survival (version 3.2-11)

# survfit.formula: Compute a Survival Curve for Censored Data

## Description

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.

## Usage

# S3 method for formula
survfit(formula, data, weights, subset, na.action,
stype=1, ctype=1, id, cluster, robust, istate, timefix=TRUE,
etype, error,  ...)

## Arguments

formula

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.

data

a data frame in which to interpret the variables named in the formula, subset and weights arguments.

weights

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.

subset

expression saying that only a subset of the rows of the data should be used in the fit.

na.action

a missing-data filter function, applied to the model frame, after any subset argument has been used. Default is options()\$na.action.

stype

the method to be used estimation of the survival curve: 1 = direct, 2 = exp(cumulative hazard).

ctype

the method to be used for estimation of the cumulative hazard: 1 = Nelson-Aalen formula, 2 = Fleming-Harrington correction for tied events.

id

identifies individual subjects, when a given person can have multiple lines of data.

cluster

used to group observations for the infinitesmal jackknife variance estimate, defaults to the value of id.

robust

logical, should the function compute a robust variance. For multi-state survival curves this is true by default. For single state data see details, below.

istate

for multi-state models, identifies the initial state of each subject or observation

timefix

process times through the aeqSurv function to eliminate potential roundoff issues.

etype

a variable giving the type of event. This has been superseded by multi-state Surv objects and is depricated; see example below.

error

this argument is no longer used

The following additional arguments are passed to internal functions called by survfit.

se.fit

logical value, default is TRUE. If FALSE then standard error computations are omitted.

conf.type

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).

conf.lower

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.

start.time

numeric value specifying a time to start calculating survival information. The resulting curve is the survival conditional on surviving to start.time.

conf.int

the level for a two-sided confidence interval on the survival curve(s). Default is 0.95.

se.fit

a logical value indicating whether standard errors should be computed. Default is TRUE.

influence

a logical value indicating whether to return the infinitesimal jackknife (influence) values for each subject. These contain the values of the derivative of each value with respect to the case weights of each subject i: $$\partial p/\partial w_i$$, evaluated at the vector of weights. The resulting object will contain influence.surv and influence.chaz components. Alternatively, options of influence=1 or influence=2 will return values for only the survival or hazard curves, respectively.

p0

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.

type

an older argument that combined stype and ctype, now depricated. 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.

## Value

an object of class "survfit". See survfit.object for details. Methods defined for survfit objects are print, plot, lines, and points.

## Details

If there is a data argument, then variables in the formula, codeweights, 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. 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 a survival curve is produced, and it does not support a robust variance.

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.

With the IJ estimate, the leverage values themselves can be returned as arrays with dimensions: number of subjects, number of unique times, and for a multi-state model, the number of unique states. Be forwarned that these arrays can be huge. If there is a cluster argument this first dimension will be the number of clusters and the variance will be a grouped IJ estimate; this can be an important tool for reducing the size. A numeric value for the influence argument allows finer control: 0= return neither (same as FALSE), 1= return the influence array for probability in state, 2= return the influence array for the cumulative hazard, 3= both (same as TRUE).

## References

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.

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, coxph, Surv.

## Examples

# NOT RUN {
#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)
# }
# NOT RUN {
# 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")
# }