1154 lines
52 KiB
Plaintext
1154 lines
52 KiB
Plaintext
|
\documentclass{article}[11pt]
|
||
|
|
\usepackage{Sweave}
|
||
|
|
\usepackage{amsmath}
|
||
|
|
\addtolength{\textwidth}{1in}
|
||
|
|
\addtolength{\oddsidemargin}{-.5in}
|
||
|
|
\setlength{\evensidemargin}{\oddsidemargin}
|
||
|
|
%\VignetteIndexEntry{Multi-state models and competing risks}
|
||
|
|
|
||
|
|
\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=compete,width=6,height=4}
|
||
|
|
\newcommand{\myfig}[1]{\includegraphics[height=!, width=\textwidth]
|
||
|
|
{compete-#1.pdf}}
|
||
|
|
\setkeys{Gin}{width=\textwidth}
|
||
|
|
<<echo=FALSE>>=
|
||
|
|
options(continue=" ", width=60)
|
||
|
|
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")
|
||
|
|
@
|
||
|
|
|
||
|
|
\title{Multi-state models and competing risks}
|
||
|
|
\author{Terry Therneau \and Cynthia Crowson \and Elizabeth Atkinson}
|
||
|
|
\newcommand{\code}[1]{\texttt{#1}}
|
||
|
|
|
||
|
|
\begin{document}
|
||
|
|
\maketitle
|
||
|
|
|
||
|
|
\section{Multi-state models}
|
||
|
|
\begin{figure}
|
||
|
|
<<sfig1, fig=TRUE, echo=FALSE>>=
|
||
|
|
oldpar <- par(mar=c(.1, .1, .1, .1), mfrow=c(2,2))
|
||
|
|
cmat1 <- matrix(c(0,0,1,0), nrow=2,
|
||
|
|
dimnames=list(c("Alive", "Dead"), c("Alive", "Dead")))
|
||
|
|
statefig(c(1,1), cmat1, cex=1.5)
|
||
|
|
|
||
|
|
state4 <- c("0", "1", "2", "...")
|
||
|
|
cmat4 <- matrix(0, 4,4, dimnames= list(state4, state4))
|
||
|
|
cmat4[1,2] <- cmat4[2,3] <- cmat4[3,4] <- 1
|
||
|
|
statefig(c(1,1,1,1), cmat4, bcol=c(1,1,1,0), cex=c(1.5, 1.5, 1.5, 3))
|
||
|
|
|
||
|
|
states <- c("A", "D1", "D2", "D3")
|
||
|
|
cmat2 <- matrix(0, 4,4, dimnames=list(states, states))
|
||
|
|
cmat2[1, -1] <- 1
|
||
|
|
statefig(c(1,3), cmat2, cex=1.5)
|
||
|
|
|
||
|
|
state3 <- c("Health", "Illness", "Death")
|
||
|
|
cmat3 <- matrix(0, 3, 3, dimnames = list(state3, state3))
|
||
|
|
cmat3[1,2] <- cmat3[2,1] <- cmat3[-3, 3] <- 1
|
||
|
|
statefig(c(1,2), cmat3, offset=.03, cex=1.5)
|
||
|
|
|
||
|
|
state4 <- c("0", "1", "2", "...")
|
||
|
|
cmat4 <- matrix(0, 4,4, dimnames= list(state4, state4))
|
||
|
|
cmat4[1,2] <- cmat4[2,3] <- cmat4[3,4] <- 1
|
||
|
|
statefig(c(1,1,1,1), cmat4, bcol=c(1,1,1,0), cex=c(1.5, 1.5, 1.5, 3))
|
||
|
|
par(oldpar)
|
||
|
|
@
|
||
|
|
\caption{Four multi-state models. The upper left panel depicts simple
|
||
|
|
survival, the upper right depicts sequential events,
|
||
|
|
the lower left is an example of competing risks, and
|
||
|
|
the lower right panel is a multi-state illness-death model.}
|
||
|
|
\label{mfig1}
|
||
|
|
\end{figure}
|
||
|
|
|
||
|
|
A multi-state model is used to model a process where subjects transition from one state to the next. For instance, a standard survival curve can be thought of as a simple multi-state model with two states (alive and dead) and one transition between those two states. A diagram illustrating this process is shown in the top left corner of figure \ref{mfig1}. In these types of diagrams, each box is a state and each arrow is a possible transition. The lower left diagram depicts a classic competing risk analysis, where
|
||
|
|
all subjects start on the left and each subject can make a single transition
|
||
|
|
to one of 3 terminal states.
|
||
|
|
The bottom right diagram shows a common multi-state situation known as
|
||
|
|
the illness-death model with recovery.
|
||
|
|
Finally, the upper right diagram represents sequential events, such as repeated
|
||
|
|
infections in the CGD study. In that case one subject had 8
|
||
|
|
events so there are 9 states corresponding to entry into the study
|
||
|
|
(0 infections) and the first, second, \ldots, eighth events.
|
||
|
|
|
||
|
|
As will be shown below, there are often multiple choices for the
|
||
|
|
state and transition diagram, and for some data sets it is revealing to
|
||
|
|
look at a problem from multiple views. In addition to deciding the diagram that best matches the research questions, the two other primary decisions are the choice of time scale for
|
||
|
|
the fits, e.g., time from entry in the study vs. time from entry in
|
||
|
|
the state, and what covariates will be used.
|
||
|
|
|
||
|
|
\section{Multi-state curves}
|
||
|
|
\subsection{Aalen-Johansen estimate}
|
||
|
|
As a starting point for the analysis, it is important to compute and plot
|
||
|
|
estimates of $p(t)$, which is a vector containing the probability of being
|
||
|
|
in each of the states at time $t$.
|
||
|
|
If there is no censoring then $p$ becomes a simple tabulation at time
|
||
|
|
$t$ of all the states.
|
||
|
|
For the general case, we compute this using the Aalen-Johansen estimate via the
|
||
|
|
\code{survfit} function.
|
||
|
|
|
||
|
|
Mathematically the estimate is simple.
|
||
|
|
For each unique time where an event occurs, form the
|
||
|
|
transition matrix $T(t)$ with elements or rates of
|
||
|
|
$\lambda_{ij}(t) =$ the fraction of subjects who transition from
|
||
|
|
state $i$ to $j$ at time $t$, among those in state $i$ just prior to
|
||
|
|
time $t$.
|
||
|
|
($T$ is equal to the identity matrix at any time point without an
|
||
|
|
observed transition.)
|
||
|
|
Then
|
||
|
|
\begin{equation}
|
||
|
|
p(t) = p(0) \prod_{s \le t} T(s) \label{AJ}
|
||
|
|
\end{equation}
|
||
|
|
where $p(0)$ is the initial distribution of subjects.
|
||
|
|
|
||
|
|
Let's work this out for the simple two-state alive $\rightarrow$ {death} model.
|
||
|
|
Let $n(t)$ be the number of subjects still at risk at time $t$ and
|
||
|
|
$d(t)$ the number of deaths at that time point.
|
||
|
|
All subjects start in the alive state and thus $p(0) = (1,0)$ and
|
||
|
|
the transition matrix is
|
||
|
|
\begin{equation*}
|
||
|
|
T(s) = \left( \begin{array}{cc}
|
||
|
|
\frac{n(s)- d(s)}{n(s)} & \frac{d(s)}{n(s)} \\\\ 0 & 1 \end{array}
|
||
|
|
\right)
|
||
|
|
\end{equation*}
|
||
|
|
The two rows are ``start in state 1 (alive)'' and ``start in state 2''
|
||
|
|
and the two colums are ``finish in state 1'', ``finish in state 2 (death)''.
|
||
|
|
The second row corresponds to the fact that death is an absorbing state:
|
||
|
|
the probability of a death $\rightarrow$ alive transition (lower left element)
|
||
|
|
is 0.
|
||
|
|
|
||
|
|
Writing out the matrices for the first few transitions and multiplying
|
||
|
|
them leads to
|
||
|
|
\begin{equation}
|
||
|
|
p_1(t) = \prod_{s \le t} \left[n(s) - d(s)\right] /n(s) \label{km}
|
||
|
|
\end{equation}
|
||
|
|
which we recognize as the Kaplan-Meier estimate of survival.
|
||
|
|
For the two state alive-dead model the Aalen-Johansen (AJ) estimate
|
||
|
|
has reprised the KM.
