# val.surv

##### Validate Predicted Probabilities Against Observed Survival Times

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.

- Keywords
- models, regression, smooth, survival

##### Usage

```
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, …)

##### Arguments

- fit
a fit object created by

`cph`

or`psm`

- newdata
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`

.- S
an

`Surv`

object- est.surv
a vector of estimated survival probabilities corresponding to times in the first column of

`S`

.- censor
a vector of censoring times. Only the censoring times for uncensored observations are used.

- u
a single numeric follow-up time

- fun
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)).- lim
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.- evaluate
the number of evenly spaced points over the range of predicted probabilities. This defines the points at which calibrated predictions are obtained for plotting.

- pred
a vector of points at which to evaluate predicted probabilities, overriding

`lim`

- maxdim
see

`hare`

- x
result of

`val.surv`

- xlab
x-axis label. For

`plot.survh`

, defaults for`xlab`

and`ylab`

come from`u`

and the units of measurement for the raw survival times.- ylab
y-axis label

- riskdist
set to

`FALSE`

to not call`scat1d`

to draw the distribution of predicted (uncalibrated) probabilities- add
set to

`TRUE`

if adding to an existing plot- scat1d.opts
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.- group
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.- g.group
number of quantile groups to use when

`group`

is given and variable is numeric.- what
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.- type
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.- xlim
- ylim
axis limits for

`plot.val.surv`

when the`censor`

variable was used.- datadensity
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`

.

##### Value

a list of class `"val.surv"`

or `"val.survh"`

##### References

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.

##### See Also

##### Examples

```
# 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
# }
```

*Documentation reproduced from package rms, version 5.1-4, License: GPL (>= 2)*