Computes an estimate of a survival curve for censored data using the Aalen-Johansen estimator. For ordinary (single event) survival this reduces to the Kaplan-Meier estimate.
# S3 method for formula
survfit(formula, data, weights, subset, na.action,
stype=1, ctype=1, id, cluster, robust, istate, timefix=TRUE,
etype, model=FALSE, error, entry=FALSE, time0=FALSE, ...)an object of class "survfit".
See survfit.object for
details. Some of the methods defined for survfit objects are
print, plot,
lines, points and residual.
a formula object, which must have a
Surv object as the
response on the left of the ~ operator and, if desired, terms
separated by + operators on the right.
One of the terms may be a strata object.
For a single survival curve the right hand side should be ~ 1.
a data frame in which to interpret the variables named in the formula,
subset and weights arguments.
The weights must be nonnegative and it is strongly recommended that
they be strictly positive, since zero weights are ambiguous, compared
to use of the subset argument.
expression saying that only a subset of the rows of the data should be used in the fit.
a missing-data filter function, applied to the model frame, after any
subset argument has been used.
Default is options()$na.action.
the method to be used estimation of the survival curve: 1 = direct, 2 = exp(cumulative hazard).
the method to be used for estimation of the cumulative hazard: 1 = Nelson-Aalen formula, 2 = Fleming-Harrington correction for tied events.
identifies individual subjects, when a given person can have multiple lines of data.
used to group observations for the infinitesimal jackknife variance estimate, defaults to the value of id.
logical, should the function compute a robust variance. For multi-state survival curves or interval censored data this is true by default. For single state data see details, below.
for multi-state models, identifies the initial state of
each subject or observation. This also forces time0 =TRUE.
process times through the aeqSurv function to
eliminate potential roundoff issues.
a variable giving the type of event. This has been superseded by multi-state Surv objects and is deprecated; see example below.
include a copy of the model frame in the output
this argument is no longer used
if TRUE, the output will contain n.enter which is
the number of observations entering the risk set at any time; extra
rows of output are created, if needed, for each unique entry time.
Only applicable if there is an id statement.
if TRUE, the output will include estimates at the starting point of the curve or `time 0'. See discussion below.
The following additional arguments are passed to internal functions
called by survfit.
a logical value indicating whether standard errors should be
computed. Default is TRUE. For a multistate model, where
the infinitesimal jackknife (robust) standar error is used, the
compute time for the standard error is O(ndp) where n = number of
observations, d = number of events and p = number of states,
while that for all other portions of the output
(pstate, cumhaz and counts) is O((n+d)p).
For a moderate to large data set the compute time difference
between nd and n+d can be huge; using se.fit = FALSE may
be a wise choice.
One of "none", "plain", "log" (the default),
"log-log", "logit" or "arcsin". Only
enough of the string to uniquely identify it is necessary.
The first option causes confidence intervals not to be
generated. The second causes the standard intervals
curve +- k *se(curve), where k is determined from
conf.int. The log option calculates intervals based on the
cumulative hazard or log(survival). The log-log option bases the
intervals on the log hazard or log(-log(survival)), the
logit option on log(survival/(1-survival))
and arcsin on arcsin(survival).
a character string to specify modified lower limits to the curve, the
upper limit remains unchanged.
Possible values are "usual" (unmodified),
"peto",
and "modified". The modified lower limit
is based on an "effective n" argument. The confidence
bands will agree with the usual calculation at each death time, but unlike
the usual bands the confidence interval becomes wider at each censored
observation. The extra width is obtained by multiplying the usual
variance by a factor m/n, where n is the number currently at risk and
m is the number at risk at the last death time. (The bands thus agree
with the un-modified bands at each death time.)
This is especially useful for survival curves with a long flat tail.
The Peto lower limit is based on the same "effective n" argument as the modified limit, but also replaces the usual Greenwood variance term with a simple approximation. It is known to be conservative.
numeric value specifying a time to start calculating survival
information.
The resulting curve is the survival conditional on surviving to
start.time.
the level for a two-sided confidence interval on the survival curve(s). Default is 0.95.
a logical value indicating whether to return the infinitesimal jackknife (influence) values for each subject. See details below.
this applies only to multi-state curves.