|
||
|
|
In the competing risks case $p(t)$
|
||
|
|
has an alternate form known as the \emph{cumulative incidence} (CI) function
|
||
|
|
\begin{equation}
|
||
|
|
CI_k(t) = \int_0^t \lambda_k(u) S(u) du \label{cuminc}
|
||
|
|
\end{equation}
|
||
|
|
where $\lambda_k$ is the incidence function for outcome $k$, and $S$ is the
|
||
|
|
overall survival curve for ``time to any endpoint''.
|
||
|
|
Repeating the same matrix exercise for the competing risks, i.e. writing
|
||
|
|
out the Aalen-Johansen computation, exactly recovers the CI formula.
|
||
|
|
The CI is also a special case of the Aalen-Johansen.
|
||
|
|
(The label ``cumulative incidence'' is one of the more unfortunate ones in
|
||
|
|
the survival lexicon,
|
||
|
|
since we normally use `incidence' and `hazard' as interchangeable synonyms but
|
||
|
|
the CI is \emph{not} a cumulative hazard.)
|
||
|
|
The AJ estimate is very flexible; subjects can visit multiple states
|
||
|
|
during the course of a study, subjects can start after time 0 (delayed
|
||
|
|
entry), and they can start in any of the states.
|
||
|
|
The \code{survfit} function implements the AJ estimate and
|
||
|
|
will handle all these cases.
|
||
|
|
|
||
|
|
The standard error of the estimates is computed using an
|
||
|
|
infinitesimal jackknife.
|
||
|
|
Let $D(t)$ be a matrix with one row per subject and one column per state.
|
||
|
|
Each row contains the \emph{change} in $p(t)$ corresponding to subject $i$,
|
||
|
|
i.e., the derivative of $p$ with respect to the $i$th subject's case
|
||
|
|
weight $dp(t)/dw_i$. Then $V(t) = D'WD$ is the estimated variance-covariance
|
||
|
|
matrix of the estimates at time $t$, where $W$ is a diagonal matrix of
|
||
|
|
observation weights.
|
||
|
|
If a single subject is represented by multiple rows in the data set, then
|
||
|
|
$D$ is first collapsed to have one row per subject, the new row for subject
|
||
|
|
$i$ is the sum of the rows for the observations that represented the
|
||
|
|
subject. This is essentially the same algorithm as the robust variance for
|
||
|
|
a Cox model.
|
||
|
|
For simple two state alive -> dead model without case weights,
|
||
|
|
the IJ estimate of variance
|
||
|
|
is identical to the traditional Greenwood estimate for the variance of the
|
||
|
|
survival curve $S$.
|
||
|
|
(This was a surprise when we first observed it; proving the equivalence
|
||
|
|
is not straightforward.)
|
||
|
|
|
||
|
|
|
||
|
|
The $p(t)$ vector obeys the obvious constraint that its sum at
|
||
|
|
any time
|
||
|
|
is equal to one; i.e., each person has to be somewhere.
|
||
|
|
We will use the phrase \emph{probability in state} or simply $p$
|
||
|
|
when referring to the vector.
|
||
|
|
|
||
|
|
In the simple two state model Pr(alive) is the
|
||
|
|
usual KM survival estimate, and we have
|
||
|
|
$p_1(t) = 1- p_2(t)$, Pr(alive) = 1 - Pr(dead).
|
||
|
|
Plots for the 2 state case sometimes choose to show Pr(alive)
|
||
|
|
and sometimes Pr(dead). Which one is used often depends on a historical
|
||
|
|
whim of the disease specialty;
|
||
|
|
cardiology journals for instance quite often use Pr(event) resulting in curves
|
||
|
|
that rise starting from zero,
|
||
|
|
while oncology journals invariably use Pr(alive) giving curves that fall
|
||
|
|
downhill from 1.
|
||
|
|
The \code{survfit} routine's historical default for the 2 state case is to
|
||
|
|
print and plot Pr(alive)= $p_1(t)$, which reflects that the
|
||
|
|
author of the routine was working primarily in cancer trials at the time
|
||
|
|
said default was chosen.
|
||
|
|
|
||
|
|
For simple survival we have gotten used to the idea of using Pr(dead) and
|
||
|
|
Pr(alive) interchangeably, but that habit needs to be left behind for
|
||
|
|
multi-state models, as curves of $1-p_k(t)$
|
||
|
|
= probability(any other state than $k$) are not useful.
|
||
|
|
In the multi-state case, some curves will rise and then fall.
|
||
|
|
For competing risks the curve for the initial state (leftmost
|
||
|
|
in the diagram) is rarely included in the final plot. Since the curves sum to
|
||
|
|
1, the full set is redundant. Pr(nothing yet) is usually the least
|
||
|
|
interesting of the set and so it is left off to make the plot less busy.
|
||
|
|
The remaining curves in the competing risks case rise from 0.
|
||
|
|
(This bothers some researchers as it `just looks wrong' to them.)
|
||
|
|
|
||
|
|
\subsection{Examples}
|
||
|
|
|
||
|
|
\begin{figure}
|
||
|
|
<<crfig2, echo=FALSE, fig=TRUE>>=
|
||
|
|
par(mar=c(.1, .1, .1, .1))
|
||
|
|
frame()
|
||
|
|
oldpar <- par(usr=c(0,100,0,100))
|
||
|
|
# first figure
|
||
|
|
xx <- c(0, 10, 10, 0)
|
||
|
|
yy <- c(0, 0, 10, 10)
|
||
|
|
polygon(xx +10, yy+70)
|
||
|
|
temp <- c(60, 80)
|
||
|
|
for (j in 1:2) {
|
||
|
|
polygon(xx + 30, yy+ temp[j])
|
||
|
|
arrows(22, 70 + 3*j, 28, temp[j] +5, length=.1)
|
||
|
|
}
|
||
|
|
text(c(15, 35, 35), c(75, 65, 85),c("Entry", "Death", "PCM"))
|
||
|
|
text(25, 55, "Competing Risk")
|
||
|
|
|
||
|
|
# Second figure
|
||
|
|
polygon(xx +60, yy+70)
|
||
|
|
for (j in 1:2) {
|
||
|
|
polygon(xx + 80, yy+ temp[j])
|
||
|
|
arrows(72, 70+ 3*j, 78, temp[j] +5, length=.1)
|
||
|
|
}
|
||
|
|
text(50+ c(15, 35, 35), c(75, 65, 85),c("Entry", "Death", "PCM"))
|
||
|
|
arrows(85, 78, 85, 72, length=.1)
|
||
|
|
text(75, 55, "Multi-state 1")
|
||
|
|
|
||
|
|
# third figure
|
||
|
|
polygon(xx+10, yy+25)
|
||
|
|
temp <- c(15, 35)
|
||
|
|
for (j in 1:2) {
|
||
|
|
polygon(2*xx +30, yy + temp[j])
|
||
|
|
arrows(22, 25 + 3*j, 28, temp[j] +5, length=.1)
|
||
|
|
}
|
||
|
|
text(c(15, 40, 40), c(30, 20, 40),c("Entry", "Death w/o PCM", "PCM"))
|
||
|
|
polygon(2*xx + 60, yy+temp[2])
|
||
|
|
arrows(52, 40, 58, 40, length=.1)
|
||
|
|
text(70, 40, "Death after PCM")
|
||
|
|
text(40, 10, "Multi-state 2")
|
||
|
|
@
|
||
|
|
\caption{Three models for the MGUS data.}
|
||
|
|
\label{mfig2}
|
||
|
|
\end{figure}
|
||
|
|
|
||
|
|
Start with a simple competing risks problem as illustrated in the first diagram of figure \ref{mfig2}.
|
||
|
|
The \code{mgus2} data set contains the time to plasma cell malignancy (PCM)
|
||
|
|
and/or death
|
||
|
|
for 1384 subjects diagnosed with monoclonal gammopathy of undetermined
|
||
|
|
significance (MGUS). Survival and progression time are in months.
|
||
|
|
The code below creates an ordinary Kaplan-Meier curve of
|
||
|
|
post-diagnosis survival for these subjects, along with a histogram of
|
||
|
|
age at diagnosis.
|
||
|
|
The mean age at diagnosis is just over 70 years.
