to males (2) and the PCM endpoint (2).
Curves are listed and plotted in the usual matrix order of R.

<<>>=
dim(crfit)
crfit$strata
crfit$states
@

One surprising aspect of multi-state data is that hazards can be estimated
independently although probabilities cannot.
If you look at the cumulative hazard estimate from the \code{pcmbad}
fit above using, for instance, \code{plot(pcmbac, cumhaz=TRUE)} you will
find that it is identical to the cumulative hazard estimate from the joint fit.
This will arise again with Cox models.  


\section{Multi-state data}
The most general multi-state data will have multiple outcomes and
multiple endpoints per subject.
In this case, we will need to use the (time1, time2) form for each subject.
The dataset structure is similar to that for time varying
covariates in a Cox model: the time variable will be intervals $(t_1, t_2]$
which are open on the left and closed on the right,
and a given subject will have multiple lines of data.
But instead of covariates changing from line to line, in this
case the status variable changes; it
contains the state that was entered at time $t_2$.
There are a few restrictions.
\begin{itemize} 
  \item An identifier variable is needed to indicate which rows of the
    dataframe belong to each subject.  If the \code{id} argument is missing,
    the code assumes that each row of data is a separate subject, which leads
    to a nonsense estimate when there are actually multiple rows per subject. 
  \item Subjects do not have to enter at time 0 or all at the same time,
    but each must traverse a connected segment of time.  Disjoint intervals
    such as the pair $(0,5]$, $(8, 10]$ are illegal.
   \item A subject cannot change groups.  Any covariates on the right hand
     side of the formula must remain constant within subject.  (This 
     function is not
     a way to create supposed `time-dependent' survival curves.)
   \item Subjects may have case weights, and these weights may change over
     time for a subject.
\end{itemize}

The \code{istate} argument can be used to designate a subject's state 
at the start of each $t_1, t_2$ time interval.
Like variables in the formula, it is searched for in the
\code{data} argument.
If it is not present,
every subject is assumed to start in a common entry state which is given
the name ``(s0)''.  The parentheses are an echo of ``(Intercept)'' in a
linear model and show a label that was provided by the program rather than
the data.
The distribution of states just prior to the first event time is 
treated as the initial distribution of states.
In common with ordinary survival, any observation which is censored before the 
first event time has no impact on the results.


\subsection{Myeloid data}
The \code{myeloid} data set contains data from a clinical trial
in subjects with  acute myeloid leukemia.  To protect patient confidentiality
the data set in the survival package has been slightly perturbed, but 
results are essentially unchanged.
In this comparison of two conditioning regimens, the
canonical path for a subject is initial therapy $\rightarrow$
complete response (CR) $\rightarrow$
hematologic stem cell transplant (SCT) $\rightarrow$
sustained remission, followed by relapse or death.
Not everyone follows this ideal path, of course.

<<overall>>=
myeloid[1:5,]
@ 
The first few rows of data are shown above.
The data set contains the follow-up time and status at last follow-up 
for each subject, along with the time to transplant
(txtime),
complete response (crtime) or relapse after CR (rltime).
Subject 1 did not receive a transplant, as shown by the NA value,
and subject 2 did not achieve CR.

\begin{figure}
  \myfig{sfit0}
  \caption{Overall survival curves for the two treatments.}
  \label{sfit0}
\end{figure}

Overall survival curves for the data are shown in figure \ref{sfit0}.
The difference between the treatment arms A and B
is substantial.  A goal of this analysis is to better 
understand this difference.  Code to generate the 
two curves is below.

<<sfit0, echo=TRUE, fig=TRUE, include=FALSE>>=
sfit0 <- survfit(Surv(futime, death) ~ trt, myeloid)
plot(sfit0, xscale=365.25, xaxs='r', col=1:2, lwd=2,
     xlab="Years post enrollment", ylab="Survival")
legend(20, .4, c("Arm A", "Arm B"),
       col=1:2, lwd=2, bty='n')
@ 

The full multi-state data set can be created with the 
\code{tmerge} routine.
<<sfit0a, echo=TRUE>>=
mdata <- tmerge(myeloid[,1:2], myeloid,  id=id,  death= event(futime, death),
                sct = event(txtime), cr = event(crtime), 
                relapse = event(rltime))
temp <- with(mdata, cr + 2*sct  + 4*relapse + 8*death)
table(temp)
@ 

Our check shows that there is one subject who had CR and stem cell transplant
on the same day (temp=3).
To avoid length 0 intervals, we break the tie so that complete response (CR)
happens first.
(Students may be surprised to see anomalies like this, since they never appear
in textbook data sets.  In real data such issues always appear.)

<<sfit0b, echo=TRUE>>=
tdata <- myeloid  # temporary working copy
tied <- with(tdata, (!is.na(crtime) & !is.na(txtime) & crtime==txtime))
tdata$crtime[tied] <- tdata$crtime[tied] -1
mdata <- tmerge(tdata[,1:2], tdata,  id=id,  death= event(futime, death),
                sct = event(txtime), cr = event(crtime), 
                relapse = event(rltime),
                priorcr = tdc(crtime), priortx = tdc(txtime))
temp <- with(mdata, cr + 2*sct  + 4*relapse + 8*death)
table(temp)
mdata$event <- factor(temp, c(0,1,2,4,8),
                       c("none", "CR", "SCT", "relapse", "death"))

mdata[1:7, c("id", "trt", "tstart", "tstop", "event", "priorcr", "priortx")]
@ 

Subject 1 has a CR on day 44, relapse on day 113, death on day 235 and
did not receive a stem cell transplant. 
The data for the first three subjects looks good.  
Check it out a little more thoroughly using survcheck.

<<>>=
survcheck(Surv(tstart, tstop, event) ~1, mdata, id=id)
@ 

The second table shows that no single subject had more than one CR, SCT,
relapse, or death; the intention of the study was to count only the first
of each of these, so this is as expected.
Several subjects visited all four intermediate states.  
The transitions table shows 11 subjects who achieved CR \emph{after} stem
cell transplant and another 106 who received a transplant before
achieving CR, both of which are deviations from the ``ideal'' pathway.
No subjects went from death to another state (which is good).

For investigating the data we would like to add a set of alternate endpoints.
\begin{enumerate}
  \item The competing risk of CR and death, ignoring other states.  This
    is used to estimate the fraction who ever achieved a complete response.
  \item The competing risk of SCT and death, ignoring other states.
  \item An endpoint that distinguishes death after SCT from death 
    before SCT.
\end{enumerate}
Each of these can be accomplished by adding further outcome variables to 
the data set, we do not need to change the time intervals.

<<newevent>>=
levels(mdata$event)
temp1        <- with(mdata, ifelse(priorcr, 0, c(0,1,0,0,2)[event]))
mdata$crstat <- factor(temp1, 0:2, c("none", "CR", "death"))

temp2        <- with(mdata, ifelse(priortx, 0, c(0,0,1,0,2)[event]))
mdata$txstat <- factor(temp2, 0:2, c("censor", "SCT", "death"))

temp3     <- with(mdata, c(0,0,1,0,2)[event] + priortx)
mdata$tx2 <- factor(temp3, 0:3,
                    c("censor", "SCT", "death w/o SCT", "death after SCT"))
@ 

Notice the use of the \code{priorcr} variable in defining \code{crstat}.
This outcome variable treats complete response as a terminal state,
which in turn means that no further transitions are allowed after
reaching CR.  

\begin{figure}
  \myfig{curve1}
  \caption{Overall survival curves: time to death, to transplant (Tx), 
    and to complete response (CR).
    Each shows the estimated fraction of subjects who have ever reached the
    given state.  The vertical line at 2 months is for reference.
    The curves were limited to the first 48 months to more clearly show 
    early events. The right hand panel shows the state-space model for each
  pair of curves.}
  \label{curve1}
\end{figure}

This data set is the basis for our first set of curves, which are shown in
figure \ref{curve1}.
The plot overlays three separate \code{survfit} calls: standard survival
until death, complete response with death as a competing risk, and
transplant with death as a competing risk.
For each fit we have shown one selected state: the fraction
who have died, fraction ever in CR, and fraction ever to receive transplant,
respectively.
Most of the CR events happen before 2 months (the green
vertical line) and nearly all the additional CRs
conferred by treatment B occur between months 2 and 8.  
Most transplants happen after 2 months, which is consistent with the
clinical guide of transplant after CR.
The survival advantage for treatment B begins between 4 and 6 months, 
which argues that it could be at least partially a consequence of the
additional CR events.

The code to draw figure \ref{curve1} is below.  It can be separated into
5 parts:
\begin{enumerate}
  \item Fits for the 3 endpoints are simple and found in the first set of lines.
    The \code{crstat} and \code{txstat} variables are factors, which causes
    multi-state curves to be generated.
  \item The \code{layout} and \code{par} commands are used to create a
    multi-part plot with curves on the left and state space diagrams on 
    the right, and to reduce the amount of white space between them.
  \item Draw a subset of the curves via subscripting. A multi-state
    survfit object appears to the user as a matrix of curves, with one row for
    each group (treatment) and one column for each state.  The CR state is
    the second column in \code{sfit2}, for instance.  
    The CR fit was drawn first simply because it has the greatest y-axis
    range, then the other curves added using the lines command.
  \item Decoration of the plots. This includes the line types, colors,
    legend, choice of x-axis labels, etc.
  \item Add the state space diagrams.  The functions for this are
    described elsewhere in the vignette.
\end{enumerate}

<<curve1, fig=TRUE, include=FALSE>>=
# I want to have the plots in months, it is simpler to fix time
#  once rather than repeat xscale many times
tdata$futime <- tdata$futime * 12 /365.25
mdata$tstart <- mdata$tstart * 12 /365.25
mdata$tstop  <- mdata$tstop * 12 /365.25


sfit1 <- survfit(Surv(futime, death)  ~ trt, tdata) # survival
sfit2 <- survfit(Surv(tstart, tstop, crstat) ~ trt, 
                 data= mdata, id = id) # CR
sfit3 <- survfit(Surv(tstart, tstop, txstat) ~ trt, 
                 data= mdata, id =id) # SCT

layout(matrix(c(1,1,1,2,3,4), 3,2), widths=2:1)
oldpar <- par(mar=c(5.1, 4.1, 1.1, .1))

mlim   <- c(0, 48) # and only show the first 4 years
plot(sfit2[,"CR"], xlim=mlim, 
         lty=3, lwd=2, col=1:2, xaxt='n',
     xlab="Months post enrollment", ylab="Fraction with the endpoint")
lines(sfit1, mark.time=FALSE, xlim=mlim,
      fun='event', col=1:2, lwd=2)

lines(sfit3[,"SCT"], xlim=mlim, col=1:2, 
          lty=2, lwd=2)

xtime <- c(0, 6, 12, 24, 36, 48)
axis(1, xtime, xtime) #axis marks every year rather than 10 months
temp <- outer(c("A", "B"), c("CR", "transplant", "death"),  paste)
temp[7] <- ""
legend(25, .3, temp[c(1,2,7,3,4,7,5,6,7)], lty=c(3,3,3, 2,2,2 ,1,1,1),
       col=c(1,2,0), bty='n', lwd=2)
abline(v=2, lty=2, col=3)

# add the state space diagrams
par(mar=c(4,.1,1,1))
crisk(c("Entry", "CR", "Death"), alty=3)
crisk(c("Entry", "Tx", "Death"), alty=2)
crisk(c("Entry","Death"))
par(oldpar)
layout(1)
@ 

The association between a particular curve and its corresponding state space
diagram is critical.  As we will see below, many different models are
possible and it is easy to get confused.  
Attachment of a diagram directly to each curve, as was done above,
will not necessarily be day-to-day practice, but the state space should
always be foremost.  If nothing else,  draw it on a scrap of paper and tape it 
to the side of the terminal when creating a data set and plots.

\begin{figure}
  \myfig{badfit}
 \caption{Correct (solid) and invalid (dashed) estimates of the number
  of subjects transplanted.}
 \label{badfit}
\end{figure}
 
Figure \ref{badfit} shows the transplant curves overlaid with the naive KM that
censors subjects at death.  There is no difference in the initial portion as
no deaths have yet intervened, but the final portion overstates the 
transplant outcome by more than 10\%. 
\begin{enumerate}
  \item The key problem with the naive estimate is that subjects who die can 
    never have a transplant.  The result of censoring them 
    is an estimate of the ``fraction who would
    be transplanted, if death before transplant were abolished''.  This is not
    a real world quantity.
  \item In order to estimate this fictional quantity one needs to assume that
    death is uninformative with respect to future disease progression.  The
    early deaths in months 0--2, before transplant begins, are however a very
    different class of patient.  Non-informative censoring is untenable.
\end{enumerate}
We are left with an unreliable estimate of an uninteresting quantity.
Mislabeling any true state as censoring is always a mistake, one that 
will not be repeated here.
Here is the code for figure \ref{badfit}.  The use of a logical (true/false)
as the status variable in the \code{Surv} call leads to ordinary survival
calculations.
<<badfit, fig=TRUE, include=FALSE>>=
badfit <- survfit(Surv(tstart, tstop, event=="SCT") ~ trt, 
                       id=id, mdata, subset=(priortx==0))

layout(matrix(c(1,1,1,2,3,4), 3,2), widths=2:1)
oldpar <- par(mar=c(5.1, 4.1, 1.1, .1))
plot(badfit, fun="event", xmax=48, xaxt='n', col=1:2, lty=2, lwd=2,
     xlab="Months from enrollment", ylab="P(state)")
axis(1, xtime, xtime)
lines(sfit3[,2], xmax=48, col=1:2, lwd=2)
legend(24, .3, c("Arm A", "Arm B"), lty=1, lwd=2,
       col=1:2, bty='n', cex=1.2)

par(mar=c(4,.1,1,1))
crisk(c("Entry", "transplant"), alty=2, cex=1.2)
crisk(c("Entry","transplant", "Death"), cex=1.2)
par(oldpar)
layout(1)
@ 

\begin{figure}
  \myfig{cr2}
  \caption{Models for `ever in CR' and `currently in CR';
    the only difference is an additional transition.
    Both models ignore transplant.}
  \label{cr2}
\end{figure}

Complete response is a goal of the initial therapy; figure \ref{cr2}
looks more closely at this.
As was noted before arm B has an increased number of late responses.
The duration of response is also increased: 
the solid curves show the number of subjects still in response, and
we see that they spread farther apart than the dotted ``ever in response''
curves. 
The figure shows only the first eight months in order to better visualize
the details, but continuing the curves out to 48 months reveals a similar
pattern.
Here is the code to create the figure.

<<cr2, fig=TRUE, include=FALSE>>=
cr2 <- mdata$event
cr2[cr2=="SCT"] <- "none" # ignore transplants
crsurv <- survfit(Surv(tstart, tstop, cr2) ~ trt,
                  data= mdata, id=id, influence=TRUE)

layout(matrix(c(1,1,2,3), 2,2), widths=2:1)
oldpar <- par(mar=c(5.1, 4.1, 1.1, .1))
plot(sfit2[,2], lty=3, lwd=2, col=1:2, xmax=12, 
     xlab="Months", ylab="CR")
lines(crsurv[,2], lty=1, lwd=2, col=1:2)
par(mar=c(4, .1, 1, 1))
crisk( c("Entry","CR", "Death"), alty=3)
state3(c("Entry", "CR", "Death/Relapse"))

par(oldpar)
layout(1)
@ 

The above code created yet another event
variable so as to ignore transitions to the transplant state.
They become a non-event, in the same way that extra lines with
a status of zero are used to create time-dependent covariates for
a Cox model fit.

