survival (version 2.38-3)

survfit.formula: Compute a Survival Curve for Censored Data

Description

Computes an estimate of a survival curve for censored data using either the Kaplan-Meier or the Fleming-Harrington method. For competing risks data it computes the cumulative incidence curve.

Usage

## S3 method for class 'formula':
survfit(formula, data, weights, subset, na.action,  
        etype, id, istate, ...)

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 objec

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.

etype

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 This has been superseded by multi-state Surv objects; see example below.

id

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

istate

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

...

The following additional arguments are passed to internal functions called by survfit. [object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Value

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

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.

Kablfleisch, 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.

Details

The estimates used are the Kalbfleisch-Prentice (Kalbfleisch and Prentice, 1980, p.86) and the Tsiatis/Link/Breslow, which reduce to the Kaplan-Meier and Fleming-Harrington estimates, respectively, when the weights are unity.

The Greenwood formula for the variance is a sum of terms d/(n*(n-m)), where d is the number of deaths at a given time point, n is the sum of weights for all individuals still at risk at that time, and m is the sum of weights for the deaths at that time. The justification is based on a binomial argument when weights are all equal to one; extension to the weighted case is ad hoc. Tsiatis (1981) proposes a sum of terms d/(n*n), based on a counting process argument which includes the weighted case.

The two variants of the F-H estimate have to do with how ties are handled. If there were 3 deaths out of 10 at risk, then the first increments the hazard by 3/10 and the second by 1/10 + 1/9 + 1/8. For the first method S(t) = exp(H), where H is the Nelson-Aalen cumulative hazard estimate, whereas the fh2 method will give results S(t) results closer to the Kaplan-Meier.

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. When the baseline method is the Kaplan-Meier, this is known to converge to the maximum likelihood estimate.

The cumulative incidence curve is an alternative to the Kaplan-Meier for competing risks data. For instance, in patients with MGUS, conversion to an overt plasma cell malignancy occurs at a nearly constant rate among those still alive. A Kaplan-Meier estimate, treating death due to other causes as censored, gives a 20 year cumulate rate of 33% for the 241 early patients of Kyle. This estimates the incidence of conversion if all other causes of death were removed, which is an unrealistic assumption given the mean starting age of 63 and a median follow up of over 21 years. The CI estimate, on the other hand, estimates the total number of conversions that will actually occur. Because the population is older, this is much smaller than the KM, 22% at 20 years for Kyle's data. If there were no censoring, then CI(t) could very simply be computed as total number of patients with progression by time t divided by the sample size n.

See Also

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

#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)

#
# Time to progression/death for patients with monoclonal gammopathy
#  Competing risk curves (cumulative incidence)
fitKM <- survfit(Surv(stop, event=='progression') ~1, data=mgus1,
                    subset=(start==0))

fitCI <- survfit(Surv(stop, status*as.numeric(event), type="mstate") ~1,
                    data=mgus1, subset=(start==0))

# CI curves are always plotted from 0 upwards, rather than 1 down
plot(fitCI, xscale=365.25, xmax=7300, mark.time=FALSE,
            col=2:3, xlab="Years post diagnosis of MGUS")
lines(fitKM, fun='event', xscale=365.25, xmax=7300, mark.time=FALSE,
            conf.int=FALSE)
text(10, .4, "Competing risk: death", col=3)
text(16, .15,"Competing risk: progression", col=2)
text(15, .30,"KM:prog")