|
||
|
|
|
||
|
|
<<mgus1, fig=TRUE>>=
|
||
|
|
oldpar <- par(mfrow=c(1,2))
|
||
|
|
hist(mgus2$age, nclass=30, main='', xlab="Age")
|
||
|
|
with(mgus2, tapply(age, sex, mean))
|
||
|
|
|
||
|
|
mfit1 <- survfit(Surv(futime, death) ~ sex, data=mgus2)
|
||
|
|
mfit1
|
||
|
|
plot(mfit1, col=c(1,2), xscale=12, mark.time=FALSE, lwd=2,
|
||
|
|
xlab="Years post diagnosis", ylab="Survival")
|
||
|
|
legend("topright", c("female", "male"), col=1:2, lwd=2, bty='n')
|
||
|
|
par(oldpar)
|
||
|
|
@
|
||
|
|
|
||
|
|
The xscale and yscale arguments to \code{plot.survfit} affect only the
|
||
|
|
axis labels, not the data. Further additions to the plot region
|
||
|
|
such as \code{legend}, \code{lines}, or \code{text} remain in the
|
||
|
|
original scale. This simplifies programmatic additions such as adding
|
||
|
|
another curve to the plot, while making interactive additions such as a
|
||
|
|
legend somewhat less simple.
|
||
|
|
|
||
|
|
As a second model for these subjects we will use competing risks (CR),
|
||
|
|
where PCM and death without malignancy
|
||
|
|
are the two terminal states, as shown in the upper left of figure \ref{mfig2}.
|
||
|
|
In a CR model we are only interested in the first event for each
|
||
|
|
subject.
|
||
|
|
Formally we are treating progression to a PCM
|
||
|
|
as an \emph{absorbing state}, i.e., one
|
||
|
|
that subjects never exit.
|
||
|
|
We create a variable \code{etime} containing the time to
|
||
|
|
the first of progression, death, or last follow-up along with an event
|
||
|
|
variable that contains the outcome.
|
||
|
|
The starting data set \code{mgus2} has two pairs of variables
|
||
|
|
\code{(ptime, pstat)} that contain the time to progression
|
||
|
|
and \code{(futime, status)} that contain the time to death or last known
|
||
|
|
alive.
|
||
|
|
The code below creates the necessary \code{etime} and \code{event}
|
||
|
|
variables, then computes and plots the competing risks estimate.
|
||
|
|
|
||
|
|
<<mgus2, echo=TRUE, fig=TRUE>>=
|
||
|
|
mgus2$etime <- with(mgus2, ifelse(pstat==0, futime, ptime))
|
||
|
|
event <- with(mgus2, ifelse(pstat==0, 2*death, 1))
|
||
|
|
mgus2$event <- factor(event, 0:2, labels=c("censor", "pcm", "death"))
|
||
|
|
table(mgus2$event)
|
||
|
|
|
||
|
|
mfit2 <- survfit(Surv(etime, event) ~ sex, data=mgus2)
|
||
|
|
print(mfit2, rmean=240, scale=12)
|
||
|
|
mfit2$transitions
|
||
|
|
|
||
|
|
plot(mfit2, col=c(1,2,1,2), lty=c(2,2,1,1),
|
||
|
|
mark.time=FALSE, lwd=2, xscale=12,
|
||
|
|
xlab="Years post diagnosis", ylab="Probability in State")
|
||
|
|
legend(240, .6, c("death:female", "death:male", "pcm:female", "pcm:male"),
|
||
|
|
col=c(1,2,1,2), lty=c(1,1,2,2), lwd=2, bty='n')
|
||
|
|
@
|
||
|
|
|
||
|
|
The \code{mfit2} call is nearly identical to that for an ordinary Kaplan-Meier,
|
||
|
|
with the exception of the \code{event} variable.
|
||
|
|
\begin{enumerate}
|
||
|
|
\item The event variable was created as a \emph{factor},
|
||
|
|
whereas for ordinary single
|
||
|
|
state survival the status is either 0/1 or TRUE/FALSE.
|
||
|
|
The first level of the factor must be censoring, which is the status code
|
||
|
|
for those whose follow-up terminated without reaching a new endpoint.
|
||
|
|
Codes for the remaining states can be in any order. The labels for the
|
||
|
|
states are unrestricted, e.g., the first one does not have to be ``censor''.
|
||
|
|
(It will be treated as `no event at this time', whatever the label.)
|
||
|
|
\item A simple print of the \code{mfit2} object shows the order in
|
||
|
|
which the curves will be displayed. This information was used to
|
||
|
|
choose the line types and colors for the curves.
|
||
|
|
\item The \code{mfit2} object contains curves for all the states, but
|
||
|
|
by default the entry state will not be plotted.
|
||
|
|
The remaining curves all start at 0.
|
||
|
|
\item The transitions component of the result is useful as a
|
||
|
|
data check, e.g., if it showed a transition from death to PCM.
|
||
|
|
\item Each subject's initial state can be specified by the \code{istate}
|
||
|
|
argument. When this is omitted all subjects
|
||
|
|
are assumed to start from an entry state named '(s0)'. This default
|
||
|
|
name is an amalgam of ``state 0'', a label sometimes used in textbooks
|
||
|
|
for the leftmost box, and an R convention of using () for constructed
|
||
|
|
names, e.g., `(Intercept)' from an lm() call.
|
||
|
|
\end{enumerate}
|
||
|
|
|
||
|
|
The printout shows that a male subject will spend, on average,
|
||
|
|
8.7 of his first 20 years post diagnosis in the entry state, 1.1
|
||
|
|
years in the PCM state and 10.3 of those 20 in the death state.
|
||
|
|
If a cutoff time is not given the default is to use the maximum observed
|
||
|
|
time for all curves, which is 424 months in this case.
|
||
|
|
|
||
|
|
The result of a multi-state \code{survfit} is a matrix of probabilities with
|
||
|
|
one row per time and one column per state.
|
||
|
|
These are contained in the probability-in-state (\code{pstate}) component of
|
||
|
|
the resulting survfit object.
|
||
|
|
Since the three MGUS states of entry/pcm/death must sum to 1 at any
|
||
|
|
given time (everyone has to be somewhere), one of the
|
||
|
|
three curves is redundant and the ``fraction still in the entry state''
|
||
|
|
curve is normally the least interesting.
|
||
|
|
By default, any state with the label `(s0)' is left off of the plot;
|
||
|
|
which is the default value of the \code{noplot} option of
|
||
|
|
\code{plot.survfit}.
|
||
|
|
One can easily plot all of the states by setting the option to NULL.
|
||
|
|
|
||
|
|
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], lwd=2, fun="event", conf.int=FALSE, xscale=12,
|
||
|
|
xlab="Years post diagnosis", ylab="Fraction with PCM")
|
||
|
|
lines(mfit2[2,"pcm"], 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 discussions 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[2,1]} corresponds to males (2) and the PCM endpoint (1).
|
||
|
|
Curves are listed and plotted in the usual matrix order of R.
|
||
|
|
|
||
|
|
A third example using the MGUS data treats it as a multi-state model
|
||
|
|
and it shown in the upper right of figure \ref{mfig2}.
|
||
|
|
In this version a subject can have multiple transitions and thus multiple
|
||
|
|
rows in the data set. In this case it is necessary to identify which data rows go
|
||
|
|
with which subject via the \code{id} argument of \code{survfit};
|
||
|
|
valid estimates of the curves and their standard errors both depend on this.
|
||
|
|
Our model looks like the illness-death model of figure \ref{mfig1} but
|
||
|
|
with ``PCM'' as the upper state and no arrow for
|
||
|
|
a return from that state to health.
|
||
|
|
The necessary data set will have two rows for any subject who has further
|
||
|
|
follow-up after a PCM and a single row for all others.
|
||
|
|
The data set is created below using the \code{tmerge} function, which is
|
||
|
|
discussed in detail in another vignette.
|
||
|
|
|
||
|
|
We need to decide what to do with the 9 subjects who have PCM
|
||
|
|
and death declared at the same month.
|
||
|
|
(Some of these were cancer cases discovered at autopsy.)
|
||
|
|
They slipped through without comment in the earlier competing risks analysis;
|
||
|
|
only when setting up this second data set did we notice the ties. Looking
|
||
|
|
back at the code, the prior example counted these subjects as a progression.
|
||
|
|
In retrospect this is a defensible choice: even though
|
||
|
|
undetected before death,
|
||
|
|
the disease must have been present for some amount of time previous and
|
||
|
|
so progression did occur first.
|
||
|
|
For the multi-state model we need to be explicit in how this is
|
||
|
|
coded since a sojourn time of 0 within a state is not allowed.
|
||
|
|
Below we push the progression time back by .1 month when there is a tie, but
|
||
|
|
that amount is entirely arbitrary.