The \code{survfit} call above included the \code{influence=TRUE}
argument, which causes the influence array to be calculated and
returned.  
It contains, for each subject, that subject's influence on the
time by state matrix of results and allows for calculation of the
standard error of the restricted mean.  We will return to this
in a later section.
<<cr2b>>=
print(crsurv, rmean=48, digits=2)
@ 

<<cr2c, echo=FALSE>>=
temp <- summary(crsurv, rmean=48)$table
delta <- round(temp[4,3] - temp[3,3], 2)
@ 

@ 
The restricted mean time in the CR state is extended by 
\Sexpr{round(temp[4,3], 1)} - \Sexpr{round(temp[3,3], 1)} =
\Sexpr{delta} months.  
A question which immediately gets asked is whether this difference
is ``significant'', to which there are two answers. 
The first and more important is to ask whether 5 months is an important gain
from either a clinical or patient perspective. 
The overall restricted mean survival for the study is approximately
30 of the first 48 months post entry (use \code{print(sfit1, rmean=48)});
on this backdrop an extra 5 months in CR might or might not be an
meaningful advantage from a patient's point of view.
The less important answer is to test whether the apparent gain is sufficiently
rare from a mathematical point of view, i.e., ``statistical'' significance.
The standard errors of the two values are
\Sexpr{round(temp[3,4],1)} and \Sexpr{round(temp[4,4],1)},
and since they are based
on disjoint subjects the values are independent, leading to a standard error
for the difference of $\sqrt{1.1^2 + 1.1^2} = 1.6$.
The 5 month difference is more than 3 standard errors, so highly significant.

\begin{figure}
  \myfig{txsurv}
  \caption{Transplant status of the subjects, broken down by whether it
    occurred before or after CR.}
  \label{txsurv}
\end{figure}

In summary
  \begin{itemize}
    \item Arm B adds late complete responses (about 4\%); there are 
      206/317 in arm A vs. 248/329 in arm B.
    \item The difference in 4 year survival is about 6\%.
    \item There is approximately 2 months longer average duration of CR (of 48).
  \end{itemize}

CR $\rightarrow$ transplant is the target treatment path for a
patient; given the improvements listed above
why does figure \ref{curve1} show no change in the number transplanted?
Figure \ref{txsurv} shows the transplants broken down by whether this
happened before or after complete response.
Most of the non-CR transplants happen by 10 months.
One possible explanation is that once it is apparent to the
patient/physician pair that CR is not going to occur, they proceed forward with
other treatment options.  
The extra CR events on arm B, which occur between 2 and 8 months, lead to 
a consequent increase in transplant as well, but at a later time of 12--24 
months: for a subject in CR we can perhaps afford to defer the transplant date.

Computation is again based on a manipulation of the event variable: in this
case dividing the transplant state into two sub-states based on the presence
of a prior CR.  The code makes use of the time-dependent covariate 
\code{priorcr}.
(Because of scheduling constraints within a hospital it is unlikely that
a CR that is within a few days prior to transplant could have affected the
decision to schedule a transplant, however.  An alternate breakdown that
might be useful would be ``transplant without CR or within 7 days after CR''
versus those that are more than a week later. 
There are many sensible questions that can be asked.)

<<txsurv, fig=TRUE, include=FALSE>>=
event2 <- with(mdata, ifelse(event=="SCT" & priorcr==1, 6,
               as.numeric(event)))
event2 <- factor(event2, 1:6, c(levels(mdata$event), "SCT after CR"))
txsurv <- survfit(Surv(tstart, tstop, event2) ~ trt, mdata, id=id,
                  subset=(priortx ==0))
dim(txsurv) # number of strata by number of states
txsurv$states # Names of states

layout(matrix(c(1,1,1,2,2,0),3,2), widths=2:1)
oldpar <- par(mar=c(5.1, 4.1, 1,.1))
plot(txsurv[,c(3,6)], col=1:2, lty=c(1,1,2,2), lwd=2, xmax=48,
     xaxt='n', xlab="Months", ylab="Transplanted")
axis(1, xtime, xtime)
legend(15, .13, c("A, transplant without CR", "B, transplant without CR",
                 "A, transplant after CR", "B, transplant after CR"),
       col=1:2, lty=c(1,1,2,2), lwd=2, bty='n')
state4()  # add the state figure
par(oldpar)
@ 

\begin{figure}
  \myfig{sfit4}
  \caption{The full multi-state curves for the two treatment arms.}
  \label{sfit4}
\end{figure}

Figure \ref{sfit4} shows the full set of state occupancy probabilities for the
cohort over the first 4 years.  At each point in time the curves
estimate the fraction of subjects currently in that state.
 The total who are in the transplant state peaks at
about 9 months and then decreases as subjects relapse or die; 
the curve rises
whenever someone receives a transplant and goes down whenever someone
leaves the state.
At 36 months treatment arm B (dashed) has a lower fraction who have died,
the survivors are about evenly split between those who have received a
transplant and those whose last state is a complete response
(only a few of the latter are post transplant). 
The fraction currently in relapse -- a transient state -- is about 5\% for
each arm.
The figure omits the curve for ``still in the entry state''.
The reason is that 
at any point in time the sum of the 5 possible states is 1 ---
everyone has to be somewhere.  Thus one of the curves
is redundant, and the fraction still in the entry state is the least 
interesting of them.


<<sfit4, fig=TRUE, include=FALSE>>=
sfit4 <- survfit(Surv(tstart, tstop, event) ~ trt, mdata, id=id)
sfit4$transitions
layout(matrix(1:2,1,2), widths=2:1)
oldpar <- par(mar=c(5.1, 4.1, 1,.1))
plot(sfit4, col=rep(1:4,each=2), lwd=2, lty=1:2, xmax=48, xaxt='n',
     xlab="Months", ylab="Current state")
axis(1, xtime, xtime)
text(c(40, 40, 40, 40), c(.51, .13, .32, .01),
     c("Death", "CR", "Transplant", "Recurrence"), col=c(4,1,2,3))

par(mar=c(5.1, .1, 1, .1))
state5()
par(oldpar)
@ 

The transitions table above shows \Sexpr{sfit4$transitions[1,4]} %$ 
direct transitions from entry to death, i.e., 
subjects who die without experiencing any of the other intermediate points,
\Sexpr{sfit4$transitions[2,2]} who go from CR to transplant (as expected),
\Sexpr{sfit4$transitions[3,1]} who go from transplant to CR, etc. %$
No one was observed to go from relapse to CR in the data set, this
serves as a data check since it should not be possible per the data entry plan.

\section{Influence matrix}

For one of the curves above we returned the influence array.
For each value in the matrix $P$ = probability in state and each subject
$i$ in the data set, this contains the effect of that subject on each
value in $P$.  Formally,
\begin{equation*}
  D_{ij}(t) = \left . \frac{\partial p_j(t)}{\partial w_i} \right|_w
\end{equation*}
where $D_{ij}(t)$ is the influence of subject $i$ on $p_j(t)$, and
$p_j(t)$ is the estimated probability for state $j$ at time $t$.
This is known as the infinitesimal jackknife (among other labels).
<<reprise>>=
crsurv <- survfit(Surv(tstart, tstop, cr2) ~ trt,
                  data= mdata, id=id, influence=TRUE)
curveA <- crsurv[1,]  # select treatment A

dim(curveA)    # P matrix for treatement A
curveA$states
dim(curveA$pstate)  # 426 time points, 5 states
dim(curveA$influence) # influence matrix for treatment A
table(myeloid$trt)
@

For treatment arm A there are \Sexpr{table(myeloid$trt)[1]} subjects and
\Sexpr{dim(curveA$pstate)[1]} time points in the $P$ matrix.
The influence array has subject as the first dimension, and for each
subject it has an image of the $P$ matrix containing that subject's
influence on each value in $P$, i.e.,
\code{influence[1, ,]} is the influence of subject 1 on $P$.
For this data set everyone starts in the entry state, so $p(0)$ = the
first row of \code{pstate} will be (1, 0, 0, 0, 0) and the influence of
each subject on this row is 0;
this does not hold if not all subjects start in the same state.

As an exercise we will calculate the mean time in state out to 48 weeks.
This is the area under the individual curves from time 0 to 48.  Since
the curves are step functions this is simple sum of rectangles, treating
any intervals after 48 months as having 0 width.
<<meantime>>=
t48 <- pmin(48, curveA$time)   
delta <- diff(c(t48, 48))  # width of intervals
rfun <- function(pmat, delta) colSums(pmat * delta)  #area under the curve
rmean <- rfun(curveA$pstate, delta)

# Apply the same calculation to each subject's influence slice
inf <- apply(curveA$influence, 1, rfun, delta=delta)
# inf is now a 5 state by 310 subject matrix, containing the IJ estimates
#  on the AUC or mean time.  The sum of squares is a variance.
se.rmean <- sqrt(rowSums(inf^2))
round(rbind(rmean, se.rmean), 2)

print(curveA, rmean=48, digits=2)
@ 

The last lines verify that this is exactly the calculation done by the
\code{print.survfitms} function; the results can also be found in
the \code{table} component returned by \code{summary.survfitms}.

In general, let $U_i$ be the influence of subject $i$.
For some function $f(P)$ of the probability in state matrix \code{pstate}, 
the influence of subject
$i$ will be $\delta_i = f(P + U_i) - f(P)$ and the infinitesimal jackknife
estimate of variance will be $\sum_i \delta^2$.
For the simple case of adding up rectangles $f(P +U_i) - f(P) = f(U_i)$ leading
to particularly simple code, but this will not always be the case.  

\section{Differences in survival}
There is a single function \code{survdiff} to test for differences between 2
or more survival curves.  It implements the $G^\rho$ family of Fleming and
Harrington \cite{FH2}.  A single parameter $\rho$ controls the weights
given to different survival times, $\rho=0$ yields the log-rank test and
$\rho=1$ the Peto-Wilcoxon.  Other values give a test that is intermediate
to these two. The default value is $\rho=0$. 
The log-rank test is equivalent to the score test from a Cox model with
the group as a factor variable.

The interpretation of the formula is the same as for \code{survfit}, i.e.,
variables on the right hand side of the equation jointly break the
patients into groups.
<<survdiff>>=
survdiff(Surv(time, status) ~ x, aml)
@

\section{Robust variance}
The \code{survfit}, \code{coxph} and \code{surveg} routines all allow for
the computation of an infintisimal jackknife variance estimate.
This estimator is widely used in statistics under several names: in 
generalized estimating equation (GEE) models is is known as the
working-independence variance; in linear models as White's estimate,
and in survey sampling as the Horvitz-Thompsen estimate.
One feature of the estimate is that it is robust to model misspecification;
the argument \code{robust=TRUE} to any of the three routines will
invoke the estimator. 
If \code{robust=TRUE} and there is no \code{cluster} or \code{id}
argument, the program will assume that each row of data is from a unique
subject, a possibly questionable assumption.  It is better to provide the
grouping explicitly.

If the robust argument is missing (the usual case), then if there is an 
\code{cluster} argument, non-integer case weights, or there is an \code{id}
argument and at least one id has multiple events, then the code
assumes that robust=TRUE, and otherwise assumes robust=FALSE.
These are cases where the robust variance is most likely to be 
desirable.  If there is an \code{id} argument but no \code{cluster} the 
default is to cluster by id.
If there are non-integer weights but no clustering
information is provided (id or cluster statement), the code will assume that
each row of data is a separate subject. 
If the response is of (time1, time2) form this assumption is almost
certainly incorrect, but the model based variance would have the same
assumption so it is a choice between two evils. Responsibility falls on the
user to clarify the proper clustering.  
(A error  or warning from the code would be defensible, but the package author
so dislikes packages that chatter warnings all the time that he is loath
to do so.)

The infinitesimal jackknife 
(IJ) matrix contains the influence of each subject on the estimator;
formally the derivative with respect to each subject's case weight.
For a single simple survival curve that has $k$ unique values, for instance, 
the IJ matrix will have $n$ rows and $k$ columns, one row per subject.
Columns of the matrix sum to zero, by definition, and the variance at
a time point $t$ will be the column sums of $(IJ)^2$.
For a competing risk problem the \code{crfit} object above will contain a
matrix \code{pstate} with $k$ rows and one column for each state,
where $k$ is the number of unique time points; and the IJ is an
array of dimension $(n, k, p)$. 
In the case of simple survival and all case weights =1, the IJ variance
collapses to the well known Greenwood variance estimate.

\section{State space figures}
\label{sect:statefig}
The state space figures in the above example were drawn with a simple
utility function \code{statefig}.  It has two primary arguments along with
standard graphical options of color, line type, etc.
\begin{enumerate}
  \item A layout vector or matrix.  A vector with values of (1, 3, 1)
    for instance will allocate one state, then a column with 3 states, then
    one more state, proceeding from left to right.  A matrix with a single
    row will do the same, whereas a matrix with one column will proceed
    from top to bottom.
  \item A $k$ by $k$ connection matrix $C$ where $k$ is the number of states.
    If $C_{ij} \ne 0$ then an arrow is drawn from state $i$ to state $j$.
    The row or column names of the matrix are used to label the states.
    The lines connecting the states can be straight or curved, see the
    help file for an example.
\end{enumerate}
 
The first few state space diagrams were competing risk models, which use
the following helper function.  It accepts a vector of state names,
where the first name is the starting state and the remainder are the
possible outcomes.
<<crisk>>=
crisk <- function(what, horizontal = TRUE, ...) {
    nstate <- length(what)
    connect <- matrix(0, nstate, nstate,
                      dimnames=list(what, what))
    connect[1,-1] <- 1  # an arrow from state 1 to each of the others
    if (horizontal) statefig(c(1, nstate-1),  connect, ...)
    else statefig(matrix(c(1, nstate-1), ncol=1), connect, ...)
}
@ 

This next function draws a variation of the illness-death model.
It has an initial state,  
an absorbing state (normally death), and an optional intermediate state.
<<state3>>=
state3 <- function(what, horizontal=TRUE, ...) {
    if (length(what) != 3) stop("Should be 3 states")
    connect <- matrix(c(0,0,0, 1,0,0, 1,1,0), 3,3,
                      dimnames=list(what, what))
    if (horizontal) statefig(1:2, connect, ...)
    else statefig(matrix(1:2, ncol=1), connect, ...)
}
@ 

The most complex of the state space figures has all 5 states.
<<state5>>=
state5 <- function(what, ...) {
    sname <- c("Entry", "CR", "Tx", "Rel", "Death")
    connect <- matrix(0, 5, 5, dimnames=list(sname, sname))
    connect[1, -1] <- c(1,1,1, 1.4)
    connect[2, 3:5] <- c(1, 1.4, 1)
    connect[3, c(2,4,5)] <- 1
    connect[4, c(3,5)]  <- 1
    statefig(matrix(c(1,3,1)), connect, cex=.8, ...)
}
@ 

For figure \ref{txsurv} I want a third row with a single
state, but don't want that state centered.
For this I need to create my own (x,y) coordinate list as
the layout parameter.  Coordinates must be between 0 and 1.
<<state4>>=
state4 <- function() {
    sname <- c("Entry", "CR", "Transplant", "Transplant")
    layout <- cbind(x =c(1/2, 3/4, 1/4, 3/4),
                    y =c(5/6, 1/2, 1/2, 1/6))
    connect <- matrix(0,4,4, dimnames=list(sname, sname))
    connect[1, 2:3] <- 1
    connect[2,4] <- 1
    statefig(layout, connect)
}
@

The statefig function was written to do ``good enough'' state space figures
quickly and easily, in the hope that users will find it simple enough that
diagrams are drawn early and often.
Packages designed for directed acyclic graphs (DAG) such as diagram, DiagrammeR,
 or dagR are far more flexible
and can create more nuanced and well decorated results.

