The val.surv function is useful for validating predicted survival
probabilities against right-censored failure times. If u is
specified, the hazard regression function hare in the
polspline package is used to relate predicted survival
probability at time u to observed survival times (and censoring
indicators) to estimate the actual survival probability at time
u as a function of the estimated survival probability at that
time, est.surv. If est.surv is not given, fit must
be specified and the survest function is used to obtain the
predicted values (using newdata if it is given, or using the
stored linear predictor values if not). hare is given the sole
predictor fun(est.surv) where fun is given by the user or
is inferred from fit. fun is the function of predicted
survival probabilities that one expects to create a linear relationship
with the linear predictors.
hare uses an adaptive procedure to find a linear spline of
fun(est.surv) in a model where the log hazard is a linear spline
in time \(t\), and cross-products between the two splines are allowed so as to
not assume proportional hazards. Thus hare assumes that the
covariate and time functions are smooth but not much else, if the number
of events in the dataset is large enough for obtaining a reliable
flexible fit. There are special print and plot methods
when u is given. In this case, val.surv returns an object
of class "val.survh", otherwise it returns an object of class
"val.surv".
If u is not specified, val.surv uses Cox-Snell (1968)
residuals on the cumulative
probability scale to check on the calibration of a survival model
against right-censored failure time data. If the predicted survival
probability at time \(t\) for a subject having predictors \(X\) is
\(S(t|X)\), this method is based on the fact that the predicted
probability of failure before time \(t\), \(1 - S(t|X)\), when
evaluated at the subject's actual survival time \(T\), has a uniform
(0,1) distribution. The quantity \(1 - S(T|X)\) is right-censored
when \(T\) is. By getting one minus the Kaplan-Meier estimate of the
distribution of \(1 - S(T|X)\) and plotting against the 45 degree line
we can check for calibration accuracy. A more stringent assessment can
be obtained by stratifying this analysis by an important predictor
variable. The theoretical uniform distribution is only an approximation
when the survival probabilities are estimates and not population values.
When censor is specified to val.surv, a different
validation is done that is more stringent but that only uses the
uncensored failure times. This method is used for type I censoring when
the theoretical censoring times are known for subjects having uncensored
failure times. Let \(T\), \(C\), and \(F\) denote respectively
the failure time, censoring time, and cumulative failure time
distribution (\(1 - S\)). The expected value of \(F(T | X)\) is 0.5
when \(T\) represents the subject's actual failure time. The expected
value for an uncensored time is the expected value of \(F(T | T \leq
C, X) = 0.5 F(C | X)\). A smooth plot of \(F(T|X) - 0.5 F(C|X)\) for
uncensored \(T\) should be a flat line through \(y=0\) if the model
is well calibrated. A smooth plot of \(2F(T|X)/F(C|X)\) for
uncensored \(T\) should be a flat line through \(y=1.0\). The smooth
plot is obtained by smoothing the (linear predictor, difference or
ratio) pairs.
val.surv(fit, newdata, S, est.surv, censor,
u, fun, lim, evaluate=100, pred, maxdim=5, ...)# S3 method for val.survh
print(x, ...)
# S3 method for val.survh
plot(x, lim, xlab, ylab,
riskdist=TRUE, add=FALSE,
scat1d.opts=list(nhistSpike=200), ...)
# S3 method for val.surv
plot(x, group, g.group=4,
what=c('difference','ratio'),
type=c('l','b','p'),
xlab, ylab, xlim, ylim, datadensity=TRUE, …)
a fit object created by cph or psm
a data frame for which val.surv should obtain predicted survival
probabilities. If omitted, survival estimates are made for all of the
subjects used in fit.
an Surv object
a vector of estimated survival probabilities corresponding to times in
the first column of S.
a vector of censoring times. Only the censoring times for uncensored observations are used.
a single numeric follow-up time
a function that transforms survival probabilities into the
scale of the linear predictor. If fit is given, and
represents either a Cox, Weibull, or exponential fit, fun is
automatically set to log(-log(p)).
