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val.surv

Validate Predicted Probabilities Against Observed Survival Times

Description

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 ≤q 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.

Usage

val.surv(fit, newdata, S, est.surv, censor,
         u, fun, lim, evaluate=100, pred, maxdim=5, ...)

## S3 method for class 'val.survh'
print(x, ...)

## S3 method for class 'val.survh'
plot(x, lim, xlab, ylab,
                         riskdist=TRUE, add=FALSE,
                         scat1d.opts=list(nhistSpike=200), ...)

## S3 method for class '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"

Author(s)

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

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.

Examples

# 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

rms

Regression Modeling Strategies

v6.2-0

GPL (>= 2)

Authors

Frank E Harrell Jr <fh@fharrell.com>

Initial release

2021-03-17