|
||
|
|
Since there are 3 possible transitions we will call the data set \code{data3}.
|
||
|
|
|
||
|
|
<<mgus4, echo=TRUE>>=
|
||
|
|
ptemp <- with(mgus2, ifelse(ptime==futime & pstat==1, ptime-.1, ptime))
|
||
|
|
data3 <- tmerge(mgus2, mgus2, id=id, death=event(futime, death),
|
||
|
|
pcm = event(ptemp, pstat))
|
||
|
|
data3 <- tmerge(data3, data3, id, enum=cumtdc(tstart))
|
||
|
|
with(data3, table(death, pcm))
|
||
|
|
@
|
||
|
|
The table above shows that there are no observations in \code{data3}
|
||
|
|
that have both a PCM and death, i.e., the ties have been resolved.
|
||
|
|
The last \code{tmerge} line above creates a variable \code{enum} which
|
||
|
|
simply counts rows for each person, which is often useful.
|
||
|
|
|
||
|
|
<<mgus4g, fig=TRUE>>=
|
||
|
|
temp <- with(data3, ifelse(death==1, 2, pcm))
|
||
|
|
data3$event <- factor(temp, 0:2, labels=c("censor", "pcm", "death"))
|
||
|
|
mfit3 <- survfit(Surv(tstart, tstop, event) ~ sex, data=data3, id=id)
|
||
|
|
print(mfit3, rmean=240, digits=2)
|
||
|
|
mfit3$transitions
|
||
|
|
plot(mfit3[,"pcm"], mark.time=FALSE, col=1:2, lty=1:2, lwd=2,
|
||
|
|
xscale=12,
|
||
|
|
xlab="Years post MGUS diagnosis", ylab="Fraction in the PCM state")
|
||
|
|
legend(40, .4, c("female", "male"), lty=1:2, col=1:2, lwd=2, bty='n')
|
||
|
|
@
|
||
|
|
This plot is quite different in that it shows the fraction of subjects
|
||
|
|
\emph{currently} in the PCM state.
|
||
|
|
Looking at our multi-state diagram this is the fraction
|
||
|
|
of subjects in the upper right PCM box.
|
||
|
|
The curve goes up whenever someone enters the box (progression) and down when
|
||
|
|
they leave (death).
|
||
|
|
Myeloma survival was quite short during the era of this study ---
|
||
|
|
most subjects died within 1 year of PCM --- and the
|
||
|
|
proportion currently in the PCM state rarely rises above 2 percent.
|
||
|
|
|
||
|
|
The result of \code{print(mfit3)} reveals, as expected, less time spent
|
||
|
|
in the PCM state. In the prior \code{mfit2} model, subjects who enter
|
||
|
|
that state remain there for the duration; in this one they quickly
|
||
|
|
pass through.
|
||
|
|
It is worthwhile to check the \code{transitions} table in the output,
|
||
|
|
simply as a data check, since
|
||
|
|
an error in creating the input data can lead to surprising counts
|
||
|
|
and an even more surprising curve.
|
||
|
|
In this case it shows subjects going from the
|
||
|
|
entry state to PCM and death along with transitions from PCM
|
||
|
|
to death. This is as expected.
|
||
|
|
|
||
|
|
|
||
|
|
We have often found the three curve display shown below useful in the case
|
||
|
|
of a transient state.
|
||
|
|
It combines the results from competing risk model used above
|
||
|
|
along with a second fit that treats death
|
||
|
|
after PCM as a separate state from death before progression,
|
||
|
|
the \emph{multi-state 2} model of figure \ref{mfig2}.
|
||
|
|
In this plot the fraction of subjects currently in the PCM state
|
||
|
|
is shown by the distance between the two curves.
|
||
|
|
Only males are shown in the plot to minimize overlap.
|
||
|
|
<<mgus5, echo=TRUE, fig=TRUE>>=
|
||
|
|
# Death after PCM will correspond to data rows with
|
||
|
|
# enum = 2 and event = death
|
||
|
|
d2 <- with(data3, ifelse(enum==2 & event=='death', 4, as.numeric(event)))
|
||
|
|
e2 <- factor(d2, labels=c("censor", "pcm", "death w/o pcm",
|
||
|
|
"death after pcm"))
|
||
|
|
mfit4 <- survfit(Surv(tstart, tstop, e2) ~ sex, data=data3, id=id)
|
||
|
|
plot(mfit2[2,], lty=c(1,2),
|
||
|
|
xscale=12, mark.time=FALSE, lwd=2,
|
||
|
|
xlab="Years post diagnosis", ylab="Probability in State")
|
||
|
|
lines(mfit4[2,4], mark.time=FALSE, col=2, lty=1, lwd=2,
|
||
|
|
conf.int=FALSE)
|
||
|
|
|
||
|
|
legend(200, .5, c("Death w/o PCM", "ever PCM",
|
||
|
|
"Death after PCM"), col=c(1,1,2), lty=c(2,1,1),
|
||
|
|
lwd=2, bty='n', cex=.82)
|
||
|
|
@
|
||
|
|
|
||
|
|
\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}
|
||
|
|
|
||
|
|
\section{Rate models}
|
||
|
|
For simple two-state survival, the Cox model leads to three relationships
|
||
|
|
\begin{align}
|
||
|
|
\lambda(t) &= \lambda_0(t) e^{X\beta} \label{hazard} \\
|
||
|
|
\Lambda(t) &= \Lambda_0(t) e^{X\beta} \label{cumhaz}\\
|
||
|
|
S(t) &= \exp(-\Lambda(t)) \label{surv}
|
||
|
|
\end{align}
|
||
|
|
where $\lambda$, $\Lambda$ and $S$ are the hazard, cumulative hazard
|
||
|
|
and survival functions, respectively.
|
||
|
|
There is a single linear predictor which governs
|
||
|
|
both the rate $\lambda$ (the arrow in figure \ref{mfig1}) and probability
|
||
|
|
of residing in the left hand box of the figure.
|
||
|
|
For multi-state models this simplicity no longer holds; proportional
|
||
|
|
hazards does not lead to proportional $p(t)$ curves.
|
||
|
|
There is a fundamental dichotomy in the analysis namely that
|
||
|
|
\begin{itemize}
|
||
|
|
\item hazards can be computed one at a time,
|
||
|
|
\item probability in state must be done for all states at once.
|
||
|
|
\end{itemize}
|
||
|
|
|
||
|
|
The analysis of multi-state data has four
|
||
|
|
key steps. In order of importance:
|
||
|
|
\begin{enumerate}
|
||
|
|
\item Draw a box and arrow figure describing the model.
|
||
|
|
\item Think through the rates (arrows).
|
||
|
|
\begin{enumerate}
|
||
|
|
\item Which covariates should be attached to each rate?
|
||
|
|
Sometimes a covariate is important for one transition, but not for
|
||
|
|
another.
|
||
|
|
\item For which transitions should one or more of the covariates be
|
||
|
|
constrained to have the same coefficient?
|
||
|
|
Sometimes there will be a biologic rationale for this. For other
|
||
|
|
studies an equivalence is forced simply because we have too many
|
||
|
|
unknowns and cannot accommodate them all.
|
||
|
|
(This is the same reason that models often
|
||
|
|
contain very few interaction terms).
|
||
|
|
\item Which, if any, of the transitions should share the same baseline
|
||
|
|
hazard? Most of the time the baseline rates are all assumed to
|
||
|
|
be different.
|
||
|
|
\item Should there be random effects, and if so what is an appropriate
|
||
|
|
correlation structure?
|
||
|
|
Do some pairs of transitions have a shared effect,
|
||
|
|
some pairs separate effects and others no random effect?
|
||
|
|
Mixed effects Cox models tend to need larger sample size --- does
|
||
|
|
the data set have enough events?
|
||
|
|
\end{enumerate}
|
||
|
|
\item Build an appropriate data set.
|
||
|
|
\item Fit the data. Examine multiple summaries of the
|
||
|
|
model fit, including the predicted occupancy curves.
|
||
|
|
\end{enumerate}
|
||
|
|
Step 1 is key to the entire endeavor.
|
||
|
|
We saw in figure \ref{mfig2} and the examples above that multiple views of a
|
||
|
|
multi-state process can be useful, and this will hold for modeling as well.
|
||
|
|
Step 3 will often
|
||
|
|
be the one that demands the most attention to detail.