\subsection{Further notes}
The Aalen-Johansen method used by \code{survfit} does not account for 
interval censoring, also known as panel data, 
where a subject's current state is recorded at some fixed time such as a
medical center visit but the actual times of transitions are unknown.
Such data requires further assumptions about the transition process in
order to model the outcomes and has a more complex likelihood. 
The \code{msm} package, for instance, deals with data of this type.
If subjects reliably come in at regular intervals then the 
difference between the two results can be small, e.g., the 
\code{msm} routine estimates time until progression \emph{occurred}
whereas \code{survfit} estimates time until progression was \emph{observed}.

\begin{itemize}
  \item When using multi-state data to create Aalen-Johansen estimates, 
    individuals
    are not allowed to have gaps in the middle of their time line.
    An example of this would be a data set with 
    (0, 30, pcm] and (50,70, death] as the two observations
    for a subject where the time from 30-70 is not accounted for.
  \item Subjects must stay in the same group over their entire observation
    time, i.e., variables on the right hand side of the equation cannot be
    time-dependent. 
  \item A transition to the same state is allowed, e.g., observations of
    (0,50, 1], (50, 75, 3], (75, 89, 4], (89, 93, 4] and (93, 100, 4] 
    for a subject who goes
    from entry to state 1, then to state 3, and finally to state 4.
    However, a warning message is issued for the data set in this case, since
    stuttering may instead be the result of a coding mistake.
    The same result is obtained if 
    the last three observations were collapsed to a single row of 
    (75, 100, 4].
\end{itemize}

%--------------------------------------------------------

\chapter{Cox model}
\index{Cox model}
The most commonly used models for survival data are those that model the
transition rate from state to state, i.e., 
the arrows of figure \ref{fig:multi}.
They are Poisson regression \eqref{eq2.1}, the Cox or proportional
hazards model \eqref{eq2.2} and the Aalen additive regression
model \eqref{eq2.3},
of which the Cox model is far and away the most popular.
As seen in the equations they are closely related.

\begin{eqnarray}
  \lambda(t) &= e^{beta_0 + \beta_1 x_1 + \beta_2 x_2 + \ldots} \
           \label{eq2.1} \\
  \lambda(t) &= e^{\beta_0(t) + \beta_1 x_1 + \beta_2 x_2 + \ldots}  \nonumber \\
             &= \lambda_0(t) e^{\beta_1 x_1 + \beta_2 x_2 + \ldots} \label{eq2.2} \\
  \lambda(t) &= \beta_0(t) + \beta_1(t) x_1 + \beta_2(t) x_2 + \ldots
                \label{eq2.3} 
\end{eqnarray}

(Textbooks on survival use $\lambda(t)$, $\alpha(t)$ and $h(t)$ in about equal
proportions.  There is no good argument for any one versus another,
but this author started his career with books that used $\lambda$ so
that is what you will get.)

% -----------------------------------------
\section{One event type, one event per subject}
Single event data is the most common use for Cox models.
We will use a data set that contains the survival of 228 patients
with advanced lung cancer.

<<lung1>>=
options(show.signif.stars=FALSE)  # display statistical intelligence
cfit1 <- coxph(Surv(time, status) ~ age + sex + wt.loss, data=lung)
print(cfit1, digits=3)
summary(cfit1, digits=3)
anova(cfit1)
@ 

As is usual with R modeling functions, the default \code{print} routine
gives a short summary and the \code{summary} routine a longer one.
The \code{anova} command shows tests for each term in a model, added
sequentially.
We purposely avoid the innane addition of ``significant stars'' to
any printout.
Age and gender are strong predictors of survival, but the amount of
recent weight loss was not influential.

The following functions can be used to extract portions of a 
\code{coxph} object.
\begin{itemize}
  \item \code{coef} or \code{coefficients}: the vector of coefficients
  \item \code{concordance}: the concordance statistic for the model fit
  \item \code{fitted}: the fitted values, also known as linear predictors
  \item \code{logLik}: the partial likelihood
  \item \code{model.frame}: the model.frame of the data used in the fit
  \item \code{model.matrix}: the $X$ matrix used in the fit
  \item \code{nobs}: the number of observations
  \item \code{predict}: a vector or matrix of predicted values
  \item \code{residuals}: a vector of residuals
  \item \code{vcov}: the variance-covariance matrix
  \item \code{weights}: the vector of case weights used in the fit
\end{itemize}
Further details about the contents of a \code{coxph} object can be
found by \code{help('coxph.object')}.

The global \code{na.action} function has an important effect on the 
returned vector of residuals, as shown below.
This can be set per fit, but is more often set globally via the
\code{options()} function.

<<na.action>>=
cfit1a <- coxph(Surv(time, status) ~ age + sex + wt.loss, data=lung,
                na.action = na.omit)
cfit1b <- coxph(Surv(time, status) ~ age + sex + wt.loss, data=lung,
                na.action = na.exclude)
r1 <- residuals(cfit1a)
r2 <- residuals(cfit1b)
length(r1)
length(r2)
@ 
The fits have excluded 14 subjects with missing values for one or more
covariates.  
The residual vector \code{r1} omits those subjects from the residuals,
while \code{r2} returns a vector of the same length as the original
data, containing NA for the omitted subjects.
Which is preferred depends on what you want to do with the residuals.
For instance \code{mean(r1)} is simpler using the first while
\code{plot(lung\$ph.ecog, r2)} is simpler with the second.

Stratified Cox models are obtained by adding one or more \code{strata}
terms to the model formula.
In a stratified model each subject is compared only to subjects within their
own stratum for computing the partial likelihood, and then the final results
are summed over the strata.
A useful rule of thumb is that a variable included as a stratum is adjusted
for in the most general way, at the price of not having an estimate of its
effect.
One common use of strata is to adjust for the enrolling institution in a
multi-center study, as below.
We see that in this case the effect of stratification is slight.
<<cox12>>=
cfit2 <- coxph(Surv(time, status) ~ age + sex + wt.loss + strata(inst),
               data=lung)
round(cbind(simple= coef(cfit1), stratified=coef(cfit2)), 4)
@ 

Predicted survival curves from a Cox model are obtained using the
\code{survfit} function.
Since these are predictions from a model, it is necessary to specify
\emph{whom} the predictions should be for, i.e., one or more
sets of covariate values.
Here is an example.

<<cox13, fig=TRUE>>=
dummy <- expand.grid(age=c(50, 60), sex=1, wt.loss=5)
dummy

csurv1 <- survfit(cfit1, newdata=dummy)
csurv2 <- survfit(cfit2, newdata=dummy)
dim(csurv1)
dim(csurv2)
plot(csurv1, col=1:2, xscale=365.25, xlab="Years", ylab="Survival")

dummy2 <- data.frame(age=c(50, 60), sex=1:2, wt.loss=5, inst=c(6,11))
csurv3 <- survfit(cfit2, newdata=dummy2)
dim(csurv3)
@ 

The simplifying aspects of the Cox model that make is so useful are
exactly those that should be verified, namely proportional hazards,
additivity, linearity, and lack of any high leverage points.
The first can be checked with the \code{cox.zph} function.

<<lung2, fig=TRUE>>=
zp1 <- cox.zph(cfit1)
zp1
plot(zp1[2], resid=FALSE)
abline(coef(fit1)[2] ,0, lty=3)
@ 

None of the test statistics for PH are remarkable. 
A simple check for linearity of age is to replace the term with a smoothing
spline. 
<<lung3, fig=TRUE>>=
cfit3 <- coxph(Surv(time, status) ~ pspline(age) + sex + wt.loss, lung)
print(cfit3, digits=2)
termplot(cfit3, term=1, se=TRUE)

cfit4 <- update(cfit1, . ~ . + age*sex)
anova(cfit1, cfit4)
@ 

The age effect appears reasonbly linear. 
Additivity can be examined by adding an age by sex interaction, and
again is not remarkable.

\section{Repeating Events}
Children with chronic granulotomous disease (CGD) are subject to repeated 
infections due to a compromised immune system.  
The \code{cgd0} data set contains results of a clinical trial of gamma
interferon as a treatment, the data set \code{cdg} contains the data
reformatted into a (tstart, tstop, status) form:
each child can have multiple rows which describe an interval of time,
and status=1 if that interval ends with an infection and 0 otherwise.
A model with a single baseline hazard, known as the Andersen-Gill model,
can be fit very simply.
The study recruited subjects from four types of institutions, and there is
an a priori belief that the four classes might recruit a different type
of subject.  Adding the hospital category as a strata allows each group
to have a different shape of baseline hazard.
<<cgd1>>=
cfit1 <- coxph(Surv(tstart, tstop, status) ~ treat + inherit + steroids +
                   age + strata(hos.cat), data=cgd)
print(cfit1, digits=2)
@ 
Further examination shows that the fit is problematic in that only 3 of 128
children have \code{steroids ==1}, so we refit without that variable.
<<cgd1b>>=
cfit2 <- coxph(Surv(tstart, tstop, status) ~ treat + inherit+
                   age + strata(hos.cat), data=cgd)
print(cfit2, digits=2)
@ 

Predicted survival and/or cumulative hazard curves can then be obtained from
the fitted model.  
Prediction requires the user to specifiy \emph{who} to predict; in this case
we will use 4 hypothetical subjects on control/interferon treatment, at ages
7 and 20 (near the quantiles).
  This creates a data frame with 4 rows.

<<cgd3, fig=TRUE>>=
dummy <- expand.grid(age=c(6,12), inherit='X-linked', 
                     treat=levels(cgd$treat))
dummy
csurv <- survfit(cfit2, newdata=dummy)
dim(csurv)

plot(csurv[1,], fun="event", col=1:2, lty=c(1,1,2,2), 
     xlab="Days on study", ylab="Pr( any infection )")
@ 

The resulting object was subscripted in order to make a plot with fewer
curves, i.e., predictions for the first level of \code{hosp.cat}.
We see that treatment is effective but the effect of age is small.

Perhaps more interesting in this situation is the expected number of
infections, rather than the probability of having at least 1.
The former is estimated by the cumulative hazard, which is also returned
by the \code{survfit} routine.

<<cfit4, fig=TRUE>>=
plot(csurv[1,], cumhaz=TRUE, col=1:2, lty=c(1,1,2,2), lwd=2,
     xlab="Days on study", ylab="E( number of infections )")
legend(20, 1.5, c("Age 6, control", "Age 12, control",
                "Age 6, gamma interferon", "Age 12, gamma interferon"),
       lty=c(2,2,1,1), col=c(1,2,1,2), lwd=2, bty='n')
@  

\section{Competing risks}
Our third category is models where there is more than one event type, but
each subject can have only one transition.
This is the setup of competing risks.

\subsection{MGUS}
As an simple multi-state example consider the monoclonal gammopathy data
 set \code{mgus2},
which contains the time to a plasma cell malignancy (PCM), usually
multiple myleoma,  and the 
time to death for 1384 subjects found to have a condition known as
monoclonal gammopathy of undetermined significance (MGUS), based on
a particular test.  
This data set has already appeared in \ref{mgusplot}.
The time values in the data set are from detection of the condition.
Here are a subset of the observations along with a simple state figure
for the data.

<<survfit-mgus1, fig=TRUE>>=
mgus2[56:59,]

sname <- c("MGUS", "Malignancy", "Death")
smat <- matrix(c(0,0,0, 1,0,0, 1,1,0), 3, 3, 
               dimnames = list(sname, sname))
statefig(c(1,2), smat)
@ 

In this data set 
subject 56 was diagnosed with a PCM 29 months after detection of MGUS and
died at 44 months. 
This subject passes through all three states.
The other three listed individuals died without a plasma cell malignancy
and traverse one of the arrows;
103 subjects (not shown) are censored before experiencing either event
and spend their entire tenure in the leftmost state.
The competing risks model will ignore the transition from malignacy to death:
the two ending states are ``malignancy before death'' and ``death without
malignancy''.  

The \code{statefig} function is designed to create simple state diagrams,
with an emphasis on ease rather than elegance. 
See more information in section \ref{sect:statefig}.

For competing risks each subject has at most one transition, so the
data set only needs one row per subject.

<<survfit-mgus2>>=
crdata <- mgus2
crdata$etime <- pmin(crdata$ptime, crdata$futime)
crdata$event <- ifelse(crdata$pstat==1, 1, 2*crdata$death)
crdata$event <- factor(crdata$event, 0:2, c("censor", "PCM", "death"))

quantile(crdata$age, na.rm=TRUE)
table(crdata$sex)
quantile(crdata$mspike, na.rm=TRUE)

cfit <- coxph(Surv(etime, event) ~ I(age/10) + sex + mspike, 
              id = id, crdata)
print(cfit, digits=1)  # narrow the printout a bit
@ 
The effect of age and sex on non-PCM mortality is profound, which is not
a surprise given the median starting age of \Sexpr{median(mgus2$age)}. %$
Death rates rise \Sexpr{round(exp(10*coef(cfit)[4]),1)} fold per decade 
of age and
the death rate for males is \Sexpr{round(exp(coef(cfit)[5]),1)} times as great
as that for females.  
The size of the serum monoclonal spike has almost no impact on non-PCM 
mortality.
A 1 unit increase changes mortality by only 2\%.

The mspike size has a major impact on progression, however; each 1 gram
change increases risk by \Sexpr{round(exp(coef(cfit)[3]) ,1)} fold.
The interquartile range of \code{mspike} is 0.9 grams so this risk increase
is clinically important.
The effect of age on the progression rate is much less pronounced,
with a coefficient only 1/4 that for mortality, while the effect of sex
on progression is completely negligible.

Estimates of the probability in state can be simply computed using 
\code{survfit}.
As with any model, estimates are always for a particular set of
covariates.  We will use 4 hypothetical subjects, male and female
with ages of 60 and 80.
<<PCMcurve, fig=TRUE>>=
dummy <- expand.grid(sex=c("F", "M"), age=c(60, 80), mspike=1.2)
csurv  <- survfit(cfit, newdata=dummy)
plot(csurv[,2], xmax=20*12, xscale=12,
     xlab="Years after MGUS diagnosis", ylab="Pr(has entered PCM state)",
     col=1:2, lty=c(1,1,2,2), lwd=2)
legend(100, .04, outer(c("female,", "male,  "), 
                     c("diagnosis at age 60", "diagnosis at age 80"), 
                      paste),
       col=1:2, lty=c(1,1,2,2), bty='n', lwd=2)
@ 

Although sex has no effect on the \emph{rate} of plasma cell malignancy,
its effect on the \emph{lifetime probability} of PCM is not zero,
however.  
As shown by the simple Poisson model below, the rate of PCM is about 1\%
per year.  Other work reveals that said rate is almost constant over
follow-up time (not shown). 
Because women in the study have an average lifetime that is 2 years
longer than men, their lifetime risk of PCM is higher as well.
Very few subjects acquire PCM more than 15 years after a MGUS diagnosis at
age 80 for the obvious reason that very few of them will still 
be alive.

<<mrate>>=
mpfit <- glm(pstat ~ sex -1 + offset(log(ptime)), data=mgus2, poisson)
exp(coef(mpfit)) * 12   # rate per year
@ 

A single outcome fit using only time to progression is instructive:
we obtain exactly the same coefficients but different absolute risks.
This is a basic property of multi-state models: hazards can be explored
separately for each transition, but absolute risk must be computed globally.
(The estimated cumulative hazards from the two models are also identical).
The incorrect curve is a vain attempt to estimate the progression rate which 
would occur if death could be abolished.  It not surprisingly ends up as about
1\% per year.  

<<msingle, fig=TRUE>>=
sfit <- coxph(Surv(etime, event=="PCM") ~ I(age/10) + sex + mspike, crdata)
rbind(single = coef(sfit),
      multi  = coef(cfit)[1:3])
#par(mfrow=c(1,2))
ssurv <- survfit(sfit, newdata=dummy)
plot(ssurv[3:4], col=1:2, lty=2, xscale=12, xmax=12*20, lwd=2, fun="event",
     xlab="Years from diagnosis", ylab= "Pr(has entered PCM state)")
lines(csurv[3:4, 2], col=1:2, lty=1, lwd=2)
legend(20, .22, outer(c("80 year old female,", "80 year old male,"),
                      c("incorrect", "correct"), paste),
                  col=1:2, lty=c(2,2,1,1), lwd=2, bty='n')
@ 

\section{Multiple event types and multiple events per subject}

Non-alcoholic fatty liver disease (NAFLD) is defined by three criteria:
presence of greater than 5\% fat in the liver (steatosis),
absence of other indications for the steatosis such as excessive
alcohol consumption or certain medications, and absence of other 
liver disease \cite{Puri12}.
NAFLD is currently responsible for almost 1/3 of
liver transplants and it's impact is growing, it is expected to be a major
driver of hepatology practice in the coming decade \cite{Tapper18},
driven at least in part by the growing obesity epidemic.
The \code{nafld} data set includes all patients with a NAFLD 
diagnosis in Olmsted County,
Minnesota between 1997 to 2014 along with up to four age and sex matched 
controls for each case \cite{Allen18}. 

We will model the onset of three important components of the metabolic
syndrome: diabetes, hypertension, and dyslipidemia, using the model shown
below.  Subjects have either 0, 1, 2, or all 3 of these metabolic comorbidities.