a 2-vector specifying limits of predicted survival
probabilities for obtaining estimated actual probabilities at time
u. Default for
val.surv is the limits for predictions from datadist,
which for large \(n\) is the 10th smallest and 10th largest
predicted survival probability. For plot.val.survh, the
default for lim is the range of the combination of predicted
probabilities and calibrated actual probabilities. lim is
used for both axes of the calibration plot.
the number of evenly spaced points over the range of predicted probabilities. This defines the points at which calibrated predictions are obtained for plotting.
a vector of points at which to evaluate predicted
probabilities, overriding lim
see hare
result of val.surv
x-axis label. For plot.survh, defaults for
xlab and ylab come from u and the units of
measurement for the raw survival times.
y-axis label
set to FALSE to not call scat1d to draw the
distribution of predicted (uncalibrated) probabilities
set to TRUE if adding to an existing plot
a list of options to pass to scat1d.
By default, the option nhistSpike=200 is passed so that a spike
histogram is used if the sample size exceeds 200.
When u is given to val.surv, … represents
optional arguments to hare. It can represent arguments to
pass to plot or lines for
plot.val.survh. Otherwise, … contains optional
arguments for plsmo or plot. For
print.val.survh, … is ignored.
a grouping variable. If numeric this variable is grouped into
g.group quantile groups (default is quartiles). group,
g.group, what, and type apply when
u is not given.
number of quantile groups to use when group is given and variable
is numeric.
the quantity to plot when censor was in effect. The default is to
show the difference between cumulative probabilities and their
expectation given the censoring time. Set what="ratio" to show the
ratio instead.
Set to the default ("l") to plot the trend line only, "b"
to plot both individual subjects ratios and trend lines, or
"p" to plot only points.
axis limits for plot.val.surv when the censor variable was used.
By default, plot.val.surv will show the data density on each curve
that is created as a result of censor being present. Set
datadensity=FALSE to suppress these tick marks drawn by scat1d.
a list of class "val.surv" or "val.survh"
Cox DR, Snell EJ (1968):A general definition of residuals (with discussion). JRSSB 30:248--275.
Kooperberg C, Stone C, Truong Y (1995): Hazard regression. JASA 90:78--94.
May M, Royston P, Egger M, Justice AC, Sterne JAC (2004):Development and validation of a prognostic model for survival time data: application to prognosis of HIV positive patients treated with antiretroviral therapy. Stat in Med 23:2375--2398.
Stallard N (2009): Simple tests for th external validation of mortality prediction scores. Stat in Med 28:377--388.
# NOT RUN {
# Generate failure times from an exponential distribution
set.seed(123) # so can reproduce results
n <- 1000
age <- 50 + 12*rnorm(n)
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'))
t <- -log(runif(n))/h
units(t) <- 'Year'
label(t) <- 'Time to Event'
ev <- ifelse(t <= cens, 1, 0)
t <- pmin(t, cens)
S <- Surv(t, ev)
# First validate true model used to generate data
# If hare is available, make a smooth calibration plot for 1-year
# survival probability where we predict 1-year survival using the
# known true population survival probability
# In addition, use groupkm to show that grouping predictions into
# intervals and computing Kaplan-Meier estimates is not as accurate.
if('polspline' %in% row.names(installed.packages())) {
s1 <- exp(-h*1)
w <- val.surv(est.surv=s1, S=S, u=1,
fun=function(p)log(-log(p)))
plot(w, lim=c(.85,1), scat1d.opts=list(nhistSpike=200, side=1))
groupkm(s1, S, m=100, u=1, pl=TRUE, add=TRUE)
}
# Now validate the true model using residuals
w <- val.surv(est.surv=exp(-h*t), S=S)
plot(w)
plot(w, group=sex) # stratify by sex
# Now fit an exponential model and validate
# Note this is not really a validation as we're using the
# training data here
f <- psm(S ~ age + sex, dist='exponential', y=TRUE)
w <- val.surv(f)
plot(w, group=sex)
# We know the censoring time on every subject, so we can
# compare the predicted Pr[T <= observed T | T>c, X] to
# its expectation 0.5 Pr[T <= C | X] where C = censoring time
# We plot a ratio that should equal one
w <- val.surv(f, censor=cens)
plot(w)
plot(w, group=age, g=3) # stratify by tertile of age
# }