psm: Parametric Survival Model

Description

psm is a modification of Therneau's survreg function for fitting the accelerated failure time family of parametric survival models. psm uses the rms class for automatic anova, fastbw, calibrate, validate, and other functions. Hazard.psm, Survival.psm, Quantile.psm, and Mean.psm create S functions that evaluate the hazard, survival, quantile, and mean (expected value) functions analytically, as functions of time or probabilities and the linear predictor values. The Nagelkerke R^2 and and adjusted Maddala-Cox-Snell R^2 are computed. For the latter the notation is R2(p,m) where p is the number of regression coefficients being adjusted for and m is the effective sample size (number of uncensored observations). See R2Measures for more information.

For the print method, format of output is controlled by the user previously running options(prType="lang") where lang is "plain" (the default), "latex", or "html".

The residuals.psm function exists mainly to compute normalized (standardized) residuals and to censor them (i.e., return them as Surv objects) just as the original failure time variable was censored. These residuals are useful for checking the underlying distributional assumption (see the examples). To get these residuals, the fit must have specified y=TRUE. A lines method for these residuals automatically draws a curve with the assumed standardized survival distribution. A survplot method runs the standardized censored residuals through npsurv to get Kaplan-Meier estimates, with optional stratification (automatically grouping a continuous variable into quantiles) and then through survplot.npsurv to plot them. Then lines is invoked to show the theoretical curve. Other types of residuals are computed by residuals using residuals.survreg.

Usage

psm(formula,
    data=environment(formula), weights,
    subset, na.action=na.delete, dist="weibull",
    init=NULL, scale=0, 
    control=survreg.control(),
    parms=NULL, 
    model=FALSE, x=FALSE, y=TRUE, time.inc, ...)
# S3 method for psm
print(x, correlation=FALSE, digits=4, r2=c(0,2,4), coefs=TRUE,
pg=FALSE, title, ...) 
Hazard(object, ...)
# S3 method for psm
Hazard(object, ...)   # for psm fit
# E.g. lambda <- Hazard(fit)
Survival(object, ...)
# S3 method for psm
Survival(object, ...) # for psm
# E.g. survival <- Survival(fit)
# S3 method for psm
Quantile(object, ...) # for psm
# E.g. quantsurv <- Quantile(fit)
# S3 method for psm
Mean(object, ...)     # for psm
# E.g. meant   <- Mean(fit)
# lambda(times, lp)   # get hazard function at t=times, xbeta=lp
# survival(times, lp) # survival function at t=times, lp
# quantsurv(q, lp)    # quantiles of survival time
# meant(lp)           # mean survival time
# S3 method for psm
residuals(object, type=c("censored.normalized",
"response", "deviance", "dfbeta",
"dfbetas", "working", "ldcase", "ldresp", "ldshape", "matrix", "score"), ...)
# S3 method for residuals.psm.censored.normalized
survplot(fit, x, g=4, col, main, ...)
# S3 method for residuals.psm.censored.normalized
lines(x, n=100, lty=1, xlim,
lwd=3, ...)
# for type="censored.normalized"

Value

psm returns a fit object with all the information survreg would store as well as what rms stores and units and time.inc.

Hazard, Survival, and Quantile return S-functions.

residuals.psm with type="censored.normalized" returns a

Surv object which has a special attribute "theoretical"

which is used by the lines

routine. This is the assumed standardized survival function as a function of time or transformed time.