An optional vector giving the initial probability across
the states. If this is missing, then p0 is estimated using the
frequency of the starting states of all observations at risk
at start.time, or if that is not specified, at
the time of the first event.
by default, the survfit routines only return
information at the event/censoring times. If entry=TRUE then
also return a n.enter component containing the number
who joined the risk set at each time; if necessary add extra
rows to the output for each unique entry time.
This is only applicable for (time1, time2) survival data, and
if there is an id statement. If a single subject had
times of (0,10), (10, 20), (25,30) with an event at 30,
then time 10 is not an entry or censoring time,
but 20 counts as censored and 25 as an entry.
an older argument that combined stype and
ctype, now deprecated. Legal values were "kaplan-meier"
which is equivalent to stype=1, ctype=1, "fleming-harrington"
which is equivalent to stype=2, ctype=1, and "fh2" which
is equivalent to stype=2, ctype=2.
If there is a data argument, then variables in the formula,
weights, subset, id, cluster and
istate arguments will be searched for in that data set.
The routine returns both an estimated probability in state and an
estimated cumulative hazard estimate.
For simple survival the probability in state = probability alive, i.e,
the estimated survival. For multi-state it will be a matrix with one
row per time and a column per state, rows sum to 1.
The cumulative hazard estimate is the Nelson-Aalen (NA) estimate or the
Fleming-Harrington (FH) estimate, the latter includes a correction for
tied event times. The estimated probability in state can estimated
either using the exponential of the cumulative hazard, or as a direct
estimate using the Aalen-Johansen approach.
For single state data the AJ estimate reduces to the Kaplan-Meier and
the probability in state to the survival curve;
for competing risks data the AJ reduces to the cumulative incidence (CI)
estimator.
For backward compatability the type argument can be used instead.
When the data set includes left censored or interval censored data (or both),
then the EM approach of Turnbull is used to compute the overall curve.
Currently this algorithm is very slow, only applies to simple survival
(not multi-state), and defaults to a robust variance. Other R
packages are available which implement the iterative convex minorant
(ICM) algorithm for
interval censored data, which is much faster than Turnbull's method.
Based on Sun (2001) the robust variance may be preferred, as the naive estimate
ignores the estimation of the weights. The standard estimate can be
obtained with robust= FALSE.
Without interval or left censored data (the usual case) the underlying algorithm for the routine is the Aalen-Johansen estimate, of which the Kaplan-Meier (for single outcome data) and the cumulative incidence (CI) estimate (for competing risks) are each a special case. For multi-state, the estimate can be written as \(p(t_0)H(t_1)H(t_2)\ldots\) where \(p(t_0)\) is the prevalance vector across the states at starting point \(t_0\), \(t_1, t_2, \ldots\) are the times at which events (transitions between states) occur, and H are square transtion matrices with a row for each state.
Starting point: When diffent subjects (id) start at different
time points, data using age as the time scale for instance,
deciding the default "time 0" can be complex. This value is the
starting point for the restricted mean estimate (area under the
curve), the initial prevalence p0, and the first
row of output if time0 = TRUE. The order of the decision is
For a 2 column response (simple survival or competing risks) use the minimum of 0 and the smallest time value (times can be negative).
If all subjects start in the same state, start at the same time,
or if p0 is specified, use the minimum observed starting
time. If there is no istate argument all observations are
assumed to start in a state "(s0)".
Use the minimum observed event time, if the number at risk at that time is >0 for every curve that will be created.
Use the minimum event time for each curve, separately.
The last two above are a failsafe to prevent the routine from basing
the initial prevalence of the states on none or only a handful of
observations. That does not mean such curves
will be scientfically sensible: when using age scale the user may wish
to specify an explicit starting time.
If time0 = TRUE the first row of output for each curve will be
at the starting time,
otherwise the first event time (for each curve separately).
Robust variance:
If a robust is TRUE, or for multi-state
curves, then the standard
errors of the results will be based on an infinitesimal jackknife (IJ)
estimate, otherwise the standard model based estimate will be used.
For single state curves, the default for robust will be TRUE
if one of: there is a cluster argument, there
are non-integer weights, or there is a id statement
and at least one of the id values has multiple events, and FALSE otherwise.
The default represents our best guess about when one would most
often desire a robust variance.