|
||
|
|
|
||
|
|
\subsection{MGUS example}
|
||
|
|
Start with the simplest model for the MGUS data:
|
||
|
|
a competing risks model (upper left diagram of figure \ref{mfig2}), distinct baseline
|
||
|
|
hazards for the two rates, no shared coefficients, and three covariates.
|
||
|
|
<<cfit1>>=
|
||
|
|
options(show.signif.stars = FALSE) # display intelligence
|
||
|
|
cfit1 <- coxph(Surv(etime, event) ~ age + sex + mspike, mgus2, id=id)
|
||
|
|
cfit1$cmap
|
||
|
|
print(cfit1, digits=2)
|
||
|
|
@
|
||
|
|
This above call fits both endpoints at once.
|
||
|
|
The resulting R object contains a single vector of
|
||
|
|
coefficients, the \code{cmap} component documents which goes with which
|
||
|
|
endpoint and is used by the print routine to organize the printout.
|
||
|
|
We see that the
|
||
|
|
effect of age and sex on non-PCM mortality is profound, which is not
|
||
|
|
a surprise given the median starting age of \Sexpr{median(mgus2$age)}.
|
||
|
|
Risk rises \Sexpr{round(exp(10*coef(cfit1)[4]),1)} fold per decade of age and
|
||
|
|
the death rate for males is \Sexpr{round(exp(coef(cfit1)[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 6\%.
|
||
|
|
|
||
|
|
The mspike size has a major impact on progression, however; each 1 gram
|
||
|
|
change increases risk by \Sexpr{round(exp(coef(cfit1)[3]) ,1)} fold.
|
||
|
|
The interquartile range of \code{mspike} is 0.9 gram 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.
|
||
|
|
|
||
|
|
The effect of sex on the \emph{lifetime} probability of PCM is not zero,
|
||
|
|
however. Because females live longer, a female with MGUS will on average
|
||
|
|
spend more total years at risk for PCM than the average male, and so has
|
||
|
|
a larger lifetime risk of PCM.
|
||
|
|
The average rate of progression is about 1\% per year, as shown below,
|
||
|
|
while the mean post diagnosis lifetime is 19 months longer for
|
||
|
|
females.
|
||
|
|
The overall effect is a 1.6\% increase in lifetime risk.
|
||
|
|
<<mpyears>>=
|
||
|
|
pfit1 <- pyears(Surv(ptime, pstat) ~ sex, mgus2, scale=12)
|
||
|
|
round(100* pfit1$event/pfit1$pyears, 1) # PCM rate per year
|
||
|
|
|
||
|
|
temp <- summary(mfit1, rmean="common") #print the mean survival time
|
||
|
|
round(temp$table[,1:6], 1)
|
||
|
|
@
|
||
|
|
|
||
|
|
The same Cox model coefficients can also be obtained by fitting separate
|
||
|
|
models for the PCM and death endpoint, censoring cases that fail due to
|
||
|
|
the other cause.
|
||
|
|
Hazards can be computed one at a time.
|
||
|
|
When computing $p(t)$, on the other hand, all the
|
||
|
|
rates must be considered at once, it is necessary to use the
|
||
|
|
joint fit \code{cfit1} above.
|
||
|
|
We create predicted curves for four hypothetical subjects below.
|
||
|
|
|
||
|
|
<<PCMcurve>>=
|
||
|
|
dummy <- expand.grid(sex=c("F", "M"), age=c(60, 80), mspike=1.2)
|
||
|
|
dummy
|
||
|
|
csurv <- survfit(cfit1, newdata=dummy)
|
||
|
|
dim(csurv)
|
||
|
|
@
|
||
|
|
The resulting set of Aalen-Johansen estimates can be indexed as though it
|
||
|
|
were an array. There is only one stratum (the Cox model did not have
|
||
|
|
strata), four hypothetical subjects, and 3 states.
|
||
|
|
(When there is only a single stratum, as here, the subscript method allows
|
||
|
|
that index to be omitted, for backwards compatability).
|
||
|
|
|
||
|
|
<<PCMcurve2, fig=TRUE>>=
|
||
|
|
plot(csurv[,'pcm'], xmax=25*12, xscale=12,
|
||
|
|
xlab="Years after MGUS diagnosis", ylab="PCM",
|
||
|
|
col=1:2, lty=c(1,1,2,2), lwd=2)
|
||
|
|
legend(10, .14, outer(c("female", "male "),
|
||
|
|
c("diagnosis at age 60", "diagnosis at age 80"),
|
||
|
|
paste, sep=", "),
|
||
|
|
col=1:2, lty=c(1,1,2,2), bty='n', lwd=2)
|
||
|
|
@
|
||
|
|
|
||
|
|
The individual survival curves that would result from fitting each
|
||
|
|
endpoint separately are not actually of interest, since
|
||
|
|
each will be a Cox model analog of the pcmbad curve we criticized earlier.
|
||
|
|
|
||
|
|
The coxph fits showed that
|
||
|
|
sex has nearly no effect on the hazard of PCM, i.e., on any given day the
|
||
|
|
risk of conversion for a male is essentially the same as for a female of the
|
||
|
|
same age.
|
||
|
|
Yet we see above that the fitted Cox models predict a higher lifetime risk
|
||
|
|
for females; this is due to a longer lifespan.
|
||
|
|
The age effect on lifetime risk that is far from proportional: the 80
|
||
|
|
year curves flatten out while predictions for the younger subjects do not.
|
||
|
|
Very few subjects acquire PCM more than 15 years after a MGUS diagnosis at
|
||
|
|
age 80 for the obvious reason: very few of them will still
|
||
|
|
be alive.
|
||
|
|
|
||
|
|
<<year20>>=
|
||
|
|
# Print out a M/F results at 20 years
|
||
|
|
temp <- summary(csurv, time=20*12)$pstate
|
||
|
|
cbind(dummy, PCM= round(100*temp[,,2], 1))
|
||
|
|
@
|
||
|
|
|
||
|
|
The above table shows that females are predicted to
|
||
|
|
have a higher risk of 20 year progression, even
|
||
|
|
though their hazard at any given moment is nearly identical to males.
|
||
|
|
The female/male difference at 20 years is on the order of our
|
||
|
|
``back of the napkin''
|
||
|
|
person-years estimate of 1\% progression per year * 1.7 more years of life
|
||
|
|
for the females, but the progression fraction varies substantially by group.
|
||
|
|
|
||
|
|
\section{Fine-Gray model}
|
||
|
|
For the competing risk case the Fine-Gray model provides an alternate way of
|
||
|
|
looking at the data.
|
||
|
|
As we saw above, the impact of a particular covariate on the final
|
||
|
|
values $P$ can be complex, even if the models for the hazards are relatively
|
||
|
|
simple.
|
||
|
|
The primary idea of the Fine-Gray approach is to turn the multi-state
|
||
|
|
problem into a collection of two-state ones.
|
||
|
|
In the upper right diagram of figure \ref{mfig2}, draw a circle around all of the
|
||
|
|
states except the chosen endpoint and collapse them into a single meta-state.
|
||
|
|
For the MGUS data these are
|
||
|
|
\begin{itemize}
|
||
|
|
\item Model 1
|
||
|
|
\begin{itemize}
|
||
|
|
\item left box: All subjects in the entry or ``death first'' state
|
||
|
|
\item right box: PCM
|
||
|
|
\end{itemize}
|
||
|
|
\item Model 2
|
||
|
|
\begin{itemize}
|
||
|
|
\item left box: All subjects in the entry or ``PCM first'' state
|
||
|
|
\item right box: Death (without PCM)
|
||
|
|
\end{itemize}
|
||
|
|
\end{itemize}
|
||
|
|
|
||
|
|
An interesting aspect of this is that the fit can be done as a
|
||
|
|
two stage process: the first stage creates a special data set while the
|
||
|
|
second fits a weighted \code{coxph} or \code{survfit} model to the data.
|
||
|
|
The data set can be created using the \code{finegray} command.
|
||
|
|
|
||
|
|
<<fg1>>=
|
||
|
|
# (first three lines are identical to an earlier section)
|
||
|
|
etime <- with(mgus2, ifelse(pstat==0, futime, ptime))
|
||
|
|
event <- with(mgus2, ifelse(pstat==0, 2*death, 1))
|
||
|
|
event <- factor(event, 0:2, labels=c("censor", "pcm", "death"))
|
||
|
|
|
||
|
|
pcmdat <- finegray(Surv(etime, event) ~ ., data=mgus2,
|
||
|
|
etype="pcm")
|
||
|
|
pcmdat[1:4, c(1:3, 11:14)]
|
||
|
|
|
||
|
|
deathdat <- finegray(Surv(etime, event) ~ ., data=mgus2,
|
||
|
|
etype="death")
|
||
|
|
dim(pcmdat)
|
||
|
|
dim(deathdat)
|
||
|
|
dim(mgus2)
|
||
|
|
@
|
||
|
|
The \code{finegray} command has been used to create two data sets:
|
||
|
|
one for the PCM endpoint and one for the death endpoint.