<<state5, fig=TRUE>>=
state5 <- c("0MC", "1MC", "2MC", "3MC", "death")
tmat <- matrix(0L, 5, 5, dimnames=list(state5, state5))
tmat[1,2] <- tmat[2,3] <- tmat[3,4] <- 1
tmat[-5,5] <- 1
statefig(rbind(4,1), tmat)
@ 

\subsection{Data}
The NAFLD data is represented as 3 data sets, \code{nafld1} has one observation
per subject containing basline information (age, sex, etc.),
nafld2 has information on repeated laboratory tests, e.g. blood pressure,
and nafld3 has information on yes/no endpoints.
After the case-control set was assembled, we removed any subjects with less
than 7 days of follow-up.  These subjects add little information, and it
prevents a particular confusion that can occur with a multi-day medical visit
where two results from the same encounter have different dates.
To protect patient confidentiality all time intervals are in days since
the index date; none of the dates from the original data were retained.
Subject age is their integer age at the index date, and the subject
identifier is an arbitrary integer. 
As a final protection, a 10\% random sample of subjects was excluded.
As a consequence analyses results will not exactly match the
original paper.

Start by building an analysis data set using \code{nafld1} and \code{nafld3}.

<<nafld1>>=
ndata <- tmerge(nafld1[,1:8], nafld1, id=id, death= event(futime, status))
ndata <- tmerge(ndata, subset(nafld3, event=="nafld"), id, 
                nafld= tdc(days))
ndata <- tmerge(ndata, subset(nafld3, event=="diabetes"), id = id,
                diabetes = tdc(days), e1= cumevent(days))
ndata <- tmerge(ndata, subset(nafld3, event=="htn"),  id = id,
                htn = tdc(days), e2 = cumevent(days))
ndata <- tmerge(ndata, subset(nafld3, event=="dyslipidemia"), id=id,
                lipid = tdc(days), e3= cumevent(days))
ndata <- tmerge(ndata, subset(nafld3, event %in% c("diabetes", "htn", 
                                                   "dyslipidemia")), 
                id=id, comorbid= cumevent(days))
summary(ndata)
@ 

<<echo=FALSE>>=
tc <- attr(ndata, 'tcount')   # shorter name for use in Sexpr below
icount <- table(table(nafld3$id)) #number with 1, 2, ... intervals
ncount <- sum(nafld3$event=="nafld")
@ 

The summary function tells us a lot about the creation process.
Each addition of a new endpoint or covariate to the data generates one 
row in the table.  Column labels are explained by figure \ref{fig:timeline}.
\begin{itemize}
  \item There are \Sexpr{tc[1,7]} last fu/death additions, 
    which by definition fall at the
    trailing end of a subject's observation interval: they define the
    interval.
  \item There are \Sexpr{tc[2,2]} nafld splits that fall after the end
    of follow-up (`late').
    These are subjects whose first NAFLD fell within a year of the end of
    their time line, and the one year delay for ``confirmed'' pushed them
    over the end.  (The time value in the \code{nafld3} data set is 1 year
    after the actual notice of NAFLD; no other endpoints have this
    offset added).  The time dependent covariate \code{nafld} never turns
    from 0 to 1 for these subjects. 
    (Why were these subjects not removed earlier by my ``at least 7 days of
    follow-up'' rule?  They are all controls for someone else and so appear
    in the data at a younger age than their NAFLD date.)
  \item \Sexpr{tc[2,4]} subjects have a NAFLD diagnosis between time 0
    and last follow-up.
    These are subjects who were selected as matched controls for another
    NAFLD case at a particular age, and later were diagnosed with NAFLD 
    themselves.
   \item \Sexpr{tc[3,1]} of the diabetes diagnoses are before entry, i.e., 
    these are the prevalent cases.  One diagnosis occurred on the day of
    entry (``leading''), and will not be counted as a post-enrollment endpoint,
    all the other fall somewhere between study entry and last follow-up.
  \item Conversely, \Sexpr{tc[5,7]} subjects were diagnosed with hypertension
    at their final visit (``trailing'').  These will be counted as an
    occurrence of a hypertension event (\code{e2}), but the time dependent 
    covariate \code{htn} will never become 1. 
  \item \Sexpr{tc[9,8]} of the total comorbidity counts are tied. These are
    subjects for whom the first diagnosis of 2 of the 3 conditions
    happened on the same office visit, the cumulative count will jump by 2.
    (We will see below that 4 subjects had all 3 on the same day.)
    Many times ties indicate a data error.
\end{itemize}

Such a detailed look at data set construction may seem over zealous.   
Our experience is that issues with covariate and event timing
occur in nearly all data sets, large or small.  The 13 NAFLD cases ``after
last follow-up'' were for instance both a surprise and a puzzle to us;
but we have learned through experience that it is best not to proceed until
such puzzles are understood.  (This particular one was benign.)
If, for instance, some condition is noted at autopsy, do we want the related
time dependent covariate to change before or after the death event?
Some sort of decision has to be made, and it is better to look and understand
than to blindly accept an arbitrary programming default.

\subsection{Fits}
Create the covariates for current state and the analysis endpoint.
It is important that data manipulations like this occur \emph{after}
the final \code{tmerge} call.  
Successive \code{tmerge} calls keep track of the time scale, time-dependent and
event covariates, passing the information forward from call to call, 
but this information is lost when the resulting data frame is manipulated.  
(The loss is intentional: we won't know if one of the tracked variables has
changed.)

The \code{tmerge} call above used the cumevent verb to count comorbidities, 
and the first
line below verifies that no subject had diabetes, for instance, coded more than
once.  For this analysis we think of the three conditions as one-time outcomes,
you can't get diabetes twice.  When the outcome data set is the result of
electronic capture one could easily have a diabetes code at every visit,
in which case the cumulative count of all events would not be
the total number of distinct comorbidities.
In this particular data set the diabetes codes had already been preprocessed
so that the data set contains only the first diabetes diagnosis, and likewise
with hypertension and dyslipidemia.  
(In counterpoint, the nafld3 data set has multiple myocardial infarctions for 
some subjects, since MI can happen more than once.)

<<nafld2>>=
with(ndata, if (any(e1>1 | e2>1 | e3>1)) stop("multiple events"))
ndata$cstate <- with(ndata, factor(diabetes + htn + lipid, 0:3, 
                                   c("0mc", "1mc", "2mc", "3mc")))
temp <- with(ndata, ifelse(death, 4, comorbid))
ndata$event <- factor(temp, 0:4, 
         c("censored", "1mc", "2mc", "3mc", "death"))
ndata$age1 <- ndata$age + ndata$tstart/365.25   # analysis on age scale
ndata$age2 <- ndata$age + ndata$tstop/365.25

check1 <- survcheck(Surv(age1, age2, event) ~ nafld + male, data=ndata, 
                   id=id, istate=cstate)
check1
@

This is a rich data set with a large number of transitions:
over 1/4 of the participants have at least one event, and there
are \Sexpr{check1$events[5,5]} subjects who transition through all 5 
possible states (4 transitions).
Unlike prior examples, subjects do not all enter the study in the same
state; about 14\% have diabetes at the time of recruitment, for instance.
Note one major difference between current state and outcome, namely that the
current state endures across intervals: it is based on \code{tdc} variables
while the outcome is based on \code{event} operators.
If a subject has time-dependent covariates, there may be intermediate
intervals where a covariate changed but an outcome did not occur;
current state will endure across intervals but the intermediate outcome will
be ``censor''.

We see a number of subjects who ``jump'' states, e.g., directly from 0 to
2 comorbidities.  
This serves to remind us that this is actually a model of time
until \emph{detected} comorbidity; which will often have such jumps even if
the underlying biology is continuous.
The data look like the figure below, where the dotted lines are 
transformations that we observe, but would not be present if the subjects were
monitored continuously.
A call to the \code{survcheck} routine is almost mandatory for a complex 
setup like this,
to ensure that the data set which has been built is what you intended to build.

Calling \code{survcheck} with \textasciitilde 1 on the right hand side or with
the covariates for the model on the right hand side will potentially give different
event counts, due to the removal of rows with a missing value. 
Both can be useful summaries.
For a multi-state coxph model neither may be exactly correct, however.  If the model
contains a covariate which applies only to certain transitions, then events that
do not depend on that covariate will be retained, 
while event occurences that do depend on the covariate
will be dropped, leading to counts that may be intermediate between
the two survcheck outputs.

<<nafld3, fig=TRUE>>=
states <- c("No comorbidity", "1 comorbidity", "2 comorbidities", 
            "3 comorbitities", "Death")
cmat <- matrix(0, 5,5)
cmat[,5] <- 1
cmat[1,2] <- cmat[2,3] <- cmat[3,4] <- 1
cmat[1,3] <- cmat[2,4] <- 1.6
cmat[1,4] <- 1.6
dimnames(cmat) <- list(states, states)
statefig(cbind(4,1), cmat, alty=c(1,2,1,2,2,1,1,1,1,1,1))
@

Since age is the dominant driver of the transitions we have chosen to 
do the fits directly on age scale rather than model the age effect.
We force common coefficients for the transitions from 0 comorbidities to
1, 2 or 3, and for transitions from 1 comorbidity to 2 or 3.
This is essentially a model of ``any progression'' from a given state.
We also force the effect of male sex to be the same for any transition
to death. 

<<nafld4>>=
nfit1 <- coxph(list(Surv(age1, age2, event) ~ nafld + male,
                    "0mc":state("1mc", "2mc", "3mc") ~ nafld+ male / common,
                     2:3 + 2:4   ~ nafld + male / common,
                     0:"death" ~ male / common),
               data=ndata, id=id, istate=cstate)
nfit1$states
nfit1$cmap
@ 

A list has been used as the formula for the \code{coxph} call.
The first element is a standard formula, and will be the default for
all of the transitions found in the model.
Elements 2--4 of the list are pseudo formulas, which specify a set of
states on the left and covariates on the right, along with the optional
modifier \code{/common}.
As shown, there are multiple ways to specify a set of transitions either by
name or by number, the value 0 is shorthand for ``any state''.
The coefficient matrix reveals that the 1:2, 1:3, and 1:4 transitions all
share the same coefficients, as intended.

The actual coefficient vector (\code{coef(fit)}) and variance covariance
matrix do not have repeats.
The fit also includes a \code{cmap} component that records the mapping
between the coefficient vector and the state transtitions. 
The result of \code{coef(nfit1)} is a vector of length 9, the first element of
which is the nafld effect for transitions 1:2, 1:3, and 1:4,
the second coefficient is the effect of male on those three transitions, etc. 
Because the coefficient vector, variance matrix, and etc. are identical
to those for a simple coxph call, downstream operations such as 
\code{predict} and \code{summary} are unchanged.

The standard printouts makes use of \code{cmap} to rearrange the output into 
a nicer format.
It is interesting, though not surprising, that the impact of NAFLD on death
is largest for those with 0mc and smallest for those with 3mc.  

<<nafld5b>>=
print(coef(nfit1), digits=3)

print(coef(nfit1, matrix=TRUE), digits=3)  # alternate form

print(nfit1)
@ 

The summary command does not rearrange.
<<>>=
options(show.signif.stars = FALSE) # display statistical maturity
summary(nfit1, digits=3)
@ 


The names attached to the coefficients in a multi-state model are a compromise,
designed to give some information to the reader, albeit imperfect. 
If a variable such as 'sex' only applies to a single coefficient, the simple name
is used, even if the coefficient corresponds to multiple transitions.
Otherwise a suffix ``\_a:b'' is appended where a:b corresponds to the first
transition that maps onto this coefficient.
(First in the sense of standard R matrices, i.e., reading the elements of
\code{cmap} in column order.)

A second available keyword is \code{shared}, which indicates that the baseline
hazards for transitions share a common shape.
Here is an example:
<<nafld5c>>=
nfit2 <- coxph(list(Surv(age1, age2, event) ~ nafld + male,
                    "0mc":state("1mc", "2mc", "3mc") ~ nafld+ male / common,
                     2:3 + 2:4   ~ nafld + male / common,
                     1:5 + 2:5 +3:5 ~ male / common + shared),
               data=ndata, id=id, istate=cstate)
nfit2$cmap
@ 

\subsection{Timeline data}
The \code{survfit} and \code{coxph} routines also accept data in what we refer
to as a ``timeline'' form.  (The option is still under development so detail
may change.) 
Timeline data contains a case identifier and a timeline variable, where this
value pair that is unique for each row.  The other covariates are any number
of variables whose value is ``what was observed at that time'', or missing if
there was no observation of that variable at that time.
In contrast to counting process data, there are no time intervals and no 
distinction between covariates and endpoints.
In this sense the data is much more straightforward; a simple description of
what was seen. 

Here is a simple example using the mgus2 data set for a competing risks
analysis.  The \code{Surv2} operation on the left-hand side indicates to the
routine that timeline data is being used.

<<timeline1>>=
ctime <- with(mgus2, ifelse(pstat==1, ptime, futime))
cstat <- with(mgus2, ifelse(pstat==1, 1, 2*death))
cstat <- factor(cstat, 0:2, c("censor", "PCM", "death"))
tdata <- data.frame(id=mgus2$id, days=ctime, cstat=cstat)

# counting process
mdata1 <- tmerge(mgus2[,1:7], tdata, id, state=event(days, cstat))
mfit1 <- coxph(Surv(tstart, tstop, state) ~ age + sex, id=id, mdata1)

# timeline
mdata2 <- data.frame(mgus2[,1:7], days=0)
mdata2 <- merge(mdata2, tdata, all=TRUE)
mfit2 <-  coxph(Surv2(days, cstat) ~ age + sex, id=id, mdata2)
all.equal(coef(mfit1), coef(mfit2))
@

The counting process data set from \code{tmerge} as fewer rows but a more complex
structure.
<<timeline1b>>=
mdata1[1:3,]
print(mdata2[1:6,], na.print='.')
@ 

Here is a reprise of the NAFLD data set using the timeline form.  
<<timeline2>>=
tldata <- data.frame(nafld1[,1:7],
                    days= 0,  death=0, iage=nafld1$age, nafld=0)
tldata <- merge(tldata, with(nafld1, data.frame(id=id, days=futime, death=status)),
               all=TRUE)

# Add in the comorbidities of interest.  None of these 4 happen to have
#  duplicates (MI does, for instance).  
# Start by removing the the 13 rows with a "confirmed NAFLD" (actual NAFLD + 1 year)
#  that is after the actual last follow-up date.
# Treat diabetes before day 0 as diabetes on day 0.
badrow <- which(nafld3$days > nafld1$futime[match(nafld3$id, nafld1$id)])
fixnf3 <- nafld3[-badrow,]

tldata <- merge(tldata, with(subset(fixnf3, event=="diabetes"),
                           data.frame(id=id, days=pmax(0,days), diabetes=1)),
                   all=TRUE, by=c("id", "days"))
tldata <- merge(tldata, with(subset(fixnf3, event=="htn"),
                           data.frame(id=id, days=pmax(0,days), htn=1)),
                     all=TRUE, by=c("id", "days"))
tldata <- merge(tldata, with(subset(fixnf3, event=="dyslipidemia"),
                           data.frame(id=id, days= pmax(0, days), dyslipid=1)),
                       all=TRUE, by=c("id", "days"))
tldata <- merge(tldata, with(subset(fixnf3, event=="nafld"),
                           data.frame(id=id, days= pmax(0,days), nafld=1)),
                    by=c("id", "days"), all=TRUE)

tldata$nafld <- with(tldata, ifelse(is.na(nafld.y), nafld.x, nafld.y))
@

We want to assume that a subject is non-NAFLD until detection, which means setting 
\code{nafld} to 0 at time 0; this was done in the first tldata line above.
Ideally, we would have a version of \code{merge} that overwrites that value for
a subject with NAFLD on day 0, but that is not how \code{merge} works;
given \code{any} tied days between tldata and fixnf3 there will be two variables
\code{nafld.x} and \code{nafld.y}.  The last line above makes one from the two.
Initial values for the number of comorbidities are handled by the cumevent function.

