R: Cox Proportional Hazards Model and Extensions

cph {rms} | R Documentation

## Cox Proportional Hazards Model and Extensions

### Description

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.

### 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,
        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 class 'cph'
    Survival(object, ...)
    # Evaluate result as g(times, lp, stratum=1, type=c("step","polygon"))

    ## S3 method for class 'cph'
    Quantile(object, ...)
    # 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, \dots)

### 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 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.
`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 satisfying a logical expression or those having row numbers in the given vector.
`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 `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")`.
`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 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.
`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. 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.
`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 computing the standard errors.
`linear.predictors` | set to `FALSE` to omit `linear.predictors` vector from fit
`residuals` | set to `FALSE` to omit `residuals` vector from fit
`nonames` | set to `TRUE` to not set `names` attribute for `linear.predictors`, `residuals`, `se.fit`, and rows of design matrix
`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` 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.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 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.
`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`) 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.
`vartype` | see `survfit.coxph`
`debug` | 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.
`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`. 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.

### 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`, `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.
`var` | variance/covariance matrix of coefficients
`linear.predictors` | values of predicted X beta for observations used in fit, normalized to have overall mean zero, then having any offsets added
`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.

### Author(s)

Frank Harrell
Department of Biostatistics, Vanderbilt University
f.harrell@vanderbilt.edu

### See Also

`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`, `survConcordance`

### 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')
    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)

[Package rms version 5.1-1 Index]