# cph

##### Cox Proportional Hazards Model and Extensions

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.

- Keywords
- models, nonparametric, survival

##### Usage

```
cph(formula = formula(data), data=parent.frame(),
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,
eps=1e-4, init, iter.max=10, tol=1e-9, surv=FALSE, time.inc,
type, vartype, ...)
```## S3 method for class 'cph':
Survival(object, \dots)
# Evaluate result as g(times, lp, stratum=1, type=c("step","polygon"))

## S3 method for class 'cph':
Quantile(object, \dots)
# Evaluate like h(q, lp, stratum=1, type=c("step","polygon"))

## S3 method for class '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, ...)

##### Arguments

- formula
- 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 - object
- an object created by
`cph`

with`surv=TRUE`

- data
- name of an S data frame containing all needed variables. Omit this to use a data frame already in the S ``search list''.
- weights
- case weights
- subset
- 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 satisfy - na.action
- 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 - method
- 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 retu - singular.ok
- 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. Fo - robust
- if
`TRUE`

a robust variance estimate is returned. Default is`TRUE`

if the model includes a`cluster()`

operative,`FALSE`

otherwise. - model
- default is
`FALSE`

(false). Set to`TRUE`

to return the model frame as element`model`

of the fit object. - x
- default is
`FALSE`

. Set to`TRUE`

to return the expanded design matrix as element`x`

(without intercept indicators) of the returned fit object. - y
- default is
`FALSE`

. Set to`TRUE`

to return the vector of response values (`Surv`

object) as element`y`

of the fit. - se.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 comp - eps
- convergence criterion - change in log likelihood.
- init
- 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`

. - iter.max
- maximum number of iterations to allow. Set to
`0`

to obtain certain null-model residuals. - tol
- tolerance for declaring singularity for matrix inversion (available only when survival5 or later package is in effect)
- surv
- 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`

- time.inc
- 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 - type
- (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`

- vartype
- see
`survfit.coxph`

- ...
- other arguments passed to
`coxph.fit`

from`cph`

. Ignored by other functions. - times
- a scalar or vector of times at which to evaluate the survival estimates
- lp
- a scalar or vector of linear predictors (including the centering constant) at which to evaluate the survival estimates
- stratum
- a scalar stratum number or name (e.g.,
`"sex=male"`

) to use in getting survival probabilities - q
- a scalar quantile or a vector of quantiles to compute
- n
- the number of points at which to evaluate the mean survival time, for
`method="approximate"`

in`Mean.cph`

. - tmax
- 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`

. F

##### Details

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`

.

##### Value

- 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. n table with one row per stratum containing number of censored and uncensored observations coef vector of regression coefficients stats vector containing the named elements `Obs`

,`Events`

,`Model L.R.`

,`d.f.`

,`P`

,`Score`

,`Score P`

, and`R2`

.var variance/covariance matrix of coefficients linear.predictors values of predicted X beta for observations used in fit, normalized to have overall mean zero resid martingale residuals loglik log likelihood at initial and final parameter values score value of score statistic at initial values of parameters times lists of times (if `surv="T"`

)surv lists of underlying survival probability estimates std.err lists of standard errors of estimate log-log survival surv.summary 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).center centering constant, equal to overall mean of X beta.

##### See Also

`coxph`

, `coxph.fit`

,
`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`

, `specs.rms`

,
`lrm`

, `which.influence`

,
`na.delete`

,
`na.detail.response`

, `print.cph`

,
`latex.cph`

, `vif`

, `ie.setup`

##### Examples

```
# Simulate data from a population model in which the log hazard
# function is linear in age and there is no age x sex interaction
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')
Srv <- Surv(dt,e)
f <- cph(Srv ~ rcs(age,4) + sex, x=TRUE, y=TRUE)
cox.zph(f, "rank") # tests of PH
anova(f)
plot(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(Srv ~ 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(Srv ~ 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$se.fit
comb
age2 <- age^2
f2 <- coxph(Srv ~ (age + age2)*strata(sex))
coef(f2)
se <- predict(f2, se.fit=TRUE)$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)
```

*Documentation reproduced from package rms, version 2.0-2, License: GPL (>= 2)*