<<timeline3>>=
#
# For cumulative events within subject we use a helper function
cumevent <- function(id, time, status, istate) {
    # do all the work on ordered data
    ord <- order(id, time)
    id2 <- id[ord]
    time2 <- time[ord]
    stat2 <- ifelse(is.na(status[ord]), 0, status[ord])
    firstid <- !duplicated(id)
    csum <- cumsum(stat2)
    indx <- match(id2, id2)
    cstat<- csum + stat2[indx] - csum[indx]  
    cstat[stat2==0] <- 0
          
    if (!missing(istate)) cstat[firstid] <- istate

    keep <- (firstid | (!is.na(stat2)& stat2 !=0))
    newdata <- data.frame(id=id2[keep], time=time2[keep], status=cstat[keep])
    newdata
}

temp1 <- rowSums(tldata[,c('diabetes', 'htn', 'dyslipid')], na.rm=TRUE)
temp2 <- with(tldata, cumevent(id, days, pmax(temp1, 4*death, na.rm=TRUE)))
state <- factor(pmin(temp2$status, 4), -1:4,
                 c("censor", paste0(0:3, "mc"), "death"))
tldata <- merge(tldata, data.frame(id=temp2$id, days=temp2$time, state=state),
               all=TRUE)

tldata$age <- with(tldata, days/365.25 + age[match(id, id)])
check2 <- survcheck(Surv2(days, state) ~ 1, id=id, tldata)
check2$transitions

nfit2 <- coxph(list(Surv2(age, state) ~ nafld + male,
                    "0mc":state("1mc", "2mc", "3mc") ~ nafld+ male / common,
                     2:3 + 2:4   ~ nafld + male / common,
                     0:"death" ~ male / common),  
               data=tldata, id=id)
round(coef(nfit2), 3)
@ 

The resulting fit it identical to the one that used the counting process data set.

There are advantages and disadvantages to the timeline data as compared to counting
process style.
\begin{itemize}
  \item Counting process style has been available for a long while --- it was first
    incorporated into the survival package in 1986 --- and it has been adopted by 
    several other packages.  The format is hence familiar to many users.
  \item Nevertheless, it contains many traps for the unwary.  The distinction between
    outcome and predictor variables is critical: the former applies at the end
    of each (time1, time2) interval and the former at the start of the interval.
    If one is fitting multiple models, one where the number of comorbid conditions
    was a predictor and one where it is the outcome, different variables and/or
    data sets are required.  In multi-state data sets separate variables are needed
    for the current state and for an event, and they behave differently.
  \item The tmerge routine simplifies some tasks, but it can be subtle. 
    The author/maintainers of the routine are often puzzled ourselves.  When there are
    multiple possible endpoints and/or multiple time scales it can get particularly
    challenging.
  \item Timeline data is simpler.  There is no distinction between covariates and
    events, or between current and next state.  Any necessary temporal orderings
    are created by the underlying survival models when processing the formula.
    This makes it easier to get the data set correct.  
  \item Timeline data sets are created using standard tools. The example above used
    only tools in base R (a restriction for vignettes in the list of `recommended'
    packages), but there is a wide range of available data manipulation tools.  The
    result need only obey the requirement of having no duplicate (id, timescale) keys.
    This is a common constraint in relational databases.
\end{itemize}

\section{Testing proportional hazards}
The usual Cox model with $p$ covariates has the form
\begin{align*}
  \lambda(t) &= \lambda_0(t) e^{\beta_1 x_1 + \beta_2 x_2 + \ldots + \beta_p x_p} \\
             &= e^{\beta_0(t) + \beta_1 x_1 + \beta_2 x_2 + \ldots + \beta_p x_p}
\end{align*}
A key simplifying assumption of the model is that all of the coefficients
except $\beta_0$ (the baseline hazard) are constant over time,
which is referred to as the \emph{proportional hazards} assumption.
Numerous approaches to verifying or testing this assumption have been
proposed, of which the three most enduring have been the addition of
an additional constructed covariate, score tests, and tests based on
cumulative martingale sums.  Each of these is normally applied to one
covariate at a time.

\subsection{Constructed variables}
For the constructed variable approach, assume that the true form of the
model for variable $x_1$ is $\beta_1(t) x_1$, with the coefficient having
the simple linear form $\beta_1(t) = a + bt$.
Then 
\begin{align}
  \beta_1(t)x_1 &=  ax_1 + b(x_1t) \nonumber \\
               & = ax_1 + b z  \label{ph1}
\end{align}
that is, we can create a special time-dependent covariate $z = x_1t$, add
add it to the data set, and then use an ordinary \code{coxph} fit.

Consider the veterans lung cancer data set, which has often been used to
illustrate non-proportional hazards.
Adding this special covariate is not quite as simple as writing
<<echo=TRUE, eval=FALSE>>=
fit2 <- coxph(Surv(time, status) ~ trt + trt*time + celltype + karno,
                data = veteran)
@ 
The problem is that \code{time} is trying to play two roles in the above 
equation,
the \emph{final} follow-up time for each subject (the left hand side of 
the formula) and the
\emph{continuous} time scale $t$ of equation \eqref{ph1}.
The \code{veteran} data set contains the first of these as an explicit variable,
and the \code{coxph} function will use that variable on both the right and
left hand sides, which is not the desired time-dependent effect.
The solution to to create the special variable, explicitly,
before calling the regression function.
Since the Cox model adds a term to the likelihood at each unique death
time, it is sufficient to create a data set with the same granularity.

<<zphcheck1>>=
dtime <- unique(veteran$time[veteran$status==1]) # unique times
newdata <- survSplit(Surv(time, status) ~ trt + celltype + karno,
                     data=veteran, cut=dtime)
nrow(veteran)
nrow(newdata)
fit0 <- coxph(Surv(time, status) ~ trt + celltype + karno, veteran)
fit1 <- coxph(Surv(tstart, time, status) ~ trt + celltype + karno, 
              data=newdata)
fit2 <- coxph(Surv(tstart, time, status) ~ trt + celltype + karno +
              time:karno, newdata)
fit2

fit2b <- coxph(Surv(tstart, time, status) ~ trt + celltype + karno +
              rank(time):karno, newdata)
@ 

The fits give a warning message about the use of the \code{time} variable
on both sides of the equation, since 
two common cases where time appears on both sides are the naive model
shown further above and frank typing mistakes.
In this particular case we can ignore the warning since the data set was 
carefully constructed for this special purpose, but it should never be treated
casually.

Alternatively, \code{coxph} has built-in functionality that will build
the expanded data set for us, behind the scenes, and then use that 
expanded data for the fit.
Here is eqivalent code to test the Karnofsky variable:
<<zph2>>=
fit2 <- coxph(Surv(tstart, time, status) ~ trt + celltype + karno +
              tt(karno), data =newdata,
              tt = function(x, t,...) x*t)
@ 
 
There are 4 issues with the constructed variable approach.
\begin{enumerate}
  \item The choice $\beta(t) \approx a + bt$ was arbitrary.  Perhaps the true
    form is $a + b\log(t)$ (fit2b above), or some other function.
  \item The intermediate data set can become huge. 
    It will be of order $O(nd)$ where $d$ is the number of unique
    event times, and $d$ grows along with $n$.
  \item The coefficients for a factor variable such as celltype can be
    confusing, since the results depend on how the 0/1 indicators
    for the variable are chosen.
  \item Outliers in time are an issue. 
    The veteran cancer data set, for instance, contains 
    a time of 999 days (a particularly
    suspicious value in any data set).
    The Cox model itself depends only on the rank order of the event times,
    so such outliers are not an issue for the base model,
    but as a covariate these values can have undue influence.
    The time-dependent coeffient for Karnofsky has $p<.01$ in fit2b
    above, which uses rank(time),
    a change that is largely due to dampening the leverage of outliers.
\end{enumerate}

\subsection{Score tests}
The \code{cox.zph} function checks proportional hazards for a fitted Cox model
directly, and tries to address the four issues discussed above.
\begin{itemize}
  \item It is easy to specify alternate time transforms such as x*log(t).  
    More importantly, the code produces both a diagnostic plot that suggests
    the shape of any non-proportionality, along with a test of the chosen
    time-transform.
  \item The test statistic is based on a score test, which does not require
    creating the expanded data set.
  \item Multi-covariate terms such as a factor or splines are by default treated
    as a single effect.
  \item The default time transform is designed to minimize outliers in time.
\end{itemize}

Shown below are results for the veterans data using \code{fit0} from above.
The score statistic for the simple term x*time (\code{zp1}) closely matches the
Wald test for the full time dependent fit found in \code{fit2} above,
which is what we would expect; score, Wald and likelihood ratio tests usually
agree quite closely for Cox models.

<<zph2, fig=TRUE>>=
zp0 <- cox.zph(fit0, transform='identity')
zp0
zp1 <- cox.zph(fit0, transform='log')
zp1
oldpar <- par(mfrow=c(2,2))
for (i in 1:3) {plot(zp1[i]); abline(0,0, lty=3)}
plot(zp0[3])
par(oldpar)
@ 

A test for zero slope, from a least squares regression using data in the 
matching plot approximates the score test.
(In versions of the package prior to survival3.0, the approximate test was
used for the formal test and printout as well.)
If proporitional hazard holds we would expect the fitted line to be
horizontal, i.e., $\beta(t)$ is constant.
Rather than show a fitted line the plot adds a general smooth curve, which
can help reveal the \emph{form} of non-proportional hazards, if it exists. 
The first three panels of the plot show curves for the three covariates on
a log(time) scale.
Since \code{celltype} is a factor, the plot shows the time dependent effect
of the portion of the linear predictor associated with cell type; 
if proportional hazards is true wrt that term a fitted line should be
horizontal with a coefficient of 1.
The effect of Karnofsky score appears to become essentially 0 after
approximately 6 months; for this cohort of subjects with advanced lung cancer,
a 6 month old assessment of Karnofsky is no longer relevant.
The corresponding plot in the lower right panel, however, shows that the outlier
time of 999 days has an undue influence on any such regression. 
A test of proportional hazards on that scale must also be treated with
caution.
The plot using log scale lacks these outliers and is more interpretable.

The default time transform is based on a Kaplan-Meier transform, i.e., that
monotone transform of the time axis that will cause the KM plot to be a 
straight line.
This is a good default for the score tests, since it essentially 
guarrantees that there wil be no outliers in the constructed $x g(time)$
variable, 
while dealing with censoring in a defensible way. 
That is, the code has opted for a \emph{safe} default. 
It is not as easily interpreted as other scales for the plots, however.

The \code{cox.zph} function does not attempt a score test for random effects 
(frailty) terms, 
in fact is not clear what the computation for such a test should be.
The function will check other covariates in a model that
contains a random effect, however; in that test
the estimated random effect per subject is essentially treated as a fixed 
offset.

\subsection{Computational details}
The score test is simple in theory, but ``the devil is in the details'' as they
say.
Consider adding the constructed variables for celltype to the fit.
That is g(t)*x2, g(t)*x3, g(t)*x4, where x2-x4 are the three dummy variables
that represent celltype.
The new model has 5 + 3 covariates, and is evaluated at $(\hat\beta, 0, 0, 0)$.
The score statistic at this coefficient value will be
$(0,0,0,0,0, u_6, u_7, u_8)$, the first 5 elements are zero since that is the
definition of model convergence for $\hat\beta$.

The information or Hessian matrix for a Cox model is 
$$ \sum_{j \in deaths} V(t_j)  = \sum_jV_j$$
where $V_j$ is the variance matrix of the weighted covariate values, over
all subjects at risk at time $t_j$.
Then the expanded information matrix for the score test is
\begin{align*}
  H &= \left(\begin{array}{cc}  H_1 & H_2 \\ H_2' & H_3 \end{array} \right) \\
  H_1 &= \sum V(t_j) \\
  H_2 &= \sum V(t_j) g(t_j) \\
  H_3 &= \sum V(t_j) g^2(t_j)
\end{align*}
The inverse of the matrix will be more numerically stable if $g(t)$ is centered
at zero, and this does not change the test statistic.
In the usual case $V(t)$ is close to constant in time --- the variance of
$X$ does not change rapidly --- and then $H_2$ is approximately zero.
The original cox.zph used an approximation, which is to assume that
$V(t)$ is exactly constant.
In that case $H_2=0$ and $H_3= \sum V(t_j) \sum g^2(t_j)$ and the test
is particularly easy to compute.
This assumption of identical components can fail badly for models with a
covariate by strata interaction, and for some models with covariate
dependent censoring.

If there are $p$ covariates, the new score vector will be of length $2p$ and
the information matrix will be $2p$ by $2p$.  These can be computed using
a simple variant of the C code for coxph; no iteration is done.  
In fact, the use of time-weighted risk sets has been proposed by several
authors, for multiple rationales.  This has not been implemented in the
coxph routine (but we have thought about it).

The score tests are done for single covariates or terms. 
Using the veteran example, a test for celltype as a term would first select
rows 1-5 and 7-9 of the score vector $U$ and information matrix $H$; i.e., 
if \code{j <- c(1:5, 7:9)} the test is \code{U[j] \%*\% inverse(H[j,j]) \%*\% U[j]}.  (This is done using the the \code{solve} function of course, rather than
taking an explicit inverse).  The result is a 3 degree of freedom chisquare
statistic. 
For a single variable test, the fact that only a single element of \code{U[j]}
is non-zero allows for a faster shortcut calculation.

A few further things need to be considered.
\begin{enumerate}
  \item There may be NA coefficients in the fit, e.g., for a model that
   has redundant variables in its $X$ matrix.  It is fairly simple to keep 
   track of these, and remove any such from our set of variables $j$.
   \item There is not a good defintion of how to test PH for a random effects
     term, e.g., from a coxme model or a copxh fit with a frailty term.
     For these, we treat the resultant random coefficients as though they were
     fixed, and test the other variable under this constraint.
   \item For a penalized model, the penalty is assumed to apply to both the
     original and to then extended coefficients.  However,
     \begin{itemize}
       \item Penalties depend only the coefficient $hat \beta$, not on the
         data or the time weights.
       \item All the penalties that we support are 0 at $\beta =0$, and have a
         first derivative of 0 there, so there is no impact on the score vector
         $U$.  $U$ is 0 for the current covariates, by definition, and the new
         ones are being evaluated at 0.
       \item There will be an impact on the second derivative, however.  But
         this will by definition be idential to the penalty for the original
         variables.
     \end{itemize}
   \item The most difficult issue is use of a robust variance in the original
     model.  This requires not just the score vector $U$, but the $n by p$
     matrix of of dfbeta residuals $D$.
     This requires a special version of the relevant C routine; there are no
     simple computaional shortcuts.
\end{enumerate}



\chapter{Accelerated Failure Time models}
\label{chap:aft}
\section{Usage}
The \co{survreg} function implements the class of parametric accelerated
failure time models.  
Assume that the survival time $y$ satisfies $\log(y) = X'\beta + \sigma W$,
for $W$ from some given distribution.
Then if $\Lambda_w(t)$ is the cumulative hazard function for $W$, the
cumulative hazard function for subject $i$ is 
$\Lambda_w[\exp(-\eta_i/\sigma)t]$, that is, the time scale for the subject
is accelerated by a constant factor.
A good description of the models is found in chapter 3 of 
Kalbfleisch and Prentice \cite{Kalbfleisch80}.

The following fits a Weibull model to the lung cancer data set included
in the package.
\begin{verbatim}
> fit <- survreg(Surv(time, status) {\twiddle} age + sex + ph.karno, data=lung,
        dist='weibull')
> fit
Call:
survreg(formula = Surv(time, status) {\twiddle} age + sex + ph.karno, data = lung, dist
         = "weibull")

Coefficients:
 (Intercept)          age       sex    ph.karno 
    5.326344 -0.008910282 0.3701786 0.009263843

Scale= 0.7551354 

Loglik(model)= -1138.7   Loglik(intercept only)= -1147.5
        Chisq= 17.59 on 3 degrees of freedom, p= 0.00053 
n=227 (1 observations deleted due to missing)
\end{verbatim}

The code for the routines has undergone substantial revision between
releases 4 and 5 of the code.
Calls to the older version are not compatable with all of the changes,
users can use the \code{survreg.old} function if desired, which
retains the old argument style (but uses the newer maximization
code).
Major additions included penalzed models, strata, user specified
distributions, and more stable maximization code.