|
||
|
|
In each, four new variables have been added containing a survival
|
||
|
|
time \code{(fgstart, fgstop, fgstatus)} with an `ordinary' status of
|
||
|
|
0/1, along with a case weight and a large number of new rows.
|
||
|
|
We can use this new data set as yet another way to compute
|
||
|
|
multi-state survival curves, though there is no good reason to use this
|
||
|
|
rather roundabout approach instead of
|
||
|
|
the simpler \code{survfit(Surv(etime, event) \textasciitilde sex)}.
|
||
|
|
<<pfit2>>=
|
||
|
|
# The PCM curves of the multi-state model
|
||
|
|
pfit2 <- survfit(Surv(fgstart, fgstop, fgstatus) ~ sex,
|
||
|
|
data=pcmdat, weights=fgwt)
|
||
|
|
# The death curves of the multi-state model
|
||
|
|
dfit2 <- survfit(Surv(fgstart, fgstop, fgstatus) ~ sex,
|
||
|
|
data=deathdat, weights=fgwt)
|
||
|
|
@
|
||
|
|
Inspection shows that the
|
||
|
|
two new curves are almost identical to the prior estimates based on
|
||
|
|
the Aalen-Johansen estimate (\code{mfit2}),
|
||
|
|
and in fact would be identical if we
|
||
|
|
had accounted for the slightly different censoring patterns in males and
|
||
|
|
females (by adding \code{strata(sex)} to the right hand side of the
|
||
|
|
\code{finegray} formulas).
|
||
|
|
|
||
|
|
|
||
|
|
A Cox model fit to the constructed data set yields the Fine-Gray
|
||
|
|
models for PCM and for death.
|
||
|
|
<<fg2, fig=TRUE>>=
|
||
|
|
fgfit1 <- coxph(Surv(fgstart, fgstop, fgstatus) ~ sex, data=pcmdat,
|
||
|
|
weights= fgwt)
|
||
|
|
summary(fgfit1)
|
||
|
|
fgfit2 <- coxph(Surv(fgstart, fgstop, fgstatus) ~ sex, data=deathdat,
|
||
|
|
weights= fgwt)
|
||
|
|
fgfit2
|
||
|
|
|
||
|
|
mfit2 <- survfit(Surv(etime, event) ~ sex, data=mgus2) #reprise the AJ
|
||
|
|
plot(mfit2[,'pcm'], col=1:2,
|
||
|
|
lwd=2, xscale=12,
|
||
|
|
conf.times=c(5, 15, 25)*12,
|
||
|
|
xlab="Years post diagnosis", ylab="Fraction with PCM")
|
||
|
|
ndata <- data.frame(sex=c("F", "M"))
|
||
|
|
fgsurv1 <- survfit(fgfit1, ndata)
|
||
|
|
lines(fgsurv1, fun="event", lty=2, lwd=2, col=1:2)
|
||
|
|
legend("topleft", c("Female, Aalen-Johansen", "Male, Aalen-Johansen",
|
||
|
|
"Female, Fine-Gray", "Male, Fine-Gray"),
|
||
|
|
col=1:2, lty=c(1,1,2,2), bty='n')
|
||
|
|
|
||
|
|
# rate models with only sex
|
||
|
|
cfitr <- coxph(Surv(etime, event) ~ sex, mgus2, id=id)
|
||
|
|
rcurve <- survfit(cfitr, newdata=ndata)
|
||
|
|
#lines(rcurve[, 'pcm'], col=6:7) # makes the plot too crowsded
|
||
|
|
@
|
||
|
|
|
||
|
|
The FG model states that males have a less \emph{observed} PCM,
|
||
|
|
by a factor of \Sexpr{round(exp(coef(fgfit1)), 2)}, and that
|
||
|
|
this hazard ratio is constant over time.
|
||
|
|
An overlaid plot of the non-parametric Aalen-Johansen estimates for the
|
||
|
|
PCM state (from \code{survfit}) along with predicted curves from the
|
||
|
|
Fine-Gray model show that proportional hazards is not unreasonable for
|
||
|
|
this particular fit.
|
||
|
|
The predicted values from the rate model, computed just above but not
|
||
|
|
plotted on the curve, also fit well with the data.
|
||
|
|
|
||
|
|
When there is only a single categorical 0/1 covariate the Fine-Gray model
|
||
|
|
reduces to Gray's test of the subdistribution function, in the same way
|
||
|
|
that a \code{coxph} fit with a single categorical predictor is equivalent
|
||
|
|
to the log-rank test.
|
||
|
|
|
||
|
|
The mathematics behind the Fine-Gray estimate starts with
|
||
|
|
the functions $F_k(t) = p_k(t)$, where $p$ is the probability in state
|
||
|
|
function estimated by the AJ estimate. This can
|
||
|
|
be thought of as the distribution function for the improper random variable
|
||
|
|
$T^*= I(\mbox{event type}=k)T + I(\mbox{event type}\ne k)\infty$.
|
||
|
|
Fine and Gray refer to $F_k$ as a subdistribution function.
|
||
|
|
In analogy to the survival probability in the two state model define
|
||
|
|
\begin{equation}
|
||
|
|
\gamma_k(t) = - d \log[1-F_k(t)]/dt \label{FG}
|
||
|
|
\end{equation}
|
||
|
|
and assume that $\gamma_k(t;x) = \gamma_{k0}(t) \exp(X\beta)$.
|
||
|
|
In a 2 state alive $\longrightarrow$ death model, $\gamma$ becomes the
|
||
|
|
usual hazard function $\lambda$.
|
||
|
|
In the same way that our multivariate Cox model \code{cfit1} made the
|
||
|
|
simplifying assumption that the impact of male sex is to increase the
|
||
|
|
hazard for death
|
||
|
|
by a factor of \Sexpr{round(exp(coef(cfit1)[5]), 2)},
|
||
|
|
independent of the subject's age or serum mspike value,
|
||
|
|
the Fine-Gray model assumes that each covariate's effect on $\log(1-F)$
|
||
|
|
is a constant,
|
||
|
|
independent of other variables.
|
||
|
|
Both model's assumptions are wonderfully simplifying with respect to
|
||
|
|
understanding
|
||
|
|
a covariate, since we can think about each covariate separately from
|
||
|
|
all the others.
|
||
|
|
This is, of course, under the assumption that the model is correct: that
|
||
|
|
additivity across covariates, linearity, and proportional hazards
|
||
|
|
all hold.
|
||
|
|
In a multi-state model, however, these assumptions cannot hold for
|
||
|
|
both the per-transition and Fine-Gray models formulations at the same time;
|
||
|
|
if true for one, they will not be true for the other.
|
||
|
|
|
||
|
|
Now consider a multivariate fit on age, sex, and serum m-spike.
|
||
|
|
<<fg3>>=
|
||
|
|
fgfit2a <- coxph(Surv(fgstart, fgstop, fgstatus) ~ age + sex + mspike,
|
||
|
|
data=pcmdat, weights=fgwt)
|
||
|
|
|
||
|
|
fgfit2b <- coxph(Surv(fgstart, fgstop, fgstatus) ~ age + sex + mspike,
|
||
|
|
data=deathdat, weights=fgwt)
|
||
|
|
round(rbind(PCM= coef(fgfit2a), death=coef(fgfit2b)), 3)
|
||
|
|
@
|
||
|
|
The Fine-Gray fits show an effect of all three variables on the subdistribution
|
||
|
|
rates. Males have a lower lifetime risk of PCM before death and a higher
|
||
|
|
risk of death before PCM, while a high serum m-spike works in the opposite
|
||
|
|
direction.
|
||
|
|
The Cox models showed no effect of sex on the instantaneous hazard
|
||
|
|
of PCM and none
|
||
|
|
for serum m-spike on the death rate.
|
||
|
|
However, as shown in the last section, the Cox models do predict
|
||
|
|
a greater lifetime risk for females.