When there are non-integer case weights and (time1, time2) survival
data the routine is at an impasse: a robust variance likely is called
for, but requires either id or cluster information to be
done correctly; it will default to robust=FALSE if they are not present.
With the IJ estimate, the leverage values themselves can be returned
as an array using the influence argument.
Be forwarned that this array can be huge. Post fit influence using the
resid method is more flexible and would normally be preferred,
in particular to get influence at only a select set of time points.
The influence option is currently used mostly in the package's
validity checks.
Let \(U(t)\) be the matrix of IJ values at time t, which has
one row per observation, one column per state. The robust variance
compuation uses the collapsed weighted matrix
rowsum(wU, cluster),
where w is the vector of weights and cluster is the grouping (most often
the id). The result for each curve is an array with dimensions
(number of clusters, number of states, number of times), or a matrix
for single state data. When there are multiple curves, the
influence is a list with one element per curve.
Dorey, F. J. and Korn, E. L. (1987). Effective sample sizes for confidence intervals for survival probabilities. Statistics in Medicine 6, 679-87.
Fleming, T. H. and Harrington, D. P. (1984). Nonparametric estimation of the survival distribution in censored data. Comm. in Statistics 13, 2469-86.
Kalbfleisch, J. D. and Prentice, R. L. (1980). The Statistical Analysis of Failure Time Data. New York:Wiley.
Kyle, R. A. (1997). Moncolonal gammopathy of undetermined significance and solitary plasmacytoma. Implications for progression to overt multiple myeloma}, Hematology/Oncology Clinics N. Amer. 11, 71-87.
Link, C. L. (1984). Confidence intervals for the survival function using Cox's proportional hazards model with covariates. Biometrics 40, 601-610.
Sun, J. (2001). Variance estimation of a survival function for interval-censored data. Stat Med 20, 1949-1957.
Turnbull, B. W. (1974). Nonparametric estimation of a survivorship function with doubly censored data. J Am Stat Assoc, 69, 169-173.
survfit.coxph for survival curves from Cox models,
survfit.object for a description of the components of a
survfit object,
print.survfit,
plot.survfit,
lines.survfit,
residuals.survfit,
coxph,
Surv.
#fit a Kaplan-Meier and plot it
fit <- survfit(Surv(time, status) ~ x, data = aml)
plot(fit, lty = 2:3)
legend(100, .8, c("Maintained", "Nonmaintained"), lty = 2:3)
#fit a Cox proportional hazards model and plot the
#predicted survival for a 60 year old
fit <- coxph(Surv(futime, fustat) ~ age, data = ovarian)
plot(survfit(fit, newdata=data.frame(age=60)),
xscale=365.25, xlab = "Years", ylab="Survival")
# Here is the data set from Turnbull
# There are no interval censored subjects, only left-censored (status=3),
# right-censored (status 0) and observed events (status 1)
#
# Time
# 1 2 3 4
# Type of observation
# death 12 6 2 3
# losses 3 2 0 3
# late entry 2 4 2 5
#
tdata <- data.frame(time =c(1,1,1,2,2,2,3,3,3,4,4,4),
status=rep(c(1,0,2),4),
n =c(12,3,2,6,2,4,2,0,2,3,3,5))
fit <- survfit(Surv(time, time, status, type='interval') ~1,
data=tdata, weight=n)
#
# Three curves for patients with monoclonal gammopathy.
# 1. KM of time to PCM, ignoring death (statistically incorrect)
# 2. Competing risk curves (also known as "cumulative incidence")
# 3. Multi-state, showing Pr(in each state, at time t)
#
fitKM <- survfit(Surv(stop, event=='pcm') ~1, data=mgus1,
subset=(start==0))
fitCR <- survfit(Surv(stop, event) ~1,
data=mgus1, subset=(start==0))
fitMS <- survfit(Surv(start, stop, event) ~ 1, id=id, data=mgus1)
if (FALSE) {
# CR curves show the competing risks
plot(fitCR, xscale=365.25, xmax=7300, mark.time=FALSE,
col=2:3, xlab="Years post diagnosis of MGUS",
ylab="P(state)")
lines(fitKM, fun='event', xmax=7300, mark.time=FALSE,
conf.int=FALSE)
text(3652, .4, "Competing risk: death", col=3)
text(5840, .15,"Competing risk: progression", col=2)
text(5480, .30,"KM:prog")
}
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