\section{Strata}
In a Cox model the \code{strata} statement is used to allow separate
baseline hazards for subgroups of the data, while retaining 
common coefficients for the other covariates across groups.
For parametric models, the statement allows for a separate
scale parameter for each subgroup, but again keeping the other
coefficients common across groups.
For instance, assume that separate ``baseline'' hazards were
desired for males and females in the lung cancer data set.
If we think of the intercept and scale as the baseline shape,
then an appropriate model is
\begin{verbatim}
> sfit <- survreg(Surv(time, status) ~ sex + age + ph.karno + strata(sex),
        data=lung)
> sfit
Coefficients:
 (Intercept)       sex          age   ph.karno 
    5.059089 0.3566277 -0.006808082 0.01094966

Scale:
     sex=1     sex=2 
 0.8165161 0.6222807

Loglik(model)= -1136.7   Loglik(intercept only)= -1146.2
        Chisq= 18.95 on 3 degrees of freedom, p= 0.00028 
\end{verbatim}
The intercept only model used for the likelihood ratio test has
3 degrees of freedom, corresponding to the intercept and two scales, as compared to the
6 degrees of freedom for the full model.

This is quite different from the effect of the \code{strata}
statement in \code{censorReg}; there it acts as a `by'
statement and causes a totally separate model to be fit
to each gender. 
The same fit (but not as nice a printout) can be obtained from
\code{survreg} by adding an explicit interaction to the formula:
\begin{verbatim}
 Surv(time,status) ~ sex + (age + ph.karno)*strata(sex)
\end{verbatim}
 
\section{Penalized models}
Let the linear predictor for a \code{survreg} model be
$\eta = X\beta + Z\omega$, and consider maximizing the penalized
log-likelihood
$$
  PLL = LL(y; \beta, \omega) - p(\omega; \theta)\,,
$$
where $\beta$ and $\omega$ are the unconstrained effects, respectively,
$X$ and $Z$ are the covariates, 
$p$ is a function that penalizes certain choices for $\omega$,
and $\theta$ is a vector of tuning parameters.

For instance, ridge regression is based on the penalty
$p = \theta \sum \omega_j^2$; it shrinks coefficients towards zero.

The penalty functions in \code{survreg} currently use the same code
as those for \code{coxph}.
This works well in the case of ridge and pspline, but frailty terms
are more problematic in that the code to automatically choose the
tuning parameter for the random effect no longer solves an MLE
equation.
The current code will not lead to the correct choice of penalty.

\section{Specifying a distribution}
The fitting routine is quite general, and can accept any distribution that
spans the real line for $W$, and any monotone transformation of $y$.
The standard set of distributions is contained in a list 
\code{survreg.distributions}. Elements of the list are of two types.
Basic elements are a description of a distribution.  Here is the entry
for the logistic family:
\begin{verbatim}
logistic = list(
    name  = "Logistic",
    variance = function(parm) pi^2/3, 
    init  = function(x, weights, ...) \{
        mean <- sum(x*weights)/ sum(weights)
        var  <- sum(weights*(x-mean)^2)/ sum(weights)
        c(mean, var/3.2)
        \},
    deviance= function(y, scale, parms) \{
        status <- y[,ncol(y)]
        width <- ifelse(status==3,(y[,2] - y[,1])/scale, 0)
        center <- y[,1] - width/2
        temp2 <- ifelse(status==3, exp(width/2), 2) #avoid a log(0) message
        temp3 <- log((temp2-1)/(temp2+1))
        best <- ifelse(status==1, -log(4*scale),
                                    ifelse(status==3, temp3, 0))
        list(center=center, loglik=best) 
        \},
    density = function(x, ...) \{
        w <- exp(x)
        cbind(w/(1+w), 1/(1+w), w/(1+w)^2, (1-w)/(1+w), (w*(w-4) +1)/(1+w)^2)
        \},
    quantile = function(p, ...) log(p/(1-p))
    )
\end{verbatim}
\begin{itemize}
\item Name is used to label the printout.
\item Variance contains the variance of the distribution. For distributions
with an optional parameter such as the $t$-distribution, the \code{parm} argument will
contain those parameters.
\item Deviance gives a function to compute the deviance residuals.  More
on this is explained below in the mathematical details.
\item The density function gives the necessary quantities to fit the
distribution.  It should return a matrix with columns $F(x)$, $1-F(x)$,
$f(x)$, $f'(x)/f(x)$ and $f''(x)/f(x)$, where $f'$ and $f''$ are the
first and second derivatives of the density function, respectively. 
\item The quantiles function returns quantiles, and is used for residuals.
\end{itemize}
The reason for returning both $F$ and $1-F$ 
in the density function is to avoid round off error
when $F(x)$ is very close to 1. 
This is quite simple for symmetric distributions, in the Gaussian case
for instance we use \code{qnorm(x)} and \code{qnorm(-x)} respectively.
(In the intermediate steps of iteration very large deviates may be
generated, and a probabilty value of zero will cause further problems.)

Here is an example of the second type of entry:
\begin{verbatim}
exponential = list(
    name  = "Exponential",
    dist  = "extreme",
    scale =1 ,
    trans = function(y) log(y),
    dtrans= function(y) 1/y ,
    itrans= function(x) exp(x)
    )
\end{verbatim}
This states that an exponential fit is computed by fitting an extreme value
distribution to the log transformation of $y$.
(The distribution pointed to must not itself be a pointer to another).
The extreme value distribution is restricted to have a scale of 1. 
The first derivative of the transformation, \code{dtrans}, is used to
adjust the final log-likelihood of the model back to the exponential's scale.
The inverse transformation \code{itrans} is used to create predicted values
on the original scale.  

The formal rules for an entry are that it must include a name,
either the ``dist" component or the set ``variance",``init",
``deviance", ``density" and ``quantile", an optional scale,
and either all or none of ``trans", ``dtrans" and ``itrans".

The \code{dist="weibull"} argument to the \code{survreg} function chooses the
appropriate list from the survreg.distributions object.  User defined
distributions of either type can be specified by supplying
the appropriate list object rather than a character string.
Distributions should, in general, be defined on the entire real
line.  If not the minimizer used is likely to fail, since it has no
provision for range restrictions.

Currently supported distributions are
\begin{itemize}
\item basic
\begin{itemize}
\item (least) Extreme value
\item Gaussian
\item Logistic
\item $t$-distribution
\end{itemize}
\item transformations
\begin{itemize}
\item Exponential
\item Weibull
\item Log-normal ('lognormal' or 'loggaussian')
\item Log-logistic ('loglogistic')
\end{itemize}
\end{itemize}

% Residuals for parametric survival models
%  and predicted values
\section{Residuals}
\subsection{Response}
The target return value is $y - \yhat$, but what should we
use for $y$ when the observation is not exact?
We will let $\yhat_0$ be the MLE for the location
parameter $\mu$ over a data set with
only the observation of interest, with $\sigma$ fixed
at the solution to the problem as a whole, 
subject to the constraint that $\mu$ be consistent with the data.
That is, for an observation right censored at $t=20$, we constain
$\mu \ge 20$, similarly for left censoring, and constrain
$\mu$ to lie within the two endpoints of an
interval censored observation.
To be consistent as the width of an interval censored observation
goes to zero, this definition does require that the mode of the
density lies at zero.

For exact, left, and right censored observations $\yhat_0 =y$, so
that this appears to be an ordinary response residual.
For interval censored observations from a symmetric distribution,
$\yhat_0 =$ the center of the censoring interval.  
The only unusual case, then, is for a non-symmetric distribution
such as the extreme value.  
As shown later in the detailed information
on distributions, for the extreme value distribution this
occurs for $\yhat_0 = y^l - \log(b/[exp(b)-1])$, where
$b = y^u - y^l$ is the length of the interval.


\subsection{Deviance}
Deviance residuals are response residuals, but transformed to the
log-likelihood scale.  
$$ d_i = sign(r_i) \sqrt{LL(y_i, \yhat_0;\sigma) -LL(y_i,\eta_i; \sigma)}$$
The definition for $\yhat_0$ used for response residuals, however, could
lead to the square root of a negative number  for left or right
censored observations, e.g., if the predicted value for a right censored
observation is less than the censoring time for that observation. 
For these observations we let $\yhat_0$ be the \emph{unconstrained}
maximum, which leads to $yhat_0 = -\infty$ and $+\infty$
for right and left censored observations, respectively,
and a log-likelihood term of 0. 

The advantages of these residuals for plotting and outlier detection
are nicely detailed in McCullagh and Nelder \cite{glim}.
However, unlike GLM models, deviance residuals for interval censored data
are not free of the scale parameter.  
This means that if there are interval censored data values and one fits 
two models A and B, say, 
that the sum of the squared deviance residuals for model A minus the
sum for model B is \emph{not} equal to the difference in log-likelihoods.
This is one reason that the current \code{survreg} function does
not inherit from class \code{glm}: \code{glm} models use the deviance
as the main summary statistic in the printout.

\subsection{Dfbeta}
 The \code{dfbeta} residuals are a matrix with one row per subject and one column
per parameter.
The $i$th row gives the approximate change in the parameter vector due to 
observation $i$, i.e., the change in $\bhat$ when observation $i$ is added
to a fit based on all observations but the $i$th.
The \code{dfbetas} residuals scale each column of this matrix by the standard
error of the respective parameter.

\subsection{Working}
As shown in section \ref{sect:irls} below, the Newton-Raphson iteration used
to solve the model can be viewed as an iteratively reweighted least squares
problem with a dependent variable of ``current prediction - correction''.
The working residual is the correction term. 

\subsection{Likelihood displacement residuals}
Escobar and Meeker \cite{Escobar92} define a matrix of likelihood displacement
residuals for the accelerated failure time model.
The full residual information is a square matrix $\ddot A$, with
dimension the number of pertubations considered.  
Three examples are developed in detail, all with dimension $n$, the number
of observations.

Case weight pertubations measure the overall effect on the parameter vector
of dropping a case.  Let $V$ be the variance matrix of the model, and
$L$ the $n$ by $p$ matrix with elements $\partial L_i/ \partial \beta_j$,
where $L_i$ is the likelihood contribution of the $i$th observation.
Then $\ddot A = LVL'$.  The residuals function returns the diagonal values
of the matrix.  Note that $LV$ equals the \code{dfbeta} residuals.

Response pertubations correspond to a change of 1 $\sigma$ unit in one of
the response values.  For a Gaussian linear model, the equivalent computation
yields the diagonal elements of the hat matrix.

Shape pertubations measure the effect of a change in the log of the scale
parameter by 1 unit. 

The \code{matrix} residual type returns the raw values that can be used to
compute these and other LD influence measures.  The result is an $n$ by
6 matrix, containing columns for
$$
   L_i \quad \frac{\partial L_i}{\partial \eta_i} 
        \quad \frac{\partial^2 L_i}{\partial \eta_i^2}
       \quad \frac{\partial L_i}{\partial \log(\sigma)}       
       \quad \frac{\partial L_i}{\partial \log(\sigma)^2} 
       \quad \frac{\partial^2 L_i}{\partial \eta \partial\log(\sigma)}
$$

\section{Predicted values}
\subsection{Linear predictor and response}
The linear predictor is $\eta_i = x'_i \bhat$, where $x_i$ is the covariate
vecor for subject $i$ and $\bhat$ is the final parameter estimate.
The standard error of the linear predictor is 
$x'_i V x_i$, where $V$ is the variance matrix for $\bhat$.

The predicted response is identical to the linear predictor for fits to
the untransformed distributions, i.e., the extreme-value, logistic and
Gaussian.  For transformed distributions such as the Weibull, for which
$\log(y)$ is from an extreme value distribution, the linear predictor is
on the transformed scale and the response is the inverse transform,
e.g. $\exp(\eta_i)$ for the Weibull.  
The standard error of the transformed response is the standard error
of $\eta_i$, times the first derivative of the inverse transformation.

\subsection{Terms}
Predictions of type \code{terms} are useful for examination of terms in
the model that expand into multiple dummy variables, such as factors
and p-splines.  
The result is a matrix with one column for each of the terms in
the model, along with an optional matrix of standard errors.
Here is an example using psplines on the 1980 Stanford data
\begin{verbatim}
> fit <- survreg(Surv(time, status) ~ pspline(age, df=3) + t5, stanford2,
        dist='lognormal')
> tt <- predict(fit, type='terms', se.fit=T)
> yy <- cbind(tt$fit[,1], tt$fit[,1] -1.96*tt$se.fit[,1],
                          tt$fit[,1] +1.96*tt$se.fit[,1])
> matplot(stanford2$age, yy, type='l', lty=c(1,2,2))

> plot(stanford2$age, stanford2$time, log='y', 
        xlab='Age', ylab='Days', ylim=c(.1, 10000))
> matlines(stanford2$age, exp(yy+ attr(tt$fit, 'constant')), lty=c(1,2,2))
\end{verbatim}
The second plot puts the fit onto the scale of the data, and thus is
similar in scope to figure 1 in Escobar and Meeker \cite{Escobar92}.
Their plot is for a quadratic fit to age, and without T5 mismatch score in
the model.

\subsection{Quantiles}
If predicted quantiles are desired, then the set of probability values
$p$ must also be given to the \code{predict} function.
A matrix of $n$ rows and $p$ columns is returned, whose $ij$ element is
the $p_j$th quantile of the predicted survival distribution, 
based on the covariates of subject $i$.  
This can be written as $X\beta + z_q \sigma$ where $z_q$ is the $q$th
quantile of the parent distribution.
The variance of the quantile estimate is then $cVc'$ where
$V$ is the variance matrix of $(\beta, \sigma)$ and $c=(X,z_q)$.

In computing confidence bands for the quantiles, it may be preferable
to add standard errors on the untransformed scale.
For instance, consider the motor reliability data of Kalbfleisch and
Prentice \cite{Kalbfleisch02}.
\begin{verbatim}
> fit <- survreg(Surv(time, status) ~ temp, data=motors)
> q1 <- predict(fit, data.frame(temp=130), type='quantile',
                p=c(.1, .5, .9), se.fit=T)
> ci1 <- cbind(q1$fit, q1$fit - 1.96*q1$se.fit, q1$fit + 1.96*q1$se.fit)
> dimnames(ci1) <- list(c(.1, .5, .9), c("Estimate", "Lower ci", "Upper ci"))
> round(ci1)
    Estimate Lower ci Upper ci 
0.1    15935     9057    22812
0.5    29914    17395    42433
0.9    44687    22731    66643

> q2 <- predict(fit, data.frame(temp=130), type='uquantile', 
                p=c(.1, .5, .9), se.fit=T)
> ci2 <- cbind(q2$fit, q2$fit - 1.96*q2$se.fit, q2$fit + 1.96*q2$se.fit)
> ci2 <- exp(ci2)  #convert from log scale to original y
> dimnames(ci2) <- list(c(.1, .5, .9), c("Estimate", "Lower ci", "Upper ci"))
> round(ci2)
    Estimate Lower ci Upper ci 
0.1    15935    10349    24535
0.5    29914    19684    45459
0.9    44687    27340    73041
\end{verbatim}
Using the (default) Weibull model, the data is fit on the $\log(y)$ scale.
The confidence bands obtained by the second method are asymmetric and may
be more reasonable.  They are also guarranteed to be $>0$.

This example reproduces figure 1 of Escobar and Meeker 
\cite{Escobar92}.
\begin{verbatim}
> plot(stanford2$age, stanford2$time, log='y', 
        xlab='Age', ylab='Days', ylim=c(.01, 10^6), xlim=c(1,65))
> fit <- survreg(Surv(time, status) ~ age + age^2, stanford2,
        dist='lognormal')
> qq <- predict(fit, newdata=list(age=1:65), type='quantile',
         p=c(.1, .5, .9))
> matlines(1:65, qq, lty=c(1,2,2))
\end{verbatim}

Note that the percentile bands on this figure are really quite a different
object than the confidence bands on the spline fit.  The latter
reflect the uncertainty of the fitted estimate and are related to the
standard error.
The quantile bands reflect the predicted distribution of a subject at
each given age (assuming no error in the quadratic estimate of the
mean), and are related to the standard deviation of the population.

