Modification of Therneau's coxph function to fit the Cox model and
its extension, the Andersen-Gill model. The latter allows for interval
time-dependent covariables, time-dependent strata, and repeated events.
The Survival method for an object created by cph returns an S
function for computing estimates of the survival function.
The Quantile method for cph returns an S function for computing
quantiles of survival time (median, by default).
The Mean method returns a function for computing the mean survival
time. This function issues a warning if the last follow-up time is uncensored,
unless a restricted mean is explicitly requested.
cph(formula = formula(data), data=environment(formula),
weights, subset, na.action=na.delete,
method=c("efron","breslow","exact","model.frame","model.matrix"),
singular.ok=FALSE, robust=FALSE,
model=FALSE, x=FALSE, y=FALSE, se.fit=FALSE,
linear.predictors=TRUE, residuals=TRUE, nonames=FALSE,
eps=1e-4, init, iter.max=10, tol=1e-9, surv=FALSE, time.inc,
type=NULL, vartype=NULL, debug=FALSE, ...)# S3 method for cph
Survival(object, ...)
# Evaluate result as g(times, lp, stratum=1, type=c("step","polygon"))
# S3 method for cph
Quantile(object, ...)
# Evaluate like h(q, lp, stratum=1, type=c("step","polygon"))
# S3 method for cph
Mean(object, method=c("exact","approximate"), type=c("step","polygon"),
n=75, tmax, ...)
# E.g. m(lp, stratum=1, type=c("step","polygon"), tmax, \dots)
For Survival, Quantile, or Mean, an S function is returned. Otherwise,
in addition to what is listed below, formula/design information and
the components
maxtime, time.inc, units, model, x, y, se.fit are stored, the last 5
depending on the settings of options by the same names.
The vectors or matrix stored if y=TRUE or x=TRUE have rows deleted according to subset and
to missing data, and have names or row names that come from the
data frame used as input data.
table with one row per stratum containing number of censored and uncensored observations
vector of regression coefficients
vector containing the named elements Obs, Events, Model L.R., d.f.,
P, Score, Score P, R2, Somers'
Dxy, g-index,
and gr, the g-index on the hazard ratio scale.
R2 is the Nagelkerke R-squared, with division by the maximum
attainable R-squared.
variance/covariance matrix of coefficients
values of predicted X beta for observations used in fit, normalized to have overall mean zero, then having any offsets added
martingale residuals
log likelihood at initial and final parameter values
value of score statistic at initial values of parameters
lists of times (if surv="T")
lists of underlying survival probability estimates
lists of standard errors of estimate log-log survival
a 3 dimensional array if surv=TRUE.
The first dimension is time ranging from 0 to
maxtime by time.inc. The second dimension refers to strata.
The third dimension contains the time-oriented matrix with
Survival, n.risk (number of subjects at risk),
and std.err (standard error of log-log
survival).
centering constant, equal to overall mean of X beta.
an S formula object with a Surv object on the
left-hand side. The terms can specify any S model
formula with up to third-order interactions. The strat
function may appear in the terms, as a main effect or an interacting
factor. To stratify on both race and sex, you would include both
terms strat(race) and strat(sex). Stratification
factors may interact with non-stratification factors;
not all stratification terms need interact with the same modeled
factors.
an object created by cph with surv=TRUE
name of an S data frame containing all needed variables. Omit this to use a data frame already in the S ``search list''.
case weights
an expression defining a subset of the observations to use in the fit. The default
is to use all observations. Specify for example age>50 & sex="male" or
c(1:100,200:300)
respectively to use the observations satisfying a logical expression or those having
row numbers in the given vector.
specifies an S function to handle missing data. The default is the function na.delete,
which causes observations with any variable missing to be deleted. The main difference
between na.delete and the S-supplied function na.omit is that
na.delete makes a list
of the number of observations that are missing on each variable in the model.
The na.action is usally specified by e.g. options(na.action="na.delete").
for cph, specifies a particular fitting method, "model.frame" instead to return the model frame
of the predictor and response variables satisfying any subset or missing value
checks, or "model.matrix" to return the expanded design matrix.
The default is "efron", to use Efron's likelihood for fitting the
model.
For Mean.cph, method is "exact" to use numerical
integration of the
survival function at any linear predictor value to obtain a mean survival
time. Specify method="approximate" to use an approximate method that is
slower when Mean.cph is executing but then is essentially instant
thereafter. For the approximate method, the area is computed for n
points equally spaced between the min and max observed linear predictor
values. This calculation is done separately for each stratum. Then the
n pairs (X beta, area) are saved in the generated S function, and when
this function is evaluated, the approx function is used to evaluate
the mean for any given linear predictor values, using linear interpolation
over the n X beta values.