Arguments

formula: an S statistical model formula. Interactions up to third order are supported. The left hand side must be a Surv object.
object: a fit created by psm. For survplot with residuals from psm, object is the result of residuals.psm.
fit: a fit created by psm
data,subset,weights,dist,scale,init,na.action,control: see survreg.
parms: a list of fixed parameters. For the $t$-distribution this is the degrees of freedom; most of the distributions have no parameters.
model: set to TRUE to include the model frame in the returned object
x: set to TRUE to include the design matrix in the object produced by psm. For the survplot method, x is an optional stratification variable (character, numeric, or categorical). For lines.residuals.psm.censored.normalized, x is the result of residuals.psm. For print it is the result of psm.
y: set to TRUE to include the Surv() matrix
time.inc: setting for default time spacing. Used in constructing time axis in survplot, and also in make confidence bars. Default is 30 if time variable has units="Day", 1 otherwise, unless maximum follow-up time $< 1$. Then max time/10 is used as time.inc. If time.inc is not given and max time/default time.inc is $> 25$, time.inc is increased.
correlation: set to TRUE to print the correlation matrix for parameter estimates
digits: number of places to print to the right of the decimal point
r2: vector of integers specifying which R^2 measures to print, with 0 for Nagelkerke R^2 and 1:4 corresponding to the 4 measures computed by R2Measures. Default is to print Nagelkerke (labeled R2) and second and fourth R2Measures which are the measures adjusted for the number of predictors, first for the raw sample size then for the effective sample size, which here is the number of uncensored observations.
coefs: specify coefs=FALSE to suppress printing the table of model coefficients, standard errors, etc. Specify coefs=n to print only the first n regression coefficients in the model.
pg: set to TRUE to print g-indexes
title: a character string title to be passed to prModFit
...: other arguments to fitting routines, or to pass to survplot from
survplot.residuals.psm.censored.normalized. Passed to the generic lines function for lines.
times: a scalar or vector of times for which to evaluate survival probability or hazard
lp: a scalar or vector of linear predictor values at which to evaluate survival probability or hazard. If both times and lp are vectors, they must be of the same length.
q: a scalar or vector of probabilities. The default is .5, so just the median survival time is returned. If q and lp are both vectors, a matrix of quantiles is returned, with rows corresponding to lp and columns to q.
type: type of residual desired. Default is censored normalized residuals, defined as (link(Y) - linear.predictors)/scale parameter, where the link function was usually the log function. See survreg for other types. type="score" returns the score residual matrix.
n: number of points to evaluate theoretical standardized survival function for
lines.residuals.psm.censored.normalized
lty: line type for lines, default is 1
xlim: range of times (or transformed times) for which to evaluate the standardized survival function. Default is range in normalized residuals.
lwd: line width for theoretical distribution, default is 3
g: number of quantile groups to use for stratifying continuous variables having more than 5 levels
col: vector of colors for survplot method, corresponding to levels of x (must be a scalar if there is no x)
main: main plot title for survplot. If omitted, is the name or label of x if x is given. Use main="" to suppress a title when you specify x.

Author

Frank Harrell
Department of Biostatistics
Vanderbilt University
fh@fharrell.com

Details

The object survreg.distributions contains definitions of properties of the various survival distributions.
psm does not trap singularity errors due to the way survreg.fit does matrix inversion. It will trap non-convergence (thus returning fit$fail=TRUE) if you give the argument failure=2 inside the control list which is passed to survreg.fit. For example, use f <- psm(S ~ x, control=list(failure=2, maxiter=20)) to allow up to 20 iterations and to set f$fail=TRUE in case of non-convergence. This is especially useful in simulation work.

Examples

Run this code

require(survival)
n <- 400
set.seed(1)
age <- rnorm(n, 50, 12)
sex <- factor(sample(c('Female','Male'),n,TRUE))
dd <- datadist(age,sex)
options(datadist='dd')
# Population hazard function:
h <- .02*exp(.06*(age-50)+.8*(sex=='Female'))
d.time <- -log(runif(n))/h
cens <- 15*runif(n)
death <- ifelse(d.time <= cens,1,0)
d.time <- pmin(d.time, cens)

f <- psm(Surv(d.time,death) ~ sex*pol(age,2), 
         dist='lognormal')
# Log-normal model is a bad fit for proportional hazards data
print(f, r2=0:4, pg=TRUE)

anova(f)
fastbw(f)  # if deletes sex while keeping age*sex ignore the result
f <- update(f, x=TRUE,y=TRUE)       # so can validate, compute certain resids
validate(f, B=10)      # ordinarily use B=300 or more
plot(Predict(f, age, sex))   # needs datadist since no explicit age, hosp.
# Could have used ggplot(Predict(...))
survplot(f, age=c(20,60))     # needs datadist since hospital not set here
# latex(f)


S <- Survival(f)
plot(f$linear.predictors, S(6, f$linear.predictors),
     xlab=expression(X*hat(beta)),
     ylab=expression(S(6,X*hat(beta))))
# plots 6-month survival as a function of linear predictor (X*Beta hat)


times <- seq(0,24,by=.25)
plot(times, S(times,0), type='l')   # plots survival curve at X*Beta hat=0
lam <- Hazard(f)
plot(times, lam(times,0), type='l') # similarly for hazard function


med <- Quantile(f)        # new function defaults to computing median only
lp <- seq(-3, 5, by=.1)
plot(lp, med(lp=lp), ylab="Median Survival Time")
med(c(.25,.5), f$linear.predictors)
                          # prints matrix with 2 columns


# fit a model with no predictors
f <- psm(Surv(d.time,death) ~ 1, dist="weibull")
f
pphsm(f)          # print proportional hazards form
g <- survest(f)
plot(g$time, g$surv, xlab='Time', type='l',
     ylab=expression(S(t)))


f <- psm(Surv(d.time,death) ~ age, 
         dist="loglogistic", y=TRUE)
r <- resid(f, 'cens') # note abbreviation
survplot(npsurv(r ~ 1), conf='none') 
                      # plot Kaplan-Meier estimate of 
                      # survival function of standardized residuals
survplot(npsurv(r ~ cut2(age, g=2)), conf='none')  
                      # both strata should be n(0,1)
lines(r)              # add theoretical survival function
#More simply:
survplot(r, age, g=2)

options(datadist=NULL)

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