\section{Fitting the model}
\label{sect:irls}
    With some care, parametric survival can be written so as to fit into the
iteratively reweighted least squares formulation used in Generalized
Linear Models of McCullagh and Nelder \cite{glim}.
A detailed description of this setup for general maximum likelihood
computation is found in Green \cite{Green84}.

Let $y$ be the data vector (possibly transformed), 
and $x_i$ be the vector of covariates for the
$i$th observation.  Assume that
$$ z_i \equiv \frac{y_i - x_i'\beta}{\sigma} \sim f $$
for some distribution $f$, where $y$ may be censored.

Then the likelihood for $y$ is
$$ l = \left( \prod_{exact} f(z_i)/\sigma \,         \right)
       \left( \prod_{right} \int_{z_i}^\infty f(u) du \,   \right)
       \left( \prod_{left}  \int_{-\infty}^{z_i} f(u) du \,\right)
       \left( \prod_{interval} \int_{z_i^l}^{z_i^u} f(u) du \right),
$$
where ``exact'', ``right'', ``left'', and ``interval'' refer to uncensored,
right censored, left censored, and interval censored observations,
respectively,
and $z_i^l$, $z_i^u$ are the lower and upper endpoints, respectively, for
an interval censored observation.
 Then the log likelihood is defined as
\begin{equation}
   LL = \sum_{exact} g_1(z_i) - log(\sigma) + \sum_{right} g_2(z_i) +
           \sum_{left} g_3(z_i)  + \sum_{interval} g_4(z_i, z_i^*)\,,
\label{ggdef} \end{equation}
where $g_1=\log(f)$, $g_2 = \log(1-F)$, etc.

Derivatives of the LL with respect to the regression parameters are:
\begin{eqnarray}
\frac{\partial LL}{\partial \beta_j} &=&
        \sum_{i=1}^n \frac{\partial g}{\partial \eta_i}
                     \frac{\partial \eta_i}{\partial \beta_j} \nonumber \\
        &=&        \sum_{i=1}^n x_{ij} \frac{\partial g}{\partial \eta_i} \\
\frac{\partial^2 LL} {\partial \beta_j \beta_k} &=&
           \sum x_{ij}x_{ik} \frac{\partial^2 g}{\partial \eta_i^2}\, ,
\end{eqnarray}
where $\eta = X'\beta$ is the vector of linear predictors.

Ignore for a moment the derivatives with respect to $\sigma$ (or treat
it as fixed). 
The Newton-Raphson step defines an update $\delta$
$$  (X^T DX) \delta = X^T  U, $$
where $D$ is the diagonal matrix formed from $-g''$,
and $U$ is the vector $g'$.
The current estimate $\beta$ satisfies $X \beta = \eta$, so that the new
estimate $\beta + \delta$ will have
\begin{eqnarray*}
    (X^T DX)(\beta + \delta) &=& X^T D \eta + X^T U \\
                           &=& (X^T D) (\eta + D^{-1}U)
\end{eqnarray*}
Thus if we treat $\sigma$ as fixed, iteration
is equivalent to IRLS with weights of $-g''$ and adjusted dependent variable
of $\eta - g'/g''$.
At the solution to the iteration we might expect that $\hat\eta \approx y$;
and a weighted regression with $y$ replacing $\eta$ gives, in general, 
good starting estimates for the iteration.
(For an interval censored observation we use the center of the interval
as `y').
Note that if all of the observations are uncensored, then this reduces
to using the linear regression of $y$ on $X$ as a starting estimate:
$y=\eta$ so $z=0$, thus $g'=0$ and $g''=$ a constant (all of the supported
densities have a mode at zero).

This clever starting estimate is introduced in Generalized Linear
Models (McCullagh and Nelder \cite{glim}), and works extremely
well in that context: convergence often occurs in 3-4 iterations.
It does not work quite so well here, since a ``good" fit to a right
censored observation might have $\eta >> y$.
Secondly,
the other coefficients are not independent of $\sigma$, and $\sigma$ often appears
to be the most touchy variable in the iteration. 

Most often, the routines will be used with $\log(y)$, which
corresponds to the set of accelerated failure time models. 
The transform can be applied implicitly or explicitly;
the following two fits give identical coefficients:
\begin{verbatim}
> fit1 <- survreg(Surv(futime, fustat)~ age + rx, fleming, dist='weibull')
> fit2 <- survreg(Surv(log(futime), fustat) ~ age + rx, data=fleming,
                dist='extreme')
\end{verbatim}
The log-likelihoods for the two fits differ by a constant, i.e.,
the sum of $d\log(y)/dy$ for the uncensored observations, and certain
predicted values and residuals will be on the $y$ versus $\log(y)$ scale.

\section {Derivatives}
  This section is very similar to the appendix of Escobar and Meeker
\cite{Escobar92}, differing only in our use of $\log(\sigma)$ rather than
$\sigma$ as the natural parameter.  
Let $f$ and $F$ denote the density and distribution functions, respectively,
of the distributions.  Using (\ref{ggdef}) as the definition of
$g1,\ldots,g4$ we have
\begin{eqnarray*}
\frac{\partial g_1}{\partial \eta} &=& - \frac{1}{\sigma}
                \left[\frac{f'(z)}{f(z)}  \right]             \\
\frac{\partial g_4}{\partial \eta} &=& - \frac{1}{\sigma} \left[
               \frac{f(z^u) - f(z^l)}{F(z^u) - F(z^l)}  \right] \\
\frac{\partial^2 g_1}{\partial \eta^2} &=&  \frac{1}{\sigma^2}
                \left[ \frac{f''(z)}{f(z)} \right]
                 - (\partial g_1 / \partial \eta)^2                   \\
\frac{\partial^2 g_4}{\partial \eta^2} &=&  \frac{1}{\sigma^2} \left[
                \frac{f'(z^u) - f'(z^l)}{F(z^u) - F(z^l)} \right]
                 - (\partial g_4 / \partial \eta)^2                 \\
\frac{\partial g_1}{\partial \log\sigma} &=& -  \left[
                \frac{zf'(z)}{f(z)}     \right]                          \\
\frac{\partial g_4}{\partial \log\sigma} &=& -  \left[
                \frac{z^uf(z^u) - z^lf(z^l)}{F(z^u) - F(z^l)} \right] \\
\frac{\partial^2 g_1}{\partial (\log\sigma)^2} &=&   \left[
                 \frac{z^2 f''(z) + zf'(z)}{f(z)} \right]
                - (\partial g_1 / \partial \log\sigma)^2                   \\
\frac{\partial^2 g_4}{\partial (\log\sigma)^2} &=&  \left[
                \frac{(z^u)^2 f'(z^u) - (z^l)^2f'(z_l) }
                {F(z^u) - F(z^l)} \right]
  - \partial g_1 /\partial \log\sigma(1+\partial g_1 / \partial \log\sigma)  \\
\frac{\partial^2 g_1}{\partial \eta \partial \log\sigma} &=&
               \frac{zf''(z)}{\sigma f(z)}
       -\partial g_1/\partial \eta (1 + \partial g_1/\partial \log\sigma) \\
\frac{\partial^2 g_4}{\partial \eta \partial \log\sigma} &=&
               \frac{z^uf'(z^u) -  z^lf'(z^l)}{\sigma [F(z^u) - F(z^l)]}
       -\partial g_4/\partial \eta (1 + \partial g_4/\partial \log\sigma) \\
\end{eqnarray*}
To obtain the derivatives for $g_2$, set the upper endpoint $z_u$ to $\infty$
in the equations for $g_4$.  To obtain the equations for $g_3$, left censored
data, set the lower endpoint to $-\infty$.

After much experimentation, a further decision was made to do the internal
iteration in terms of $\log(\sigma)$.  This avoids the boundary condition at
zero, and also helped the iteration speed considerably for some test
cases.
The changes to the code were not too great.  By the chain rule
\begin{eqnarray*}
\frac{\partial LL}{\partial \log\sigma} &=&
          \sigma \frac{\partial LL}{\partial \sigma}           \\
\frac{\partial^2 LL}{\partial (\log\sigma)^2} &=&
          \sigma^2 \frac{\partial^2 LL}{\partial \sigma^2} +
          \sigma \frac{\partial LL}{\partial \sigma}               \\
\frac{\partial^2 LL} {\partial \eta \partial \log\sigma} &=&
          \sigma \frac{\partial^2}{\partial \eta \partial \sigma}
\end{eqnarray*}
At the solution $\partial LL/\partial \sigma =0$, so the variance matrix
for $\sigma$ is a simple scale change of the returned matrix for
$\log(\sigma)$.


\section{Distributions}
\subsection {Gaussian}
 Everyone's favorite distribution.  The continual calls to $\Phi$ may make
it slow on censored data, however.  Because there is no closed form for
$\Phi$, only the equations for $g_1$ simplify from the general form given
in section 2 above.
\begin{eqnarray*}
  \mu=0 &,& \sigma^2=1 \\
   F(z)&=& \Phi(z) \\
    f(z)&=&\exp(-z^2/2) / \sqrt{2 \pi} \\
   f'(z) &=& -zf(z) \\
   f''(z) &=& (z^2-1)f(z)
\end{eqnarray*}
For uncensored data, the standard glm results are clear by substituting
$g_1= -z/\sigma$ into equations  1-5.  The first derivative vector is equal to
$X'r$ where $r= -z/\sigma$ is a scaled residual, the update step $I^{-1}U$
is independent of the estimate of $\sigma$, and the maximum likelihood
estimate of $n\sigma^2$ is the sum of squared residuals.
None of these hold so neatly for right censored data.

\subsection{Extreme value}


    If $y$ is Weibull then $\log(y)$ is distributed according to the
(least) extreme value distribution.  
As stated above, fits on the latter scale are numerically preferable 
because it removes the range restriction on $y$.
A Weibull distribution with the scale restricted to 1 gives
an exponential model.
$$     \mu= -\gamma = .5722\ldots,\; \sigma^2=\pi^2/6 $$
\begin{eqnarray*}
  F(z)&=&1 - \exp(-w)\\
  f(z)&=& we^{-w}    \\
  f'(z) &=& (1-w)f(z)   \\
  f''(z) &=& (w^2 - 3w+1)f(z)
\end{eqnarray*}
where $w \equiv exp(z)$.

The mode of the distribution is at $f(0) = 1/e$, 
so for an exact observation the
deviance term has $\hat y = y$.  For interval censored data
where the interval is
of length $b = z^u - z^l$, it turns out that we cover the most mass if the
interval has a lower endpoint of $a=\log(b/(\exp(b)-1)))$, and
the resulting log-likelihood is
$$
    \log(e^{-e^a} - e^{-e^{a+b}}).
$$
Proving this is left as an exercise for the reader.

The cumulative hazard for the Weibull is usually written as
$\Lambda(t) = (at)^p$.  
Comparing this to the extreme value we see that
$p = 1/\sigma$ and
$a= \exp(-\eta)$.  
(On the hazard
scale the change of variable from $t$ to $\log(t)$ adds another
term).  
The Weibull can be thought of as both an accelerated failure time
model, with acceleration factor $a$ or as a proportional hazards
model with constant of proportionality $a^p$.
If a Weibull model holds, the coefficients of a Cox model will be
approximately equal to $-\beta/\sigma$, the latter coming from
a \code{survreg} fit.
The change in sign reflects a change in perspective: in a proportional hazards
model a positive coefficient corresponds to an increase in the
death rate (bad), 
whereas in an accelerated failure time model a positive coefficient 
corresponds to an increase in lifetime (good).

\subsection{Logistic}
  This distribution is very close to the Gaussian except in the extreme tails,
but it is easier to compute.
However, some data sets may contain survival times close to zero,
leading to differences in fit between the lognormal and log-logistic
choices.
(In such cases the rationality of a Gaussian fit may also be in question).
Again let $w= \exp(z)$.
$$
\mu=0,\; \sigma^2=\pi^2/3
$$
\begin{eqnarray*}
  F(z) &=& w/(1+w)      \\
  f(z) &=& w/(1+w)^2      \\
  f'(z) &=& f(z)\,(1-w)/(1+w)  \\
  f''(z)&=& f(z)\,(w^2 -4w+1)/(1+w)^2
\end{eqnarray*}

The distribution is symmetric about 0, so for an exact observation the
contribution to the deviance term is $-\log(4)$.  For an interval censored
observation with span $2b$ the contribution is
$$
    \log\left(F(b) - F(-b)\right) = \log \left( \frac{e^b-1}{e^b+1} \right).
$$

\subsection{Other distributions}
Some other population hazards can be fit into this location-scale
framework, some can not.
\begin{center}
\begin{tabular}{ll}
Distribution & \multicolumn{1}{c}{Hazard} \\ \hline 
    Weibull&           $p\lambda (\lambda t)^{p-1}$ \\
    Extreme value&     $(1/ \sigma) e^{ (t- \eta)/ \sigma}$\\
    Rayleigh&          $a + bt$\\
    Gompertz&          $b c^t$\\
    Makeham&          $ a + b c^t$  \\
\end{tabular}
\end{center}
         

The Makeham hazard seems to fit human mortality experience beyond
infancy quite well, where $a$ is a constant mortality which is
independent of the health of the subject (accidents, homicide, etc)
and the second term models the Gompertz assumption that ``the average
exhaustion of a man's power to avoid death is such that at the end
of equal infinitely small itervals of time he has lost equal portions of
his remaining power to oppose destruction which he had at the
commencement of these intervals".  For older ages $a$ is a neglible 
portion of the death rate and the Gompertz model holds.

Clearly \begin{itemize}
\item The Wiebull distribution with $p=2$ ($\sigma=.5$) 
is the same as a Rayleigh
distribution with $a=0$.  It is not, however, the most general form of a 
Rayleigh.  
\item The extreme value and Gompertz distributions have the same
hazard function, with
$ \sigma = 1/ \log(c)$, and $\exp(-\eta/ \sigma) = b$.
\end{itemize}
It would appear that the Gompertz can be fit with an identity link function
combined with the extreme value distribution.  However, this ignores a
boundary restriction.  If $f(x; \eta, \sigma)$ is the extreme value 
distribution
with paramters $\eta$ and $\sigma$, 
then the definition of the Gompertz densitiy
is 
\begin{displaymath}
\begin{array}{ll}
        g(x; \eta, \sigma)  = 0 &                    x< 0 \\
        g(x; \eta, \sigma)  = c f(x; \eta, \sigma) &  x>=0
\end{array}
\end{displaymath}
where $c= \exp(\exp(-\eta / \sigma))$ is the necessary 
constant so that $g$ integrates
to 1.  If $\eta / \sigma$ is far from 1, then the correction term will be
minimal and the above fit will be a good approximation to the Gompertz.

   The Makeham distribution falls into the gamma family (equation 2.3 of
Kalbfleisch and Prentice, Survival Analysis), but with the same range
restriction problem. 

\chapter{Tied event times}
\label{chap:tied}
\section{Cox model estimates}
The theory for the Cox model has always been worked out for the case of
continuous time, which implies that there will be no tied event or censoring
times in the data set.
With respect to censoring times, all of the survival library commands treat
censoring as occuring ``just after'' the recorded time point.
The rationale is that if a subject was last observed alive on day 231, say, 
then their death time, whatever it is, must be strictly greater than 231.  
For formally, a subject who was censored at time $t$ is considered to have
been at risk for any events that occured at time $t$.

The issue with tied event times is more complex, and the software supports 
three different algorithms for dealing with this.
The Cox partial likelihood is a sum of terms, one for each event time, each
of which compares the subject who had the event to the set of subjects who 
``could have had the event": the \emph{risk set}.
The overall view is essentially a lottery model: at each event time there
was a drawing to select one subject for the event,
the risk score for each subject $\exp(X_i\beta)$ tells how many ``tickets'' 
each subject had in the drawing.
The software implements three different algorithms for dealing with tie
event times.

The Breslow approximation (\code{ties='breslow'}) in essence ignores the ties.
If two subjects A and B died on day 100, say, 
the partial likelihood will have separate terms for the two deaths.
Subject A will be at risk for the death that happened to B, and B will be at 
risk for the death that happened to A.
In life this is not technically possible of course: whoever died first will
not be at risk for the second death.

