R: Compute a Survival Curve for Censored Data

survfit.formula {survival}

R Documentation

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 class '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, ...)

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 infinitesimal jackknife variance estimate, defaults to the value of id.
`robust`	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.
`istate`	for multi-state models, identifies the initial state of each subject or observation. This also forces `time0 =TRUE`.
`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 This has been superseded by multi-state Surv objects and is deprecated; see example below.
`model`	include a copy of the model frame in the output
`error`	this argument is no longer used
`entry`	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.
`time0`	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`. se.fit 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. 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. influence a logical value indicating whether to return the infinitesimal jackknife (influence) values for each subject. See details below. 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. entry 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. type 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.`

Details

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.

Value

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.

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.

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.

Examples

#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")

## End(Not run)

[Package survival version 3.7-0 Index]