If TRUE, the program will automatically skip over columns of the X matrix
that are linear combinations of earlier columns. In this case the
coefficients for such columns will be NA, and the variance matrix will contain
zeros. For ancillary calculations, such as the linear predictor, the missing
coefficients are treated as zeros. The singularities will prevent many of
the features of the rms library from working.
if TRUE a robust variance estimate is returned. Default is TRUE if the
model includes a cluster() operative, FALSE otherwise.
default is FALSE(false). Set to TRUE to return the model frame as element
model of the fit object.
default is FALSE. Set to TRUE to return the expanded design matrix as element x
(without intercept indicators) of the
returned fit object.
default is FALSE. Set to TRUE to return the vector of
response values (Surv object) as element y of the
fit.
default is FALSE. Set to TRUE to compute the estimated standard errors of
the estimate of X beta and store them in element se.fit
of the fit. The predictors are first centered to their means
before computing the standard errors.
set to FALSE to omit
linear.predictors vector from fit
set to FALSE to omit residuals vector
from fit
set to TRUE to not set names attribute
for linear.predictors, residuals, se.fit, and
rows of design matrix
convergence criterion - change in log likelihood.
vector of initial parameter estimates. Defaults to all zeros.
Special residuals can be obtained by setting some elements of init
to MLEs and others to zero and specifying iter.max=1.
maximum number of iterations to allow. Set to 0 to obtain certain
null-model residuals.
tolerance for declaring singularity for matrix inversion (available only when survival5 or later package is in effect)
set to TRUE to compute underlying survival estimates for each
stratum, and to store these along with standard errors of log Lambda(t),
maxtime (maximum observed survival or censoring time),
and surv.summary in the returned object. Set surv="summary"
to only compute and store surv.summary, not survival estimates
at each unique uncensored failure time. If you specify x=TRUE
and y=TRUE,
you can obtain predicted survival later, with accurate confidence
intervals for any set of predictor values. The standard error information
stored as a result of surv=TRUE are only accurate at the mean of all
predictors. If the model has no covariables, these are of course OK.
The main reason for using surv is to greatly speed up the computation
of predicted survival probabilities as a function of the covariables,
when accurate confidence intervals are not needed.
time increment used in deriving surv.summary. Survival,
number at risk, and standard error will be stored for
t=0, time.inc, 2 time.inc, ..., maxtime,
where maxtime is the maximum survival time over all strata.
time.inc is also used in constructing the time axis in the
survplot function (see below). The default value for
time.inc is 30 if units(ftime) = "Day" or no units
attribute has been attached to the survival time variable. If
units(ftime) is a word other than "Day", the default
for time.inc is 1 when it is omitted, unless maxtime<1, then
maxtime/10 is used as time.inc. If time.inc is not given and
maxtime/ default time.inc > 25, time.inc is increased.
(for cph) applies if surv is TRUE or "summary".
If type is omitted, the method consistent with method is used.
See survfit.coxph (under survfit) or survfit.cph for details and for the
definitions of values of type
For Survival, Quantile, Mean set to "polygon" to use linear
interpolation instead of the usual step function. For Mean, the default
of step will yield the sample mean in the case of no censoring and no
covariables, if type="kaplan-meier" was specified to cph.
For method="exact", the value of type is passed to the
generated function, and it can be overridden when that function is
actually invoked. For method="approximate", Mean.cph
generates the function different ways according to type, and this
cannot be changed when the function is actually invoked.
see survfit.coxph
set to TRUE to print debugging information related
to model matrix construction. You can also use options(debug=TRUE).
other arguments passed to coxph.fit from cph. Ignored by
other functions.
a scalar or vector of times at which to evaluate the survival estimates
a scalar or vector of linear predictors (including the centering constant) at which to evaluate the survival estimates
a scalar stratum number or name (e.g., "sex=male") to use in getting
survival probabilities
a scalar quantile or a vector of quantiles to compute
the number of points at which to evaluate the mean survival time, for
method="approximate" in Mean.cph.
For Mean.cph, the default is to compute the overall mean (and produce
a warning message if there is censoring at the end of follow-up).
To compute a restricted mean life length, specify the truncation point as tmax.
For method="exact", tmax is passed to the generated function and it
may be overridden when that function is invoked. For method="approximate",
tmax must be specified at the time that Mean.cph is run.
Frank Harrell
Department of Biostatistics, Vanderbilt University
fh@fharrell.com
If there is any strata by covariable interaction in the model such that
the mean X beta varies greatly over strata, method="approximate" may
not yield very accurate estimates of the mean in Mean.cph.
For method="approximate" if you ask for an estimate of the mean for
a linear predictor value that was outside the range of linear predictors
stored with the fit, the mean for that observation will be NA.