The Efron approximation can be motivated by the idea of \emph{coarsening}:
time is on a continuous scale but we only observe a less precise version of
it.  For example consider the acute myelogenous leukemia data set that is
part of the package, which was a clinical trial to test if extendend
chemotherapy (``Maintained'') was superior to standard.
The time to failure was recorded in months, and in the non-maintained arm
there are two pairs of failures, at 5 and at 8 months.
It might be reasonable to assume that if the data had been recored in days
these ties might not have occured.
<<>>=
with(subset(aml, x=="Nonmaintained"), Surv(time, status))
@ 

Let the risk scores $\exp(X\beta)$ for the 12 subjects be $r_1$--$r_{12}$,
and assume that the two failures actually at month 5 are not tied on a finer
time scale. 
For the first event, whichever it is, the risk set with be all subjects 1--12
and the denominator of the partial likelihood term is $\sum r_i$.
For the second event, the denominator is either
$r_1 + r_3 + \ldots r_{12}$ or $r_2 + r_3 + \ldots r_{12}$;
the Efron approximation is to use the average of the two as the denominator
term.
In the software this is easily done by using temporary case weights:
if there were $k$ tied events then one of the denomiators gives each of those
$k$ subjects a weight of 1, then next gives each a weight of $(k-1)/k$, then
next a weight of $(k-2)/k$, etc.  
The Efron approximation imposes a tiny bit more bookkeeping, but
the the computational burden is no different that for case weights; 
i.e., it effectively takes no more computational time than the Breslow
approximation.

The third possiblility is the exact partial likelihood due to Cox, which treats
the underlying time scale as discrete rather than continuous.
When taking this view the denominator of the partial likelihood term is
again an average, but over a much larger subset.
If there are $k$ events and $n$ subjects at risk, the EPL sum is over all
$k$ choose $n$ possible choices.  
In the AML example above, the event at time 5 will be a sum over 12(11)/2= 33
terms.  If the number of ties is large this quickly grows unreasonable:
for 20 ties out of 1000 the sum has over 39 billion terms.
A clever algorithm by Gail makes this sum barely possible, but it does not
extend to the case of (tstart, tstop) style data.

An important aside is that the log-likelihood for matched logistic
regression is identical to the Cox partial likelihood for a particular
data set, when the EPL is used.
Namely, set time=1 (or any other constant), status = 0 for controls and 1 for 
cases, and fit a coxph model with each matched set as a separate stratum.
In most instances a matched set will consist of a single case along with one
or more controls, however, which is the case where the Breslow, Efron, and EPL
are identical.  (The EPL will still take slightly longer to run due to 
setting up the
necessary structure for all those sums.)

How important are the ties, actually?
Below we show a small computation in which a larger data set is successively
coarsened and compare the results.
The colon cancer data set has 929 subjects with stage B/C colon cancer
who were randomized to three treatment arms and then followed for 5
years; the time to death or progression is in days.
In the example below we successively coarsen the time scale to be
monthly, bimonthly, \ldots, bi-annual; the last of which generates an
very large number of ties. 
What we see is that
\begin{itemize}
  \item The Efron approximation is quite good at dealing with the
    coarsened data, producing nearly the same coefficient as the
    original data even when the coarsening is extreme.
  \item The Breslow approximation is biased somewhat towards 0,
    the exact paritial likelihood somewhat away from 0.
  \item The differences are very small. With monthly coarsening,
    which is itself fairly large, the 3 estimates differ by about .01
    while the standard error of the original coefficient is 0.96;
    i.e. a shift that is statistically immaterial.
\end{itemize}
    
<<coarsen,fig=TRUE>>=
tdata <- subset(colon, etype==1)   # progression or death
cmat <- matrix(0, 7, 6)
for( i in 1:7) {
    if (i==1) scale <-1 else scale <- (i-1)*365/12
    temp <- floor(tdata$time/scale)
    tfit <-  coxph(Surv(temp, status) ~ node4 + extent, tdata)
    tfit2 <- coxph(Surv(temp, status) ~ node4 + extent, tdata,
                   ties='breslow')
    tfit3 <- coxph(Surv(temp, status) ~ node4 + extent, tdata,
                   ties='exact')
    cmat[i,] <- c(coef(tfit2), coef(tfit), coef(tfit3))
}
matplot(1:7, cmat[,c(1,3,5)], xaxt='n', pch='bec',
        xlab="Time divisor", ylab="Coefficient for node4")
axis(1, 1:7, c(1, floor(1:6 *365/12)))
@ 

Early on in the package the decision was made to make the Efron approximation
the default. 
The reasoning was simply that is \emph{is} more accurate, even if only a little,
and the author's early background in numerical analysis argued strongly to
always use the best approximation available.
The second reason is that the computational cost is low. 
Most of us would pick up a 1 Euro coin on the sidewalk, even though
it will not make any real change in our income.
One downside is that no other package did this, leading to a very common
complaint/question that R ``gives different results''.
A second is that it leads to further downstream programming as discussed in 
following sections.

\section{Cumulative hazard and survival}
The coarsening argument can also be applied to the cumulative hazard
$\Lambda(t)$.
Say that there were 3 deaths with 10 subjects at risk.
The increment to the Nelson-Aalen cumulative hazard estimate would
then be 3/10.
If the data had been observed in continuous time, however, there would have
been 3 increments of 1/10 + 1/9 + 1/8.
This estimate was explored by Fleming and Harrington \cite{Fleming84}.
In the \code{survfit} function the \code{ctype} option selects for
1=Nelson-Aalen and 2= Fleming-Harrington.

The Kaplan-Meier estimate is not subject to the coarsening phenominon.
In our example, the observed data will lead to a multipilicative increment
of 7/10 and the continuous data to one of (9/10)(8/9)(7/8), which are the 
same.
An alternate estimate of the survival is $S(t) = \exp(-\Lambda(t))$.
Basing this on the FH estimate of hazard will more closely track the KM 
when there are tied event times.
The direct (KM) vs. exponential estimates of survival are obtained with
the \code{ctype=1} and \code{ctype=2} arguments; however, the
exponential estimate is quite uncommon outside of the Cox model.

\section{Predicted cumulative hazard and survival from a Cox model}

Predictions from a coxph model must always be for \emph{someone}, i.e.,
some particular set of covariate values.
Let $r_i = \exp(X_i\hat\beta -c)$ be the recentered risk scores for each
subject $i$, where the recentering constant $c = X_n\hat\beta$,
$X_n$ being the covariates of the ``new'' subject for which prediction
is desired.  
(We don't want to create a prediction for a baseline subject with
X=0, what textbooks often call a ``baseline hazard'', since if 0 is too
far from the center of the data the exp function can easily overflow.)
The estimated cumulative hazard at any event time is then
\begin{equation}
  \Lhat(t) = \int_0^t \frac{\sum_i dN_i(t)}{\sum_i Y_i(t) r_i} 
  \label{haz:breslow}
\end{equation}
Equation \eqref{haz:breslow} is known as the Breslow estimate;
if $\hat\beta=0$ then $r_i=1$ and it becomes equal to
the Nelson-Aalen estimator.

If the Efron estimate is used for ties, then the software uses an Efron
estimate of the cumulative hazard; which reduces to the Fleming-Harrington
if $\hat\beta =0$.  
Using the hazard estimate that matches the parial likelihood estimate causes
an important property of the vector of martingale residuals $m$ to hold, 
namely that $mX$ is equal to the first derivative of the partial likelihood.
residuals hold for both

The Cox model is a case where the default estimate of survival is based on
the exponent of the cumulative hazard, rather than a 'direct' one such as
the Kaplan-Meier.  There are three reasons for this.
\begin{enumerate}
  \item The most obvious 'direct' estimate is to use 
    $(\sum dN_i(t) -\sum Y_i(t)r_i)/ (\sum Y_i(t)r_i)$ as a multiplicative
    update at each event time $t$.  This expression in not guarranteed to
    be between 0 and 1, however, particular for new subjects who are near
    or past the boundaries of the original data set.  This leads to using
    some sort of ad hoc correction to avoid failure.
  \item The direct estimate of Kalbfleisch and Prentice avoids this, but it
    does not extend to delayed entry, multi-state models or other extensions
    of the basic model.
  \item The KP estimate reqires iteration so the code is more complex.
\end{enumerate}

\chapter{Multi-state models}
Multi-state hazards models have a very interesting (and useful)
property,
which is that hazards can be estimated singly (without reference to any
other transition) but probability-in-state estimators must be computed 
globally.
Thus, one can estimate non-parametric cumulative hazard estimates 
(Nelson-Aalen), the hazard ratios for any given transtion (Cox model)
or the predicted cumulative hazard function based on a per transition
Cox model without incurring any issues with respect to competing risks.
(If there is informative censoring the overall and individual estimates
still agree, but they will both be wrong.
An example of informative censoring would be subjects who are removed from
the data because of an impending event, e.g., censoring subjects who enter
hospice care would underestimate death rates.)

Now say that we had a simple competing risks problem, 10 subjects are
alive and in the initial state on day 100, at which time two of them
transition to
two different endpoints.  
A coarsening argument would say that on the underlying continuous time
scale these two subjects would not be tied, and then would use  9.5 
as the denominator for
each of the two cumulative hazard increments.
Such an estimate would however then be at variance with the two indiviually 
computed hazards: global coarsening removes the separability.
The survival package takes a moderated view and will apply the coarsening
argument separately to each hazard, i.e., it chose to retain the separation
policy.



\appendix
\chapter{Changes from version 2.44 to 3.1}

\section{Changes in version 3}
\label{sect:changes}
Some common concepts had appeared piecemeal in
more than one function, but not using the same keywords.  Two particular areas
are survival curves and multiple observations per subject.

Survival and cumulative hazard curves are generated by the 
\code{survfit} function, either from
raw data (survfit.formula), or a fitted Cox or parametric survival model
(survfit.coxph, survfit.survreg). 
Two choices that appear are:
\begin{enumerate}
  \item If there are tied event times, to estimate the hazard using a 
    straightforward increment of (number of events)/(number at risk), or
    make a correction for the ties.  The simpler method is known variously
    as the Nelson, Aalen, Breslow, and Tsiatis estimate, along with hyphenated
    forms combining 2 or 3 of these labels.
    One of the simpler corrections for ties is known as the Fleming-Harrington
    approximation when used with raw data, and the Efron when used 
    in a Cox model.
  \item The survival curve $S(t)$ can be estimated directly or as the
    exponential of the cumulative hazard estimate.  The first of these is
    known as the Kaplan-Meier, cumulative incidence (CI), Aalen-Johansen,
    and Kalbfleisch-Prentice estimate, depending on context, 
    the second as a Fleming-Harrington, Breslow, or Efron estimate, again
    depending on context.
\end{enumerate}

With respect to the two above, subtypes of the \code{survfit} routine have
had either a \code{type} or \code{method} argument over the years which tried
to capture both of these at the same time, 
and consequently have had a bewildering number of options,
for example ``fleming-harrington'' in \code{survfit.formula} 
stood for the simple cumulative hazard
estimate plus the exponential survival estimate, 
``fh2'' specified the tie-corrected cumulative hazard plus exponential survival,
while \code{survfit.coxph} used ``breslow'' and ``efron'' for the same two
combinations.
The updated routines now have separate \code{stype} and \code{ctype}
arguments.  For the first, 1= direct and 2=exponent of the cumulative hazard
and for the second, 1=simple and 2= corrected for ties. 

The Cox model is a special case in two ways: 
1. the the way in which ties are treated
in the likelihood should match the way they are treated in creating the hazard
and 2. the direct estimate of survival can be very difficult to compute.
The survival package's default is to use the \code{ctype} option 
which matches the ties option
of the \code{coxph} call along with an exponential estimate of survival.
This \code{ctype} choice preserves some useful properties of the martingale
residuals.

A second issue is multiple observations per subject, and how those impact
the computations.  This leads to 3 common arguments:
\begin{itemize}
  \item id: an identifier in each row of the data, which allows the routines
    to identify multiple rows for a subject
  \item cluster: identify correlated rows, which should be combined when
    creating the robust variance
  \item robust: TRUE or FALSE, to compute a robust variance.
\end{itemize}


These arguments have been inconsistent in the past, partly because of the
sequential appearance of multiple use cases.  The package started with
only the simplest data form: one observation per subject, one endpoint.
To this has been added:
\begingroup
\renewcommand{\theenumi}{\alph{enumi}}
\begin{enumerate}
  \item Multiple observations per subject
  \item Multiple endpoints per subject
  \item Multiple types of endpoints
\end{enumerate}
\endgroup

Case (a) arises as a way to code time-dependent covariates, and in this
case an \code{id} statement is not needed, and in fact you will get the
same estimates and standard errors with or without it. 
(There will be a change in the counts of subjects who leave or enter an
interval, since an observation pair (0, 10), (10, 20) for the same subject
will not count as an exit (censor) at 10 along with an entry at 10.)
If (b) is true then the robust variance is called for and one will want to 
have either a \code{cluster} argument or the \code{robust=TRUE} argument.
In the coxph routine, a \code{cluster(group)} term in the model statement
can be used instead of the cluster argument,
but this is no longer the preferred form. 
When (b) and (c) are true then the \code{id} statement is required in order
to obtain a correct \emph{estimate} of the result. 
This is also the case for (c) alone when subjects do not all start in the
same state.  
For competing risks data --- multiple endpoints, 
everyone starts in the same state, only one transition per subject --- 
the \code{id} statement is not necessary nor (I think) is a robust variance.

When there is an \code{id} statement but no \code{cluster} or \code{robust}
directive, then the programs will use (b) as a litmus test to decide
between model based or robust variance, if possible.
(There are edge cases where only one of the two variance estimates has 
been implemented, however).
If there is a \code{cluster} argument then \code{robust=TRUE} is assumed.
If only a \code{robust=TRUE} argument is given
then the code treats each line of data as independent.

\section{Survfit}
  There has been a serious effort to harmonize the various survfit methods.
Not all paths had the same options or produced the same outputs.

\begin{itemize}
  \item Common arguments of id, cluster, influence, stype and ctype.
    \begin{itemize} 
      \item If stype=1 then the survival curve S(t) is produced directly,
        if stype=2 it is created as the exp(-H) where H is the cumulative hazard.
      \item If ctype=1 the Nelson-Aalen formula is used, and for ctype=2 there
        is a correction for ties.
      \item The usual curve for a Cox model using the Efron approximate is
        (2,2), for instance, while the ordinary non-parametric KM is (1, 1).
    \end{itemize}
  \item The routines now produce both the estimated survival and the
    estimated cumulative hazard, along with their errors 
  \item Some code paths produce std(S) and some std(log(S)), the object now
    contains a \code{log.se} flag telling which.  (Before, downstream routines
    just ``had to know'').
  \item using a single subscript on a survfit object now behaves like the
    use of a single subscript on an array or matrix, in that the result has
    only one dimension.
\end{itemize}

A utility function \code{survfit0} is used by the print and plot methods to add
a starting ``time 0'' value, normally x=0, y=1, to the survival curve(s).  
It also aligns
all the matrices so that they correspond to the time vector, inserts the
correct standard errors, etc.  
This may be useful to other programs.

\section{Coxph}
 The multi-state objects include a \code{states} vector, which is a simple
list of the state names.
The \code{cmap} component is an integer matrix  with one row for each term in the
model and one column for each transition. 
Each element indexes a position in the coefficient vector and variance matrix.
\begin{itemize}
  \item Column labels are of the form 1:2, which denotes a transition from
    \code{state[1]} to \code{state[2]}.
  \item If a particular term in the data, ``age'' say, was not part of the model
    for a particular transition then a 0 will appear in that position 
    of \code{cmap}.
   \item If two transitions share a common coefficient, both those element of
    \code{cmap} will point to the same location.
   \item Following the coefficient information will be a row labeled 
     \code{(Baseline)}, which contains integers identifing which transitions do
     or do not share their baseline hazard.
   \item Following this are rows for each strata term (if any) in the model, 
     each a 0/1 vector which marks transitions to which this strata applies.
\end{itemize}

\bibliographystyle{plain}
\bibliography{refer}

\end{document}