|
||
|
|
We had also seen that older subjects are less likely to experience PCM
|
||
|
|
due to the competing risk of death;
|
||
|
|
this is reflected in the FG model as a negative coefficient for age.
|
||
|
|
|
||
|
|
Now compute predicted survival curves for the model, and show them
|
||
|
|
alongside the predictions from the multi-state Cox model.
|
||
|
|
|
||
|
|
<<finegray2, fig=TRUE>>=
|
||
|
|
oldpar <- par(mfrow=c(1,2))
|
||
|
|
dummy <- expand.grid(sex= c("F", "M"), age=c(60, 80), mspike=1.2)
|
||
|
|
fsurv1 <- survfit(fgfit2a, dummy) # time to progression curves
|
||
|
|
plot(fsurv1, xscale=12, col=1:2, lty=c(1,1,2,2), lwd=2, fun='event',
|
||
|
|
xlab="Years", ylab="Fine-Gray predicted",
|
||
|
|
xmax=12*25, ylim=c(0, .15))
|
||
|
|
legend(1, .15, c("Female, 60", "Male, 60","Female: 80", "Male, 80"),
|
||
|
|
col=c(1,2,1,2), lty=c(1,1,2,2), lwd=2, bty='n')
|
||
|
|
|
||
|
|
plot(csurv[,2], xscale=12, col=1:2, lty=c(1,1,2,2), lwd=2,
|
||
|
|
xlab="Years", ylab="Multi-state predicted",
|
||
|
|
xmax=12*25, ylim=c(0, .15))
|
||
|
|
legend(1, .15, c("Female, 60", "Male, 60","Female: 80", "Male, 80"),
|
||
|
|
col=c(1,2,1,2), lty=c(1,1,2,2), lwd=2, bty='n')
|
||
|
|
par(oldpar)
|
||
|
|
@
|
||
|
|
|
||
|
|
The predictions as a function of age group are quite different for the
|
||
|
|
Fine-Gray model: new PCM cases are predicted 20+ years after diagnosis
|
||
|
|
in both the old and young age groups, while they are predicted to cease
|
||
|
|
in the multi-state fit.
|
||
|
|
The average of all four curves is nearly the same at each age, but
|
||
|
|
the global proportional hazards assumption of
|
||
|
|
the FG model forces the curves to remain parallel.
|
||
|
|
|
||
|
|
We can check the proportional hazards assumption of the models using
|
||
|
|
the \code{cox.zph} function,
|
||
|
|
linearity of the continuous variables age and mspike by using
|
||
|
|
non-linear terms such as \code{pspline} or \code{ns}, and
|
||
|
|
additivity by exploring interactions.
|
||
|
|
All are obvious and important next steps. For instance, the proportional hazards assumption for age shows clear violations.
|
||
|
|
|
||
|
|
<<finegray-check, fig=TRUE>>=
|
||
|
|
zph.fgfit2a <- cox.zph(fgfit2a)
|
||
|
|
zph.fgfit2a
|
||
|
|
plot(zph.fgfit2a[1])
|
||
|
|
abline(h=coef(fgfit2a)[1], lty=2, col=2)
|
||
|
|
@
|
||
|
|
|
||
|
|
A further weakness of the Fine-Gray approach is that since the two endpoints
|
||
|
|
are modeled separately, the results do not have to be consistent.
|
||
|
|
Below is a graph of the predicted fraction who have experienced neither
|
||
|
|
endpoint. For subjects diagnosed at age 80 the Fine-Gray models predict that more than
|
||
|
|
100\% will either progress or die by 30 years.
|
||
|
|
Predictions based on the Aalen-Johansen approach do not have this issue.
|
||
|
|
<<finegray3, fig=TRUE>>=
|
||
|
|
fsurv2 <- survfit(fgfit2b, dummy) # time to progression curves
|
||
|
|
xtime <- 0:(30*12) #30 years
|
||
|
|
y1a <- 1 - summary(fsurv1, times=xtime)$surv #predicted pcm
|
||
|
|
y1b <- 1 - summary(fsurv2, times=xtime)$surv #predicted deaths before pcm
|
||
|
|
y1 <- (y1a + y1b) #either
|
||
|
|
|
||
|
|
matplot(xtime/12, y1, col=1:2, lty=c(1,1,2,2), type='l',
|
||
|
|
xlab="Years post diagnosis", ylab="FG: either endpoint")
|
||
|
|
abline(h=1, col=3)
|
||
|
|
legend("bottomright", c("Female, 60", "Male, 60","Female: 80", "Male, 80"),
|
||
|
|
col=c(1,2,1,2), lty=c(1,1,2,2), lwd=2, bty='n')
|
||
|
|
@
|
||
|
|
|
||
|
|
|
||
|
|
The primary strength of the Fine-Gray model with respect to the Cox model
|
||
|
|
approach is that if lifetime risk is a primary question, then
|
||
|
|
the model has given
|
||
|
|
us a simple and digestible answer to that question:
|
||
|
|
``females have a 1.2 fold higher lifetime risk of PCM, after adjustment for age
|
||
|
|
and serum m-spike''.
|
||
|
|
This simplicity is not without a price, however, and these authors are
|
||
|
|
not proponents of the approach.
|
||
|
|
There are five issues.
|
||
|
|
\begin{enumerate}
|
||
|
|
\item The attempt to capture a complex process as a single value is
|
||
|
|
grasping for a simplicity that does not exist for many (perhaps most)
|
||
|
|
data sets.
|
||
|
|
The necessary assumptions in a multivariate Cox model of proportional
|
||
|
|
hazards, linearity of continuous variables, and no interactions are
|
||
|
|
strong ones. For the FG model these need to hold for a combined process
|
||
|
|
--- the mixture of transition rates to each endpoint --- which turns out
|
||
|
|
to be a more difficult barrier.
|
||
|
|
\item The sum of predictions need not be consistent.
|
||
|
|
\item From the per-transition Cox model one can work forward and compute
|
||
|
|
$p(t)$, the occupancy probabilities for each state over time;
|
||
|
|
both the hazard ratios and $p$ are useful summaries of the data.
|
||
|
|
We don't have tools to work backwards from a Fine-Gray fit to the
|
||
|
|
per transition hazards.
|
||
|
|
\item The approach is viable only for competing risks and not for
|
||
|
|
other multi-state models.
|
||
|
|
\item The risk sets are odd.
|
||
|
|
\end{enumerate}
|
||
|
|
The last of these is perhaps the most frequently listed issue with the Fine-Gray
|
||
|
|
model, but it is actually a minor complaint.
|
||
|
|
The state probabilities $p(t)$ in a multi-state model are implicitly
|
||
|
|
fractions of the total population
|
||
|
|
we started with: someone who dies in month 1 is still a part of the denominator
|
||
|
|
for the fraction of subjects with PCM at 20 years.
|
||
|
|
In the Fine-Gray formulas this subject explicitly appears in risk set
|
||
|
|
denominators at a later time, which looks odd but is more of an artifact.
|
||
|
|
|
||
|
|
The first issue is substantial, however,
|
||
|
|
and checking the model assumptions of a Fine-Gray fit is
|
||
|
|
mandatory. The second point is alarming, but it
|
||
|
|
normally does not have a practical
|
||
|
|
impact unless there is long follow-up.
|
||
|
|
|
||
|
|
\section{Shared coefficients}
|
||
|
|
To fit risk models that have shared coefficients or baseline hazards
|
||
|
|
we use an extended formula notation for the \code{coxph} function.
|
||
|
|
For the simple competing risks MGUS fit above, assume that
|
||
|
|
we wanted to add hemoglobin to the fit, with a common coefficient for
|
||
|
|
both the PCM and death endpoint. (Anemia is a feature of both PCM and old
|
||
|
|
age.)
|
||
|
|
|
||
|
|
<<hgfit>>=
|
||
|
|
hgfit <- coxph(list(Surv(etime, event) ~ age + sex + mspike,
|
||
|
|
1:2 + 1:3 ~ hgb / common),
|
||
|
|
data = mgus2, id = id)
|
||
|
|
hgfit
|
||
|
|
@
|
||
|
|
Notice that the hemoglobin coefficient is identical for the PCM and death
|
||
|
|
models.
|
||
|
|
A closer look at the result shows that the coefficient vector is of length
|
||
|
|
7 and not 8, orchestrated by the \code{cmap} component.
|
||
|
|
The variance matrix likewise is 7 by 7.