coxph, survival-internal,
Surv, residuals.cph,
cox.zph, survfit.cph,
survest.cph, survfit.coxph,
survplot, datadist,
rms, rms.trans, anova.rms,
summary.rms, Predict,
fastbw, validate, calibrate,
plot.Predict, ggplot.Predict,
specs.rms, lrm, which.influence,
na.delete,
na.detail.response, print.cph,
latex.cph, vif, ie.setup,
GiniMd, dxy.cens,
concordance
# Simulate data from a population model in which the log hazard
# function is linear in age and there is no age x sex interaction
require(survival)
require(ggplot2)
n <- 1000
set.seed(731)
age <- 50 + 12*rnorm(n)
label(age) <- "Age"
sex <- factor(sample(c('Male','Female'), n,
rep=TRUE, prob=c(.6, .4)))
cens <- 15*runif(n)
h <- .02*exp(.04*(age-50)+.8*(sex=='Female'))
dt <- -log(runif(n))/h
label(dt) <- 'Follow-up Time'
e <- ifelse(dt <= cens,1,0)
dt <- pmin(dt, cens)
units(dt) <- "Year"
dd <- datadist(age, sex)
options(datadist='dd')
S <- Surv(dt,e)
f <- cph(S ~ rcs(age,4) + sex, x=TRUE, y=TRUE)
cox.zph(f, "rank") # tests of PH
anova(f)
ggplot(Predict(f, age, sex)) # plot age effect, 2 curves for 2 sexes
survplot(f, sex) # time on x-axis, curves for x2
res <- resid(f, "scaledsch")
time <- as.numeric(dimnames(res)[[1]])
z <- loess(res[,4] ~ time, span=0.50) # residuals for sex
plot(time, fitted(z))
lines(supsmu(time, res[,4]),lty=2)
plot(cox.zph(f,"identity")) #Easier approach for last few lines
# latex(f)
f <- cph(S ~ age + strat(sex), surv=TRUE)
g <- Survival(f) # g is a function
g(seq(.1,1,by=.1), stratum="sex=Male", type="poly") #could use stratum=2
med <- Quantile(f)
plot(Predict(f, age, fun=function(x) med(lp=x))) #plot median survival
# Fit a model that is quadratic in age, interacting with sex as strata
# Compare standard errors of linear predictor values with those from
# coxph
# Use more stringent convergence criteria to match with coxph
f <- cph(S ~ pol(age,2)*strat(sex), x=TRUE, eps=1e-9, iter.max=20)
coef(f)
se <- predict(f, se.fit=TRUE)$se.fit
require(lattice)
xyplot(se ~ age | sex, main='From cph')
a <- c(30,50,70)
comb <- data.frame(age=rep(a, each=2),
sex=rep(levels(sex), 3))
p <- predict(f, comb, se.fit=TRUE)
comb$yhat <- p$linear.predictors
comb$se <- p$se.fit
z <- qnorm(.975)
comb$lower <- p$linear.predictors - z*p$se.fit
comb$upper <- p$linear.predictors + z*p$se.fit
comb
age2 <- age^2
f2 <- coxph(S ~ (age + age2)*strata(sex))
coef(f2)
se <- predict(f2, se.fit=TRUE)$se.fit
xyplot(se ~ age | sex, main='From coxph')
comb <- data.frame(age=rep(a, each=2), age2=rep(a, each=2)^2,
sex=rep(levels(sex), 3))
p <- predict(f2, newdata=comb, se.fit=TRUE)
comb$yhat <- p$fit
comb$se <- p$se.fit
comb$lower <- p$fit - z*p$se.fit
comb$upper <- p$fit + z*p$se.fit
comb
# g <- cph(Surv(hospital.charges) ~ age, surv=TRUE)
# Cox model very useful for analyzing highly skewed data, censored or not
# m <- Mean(g)
# m(0) # Predicted mean charge for reference age
#Fit a time-dependent covariable representing the instantaneous effect
#of an intervening non-fatal event
rm(age)
set.seed(121)
dframe <- data.frame(failure.time=1:10, event=rep(0:1,5),
ie.time=c(NA,1.5,2.5,NA,3,4,NA,5,5,5),
age=sample(40:80,10,rep=TRUE))
z <- ie.setup(dframe$failure.time, dframe$event, dframe$ie.time)
S <- z$S
ie.status <- z$ie.status
attach(dframe[z$subs,]) # replicates all variables
f <- cph(S ~ age + ie.status, x=TRUE, y=TRUE)
#Must use x=TRUE,y=TRUE to get survival curves with time-dep. covariables
#Get estimated survival curve for a 50-year old who has an intervening
#non-fatal event at 5 days
new <- data.frame(S=Surv(c(0,5), c(5,999), c(FALSE,FALSE)), age=rep(50,2),
ie.status=c(0,1))
g <- survfit(f, new)
plot(c(0,g$time), c(1,g$surv[,2]), type='s',
xlab='Days', ylab='Survival Prob.')
# Not certain about what columns represent in g$surv for survival5
# but appears to be for different ie.status
#or:
#g <- survest(f, new)
#plot(g$time, g$surv, type='s', xlab='Days', ylab='Survival Prob.')
#Compare with estimates when there is no intervening event
new2 <- data.frame(S=Surv(c(0,5), c(5, 999), c(FALSE,FALSE)), age=rep(50,2),
ie.status=c(0,0))
g2 <- survfit(f, new2)
lines(c(0,g2$time), c(1,g2$surv[,2]), type='s', lty=2)
#or:
#g2 <- survest(f, new2)
#lines(g2$time, g2$surv, type='s', lty=2)
detach("dframe[z$subs, ]")
options(datadist=NULL)
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