|
||
|
|
<<hgfit2>>=
|
||
|
|
hgfit$coef
|
||
|
|
|
||
|
|
hgfit$cmap
|
||
|
|
@
|
||
|
|
|
||
|
|
In a \code{coxph} formula list, the first element is the default formula
|
||
|
|
which will be applied to all the transitions.
|
||
|
|
The second, third, etc. elements are pseudo-formulas that have a set
|
||
|
|
of transitions as the ``response'', and usual covariate formulas on the
|
||
|
|
right.
|
||
|
|
For instance, if we wanted creatinine to be a covariate for only the transition
|
||
|
|
to PCM a third line of \code{1:2 \textasciitilde creat} could be added.
|
||
|
|
|
||
|
|
There is flexibility in how the left hand side is written. The following are
|
||
|
|
all valid forms for the second line of our formula.
|
||
|
|
<<echo=TRUE, eval=FALSE>>=
|
||
|
|
list(1: c("pcm", "death") ~ hgb / common,
|
||
|
|
1:0 ~ hgb / common,
|
||
|
|
state("(s0)"): (2:3) ~ hgb / common)
|
||
|
|
@
|
||
|
|
A '0' is shorthand for ``any state''. Transitions in the formula that do not
|
||
|
|
occur in the data are silently ignored, e.g., the implicit 1:1 transition
|
||
|
|
of our second line above.
|
||
|
|
|
||
|
|
For the three transition model based on data3, the following fits first a
|
||
|
|
model with 3 baseline hazards and 6 coefficients.
|
||
|
|
The second fit assumes a common baseline hazard for the two transitions to
|
||
|
|
death.
|
||
|
|
<<c3>>=
|
||
|
|
cox3a <- coxph(Surv(tstart, tstop, event) ~ age + sex, data3, id=id)
|
||
|
|
cox3b <- coxph(list(Surv(tstart, tstop, event) ~ age + sex,
|
||
|
|
1:3 + 2:3 ~ 1/ common),
|
||
|
|
data= data3, id= id)
|
||
|
|
cox3b$cmap
|
||
|
|
@
|
||
|
|
For this data set model \code{cox3b} makes very little medical sense, since
|
||
|
|
death rates after PCM do not even remotely resemble those for patients
|
||
|
|
without a plasma cell malignancy.
|
||
|
|
(The fact that the package allows something does not make it a good idea.)
|
||
|
|
In other data sets a shared baseline for all the transitions to death can
|
||
|
|
be an effective model.
|
||
|
|
Be aware that though shared baselines for two transitions that end in the
|
||
|
|
same state is acceptable, a shared hazard for two transitions that
|
||
|
|
originate in the same state is not compatable with the partial likelihood
|
||
|
|
formulation of a Cox model, and will lead to surprising results (often
|
||
|
|
a large number of NA coefficients).
|
||
|
|
|
||
|
|
An alternative approach to the above is to fit the joint model using
|
||
|
|
a special data set.
|
||
|
|
To reproduce the \code{hgfit} model above,
|
||
|
|
create a stacked data set with $2n$ observations. The first $n$
|
||
|
|
rows are the data set we would use for a time to PCM analysis,
|
||
|
|
with a simple 0/1 status variable encoding the PCM outcome.
|
||
|
|
The second $n$ rows are the data set we would have used for the
|
||
|
|
`death before PCM' fits, with status encoding the death-before-PCM endpoint.
|
||
|
|
A last variable, \code{group}, is `pcm' for the first $n$
|
||
|
|
observations and
|
||
|
|
`death' for the remainder. Then fit a model
|
||
|
|
<<pcmstack, echo=TRUE>>=
|
||
|
|
temp1 <- data.frame(mgus2, time=etime, status=(event=="pcm"), group='pcm')
|
||
|
|
temp2 <- data.frame(mgus2, time=etime, status=(event=="death"), group="death")
|
||
|
|
stacked <- rbind(temp1, temp2)
|
||
|
|
allfit <- coxph(Surv(time, status) ~ hgb + (age + sex+ mspike)*strata(group),
|
||
|
|
data=stacked)
|
||
|
|
all.equal(allfit$loglik, hgfit$loglik)
|
||
|
|
@
|
||
|
|
|
||
|
|
This fits a common effect for hemoglobin (hgb) but separate age and sex
|
||
|
|
effects for the two endpoints, along with separate baseline hazards.
|
||
|
|
|
||
|
|
|
||
|
|
\section{Other software}
|
||
|
|
\subsection{The mstate package}
|
||
|
|
As the number of states + transitions (arrows + boxes) gets larger then
|
||
|
|
the `by hand' approach used above for creating a stacked data set
|
||
|
|
becomes a challenge.
|
||
|
|
(It is still fairly easy to do, just not as easy to ensure it has
|
||
|
|
been done \emph{correctly}.)
|
||
|
|
The \code{mstate} package starts with a definition of the matrix of possible
|
||
|
|
transitions and uses that to drive tools that build and analyze a stacked
|
||
|
|
data set in a more automated fashion.
|
||
|
|
We find it a little more difficult to use than a \code{coxph} model with
|
||
|
|
a multistate status variable.
|
||
|
|
(The fact that we like our own child best should be no surprise, however).
|
||
|
|
One current disadvantage of the survival package is that the Aalen-Johansen
|
||
|
|
curves from a multi-state coxph model currently do not include a variance
|
||
|
|
estimate, whereas those from mstate do have a variance.
|
||
|
|
|
||
|
|
A current restriction in R is that packages on the \emph{recommended} list
|
||
|
|
(of which survival is one) should not depend on any packages outside that
|
||
|
|
list.
|
||
|
|
This precludes adding an mstate example to this vignette, even though it is
|
||
|
|
a natural fit.
|
||
|
|
|
||
|
|
\subsection{The \code{msm} package}
|
||
|
|
There are two broad classes of multi-state data:
|
||
|
|
\begin{itemize}
|
||
|
|
\item Panel data arises when subjects have regular visits, with the
|
||
|
|
current state assessed at each visit. We don't know when the transitions
|
||
|
|
between states occur, or if other states may have been visited in the
|
||
|
|
interim --- only the subject's state at specific times. This is also
|
||
|
|
known as interval censored data.
|
||
|
|
\item Survival data arises when we observe the transition times;
|
||
|
|
death, for example.
|
||
|
|
\end{itemize}
|
||
|
|
|
||
|
|
The overall model (boxes and arrows), the quantities of interest (transition
|
||
|
|
rates and $p(t)$), and the desired printout and graphs are identical for
|
||
|
|
the two cases. Much of the work in
|
||
|
|
creating a data set is also nearly the same.
|
||
|
|
The underlying likelihood equations and resulting analytical methods for
|
||
|
|
solving the problem are, however, completely different.
|
||
|
|
The \code{msm} package addresses panel data, while \code{survival}, \code{mstate}, and many
|
||
|
|
others are devoted to survival data.
|
||
|
|
|
||
|
|
\section{Conclusions}
|
||
|
|
When working with acute diseases, such as advanced cancer or end-stage liver
|
||
|
|
disease, there is often a single dominating endpoint.
|
||
|
|
Ordinary single event Kaplan-Meier curves and Cox models are then
|
||
|
|
efficient and sufficient tools for much of the analysis.
|
||
|
|
Such data was the primary use case for survival analysis earlier in the
|
||
|
|
authors' careers.
|
||
|
|
Data with multiple important endpoints is now common, and multi-state
|
||
|
|
methods are an important addition to the statistical toolbox.
|
||
|
|
As shown above, they are now readily available and easy to use.
|
||
|
|
|
||
|
|
It is sometimes assumed that the presence of competing risks
|
||
|
|
\emph{requires} the use of a Fine-Gray model (we have seen it in referee
|
||
|
|
reports), but this is not correct.
|
||
|
|
The model may often be useful, but is one available option among many.
|
||
|
|
Grasping the big picture for a multi-state data set is always a challenge
|
||
|
|
and we should make use of as many tools as possible.
|
||
|
|
It is not always easy to jump between observed
|
||
|
|
deaths, hazard rates, and lifetime risk.
|
||
|
|
We are often reminded of the story of the gentleman on his 100th birthday
|
||
|
|
who proclaimed that he was looking forward to many more years ahead because
|
||
|
|
``I read the obituaries every day, and you almost never see someone
|
||
|
|
over 100 listed there''.
|
||
|
|
|
||
|
|
%\bibliographystyle{plain}
|
||
|
|
%\bibliography{refer}
|
||
|
|
|
||
|
|
\end{document}
|